From 4fe81875f185f0a9f77153e36d650e59cb95347b Mon Sep 17 00:00:00 2001 From: mitchkeller Date: Fri, 2 Aug 2024 17:33:11 -0500 Subject: [PATCH 01/10] Set up worksheets and structure activity with task --- requirements.txt | 2 +- source/activities/act-0-0-0.xml | 55 +- .../activities/act-changing-aroc-graphs.xml | 106 ++-- .../act-changing-aroc-population.xml | 187 +++--- .../activities/act-changing-aroc-trends.xml | 121 ++-- .../act-changing-combining-arithmetic.xml | 127 ++-- .../act-changing-combining-context.xml | 99 ++-- .../act-changing-combining-piecewise.xml | 118 ++-- .../act-changing-composite-aroc.xml | 87 +-- ...ct-changing-composite-crickets-celsius.xml | 105 ++-- .../act-changing-composite-tables-graphs.xml | 236 ++++---- .../act-changing-functions-is-it.xml | 172 +++--- ...ging-functions-spherical-tank-draining.xml | 112 ++-- .../act-changing-functions-spherical-tank.xml | 95 +-- .../act-changing-inverse-Dolbear.xml | 84 +-- .../act-changing-inverse-does-it.xml | 212 +++---- .../act-changing-inverse-rainfall.xml | 95 +-- .../act-changing-linear-Kilimanjaro.xml | 102 ++-- .../act-changing-linear-finding-eqs.xml | 182 +++--- .../act-changing-linear-in-context.xml | 95 +-- .../act-changing-quadratic-falling-ball.xml | 112 ++-- .../act-changing-quadratic-parameters.xml | 99 ++-- .../act-changing-quadratic-properties.xml | 99 ++-- .../act-changing-tandem-conical-tank.xml | 196 ++++--- .../act-changing-tandem-spherical-tank.xml | 206 ++++--- .../act-changing-transformations-combined.xml | 113 ++-- ...-changing-transformations-translations.xml | 99 ++-- ...-changing-transformations-vert-stretch.xml | 114 ++-- .../act-circular-sine-cosine-computing.xml | 117 ++-- .../act-circular-sine-cosine-incr-CCU.xml | 117 ++-- .../activities/act-circular-sine-cosine.xml | 271 +++++---- .../act-circular-sinusoidal-horiz-stretch.xml | 114 ++-- .../act-circular-sinusoidal-model.xml | 37 +- .../act-circular-sinusoidal-oscillator.xml | 37 +- .../act-circular-sinusoidal-period.xml | 115 ++-- .../act-circular-traversing-2nd-ex.xml | 290 ++++++---- ...ct-circular-traversing-oscillator-aroc.xml | 342 +++++++---- .../act-circular-traversing-oscillator.xml | 317 ++++++---- .../act-circular-unit-circle-non-unit.xml | 87 +-- ...t-circular-unit-circle-radians-degrees.xml | 126 ++-- ...circular-unit-circle-special-triangles.xml | 137 ++--- source/activities/act-exp-e-aroc-e.xml | 101 ++-- source/activities/act-exp-e-graph-solve.xml | 130 +++-- source/activities/act-exp-growth-a-b-t.xml | 114 ++-- source/activities/act-exp-growth-find-a-b.xml | 84 +-- source/activities/act-exp-growth-rates.xml | 95 +-- source/activities/act-exp-log-base-10.xml | 145 +++-- source/activities/act-exp-log-equations.xml | 160 +++--- .../act-exp-log-exponential-equations.xml | 136 +++-- source/activities/act-exp-log-natural.xml | 229 +++++--- .../act-exp-log-properties-exp-or-log.xml | 158 ++--- .../act-exp-log-properties-find-k.xml | 100 ++-- .../activities/act-exp-modeling-behavior.xml | 170 +++--- source/activities/act-exp-modeling-potato.xml | 103 ++-- source/activities/act-exp-modeling-soda.xml | 103 ++-- source/activities/act-exp-temp-pop-NLOC1.xml | 88 +-- .../act-exp-temp-pop-logistic-Desmos.xml | 93 +-- .../act-exp-temp-pop-logistic-exact.xml | 73 +-- source/activities/act-poly-infty-limit.xml | 224 +++++--- .../act-poly-infty-natural-powers.xml | 89 +-- .../act-poly-infty-negative-powers.xml | 100 ++-- ...ct-poly-polynomial-applications-Taylor.xml | 98 ++-- ...ct-poly-polynomial-applications-postal.xml | 107 ++-- .../act-poly-polynomial-applications-soup.xml | 87 +-- .../activities/act-poly-polynomials-find.xml | 84 +-- .../act-poly-polynomials-multiple-zeros.xml | 119 ++-- .../act-poly-polynomials-sign-chart.xml | 106 ++-- .../act-poly-rational-application.xml | 106 ++-- .../activities/act-poly-rational-domain.xml | 130 +++-- .../act-poly-rational-features-ZAH.xml | 130 +++-- .../activities/act-poly-rational-formula.xml | 116 ++-- .../act-poly-rational-long-term-1.xml | 87 +-- .../act-poly-rational-long-term-2.xml | 84 +-- .../act-trig-finding-angles-baseball.xml | 38 +- .../act-trig-finding-angles-exactly.xml | 73 +-- .../act-trig-finding-angles-rocket.xml | 38 +- .../act-trig-finding-angles-roof.xml | 35 +- source/activities/act-trig-inverse-arccos.xml | 190 +++--- source/activities/act-trig-inverse-arcsin.xml | 93 +-- source/activities/act-trig-inverse-arctan.xml | 95 +-- .../activities/act-trig-other-aroc-sine.xml | 87 +-- source/activities/act-trig-other-cot.xml | 542 ++++++++++++------ source/activities/act-trig-other-csc.xml | 389 ++++++++----- source/activities/act-trig-other-special.xml | 35 +- source/activities/act-trig-right-SOH-CAH.xml | 106 ++-- source/activities/act-trig-right-similar.xml | 93 +-- .../act-trig-right-sine-cosine-ratio.xml | 48 +- .../activities/act-trig-tangent-mountain.xml | 115 ++-- source/activities/act-trig-tangent-tower.xml | 34 +- .../act-trig-tangent-two-towers.xml | 52 +- source/activities/z-act-inverse-rainfall.xml | 55 +- .../z-act-transformations-translations.xml | 105 ++-- source/previews/PA-0-0.xml | 19 +- source/previews/PA-changing-aroc.xml | 100 ++-- source/previews/PA-changing-combining.xml | 167 +++--- source/previews/PA-changing-composite.xml | 67 ++- .../PA-changing-functions-crickets.xml | 78 ++- source/previews/PA-changing-inverse-F-C.xml | 73 ++- source/previews/PA-changing-linear-3-ex.xml | 206 ++++--- source/previews/PA-changing-quadratic.xml | 233 +++++--- .../previews/PA-changing-tandem-aquarium.xml | 85 +-- .../PA-changing-transformations-quadratic.xml | 76 +-- source/previews/PA-circular-sine.xml | 232 +++++--- source/previews/PA-circular-sinusoidal.xml | 74 +-- source/previews/PA-circular-traversing.xml | 92 +-- source/previews/PA-circular-unit-circle.xml | 94 +-- source/previews/PA-exp-e.xml | 78 ++- source/previews/PA-exp-growth.xml | 80 +-- source/previews/PA-exp-log-properties.xml | 83 ++- source/previews/PA-exp-log.xml | 235 +++++--- source/previews/PA-exp-modeling.xml | 70 ++- source/previews/PA-exp-temp-pop.xml | 69 ++- source/previews/PA-poly-infty.xml | 114 ++-- .../PA-poly-polynomial-applications.xml | 78 ++- source/previews/PA-poly-polynomials.xml | 97 ++-- source/previews/PA-poly-rational-features.xml | 374 +++++++----- source/previews/PA-poly-rational.xml | 89 +-- source/previews/PA-trig-finding-angles.xml | 78 ++- source/previews/PA-trig-inverse.xml | 106 ++-- source/previews/PA-trig-other.xml | 17 +- source/previews/PA-trig-right.xml | 89 +-- source/previews/PA-trig-tangent.xml | 99 ++-- xsl/activity-task.xsl | 107 ++++ 123 files changed, 8761 insertions(+), 6207 deletions(-) create mode 100644 xsl/activity-task.xsl diff --git a/requirements.txt b/requirements.txt index e50308a9..4b96bf58 100644 --- a/requirements.txt +++ b/requirements.txt @@ -1 +1 @@ -pretextbook == 2.3.4 +pretextbook == 2.6.0 diff --git a/source/activities/act-0-0-0.xml b/source/activities/act-0-0-0.xml index 2a4dac78..bc459d30 100755 --- a/source/activities/act-0-0-0.xml +++ b/source/activities/act-0-0-0.xml @@ -1,4 +1,4 @@ - + @@ -12,34 +12,25 @@ - - - - -

- -

- -

-

    -
  1. -

    - -

    -
    2. -
-

- - -

- -

- - -

- -

- - - - + + + +

+

+ + + +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-aroc-graphs.xml b/source/activities/act-changing-aroc-graphs.xml index fd18e347..690a6de6 100755 --- a/source/activities/act-changing-aroc-graphs.xml +++ b/source/activities/act-changing-aroc-graphs.xml @@ -1,4 +1,4 @@ - + @@ -12,62 +12,58 @@ - - - - -

+ + + +

Sketch at least two different possible graphs that satisfy the criteria for the function stated in each part. Make your graphs as significantly different as you can. If it is impossible for a graph to satisfy the criteria, explain why.

- -

-

    -
  1. -

    - f is a function defined on [-1,7] such that f(1) = 4 and AV_{[1,3]} = -2. +

    + + + +

    f is a function defined on [-1,7] such that f(1) = 4 and AV_{[1,3]} = -2.

    - - - - - - -
    2. -
  2. -

    - g is a function defined on [-1,7] such that g(4) = 3, AV_{[0,4]} = 0.5, and g is not always increasing on (0,4). + + + + + + + + + + + +

    g is a function defined on [-1,7] such that g(4) = 3, AV_{[0,4]} = 0.5, and g is not always increasing on (0,4).

    - - - - - - -
    3. -
  3. -

    - h is a function defined on [-1,7] such that h(2) = 5, h(4) = 3 and AV_{[2,4]} = -2. + + + + + + + + + + + +

    h is a function defined on [-1,7] such that h(2) = 5, h(4) = 3 and AV_{[2,4]} = -2.

    - - - - - - -
    4. -
-

- - -

- -

- - -

- -

- - - - + + + + + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-aroc-population.xml b/source/activities/act-changing-aroc-population.xml index 3acc035f..a1287d9a 100755 --- a/source/activities/act-changing-aroc-population.xml +++ b/source/activities/act-changing-aroc-population.xml @@ -1,4 +1,4 @@ - + @@ -12,108 +12,119 @@ - - - - -

+ + + +

According to the US census, the populations of Kent and Ottawa Counties in Michigan where GVSU is located Grand Rapids is in Kent, Allendale in Ottawa. from 1960 to 2010 measured in 10-year intervals are given in the following tables.

- - - Kent County population data. - - - 1960 - 1970 - 1980 - 1990 - 2000 - 2010 - - - 363,187 - 411,044 - 444,506 - 500,631 - 574,336 - 602,622 - - -
- - - Ottawa county population data. - - - 1960 - 1970 - 1980 - 1990 - 2000 - 2010 - - - 98,719 - 128,181 - 157,174 - 187,768 - 238,313 - 263,801 - - -
- -

+ + Kent County population data. + + + 1960 + 1970 + 1980 + 1990 + 2000 + 2010 + + + 363,187 + 411,044 + 444,506 + 500,631 + 574,336 + 602,622 + + +
+ + Ottawa county population data. + + + 1960 + 1970 + 1980 + 1990 + 2000 + 2010 + + + 98,719 + 128,181 + 157,174 + 187,768 + 238,313 + 263,801 + + +
+

Let K(Y) represent the population of Kent County in year Y and W(Y) the population of Ottawa County in year Y.

- -

-

    -
  1. -

    +

    + + + +

    Compute AV_{[1990,2010]} for both K and W.

    -
    2. -
  2. -

    + + + + + + + +

    What are the units on each of the quantities you computed in (a.)?

    -
    3. -
  3. -

    + + + + + + + +

    Write a careful sentence that explains the meaning of the average rate of change of the Ottawa county population on the time interval [1990,2010]. Your sentence should begin something like - In an average year between 1990 and 2010, the population of Ottawa County was \ldots -

    -
    4. -
  4. -

    + In an average year between 1990 and 2010, the population of Ottawa County was \ldots

    + + + + + + + +

    Which county had a greater average rate of change during the time interval [2000,2010]? Were there any intervals in which one of the counties had a negative average rate of change?

    -
    5. -
  5. -

    + + + + + + + +

    Using the given data, what do you predict will be the population of Ottawa County in 2018? Why?

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-aroc-trends.xml b/source/activities/act-changing-aroc-trends.xml index 24052002..557a4e8d 100755 --- a/source/activities/act-changing-aroc-trends.xml +++ b/source/activities/act-changing-aroc-trends.xml @@ -1,4 +1,4 @@ - + @@ -12,79 +12,88 @@ - - - - -

+ + + +

Let's consider two different functions and see how different computations of their average rate of change tells us about their respective behavior. Plots of q and h are shown in Figures and .

- -

-

    -
  1. -

    +

    + + + +

    Consider the function q(x) = 4-(x-2)^2. Compute AV_{[0,1]}, AV_{[1,2]}, AV_{[2,3]}, and AV_{[3,4]}. What do your last two computations tell you about the behavior of the function q on [2,4]?

    - -
    2. - -
  2. -

    + + + + + + + +

    Consider the function h(t) = 3 - 2(0.5)^t. Compute AV_{[-1,1]}, AV_{[1,3]}, and AV_{[3,5]}. What do your computations tell you about the behavior of the function h on [-1,5]?

    -
    3. - -
  3. -

    + + + + + + + +

    On the graphs in Figures and , plot the line segments whose respective slopes are the average rates of change you computed in (a) and (b).

    - -
    - Plot of q from part (a). - -
    -
    - Plot of h from part (b). - -
    -     - -
    4. - -
  4. -

    + +

    + Plot of q from part (a). + +
    +
    + Plot of h from part (b). + +
    + + + + + + + + +

    True or false: Since AV_{[0,3]} = 1, the function q is increasing on the interval (0,3). Justify your decision.

    -
    5. - -
  5. -

    + + + + + + + +

    Give an example of a function that has the same average rate of change no matter what interval you choose. You can provide your example through a table, a graph, or a formula; regardless of your choice, write a sentence to explain.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-combining-arithmetic.xml b/source/activities/act-changing-combining-arithmetic.xml index b25d96e5..aa642957 100755 --- a/source/activities/act-changing-combining-arithmetic.xml +++ b/source/activities/act-changing-combining-arithmetic.xml @@ -1,4 +1,4 @@ - + @@ -12,70 +12,89 @@ - - - - -

+ + + +

Consider the functions f and g defined by Figure and Figure. Assume that the given lines and curves pass through intersection points on the grid when it looks plausible. For instance, (0,2.5) and (3,-0.5) lie on the graph of f, and (-1,3) and (1.5, 1.5) lie on the graph of g.

- - -
- The function f. - -
-
- The function g. - -
- - -

-

    -
  1. -

    + +

    + The function f. + +
    +
    + The function g. + +
    + +

    + + + +

    Determine the exact value of (f+g)(0).

    -
    2. -
  2. -

    + + + + + + + +

    Determine the exact value of (g-f)(1).

    -
    3. -
  3. -

    + + + + + + + +

    Determine the exact value of (f \cdot g)(-1).

    -
    4. -
  4. -

    + + + + + + + +

    Are there any values of x for which \left( \frac{f}{g} \right)(x) is undefined? If not, explain why. If so, determine the values and justify your answer.

    -
    5. -
  5. -

    + + + + + + + +

    For what values of x is (f \cdot g)(x) = 0? Why?

    -
    6. -
  6. -

    + + + + + + + +

    Are there any values of x for which (f-g)(x) = 0? Why or why not?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-combining-context.xml b/source/activities/act-changing-combining-context.xml index d53a11bc..268e8ae8 100755 --- a/source/activities/act-changing-combining-context.xml +++ b/source/activities/act-changing-combining-context.xml @@ -1,4 +1,4 @@ - + @@ -12,75 +12,86 @@ - - - - -

+ + + +

Let f be a function that measures a car's fuel economy in the following way. Given an input velocity v in miles per hour, f(v) is the number of gallons of fuel that the car consumes per mile (i.e., gallons per mile). We know that f(60) = 0.04.

- -

-

    -
  1. -

    +

    + + + +

    What is the meaning of the statement f(60) = 0.04 in the context of the problem? That is, what does this say about the car's fuel economy? Write a complete sentence to explain.

    -
    2. - -
  2. -

    + + + + + + + +

    Consider the function g(v) = \frac{1}{f(v)}. What is the value of g(60)? What are the units on g? What does g measure?

    -
    3. - -
  3. -

    + + + + + + + +

    Consider the function h(v) = v \cdot f(v). What is the value of h(60)? What are the units on h? What does h measure?

    -
    4. - -
  4. -

    + + + + + + + +

    Do f(60), g(60), and h(60) tell us fundamentally different information, or are they all essentially saying the same thing? Explain.

    -
    5. - -
  5. -

    + + + + + + + +

    Suppose we also know that f(70) = 0.045. Find the average rate of change of f on the interval [60,70]. What are the units on the average rate of change of f? What does this quantity measure? Write a complete sentence to explain.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-combining-piecewise.xml b/source/activities/act-changing-combining-piecewise.xml index b090d7f4..2adff79d 100755 --- a/source/activities/act-changing-combining-piecewise.xml +++ b/source/activities/act-changing-combining-piecewise.xml @@ -1,4 +1,4 @@ - + @@ -12,19 +12,18 @@ - - - - -

+ + + +

In what follows, we work to understand two different piecewise functions entirely by hand based on familiar properties of linear and quadratic functions.

- -

-

    -
  1. -

    +

    + + + +

    Consider the function p defined by the following rule: p(x) = @@ -36,53 +35,66 @@ What are the values of p(-4), p(-2), p(0), p(2), and p(4)?

    -
    2. -
  2. -

    + + + + + + + +

    What point is the vertex of the quadratic part of p that is valid for x \lt 0? What point is the vertex of the quadratic part of p that is valid for x \ge 0?

    -
    3. - -
  3. -

    + + + + + + + +

    For what values of x is p(x) = 0? In addition, what is the y-intercept of p?

    -
    4. -
  4. -

    + + + + + + + +

    Sketch an accurate, labeled graph of y = p(x) on the axes provided in Figure.

    - - -
    - Axes to plot y = p(x). - -
    -
    - Graph of y = f(x). - -
    -     - -
    5. -
  5. -

    + +

    + Axes to plot y = p(x). + +
    +
    + Graph of y = f(x). + +
    + + + + + + + + +

    For the function f defined by Figure, determine a piecewise-defined formula for f that is expressed in bracket notation similar to the definition of y = p(x) above.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-composite-aroc.xml b/source/activities/act-changing-composite-aroc.xml index bc95196c..e02d45d5 100755 --- a/source/activities/act-changing-composite-aroc.xml +++ b/source/activities/act-changing-composite-aroc.xml @@ -1,4 +1,4 @@ - + @@ -12,55 +12,62 @@ - - - - -

+ + + +

Let f(x) = 2x^2 - 3x + 1 and g(x) = \frac{5}{x}.

- -

-

    -
  1. -

    +

    + + + +

    Compute f(1+h) and expand and simplify the result as much as possible by combining like terms.

    -
    2. - -
  2. -

    + + + + + + + +

    Determine the most simplified expression you can for the average rate of change of f on the interval [1,1+h]. That is, determine AV_{[1,1+h]} for f and simplify the result as much as possible.

    -
    3. - -
  3. -

    + + + + + + + +

    Compute g(1+h). Is there any valid algebra you can do to write g(1+h) more simply?

    -
    4. - -
  4. -

    + + + + + + + +

    Determine the most simplified expression you can for the average rate of change of g on the interval [1,1+h]. That is, determine AV_{[1,1+h]} for g and simplify the result.

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-composite-crickets-celsius.xml b/source/activities/act-changing-composite-crickets-celsius.xml index dfc33b0d..51534dc5 100755 --- a/source/activities/act-changing-composite-crickets-celsius.xml +++ b/source/activities/act-changing-composite-crickets-celsius.xml @@ -1,4 +1,4 @@ - + @@ -12,60 +12,69 @@ - - - - -

+ + + +

Let F = D(N) = 40 + 0.25N be Dolbear's function that converts an input of number of chirps per minute to degrees Fahrenheit, and let C = G(F) = \frac{5}{9}(F-32) be the function that converts an input of degrees Fahrenheit to an output of degrees Celsius.

- -

-

    -
  1. -

    +

    + + + +

    Determine a formula for the new function H = (G \circ D) that depends only on the variable N.

    -
    2. -
  2. -

    + + + + + + + +

    What is the meaning of the function you found in (a)?

    -
    3. -
  3. -

    + + + + + + + +

    How does a plot of the function H = (G \circ D) compare to that of Dolbear's function? Sketch a plot of y = H(N) = (G \circ D)(N) on the blank axes to the right of the plot of Dolbear's function, and discuss the similarities and differences between them. Be sure to label the vertical scale on your axes.

    - - -
    - Dolbear's function. - -
    -
    - Blank axes to plot H = (G \circ D)(N). - -
    -     -
    4. -
  4. -

    + +

    + Dolbear's function. + +
    +
    + Blank axes to plot H = (G \circ D)(N). + +
    + + + + + + + + +

    What is the domain of the function H = G \circ D? What is its range?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-composite-tables-graphs.xml b/source/activities/act-changing-composite-tables-graphs.xml index 96b5879d..6ba36325 100755 --- a/source/activities/act-changing-composite-tables-graphs.xml +++ b/source/activities/act-changing-composite-tables-graphs.xml @@ -1,4 +1,4 @@ - + @@ -12,117 +12,143 @@ - - - - -

+ + + +

Let functions p and q be given by the graphs in Figure (which are each piecewise linear - that is, parts that look like straight lines are straight lines) and let f and g be given by Table.

- - - - Table that defines <m>f</m> and <m>g</m>. - - - x - 0 - 1 - 2 - 3 - 4 - - - f(x) - 6 - 4 - 3 - 4 - 6 - - - g(x) - 1 - 3 - 0 - 4 - 2 - - -
- -
- The graphs of p and q. - -
- - -

+ + + Table that defines <m>f</m> and <m>g</m>. + + + + x + + 0 + 1 + 2 + 3 + 4 + + + + f(x) + + 6 + 4 + 3 + 4 + 6 + + + + g(x) + + 1 + 3 + 0 + 4 + 2 + + +
+

+ The graphs of p and q. + +
+ +

Compute each of the following quantities or explain why they are not defined.

- -

-

    -
  1. -

    - p(q(0)) -

    -
    2. - -
  2. -

    - q(p(0)) -

    -
    3. - -
  3. - (p \circ p)(-1) -
    4. - -
  4. -

    - (f \circ g)(2) -

    -
    5. - -
  5. -

    - (g \circ f)(3) -

    -
    6. - -
  6. -

    - g(f(0)) -

    -
    7. - -
  7. -

    +

    + + + +

    + p(q(0)) +

    + + + + + + + +

    + q(p(0)) +

    +     + + + +     + + + (p \circ p)(-1) + + + + + + + +

    + (f \circ g)(2) +

    +     + + + +     + + +

    + (g \circ f)(3) +

    +     + + + +     + + +

    + g(f(0)) +

    +     + + + +     + + +

    For what value(s) of x is f(g(x)) = 4?

    -
    8. - -
  8. -

    + + + + + + + +

    For what value(s) of x is q(p(x)) = 1?

    -
    9. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-functions-is-it.xml b/source/activities/act-changing-functions-is-it.xml index f703b938..28d94c7b 100755 --- a/source/activities/act-changing-functions-is-it.xml +++ b/source/activities/act-changing-functions-is-it.xml @@ -1,4 +1,4 @@ - + @@ -12,82 +12,114 @@ - - - - -

+ + + +

Each of the following prompts describes a relationship between two quantities. For each, your task is to decide whether or not the relationship can be thought of as a function. If not, explain why. If so, state the domain and codomain of the function and write at least one sentence to explain the process that leads from the collection of inputs to the collection of outputs.

- -

-

    -
  1. -

    +

    + + + +

    The relationship between x and y in each of the graphs below (address each graph separately as a potential situation where y is a function of x). In Figure, any point on the circle relates x and y. For instance, the y-value \sqrt{7} is related to the x-value -3. In Figure, any point on the blue curve relates x and y. For instance, when x = -1, the corresponding y-value is y = 3. An unfilled circle indicates that there is not a point on the graph at that specific location.

    - -
    - A circle of radius 4 centered at (0,0). - -
    -
    - A graph of a possible function g. - -
    -     -
    2. -
  2. -

    + +

    + A circle of radius 4 centered at (0,0). + +
    +
    + A graph of a possible function g. + +
    + + + + + + + + +

    The relationship between the day of the year and the value of the S&P500 stock index (at the close of trading on a given day), where we attempt to consider the index's value (at the close of trading) as a function of the day of the year.

    -
    3. -
  3. -

    + + + + + + + +

    The relationship between a car's velocity and its odometer, where we attempt to view the car's odometer reading as a function of its velocity.

    -
    4. -
  4. -

    + + + + + + + +

    The relationship between x and y that is given in the following table where we attempt to view y as depending on x.

    - - - A table that relates <m>x</m> and <m>y</m> values. - - - x - 1 - 2 - 3 - 2 - 1 - - - y - 11 - 12 - 13 - 14 - 15 - - -
    -
    5. - -
-

- - -

- -

- - -

- -

- - - - + + A table that relates <m>x</m> and <m>y</m> values. + + + + x + + + 1 + + + 2 + + + 3 + + + 2 + + + 1 + + + + + y + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + +
+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-functions-spherical-tank-draining.xml b/source/activities/act-changing-functions-spherical-tank-draining.xml index 7896bf73..6a51bc7c 100755 --- a/source/activities/act-changing-functions-spherical-tank-draining.xml +++ b/source/activities/act-changing-functions-spherical-tank-draining.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

Consider a spherical tank of radius 4 m that is completely full of water. Suppose that the tank is being drained by regulating an exit valve in such a way that the height of the water in the tank is always decreasing at a rate of 0.5 meters per minute. Let V be the volume of water in the tank (in cubic meters) at a given time t (in minutes), and h the depth of the water (in meters) at the same time. It can be shown using calculus @@ -26,56 +25,75 @@ . In addition, let h = q(t) be the function whose output is the depth of the water in the tank at time t.

- -

-

    -
  1. -

    +

    + + + +

    What is the height of the water when t = 0? When t = 1? When t = 2? How long will it take the tank to completely drain? Why?

    -
    2. -
  2. -

    + + + + + + + +

    What is the domain of the model h = q(t)? What is the domain of the model V = p(t)?

    -
    3. -
  3. -

    + + + + + + + +

    How much water is in the tank when the tank is full? What is the range of the model h = q(t)? What is the range of the model V = p(t)?

    -
    4. -
  4. -

    - We will frequently use a graphing utility to help us understand function behavior, and strongly recommend Desmos because it is intuitive, online, and free.To learn more about Desmos, see their outstanding online tutorials. -

    - -

    + + + + + + + +

    + We will frequently use a graphing utility to help us understand function behavior, and strongly recommend Desmos because it is intuitive, online, and free.To learn more about Desmos, see their outstanding online tutorials.

    +

    In this prepared Desmos worksheet, you can see how we enter the (abstract) function V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t), as well as the corresponding graph the program generates. Make as many observations as you can about the model V = p(t). You should discuss its shape and overall behavior, its domain, its range, and more.

    -
    5. -
  5. -

    + + + + + + + +

    How does the model V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t) differ from the abstract function y = r(x) = \frac{256\pi}{3} - \frac{\pi}{24} x^2(24-x)? In particular, how do the domain and range of the model differ from those of the abstract function, if at all?

    -
    6. -
  6. -

    + + + + + + + +

    How should the graph of the height function h = q(t) appear? Can you determine a formula for q? Explain your thinking.

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-functions-spherical-tank.xml b/source/activities/act-changing-functions-spherical-tank.xml index 3c1d3ba5..15bde161 100755 --- a/source/activities/act-changing-functions-spherical-tank.xml +++ b/source/activities/act-changing-functions-spherical-tank.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

Consider a spherical tank of radius 4 m that is filling with water. Let V be the volume of water in the tank (in cubic meters) at a given time, and h the depth of the water (in meters) at the same time. It can be shown using calculus @@ -25,48 +24,64 @@ V = f(h) = \frac{\pi}{3} h^2(12-h) .

- -

-

    -
  1. -

    +

    + + + +

    What values of h make sense to consider in the context of this function? What values of V make sense in the same context?

    -
    2. -
  2. -

    + + + + + + + +

    What is the domain of the function f in the context of the spherical tank? Why? What is the corresponding codomain? Why?

    -
    3. -
  3. -

    + + + + + + + +

    Determine and interpret (with appropriate units) the values f(2), f(4), and f(8). What is important about the value of f(8)?

    -
    4. -
  4. -

    + + + + + + + +

    Consider the claim: since f(9) = \frac{\pi}{3} 9^2(12-9) = 81\pi \approx 254.47, when the water is 9 meters deep, there is about 254.47 cubic meters of water in the tank. Is this claim valid? Why or why not? Further, does it make sense to observe that f(13) = -\frac{169\pi}{3}? Why or why not?

    -
    5. -
  5. -

    + + + + + + + +

    Can you determine a value of h for which f(h) = 300 cubic meters?

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-inverse-Dolbear.xml b/source/activities/act-changing-inverse-Dolbear.xml index 7b18709e..2b3a5ca8 100755 --- a/source/activities/act-changing-inverse-Dolbear.xml +++ b/source/activities/act-changing-inverse-Dolbear.xml @@ -1,4 +1,4 @@ - + @@ -12,49 +12,59 @@ - - - - -

+ + + +

Recall Dolbear's function F = D(N) = 40 + \frac{1}{4}N that converts the number, N, of snowy tree cricket chirps per minute to a corresponding Fahrenheit temperature. We have earlier established that the domain of D is [40,180] and the range of D is [50,85], as seen in Figure.

- -

-

    -
  1. -

    +

    + + + +

    Solve the equation F = 40 + \frac{1}{4}N for N in terms of F. Call the resulting function N = E(F).

    -
    2. -
  2. -

    + + + + + + + +

    Explain in words the process or effect of the function N = E(F). What does it take as input? What does it generate as output?

    -
    3. -
  3. -

    + + + + + + + +

    Use the function E that you found in (a.) to compute j(N) = E(D(N)). Simplify your result as much as possible. Do likewise for k(F) = D(E(F)). What do you notice about these two composite functions j and k?

    -
    4. -
  4. -

    + + + + + + + +

    Consider the equations F = 40 + \frac{1}{4}N and N = 4(F-40). Do these equations express different relationships between F and N, or do they express the same relationship in two different ways? Explain.

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-inverse-does-it.xml b/source/activities/act-changing-inverse-does-it.xml index bec6586e..45dbb577 100755 --- a/source/activities/act-changing-inverse-does-it.xml +++ b/source/activities/act-changing-inverse-does-it.xml @@ -1,4 +1,4 @@ - + @@ -12,111 +12,129 @@ - - - - -

+ + + +

Determine, with justification, whether each of the following functions has an inverse function. For each function that has an inverse function, give two examples of values of the inverse function by writing statements such as s^{-1}(3) = 1.

- -

-

    -
  1. -

    +

    + + + +

    The function f : S \to S given by Table, where S = \{0, 1, 2, 3, 4 \}.

    - - - Values of <m>y = f(x)</m>. - - - x - 0 - 1 - 2 - 3 - 4 - - - f(x) - 1 - 2 - 4 - 3 - 2 - - -
    - -
    2. -
  2. -

    + + Values of <m>y = f(x)</m>. + + + + x + + 0 + 1 + 2 + 3 + 4 + + + + f(x) + + 1 + 2 + 4 + 3 + 2 + + +
    + + + + + + + +

    The function g : S \to S given by Table, where S = \{0, 1, 2, 3, 4 \}.

    - - - Values of <m>y = g(x)</m>. - - - x - 0 - 1 - 2 - 3 - 4 - - - g(x) - 4 - 0 - 3 - 1 - 2 - - -
    -
    3. -
  3. -

    + + Values of <m>y = g(x)</m>. + + + + x + + 0 + 1 + 2 + 3 + 4 + + + + g(x) + + 4 + 0 + 3 + 1 + 2 + + +
    + + + + + + + +

    The function p given by p(t) = 7 - \frac{3}{5}t. Assume that the domain and codomain of p are both all real numbers.

    -
    4. -
  4. -

    + + + + + + + +

    The function q given by q(t) = 7 - \frac{3}{5}t^4. Assume that the domain and codomain of q are both all real numbers.

    -
    5. -
  5. -

    + + + + + + + +

    The functions r and s given by the graphs in Figure and Figure. Assume that the graphs show all of the important behavior of the functions and that the apparent trends continue beyond what is pictured.

    - - -
    - The graph of y = r(t). - -
    -
    - The graph of y = s(t). - -
    -     -
    6. -
-

- - - -

- -

- - -

- -

- - - - + +
+ The graph of y = r(t). + +
+
+ The graph of y = s(t). + +
+ + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-inverse-rainfall.xml b/source/activities/act-changing-inverse-rainfall.xml index 98d1c2f2..7d7e0001 100755 --- a/source/activities/act-changing-inverse-rainfall.xml +++ b/source/activities/act-changing-inverse-rainfall.xml @@ -1,4 +1,4 @@ - + @@ -12,57 +12,72 @@ - - - - -

+ + + +

During a major rainstorm, the rainfall at Gerald R. Ford Airport is measured on a frequent basis for a 10-hour period of time. The following function g models the rate, R, at which the rain falls (in cm/hr) on the time interval t = 0 to t = 10: R = g(t) = \frac{4}{t+2} + 1 .

- -

-

    -
  1. -

    +

    + + + +

    Compute g(3) and write a complete sentence to explain its meaning in the given context, including units.

    -
    2. -
  2. -

    + + + + + + + +

    Compute the average rate of change of g on the time interval [3,5] and write two careful complete sentences to explain the meaning of this value in the context of the problem, including units. Explicitly address what the value you compute tells you about how rain is falling over a certain time interval, and what you should expect as time goes on.

    -
    3. -
  3. -

    + + + + + + + +

    Plot the function y = g(t) using a computational device. On the domain [0,10], what is the corresponding range of g? Why does the function g have an inverse function?

    -
    4. -
  4. -

    + + + + + + + +

    Determine g^{-1} \left( \frac{9}{5} \right) and write a complete sentence to explain its meaning in the given context.

    -
    5. -
  5. -

    + + + + + + + +

    According to the model g, is there ever a time during the storm that the rain falls at a rate of exactly 1 centimeter per hour? Why or why not? Provide an algebraic justification for your answer.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-linear-Kilimanjaro.xml b/source/activities/act-changing-linear-Kilimanjaro.xml index d2a99269..5c90cf2e 100755 --- a/source/activities/act-changing-linear-Kilimanjaro.xml +++ b/source/activities/act-changing-linear-Kilimanjaro.xml @@ -1,4 +1,4 @@ - + @@ -12,24 +12,23 @@ - - - - -

+ + + +

The summit of Africa's largest peak, Mt. Kilimanjaro - The main context of this problem comes from Exercise 30 on p.27 of Connally's + The main context of this problem comes from Exercise 30 on p.27 of Connally's Functions Modeling Change, 5th ed. , has two main ice fields and a glacier at its peak. Geologists measured the ice cover in the year 2000 (t = 0) to be approximately 1951 m^2; in the year 2007, the ice cover measured 1555 m^2.

- -

-

    -
  1. -

    +

    + + + +

    Suppose that the amount of ice cover at the peak of Mt. Kilimanjaro is changing at a constant average rate from year to year. Find a linear model A = f(t) whose output is the area of the ice cover, @@ -37,50 +36,61 @@ in square meters in year t (where t is the number of years after 2000).

    -
    2. - -
  2. -

    + + + + + + + +

    What do the slope and A-intercept mean in the model you found in (a)? In particular, what are the units on the slope?

    -
    3. - -
  3. -

    + + + + + + + +

    Compute f(17). What does this quantity measure? Write a complete sentence to explain.

    -
    4. - -
  4. -

    + + + + + + + +

    If the model holds further into the future, when do we predict the ice cover will vanish?

    -
    5. - -
  5. -

    + + + + + + + +

    In light of your work above, what is a reasonable domain to use for the model A = f(t)? What is the corresponding range?

    -
    6. -
-

- - - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-linear-finding-eqs.xml b/source/activities/act-changing-linear-finding-eqs.xml index 451e255c..ee31bbc3 100755 --- a/source/activities/act-changing-linear-finding-eqs.xml +++ b/source/activities/act-changing-linear-finding-eqs.xml @@ -1,4 +1,4 @@ - + @@ -12,91 +12,121 @@ - - - - -

+ + + +

Find an equation for the line that is determined by the following conditions; write your answer in point-slope form wherever possible.

-

-

    -
  1. -

    +

    + + + +

    The line with slope \frac{3}{7} that passes through (-11, -17).

    -
    2. - -
  2. -

    + + + + + + + +

    The line passing through the points (-2,5) and (3,-1).

    -
    3. - -
  3. -

    + + + + + + + +

    The line passing through (4,9) that is parallel to the line 2x - 3y = 5.

    -
    4. - -
  4. -

    + + + + + + + +

    Explain why the function f given by Table appears to be linear and find a formula for f(x).

    - - - Data for a linear function <m>f</m>. - - - x - f(x) - - - 1 - 7 - - - 3 - 3 - - - 4 - 1 - - - 7 - -5 - - -
    - -
    - Plot of a linear function h. - -
    -     -
    5. -
  5. -

    + + + Data for a linear function <m>f</m>. + + + + x + + + f(x) + + + + + 1 + + + 7 + + + + + 3 + + + 3 + + + + + 4 + + + 1 + + + + + 7 + + + -5 + + + +
    +

    + Plot of a linear function h. + +
    + + + + + + + + +

    Find a formula for the linear function shown in Figure.

    -
    6. -
-

- - - - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-linear-in-context.xml b/source/activities/act-changing-linear-in-context.xml index 8aeae798..c55c28c2 100755 --- a/source/activities/act-changing-linear-in-context.xml +++ b/source/activities/act-changing-linear-in-context.xml @@ -1,4 +1,4 @@ - + @@ -12,55 +12,70 @@ - - - - -

+ + + +

In each of the following prompts, we investigate linear functions in context.

- -

-

    -
  1. -

    +

    + + + +

    A town's population initially has 28750 people present and then grows at a constant rate of 825 people per year. Find a linear model P = f(t) for the number of people in the town in year t.

    -
    2. -
  2. -

    + + + + + + + +

    A different town's population Q is given by the function Q = g(t) = 42505 - 465t. What is the slope of this function and what is its meaning in the model? Write a complete sentence to explain.

    -
    3. -
  3. -

    + + + + + + + +

    A spherical tank is being drained with a pump. Initially the tank is full with \frac{32\pi}{3} cubic feet of water. Assume the tank is drained at a constant rate of 1.2 cubic feet per minute. Find a linear model V = p(t) for the total amount of water in the tank at time t. In addition, what is a reasonable approximate domain for the model?

    -
    4. -
  4. -

    + + + + + + + +

    A conical tank is being filled in such a way that the height of the water in the tank, h (in feet), at time t (in minutes) is given by the function h = q(t) = 0.65t. What can you say about how the water level is rising? Write at least one careful sentence to explain.

    -
    5. -
  5. -

    + + + + + + + +

    Suppose we know that a 5-year old car's value is $10200, and that after 10 years its value is $4600. Assuming that the car's value depreciates linearly, find a function C = L(t) whose output is the value of the car in year t. What is a reasonable domain for the model? What is the value and meaning of the slope of the line? Write at least one careful sentence to explain.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-quadratic-falling-ball.xml b/source/activities/act-changing-quadratic-falling-ball.xml index 6e0a0e1b..685a47c0 100755 --- a/source/activities/act-changing-quadratic-falling-ball.xml +++ b/source/activities/act-changing-quadratic-falling-ball.xml @@ -1,4 +1,4 @@ - + @@ -12,71 +12,85 @@ - - - - -

+ + + +

A water balloon is tossed vertically from a window at an initial height of 37 feet and with an initial velocity of 41 feet per second.

- -

-

    -
  1. -

    +

    + + + +

    Determine a formula, s(t), for the function that models the height of the water balloon at time t.

    -
    2. - -
  2. -

    + + + + + + + +

    Plot the function in Desmos in an appropriate window.

    -
    3. - -
  3. -

    + + + + + + + +

    Use the graph to estimate the time the water balloon lands.

    -
    4. - -
  4. -

    + + + + + + + +

    Use algebra to find the exact time the water balloon lands.

    -
    5. - -
  5. -

    + + + + + + + +

    Determine the exact time the water balloon reaches its highest point and its height at that time.

    -
    6. - -
  6. -

    + + + + + + + +

    Compute the average rate of change of s on the intervals [1.5, 2], [2, 2.5], [2.5,3]. Include units on your answers and write one sentence to explain the meaning of the values you found. Sketch appropriate lines on the graph of s whose respective slopes are the values of these average rates of change.

    -
    7. -
-

- - - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-quadratic-parameters.xml b/source/activities/act-changing-quadratic-parameters.xml index 99c59144..ec754bea 100755 --- a/source/activities/act-changing-quadratic-parameters.xml +++ b/source/activities/act-changing-quadratic-parameters.xml @@ -1,4 +1,4 @@ - + @@ -12,65 +12,76 @@ - - - - -

+ + + +

Open a browser and point it to Desmos. In Desmos, enter q(x) = ax^2 + bx + c; you will be prompted to add sliders for a, b, and c. Do so. Then begin exploring with the sliders and respond to the following questions.

- -

-

    -
  1. -

    +

    + + + +

    Describe how changing the value of a affects the graph of q.

    -
    2. - -
  2. -

    + + + + + + + +

    Describe how changing the value of b affects the graph of q.

    -
    3. - -
  3. -

    + + + + + + + +

    Describe how changing the value of c affects the graph of q.

    -
    4. - -
  4. -

    + + + + + + + +

    Which parameter seems to have the simplest effect? Which parameter seems to have the most complicated effect? Why?

    -
    5. - -
  5. -

    + + + + + + + +

    Is it possible to find a formula for a quadratic function that passes through the points (0,8), (1,12), (2,12)? If yes, do so; if not, explain why not.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-quadratic-properties.xml b/source/activities/act-changing-quadratic-properties.xml index d16d4d79..2ec28fc7 100755 --- a/source/activities/act-changing-quadratic-properties.xml +++ b/source/activities/act-changing-quadratic-properties.xml @@ -1,4 +1,4 @@ - + @@ -12,54 +12,69 @@ - - - - -

+ + + +

Reason algebraically using appropriate properties of quadratic functions to answer the following questions. Use Desmos to check your results graphically.

- -

-

    -
  1. -

    +

    + + + +

    How many quadratic functions have x-intercepts at (-5,0) and (10,0) and a y-intercept at (0,-1)? Can you determine an exact formula for such a function? If yes, do so. If not, explain why.

    -
    2. -
  2. -

    + + + + + + + +

    Suppose that a quadratic function q has vertex (-3,-4) and opens upward. How many x-intercepts can you guarantee the function has? Why?

    -
    3. -
  3. -

    + + + + + + + +

    In addition to the information in (b), suppose you know that q(-1) = -3. Can you determine an exact formula for q? If yes, do so. If not, explain why.

    -
    4. -
  4. -

    - Does the quadratic function p(x) = -3(x+1)^2 + 9 have 0, 1, or 2 x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist. + + + + + + + +

    + Does the quadratic function p(x) = -3(x+1)^2 + 9 have 0, 1, or 2x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.

    -
    5. -
  5. -

    - Does the quadratic function w(x) = -2x^2 + 10x - 20 have 0, 1, or 2 x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist. + + + + + + + +

    + Does the quadratic function w(x) = -2x^2 + 10x - 20 have 0, 1, or 2x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-tandem-conical-tank.xml b/source/activities/act-changing-tandem-conical-tank.xml index de7f0ac6..ae6f4d12 100755 --- a/source/activities/act-changing-tandem-conical-tank.xml +++ b/source/activities/act-changing-tandem-conical-tank.xml @@ -1,4 +1,4 @@ - + @@ -12,90 +12,130 @@ - - - - -

+ + + +

Consider a tank in the shape of an inverted circular cone (point down) where the tank's radius is 2 feet and its depth is 4 feet. Suppose that the tank is being filled with water that is entering at a constant rate of 0.75 cubic feet per minute.

- -

-

    -
  1. -

    +

    + + + +

    Sketch a labeled picture of the tank, including a snapshot of there being water in the tank prior to the tank being completely full.

    -
    2. -
  2. -

    + + + + + + + +

    What are some quantities that are changing in this scenario? What are some quantities that are not changing?

    -
    3. -
  3. -

    + + + + + + + +

    Fill in the following table of values to determine how much water, V, is in the tank at a given time in minutes, t, and thus generate a graph of the relationship between volume and time by plotting the data on the provided axes.

    - - - Table to record data on volume and time in the conical tank. - - - t - V - - - 0 - - - - 1 - - - - 2 - - - - 3 - - - - 4 - - - - 5 - - - -
    -
    - How volume and time change in tandem in the conical tank. - -
    -     -
    4. -
  4. -

    + + + Table to record data on volume and time in the conical tank. + + + + t + + + V + + + + + 0 + + + + + + + + 1 + + + + + + + + 2 + + + + + + + + 3 + + + + + + + + 4 + + + + + + + + 5 + + + + + + +
    +

    + How volume and time change in tandem in the conical tank. + +
    + + + + + + + + +

    Finally, think about how the height, h, of the water changes in tandem with time. Without attempting to determine specific values of h at particular values of t, how would you expect the data for the relationship between h and t to appear? Use the provided axes to sketch at least two possibilities; write at least one sentence to explain how you think the graph should appear.

    - - - -
    5. -
-

- - -

- -

- - -

- -

- - - \ No newline at end of file + + + + + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-tandem-spherical-tank.xml b/source/activities/act-changing-tandem-spherical-tank.xml index b72cc01c..28cfd104 100755 --- a/source/activities/act-changing-tandem-spherical-tank.xml +++ b/source/activities/act-changing-tandem-spherical-tank.xml @@ -1,4 +1,4 @@ - + @@ -12,100 +12,138 @@ - - - - -

+ + + +

Consider a tank in the shape of a sphere where the tank's radius is 3 feet. Suppose that the tank is initially completely full and that it is being drained by a pump at a constant rate of 1.2 cubic feet per minute.

- -

-

    -
  1. -

    +

    + + + +

    Sketch a labeled picture of the tank, including a snapshot of some water remaining in the tank prior to the tank being completely empty.

    -
    2. -
  2. -

    + + + + + + + +

    What are some quantities that are changing in this scenario? What are some quantities that are not changing?

    -
    3. -
  3. -

    + + + + + + + +

    Recall that the volume of a sphere of radius r is V = \frac{4}{3} \pi r^3. When the tank is completely full at time t = 0 right before it starts being drained, how much water is present?

    -
    4. -
  4. -

    + + + + + + + +

    How long will it take for the tank to drain completely?

    -
    5. -
  5. -

    + + + + + + + +

    Fill in the following table of values to determine how much water, V, is in the tank at a given time in minutes, t, and thus generate a graph of the relationship between volume and time. Write a sentence to explain why the data's graph appears the way that it does.

    - - - Data for how volume and time change together. - - - t - V - - - 0 - - - - 20 - - - - 40 - - - - 60 - - - - 80 - - - - 94.24 - - - -
    -
    - A plot of how volume and time change in tandem in a draining spherical tank. - -
    -     -
    6. -
  6. -

    + + + Data for how volume and time change together. + + + + t + + + V + + + + + 0 + + + + + + 20 + + + + + + 40 + + + + + + 60 + + + + + + 80 + + + + + + 94.24 + + + + +
    +

    + A plot of how volume and time change in tandem in a draining spherical tank. + +
    + + + + + + + + +

    Finally, think about how the height of the water changes in tandem with time. What is the height of the water when t = 0? What is the height when the tank is empty? How would you expect the data for the relationship between h and t to appear? Use the provided axes to sketch at least two possibilities; write at least one sentence to explain how you think the graph should appear.

    - - - -
    7. -
-

- - -

- -

- - -

- -

- - - \ No newline at end of file + + + + + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-transformations-combined.xml b/source/activities/act-changing-transformations-combined.xml index 48a324f3..600679b9 100755 --- a/source/activities/act-changing-transformations-combined.xml +++ b/source/activities/act-changing-transformations-combined.xml @@ -1,4 +1,4 @@ - + @@ -12,65 +12,72 @@ - - - - -

+ + + +

Consider the functions f and g given in Figure and Figure.

- - -
- A parent function f. - -
-
- A parent function g. - -
- - -

-

    -
  1. -

    + +

    + A parent function f. + +
    +
    + A parent function g. + +
    + +

    + + + +

    Sketch an accurate graph of the transformation y = p(x) = -\frac{1}{2}f(x-1)+2. Write at least one sentence to explain how you developed the graph of p, and identify the point on p that corresponds to the original point (-2,2) on the graph of f.

    -
    2. -
  2. -

    + + + + + + + +

    Sketch an accurate graph of the transformation y = q(x) = 2g(x+0.5)-0.75. Write at least one sentence to explain how you developed the graph of q, and identify the point on q that corresponds to the original point (1.5,1.5) on the graph of g.

    -
    3. -
  3. -

    + + + + + + + +

    Is the function y = r(x) = \frac{1}{2}(-f(x-1) - 4) the same function as p or different? Why? Explain in two different ways: discuss the algebraic similarities and differences between p and r, and also discuss how each is a transformation of f.

    -
    4. -
  4. -

    + + + + + + + +

    Find a formula for a function y = s(x) (in terms of g) that represents this transformation of g: a horizontal shift of 1.25 units left, followed by a reflection across the x-axis and a vertical stretch by a factor of 2.5 units, followed by a vertical shift of 1.75 units. Sketch an accurate, labeled graph of s on the following axes along with the given parent function g.

    - - - - - -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-transformations-translations.xml b/source/activities/act-changing-transformations-translations.xml index 37ce015b..d3ccce80 100755 --- a/source/activities/act-changing-transformations-translations.xml +++ b/source/activities/act-changing-transformations-translations.xml @@ -1,4 +1,4 @@ - + @@ -12,60 +12,59 @@ - - - - -

+ + + +

Consider the functions r and s given in Figure and Figure.

- - -
- A parent function r. - -
-
- A parent function s. - -
- - -

-

    -
  1. -

    + +

    + A parent function r. + +
    +
    + A parent function s. + +
    + +

    + + + +

    On the same axes as the plot of y = r(x), sketch the following graphs: y = g(x) = r(x) + 2, y = h(x) = r(x+1), and y = f(x) = r(x+1) + 2. Be sure to label the point on each of g, h, and f that corresponds to (-2,-1) on the original graph of r. In addition, write one sentence to explain the overall transformations that have resulted in g, h, and f.

    -
    2. - -
  2. -

    + + + + + + + +

    On the same axes as the plot of y = s(x), sketch the following graphs: y = k(x) = s(x) - 1, y = j(x) = s(x-2), and y = m(x) = s(x-2) - 1. Be sure to label the point on each of k, j, and m that corresponds to (-2,-3) on the original graph of s. In addition, write one sentence to explain the overall transformations that have resulted in k, j, and m.

    -
    3. -
  3. -

    + + + + + + + +

    Now consider the function q(x) = x^2. Determine a formula for the function that is given by p(x) = q(x+3) - 4. How is p a transformation of q?

    -
    4. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-changing-transformations-vert-stretch.xml b/source/activities/act-changing-transformations-vert-stretch.xml index 866d5da8..e90de288 100755 --- a/source/activities/act-changing-transformations-vert-stretch.xml +++ b/source/activities/act-changing-transformations-vert-stretch.xml @@ -1,4 +1,4 @@ - + @@ -12,65 +12,73 @@ - - - - -

+ + + +

Consider the functions r and s given in Figure and Figure.

- - -
- A parent function r. - -
-
- A parent function s. - -
- - -

-

    -
  1. -

    + +

    + A parent function r. + +
    +
    + A parent function s. + +
    + +

    + + + +

    On the same axes as the plot of y = r(x), sketch the following graphs: y = g(x) = 3r(x) and y = h(x) = \frac{1}{3}r(x). Be sure to label the point on g and h that corresponds to the point (-2,-1) on the original graph of r. In addition, write one sentence to explain the overall transformations that have resulted in g and h from r.

    -
    2. -
  2. -

    + + + + + + + +

    On the same axes as the plot of y = s(x), sketch the following graphs: y = k(x) = -s(x) and y = j(x) = -\frac{1}{2}s(x). Be sure to label the point on k and j that corresponds to the point (-2,-3) on the original graph of s. In addition, write one sentence to explain the overall transformations that have resulted in k and j from s.

    -
    3. -
  3. -

    + + + + + + + +

    On the additional copies of the two figures below, sketch the graphs of the following transformed functions: y = m(x) = 2r(x+1)-1 (at left) and y = n(x) = \frac{1}{2}s(x-2)+2. As above, be sure to label a key point on each graph that corresonds to the labeled point on the original parent function.

    - - - - - -
    4. -
  4. -

    + + + + + + + + + + + +

    Describe in words how the function y = m(x) = 2r(x+1)-1 is the result of three elementary transformations of y = r(x). Does the order in which these transformations occur matter? Why or why not?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-sine-cosine-computing.xml b/source/activities/act-circular-sine-cosine-computing.xml index 710082ad..0bc4d3f6 100755 --- a/source/activities/act-circular-sine-cosine-computing.xml +++ b/source/activities/act-circular-sine-cosine-computing.xml @@ -1,4 +1,4 @@ - + @@ -12,64 +12,89 @@ - - - - -

+ + + +

Answer the following questions exactly wherever possible. If you estimate a value, do so to at least 5 decimal places of accuracy.

- -

-

    -
  1. -

    +

    + + + +

    The x coordinate of the point on the unit circle that lies in the third quadrant and whose y-coordinate is y = -\frac{3}{4}.

    -
    2. -
  2. -

    + + + + + + + +

    The y-coordinate of the point on the unit circle generated by a central angle opening counterclockwise with one side on the positive x-axis that measures t = 2 radians.

    -
    3. -
  3. -

    + + + + + + + +

    The x-coordinate of the point on the unit circle generated by a central angle with one side on the positive x-axis that measures t = -3.05 radians. (With the negative radian measure, we view the angle as opening counterclockwise from its initial side on the positive x-axis.)

    -
    4. -
  4. -

    + + + + + + + +

    The value of \cos(t) where t is an angle in Quadrant II that satisfies \sin(t) = \frac{1}{2}.

    -
    5. -
  5. -

    + + + + + + + +

    The value of \sin(t) where t is an angle in Quadrant III for which \cos(t) = -0.7.

    -
    6. -
  6. -

    + + + + + + + +

    The average rate of change of f(t) = \sin(t) on the intervals [0.1,0.2] and [0.8,0.9].

    -
    7. -
  7. -

    + + + + + + + +

    The average rate of change of g(t) = \cos(t) on the intervals [0.1,0.2] and [0.8,0.9].

    -
    8. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-sine-cosine-incr-CCU.xml b/source/activities/act-circular-sine-cosine-incr-CCU.xml index bba10416..6c9fcd61 100755 --- a/source/activities/act-circular-sine-cosine-incr-CCU.xml +++ b/source/activities/act-circular-sine-cosine-incr-CCU.xml @@ -1,4 +1,4 @@ - + @@ -12,68 +12,93 @@ - - - - -

+ + + +

Use Figure to assist in answering the following questions.

- -

-

    -
  1. -

    +

    + + + +

    Give an example of the largest interval you can find on which f(t) = \sin(t) is decreasing.

    -
    2. -
  2. -

    + + + + + + + +

    Give an example of the largest interval you can find on which f(t) = \sin(t) is decreasing and concave down.

    -
    3. -
  3. -

    + + + + + + + +

    Give an example of the largest interval you can find on which g(t) = \cos(t) is increasing.

    -
    4. -
  4. -

    + + + + + + + +

    Give an example of the largest interval you can find on which g(t) = \cos(t) is increasing and concave up.

    -
    5. -
  5. -

    + + + + + + + +

    Without doing any computation, on which interval is the average rate of change of g(t) = \cos(t) greater: [\pi, \pi+0.1] or [\frac{3\pi}{2}, \frac{3\pi}{2} + 0.1]? Why?

    -
    6. -
  6. -

    + + + + + + + +

    In general, how would you characterize the locations on the sine and cosine graphs where the functions are increasing or decreasingly most rapidly?

    -
    7. -
  7. -

    + + + + + + + +

    Thinking from the perspective of the unit circle, for which quadrants of the x-y plane is \cos(t) negative for an angle t that lies in that quadrant?

    -
    8. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-sine-cosine.xml b/source/activities/act-circular-sine-cosine.xml index 4886d731..87761cb3 100755 --- a/source/activities/act-circular-sine-cosine.xml +++ b/source/activities/act-circular-sine-cosine.xml @@ -1,4 +1,4 @@ - + @@ -12,120 +12,183 @@ - - - - -

+ + + +

Let k = g(t) be the function that tracks the x-coordinate of a point traversing the unit circle counterclockwise from (1,0). That is, g(t) = \cos(t). Use the information we know about the unit circle that is summarized in Figure to respond to the following questions.

- -

-

    -
  1. -

    +

    + + + +

    What is the exact value of \cos(\frac{\pi}{6})? of \cos(\frac{5\pi}{6})? \cos(-\frac{\pi}{3})?

    -
    2. -
  2. -

    + + + + + + + +

    Complete the following table with the exact values of k that correspond to the stated inputs.

    - - Exact values of <m>k = g(t) = \cos(t)</m>. - - - t - 0 - \frac{\pi}{6} - \frac{\pi}{4} - \frac{\pi}{3} - \frac{\pi}{2} - \frac{2\pi}{3} - \frac{3\pi}{4} - \frac{5\pi}{6} - \pi - - - k - - - - - - - - - - - - - - - t - \pi - \frac{7\pi}{6} - \frac{5\pi}{4} - \frac{4\pi}{3} - \frac{3\pi}{2} - \frac{5\pi}{3} - \frac{7\pi}{4} - \frac{11\pi}{6} - 2\pi - - - k - - - - - - - - - - - -
    -
    3. -
  3. -

    + + Exact values of <m>k = g(t) = \cos(t)</m>. + + + + t + + + 0 + + + \frac{\pi}{6} + + + \frac{\pi}{4} + + + \frac{\pi}{3} + + + \frac{\pi}{2} + + + \frac{2\pi}{3} + + + \frac{3\pi}{4} + + + \frac{5\pi}{6} + + + \pi + + + + + k + + + + + + + + + + + + + + + + + t + + + \pi + + + \frac{7\pi}{6} + + + \frac{5\pi}{4} + + + \frac{4\pi}{3} + + + \frac{3\pi}{2} + + + \frac{5\pi}{3} + + + \frac{7\pi}{4} + + + \frac{11\pi}{6} + + + 2\pi + + + + + k + + + + + + + + + + + + +
    + + + + + + + +

    On the axes provided in Figure, sketch an accurate graph of k = \cos(t). Label the exact location of several key points on the curve.

    - -
    - Axes for plotting k = \cos(t). - -
    -
    4. -
  4. -

    +

    + Axes for plotting k = \cos(t). + +
    + + + + + + + +

    What is the exact value of \cos( \frac{11\pi}{4} )? of \cos( \frac{14\pi}{3} )?

    -
    5. -
  5. -

    + + + + + + + +

    Give four different values of t for which \cos(t) = -\frac{\sqrt{3}}{2}.

    -
    6. -
  6. -

    + + + + + + + +

    How is the graph of k = \cos(t) different from the graph of h = \sin(t)? How are the graphs similar?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-sinusoidal-horiz-stretch.xml b/source/activities/act-circular-sinusoidal-horiz-stretch.xml index 50f7df83..7f76a6d5 100755 --- a/source/activities/act-circular-sinusoidal-horiz-stretch.xml +++ b/source/activities/act-circular-sinusoidal-horiz-stretch.xml @@ -1,4 +1,4 @@ - + @@ -12,65 +12,73 @@ - - - - -

+ + + +

Consider the functions f and g given in Figure and Figure.

- - -
- A parent function f. - -
-
- A parent function g. - -
- - -

-

    -
  1. -

    + +

    + A parent function f. + +
    +
    + A parent function g. + +
    + +

    + + + +

    On the same axes as the plot of y = f(t), sketch the following graphs: y = h(t) = f(\frac{1}{3}t) and y = j(t) = f(4t). Be sure to label several points on each of f, h, and j with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in h and j from f.

    -
    2. -
  2. -

    + + + + + + + +

    On the same axes as the plot of y = g(t), sketch the following graphs: y = k(t) = g(2t) and y = m(t) = g(\frac{1}{2}t). Be sure to label several points on each of g, k, and m with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in k and m from g.

    -
    3. -
  3. -

    + + + + + + + +

    On the additional copies of the two figures below, sketch the graphs of the following transformed functions: y = r(t) = 2f(\frac{1}{2}t) (at left) and y = s(t) = \frac{1}{2}g(2t). As above, be sure to label several points on each graph and indicate their correspondence to points on the original parent function.

    - - - - - -
    4. -
  4. -

    + + + + + + + + + + + +

    Describe in words how the function y = r(t) = 2f(\frac{1}{2}t) is the result of composing two elementary transformations of y = f(t). Does the order in which these transformations are composed matter? Why or why not?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-sinusoidal-model.xml b/source/activities/act-circular-sinusoidal-model.xml index 73d95450..2b40df48 100755 --- a/source/activities/act-circular-sinusoidal-model.xml +++ b/source/activities/act-circular-sinusoidal-model.xml @@ -1,4 +1,4 @@ - + @@ -12,34 +12,27 @@ - - - - -

+ + + +

Consider a spring-mass system where the weight is hanging from the ceiling in such a way that the following is known: we let d(t) denote the distance from the ceiling to the weight at time t in seconds and know that the weight oscillates periodically with a minimum value of 1.5 feet and a maximum value of 4 feet, with a period of 3, and you know d(0.5) = 2.75 and d\left(1.25\right) = 4.

- -

+

State the midline, amplitude, range, and an anchor point for the function, and hence determine a formula for d(t) in the form a\cos(k(t-b))+c or a\sin(k(t-b))+c. Show your work and thinking, and use Desmos appropriately to check that your formula generates the desired behavior.

- - -

- -

- - -

- -

- - - - + + +

+ + +

+ + + diff --git a/source/activities/act-circular-sinusoidal-oscillator.xml b/source/activities/act-circular-sinusoidal-oscillator.xml index cc9f4e45..7792b318 100755 --- a/source/activities/act-circular-sinusoidal-oscillator.xml +++ b/source/activities/act-circular-sinusoidal-oscillator.xml @@ -1,4 +1,4 @@ - + @@ -12,28 +12,21 @@ - - - - -

+ + + +

Consider a spring-mass system where a weight is resting on a frictionless table. We let d(t) denote the distance from the wall (where the spring is attached) to the weight at time t in seconds and know that the weight oscillates periodically with a minimum value of 2 feet and a maximum value of 7 feet with a period of 2 \pi. We also know that d(0) = 4.5 and d\left(\frac{\pi}{2}\right) = 2.

- -

+

Determine a formula for d(t) in the form d(t) = a\cos(t-b)+c or d(t) = a\sin(t-b)+c. Is it possible to find two different formulas that work? For any formula you find, identify the anchor point.

- - -

- -

- - -

- -

- - - - + + +

+ + +

+ + + diff --git a/source/activities/act-circular-sinusoidal-period.xml b/source/activities/act-circular-sinusoidal-period.xml index 81c42ab0..d1bac6b8 100755 --- a/source/activities/act-circular-sinusoidal-period.xml +++ b/source/activities/act-circular-sinusoidal-period.xml @@ -1,4 +1,4 @@ - + @@ -12,54 +12,69 @@ - - - - -

+ + + +

Determine the exact period, amplitude, and midline of each of the following functions. In addition, state the range of each function, any horizontal shift that has been introduced to the graph, and identify an anchor point. Make your conclusions without consulting Desmos, and then use the program to check your work.

- -

-

    -
  1. -

    - p(x) = \sin(10x) + 2 -

    -
    2. -
  2. -

    - q(x) = -3\cos(0.25x) - 4 -

    -
    3. -
  3. -

    - r(x) = 2\sin\left( \frac{\pi}{4} x\right) + 5 -

    -
    4. -
  4. -

    - w(x) = 2\cos\left( \frac{\pi}{2} (x-3) \right) + 5 -

    -
    5. -
  5. -

    - u(x) = -0.25\sin\left(3x-6\right) + 5 -

    -
    6. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ p(x) = \sin(10x) + 2 +

+ + + + + + + +

+ q(x) = -3\cos(0.25x) - 4 +

+ + + + + + + +

+ r(x) = 2\sin\left( \frac{\pi}{4} x\right) + 5 +

+ + + + + + + +

+ w(x) = 2\cos\left( \frac{\pi}{2} (x-3) \right) + 5 +

+ + + + + + + +

+ u(x) = -0.25\sin\left(3x-6\right) + 5 +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-traversing-2nd-ex.xml b/source/activities/act-circular-traversing-2nd-ex.xml index 4d0d7d6e..6f41a075 100755 --- a/source/activities/act-circular-traversing-2nd-ex.xml +++ b/source/activities/act-circular-traversing-2nd-ex.xml @@ -1,4 +1,4 @@ - + @@ -12,116 +12,206 @@ - - - - -

+ + + +

Consider the circle pictured in Figure that is centered at the point (2,2) and that has circumference 8. Assume that we track the y-coordinate (that is, the height, h) of a point that is traversing the circle counterclockwise and that it starts at P_0 as pictured.

- - -
- A point traversing the circle. - -
-
- Axes for plotting h as a function of d. - -
- - -

-

    -
  1. -

    + +

    + A point traversing the circle. + +
    +
    + Axes for plotting h as a function of d. + +
    + +

    + + + +

    How far along the circle is the point P_1 from P_0? Why?

    -
    2. -
  2. -

    + + + + + + + +

    Label the subsequent points in the figure P_2, P_3, \ldots as we move counterclockwise around the circle. What is the exact y-coordinate of the point P_2? of P_4? Why?

    -
    3. -
  3. -

    + + + + + + + +

    Determine the y-coordinates of the remaining points on the circle (exactly where possible, otherwise approximately) and hence complete the entries in Table that track the height, h, of the point traversing the circle as a function of distance traveled, d. Note that the d-values in the table correspond to the point traversing the circle more than once.

    - - - Data for <m>h</m> as a function of <m>d</m>. - - - d - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 16 - - - h - 2 - - - - - - - - - - - - - - - - - - -
    -
    4. -
  4. -

    + + Data for <m>h</m> as a function of <m>d</m>. + + + + d + + + 0 + + + 1 + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + + + h + + + 2 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + + + + + + +

    By plotting the points in Table and connecting them in an intuitive way, sketch a graph of h as a function of d on the axes provided in Figure over the interval 0 \le d \le 16. Clearly label the scale of your axes and the coordinates of several important points on the curve.

    -
    5. -
  5. -

    + + + + + + + +

    What is similar about your graph in comparison to the one in Figure? What is different?

    -
    6. -
  6. -

    + + + + + + + +

    What will be the value of h when d = 51? How about when d = 102?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-traversing-oscillator-aroc.xml b/source/activities/act-circular-traversing-oscillator-aroc.xml index ebd14f0b..9485d850 100755 --- a/source/activities/act-circular-traversing-oscillator-aroc.xml +++ b/source/activities/act-circular-traversing-oscillator-aroc.xml @@ -1,4 +1,4 @@ - + @@ -12,142 +12,230 @@ - - - - -

+ + + +

Consider the same setting as Activity: a weight oscillates back and forth on a frictionless table with distance from the wall given by, h = f(t) (in inches) at any given time, t (in seconds). A graph of f and a table of select values are given below.

- - - - - t - f(t) - - - 0.25 - 6.087 - - - 0.5 - 4.464 - - - 0.75 - 3.381 - - - 1 - 3.000 - - - 1.25 - 3.381 - - - 1.5 - 4.464 - - - 1.75 - 6.087 - - - 2 - 8.000 - - - - - - t - f(t) - - - 2.25 - 9.913 - - - 2.5 - 11.536 - - - 2.75 - 12.619 - - - 3 - 13.000 - - - 3.25 - 12.619 - - - 3.5 - 11.536 - - - 3.75 - 9.913 - - - 4 - 8.000 - - - - - - -

-

    -
  1. -

    + + + + + t + + + f(t) + + + + + 0.25 + + + 6.087 + + + + + 0.5 + + + 4.464 + + + + + 0.75 + + + 3.381 + + + + + 1 + + + 3.000 + + + + + 1.25 + + + 3.381 + + + + + 1.5 + + + 4.464 + + + + + 1.75 + + + 6.087 + + + + + 2 + + + 8.000 + + + + + + + t + + + f(t) + + + + + 2.25 + + + 9.913 + + + + + 2.5 + + + 11.536 + + + + + 2.75 + + + 12.619 + + + + + 3 + + + 13.000 + + + + + 3.25 + + + 12.619 + + + + + 3.5 + + + 11.536 + + + + + 3.75 + + + 9.913 + + + + + 4 + + + 8.000 + + + + + +

    + + + +

    Determine AV_{[2,2.25]}, AV_{[2.25,2.5]}, AV_{[2.5,2.75]}, and AV_{[2.75,3]}. What do these four values tell us about how the weight is moving on the interval [2,3]?

    -
    2. -
  2. -

    + + + + + + + +

    Give an example of an interval of length 0.25 units on which f has its most negative average rate of change. Justify your choice.

    -
    3. -
  3. -

    + + + + + + + +

    Give an example of the longest interval you can find on which f is decreasing.

    -
    4. -
  4. -

    - Give an example of an interval on which f is concave up.Recall that a function is concave up on an interval provided that throughout the interval, the curve bends upward, similar to a parabola that opens up. -

    -
    5. -
  5. -

    + + + + + + + +

    + Give an example of an interval on which f is concave up.Recall that a function is concave up on an interval provided that throughout the interval, the curve bends upward, similar to a parabola that opens up.

    + + + + + + + +

    On an interval where f is both decreasing and concave down, what does this tell us about how the weight is moving on that interval? For instance, is the weight moving toward or away from the wall? is it speeding up or slowing down?

    -
    6. -
  6. -

    + + + + + + + +

    What general conclusions can you make about the average rate of change of a circular function on intervals near its highest or lowest points? about its average rate of change on intervals near the function's midline?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-traversing-oscillator.xml b/source/activities/act-circular-traversing-oscillator.xml index ee7d66cb..b90479f6 100755 --- a/source/activities/act-circular-traversing-oscillator.xml +++ b/source/activities/act-circular-traversing-oscillator.xml @@ -1,4 +1,4 @@ - + @@ -12,134 +12,213 @@ - - - - -

+ + + +

A weight is placed on a frictionless table next to a wall and attached to a spring that is fixed to the wall. From its natural position of rest, the weight is imparted an initial velocity that sets it in motion. The weight then oscillates back and forth, and we can measure its distance, h = f(t) (in inches) from the wall at any given time, t (in seconds). A graph of f and a table of select values are given below.

- - - - - t - f(t) - - - 0.25 - 6.087 - - - 0.5 - 4.464 - - - 0.75 - 3.381 - - - 1 - 3.000 - - - 1.25 - 3.381 - - - 1.5 - 4.464 - - - 1.75 - 6.087 - - - 2 - 8.000 - - - - - - t - f(t) - - - 2.25 - 9.913 - - - 2.5 - 11.536 - - - 2.75 - 12.619 - - - 3 - 13.000 - - - 3.25 - 12.619 - - - 3.5 - 11.536 - - - 3.75 - 9.913 - - - 4 - 8.000 - - - - - - -

-

    -
  1. -

    + + + + + t + + + f(t) + + + + + 0.25 + + + 6.087 + + + + + 0.5 + + + 4.464 + + + + + 0.75 + + + 3.381 + + + + + 1 + + + 3.000 + + + + + 1.25 + + + 3.381 + + + + + 1.5 + + + 4.464 + + + + + 1.75 + + + 6.087 + + + + + 2 + + + 8.000 + + + + + + + t + + + f(t) + + + + + 2.25 + + + 9.913 + + + + + 2.5 + + + 11.536 + + + + + 2.75 + + + 12.619 + + + + + 3 + + + 13.000 + + + + + 3.25 + + + 12.619 + + + + + 3.5 + + + 11.536 + + + + + 3.75 + + + 9.913 + + + + + 4 + + + 8.000 + + + + + +

    + + + +

    Determine the period p, midline y = m, and amplitude a of the function f.

    -
    2. -
  2. -

    + + + + + + + +

    What is the greatest distance the weight is displaced from the wall? What is the least distance the weight is displaced from the wall? What is the range of f?

    -
    3. -
  3. -

    + + + + + + + +

    Determine the average rate of change of f on the intervals [4,4.25] and [4.75,5]. Write one careful sentence to explain the meaning of each (including units). In addition, write a sentence to compare the two different values you find and what they together say about the motion of the weight.

    -
    4. -
  4. -

    + + + + + + + +

    Based on the periodicity of the function, what is the value of f(6.75)? of f(11.25)?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-unit-circle-non-unit.xml b/source/activities/act-circular-unit-circle-non-unit.xml index 710bce49..b67e9c13 100755 --- a/source/activities/act-circular-unit-circle-non-unit.xml +++ b/source/activities/act-circular-unit-circle-non-unit.xml @@ -1,4 +1,4 @@ - + @@ -12,53 +12,60 @@ - - - - -

+ + + +

Determine each of the following values or points exactly.

- -

-

    -
  1. -

    +

    + + + +

    In a circle of radius 11, the arc length intercepted by a central angle of \frac{5\pi}{3}.

    -
    2. - -
  2. -

    + + + + + + + +

    In a circle of radius 3, the central angle measure that intercepts an arc of length \frac{\pi}{4}.

    -
    3. - -
  3. -

    + + + + + + + +

    The radius of the circle in which an angle of \frac{7\pi}{6} intercepts an arc of length \frac{\pi}{2}.

    -
    4. - -
  4. -

    + + + + + + + +

    The exact coordinates of the point on the circle of radius 5 that lies \frac{25\pi}{6} units counterclockwise along the circle from (5,0).

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-unit-circle-radians-degrees.xml b/source/activities/act-circular-unit-circle-radians-degrees.xml index 2c0a6b0e..5ffb8880 100755 --- a/source/activities/act-circular-unit-circle-radians-degrees.xml +++ b/source/activities/act-circular-unit-circle-radians-degrees.xml @@ -1,4 +1,4 @@ - + @@ -12,64 +12,76 @@ - - - - -

+ + + +

Convert each of the following quantities to the alternative measure: degrees to radians or radians to degrees.

- -

-

    -
  1. -

    - 30^\circ +

    + + + +

    + 30^\circ +

    + + + + + + + +

    \frac{2\pi}{3} radians

    -
    2. - -
  2. -

    - \frac{2\pi}{3} radians + + + + + + + +

    \frac{5\pi}{4} radians

    -
    3. - -
  3. -

    - \frac{5\pi}{4} radians + + + + + + + +

    + 240^\circ +

    + + + + + + + +

    + 17^\circ +

    +     + + + +     + + +

    2 radians

    -
    4. - -
  4. -

    - 240^\circ -

    -
    5. - -
  5. -

    - 17^\circ -

    -
    6. - -
  6. -

    - 2 radians -

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-circular-unit-circle-special-triangles.xml b/source/activities/act-circular-unit-circle-special-triangles.xml index 089120af..3e106279 100755 --- a/source/activities/act-circular-unit-circle-special-triangles.xml +++ b/source/activities/act-circular-unit-circle-special-triangles.xml @@ -1,4 +1,4 @@ - + @@ -12,81 +12,88 @@ - - - - -

+ + + +

In what follows, we work to understand key relationships in 45^\circ-45^\circ-90^\circ and 30^\circ-60^\circ-90^\circ triangles.

- - -
- A right triangle with two 45^\circ angles. - -
-
- A right triangle with a 30^\circ angle. - -
- - -

-

    -
  1. -

    + +

    + A right triangle with two 45^\circ angles. + +
    +
    + A right triangle with a 30^\circ angle. + +
    + +

    + + + +

    For the 45^\circ-45^\circ-90^\circ triangle with legs of length x and y and hypotenuse of length 1, what does the fact that the triangle is isosceles tell us about the relationship between x and y? What are their exact values?

    -
    2. -
  2. -

    + + + + + + + +

    Now consider the 30^\circ-60^\circ-90^\circ triangle with hypotenuse of length 1 and the longer leg (of length x) lying along the positive x-axis. What special kind of triangle is formed when we reflect this triangle across the x-axis? How can we use this perspective to determine the exact values of x and y?

    -
    3. -
  3. -

    + + + + + + + +

    Suppose we consider the related 30^\circ-60^\circ-90^\circ triangle with hypotenuse of length 1 and the shorter leg (of length x) lying along the positive x-axis. What are the exact values of x and y in this triangle?

    -
    4. -
  4. -

    + + + + + + + +

    We know from the conversion factor from degrees to radians that an angle of 30^\circ corresponds to an angle measuring \frac{\pi}{6} radians, an angle of 45^\circ corresponds to \frac{\pi}{4} radians, and 60^\circ corresponds to \frac{\pi}{3} radians.

    - - -
    - An angle measuring \frac{\pi}{6} radians. - -
    -
    - An angle measuring \frac{\pi}{4} radians. - -
    -
    - An angle measuring \frac{\pi}{3} radians. - -
    -     - -

    + +

    + An angle measuring \frac{\pi}{6} radians. + +
    +
    + An angle measuring \frac{\pi}{4} radians. + +
    +
    + An angle measuring \frac{\pi}{3} radians. + +
    + +

    Use your work in (a), (b), and (c) to label the noted point in each of Figure, Figure, and Figure, respectively, with its exact coordinates.

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-e-aroc-e.xml b/source/activities/act-exp-e-aroc-e.xml index 665bd0a0..9f7d5599 100755 --- a/source/activities/act-exp-e-aroc-e.xml +++ b/source/activities/act-exp-e-aroc-e.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

Recall from Section that the average rate of change of a function f on an interval [a,b] is AV_{[a,b]} = \frac{f(b)-f(a)}{b-a} @@ -27,58 +26,72 @@ . In this activity we explore the average rate of change of f(t) = e^t near the points where t = 1 and t = 2.

- -

+

In a new Desmos worksheet, let f(t) = e^t and define the function A by the rule A(h) = \frac{f(1+h)-f(1)}{h} .

- -

-

    -
  1. -

    +

    + + + +

    What is the meaning of A(0.5) in terms of the function f and its graph?

    -
    2. -
  2. -

    + + + + + + + +

    Compute the value of A(h) for at least 6 different small values of h, both positive and negative. For instance, one value to try might be h = 0.0001. Record a table of your results.

    -
    3. -
  3. -

    + + + + + + + +

    What do you notice about the values you found in (b)? How do they compare to an important number?

    -
    4. -
  4. -

    - Explain why the following sentence makes sense: The function e^t is increasing at an average rate that is about the same as its value on small intervals near t = 1. -

    -
    5. -
  5. -

    + + + + + + + +

    + Explain why the following sentence makes sense: The function e^t is increasing at an average rate that is about the same as its value on small intervals near t = 1.

    + + + + + + + +

    Adjust your definition of A in Desmos by changing 1 to 2 so that A(h) = \frac{f(2+h)-f(2)}{h} . How does the value of A(h) compare to f(2) for small values of h?

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-e-graph-solve.xml b/source/activities/act-exp-e-graph-solve.xml index c8c63606..a7b9aa51 100755 --- a/source/activities/act-exp-e-graph-solve.xml +++ b/source/activities/act-exp-e-graph-solve.xml @@ -1,4 +1,4 @@ - + @@ -12,59 +12,79 @@ - - - - -

+ + + +

By graphing f(t) = e^t and appropriate horizontal lines, estimate the solution to each of the following equations. Note that in some parts, you may need to do some algebraic work in addition to using the graph.

- -

-

    -
  1. -

    - e^t = 2 -

    -
    2. -
  2. -

    - e^{3t} = 5 -

    -
    3. -
  3. -

    - 2e^t - 4 = 7 -

    -
    4. -
  4. -

    - 3e^{0.25t} + 2 = 6 -

    -
    5. -
  5. -

    - 4 - 2e^{-0.7t} = 3 -

    -
    6. -
  6. -

    - 2e^{1.2t} = 1.5e^{1.6t} -

    -
    7. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ e^t = 2 +

+ + + + + + + +

+ e^{3t} = 5 +

+ + + + + + + +

+ 2e^t - 4 = 7 +

+ + + + + + + +

+ 3e^{0.25t} + 2 = 6 +

+ + + + + + + +

+ 4 - 2e^{-0.7t} = 3 +

+ + + + + + + +

+ 2e^{1.2t} = 1.5e^{1.6t} +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-growth-a-b-t.xml b/source/activities/act-exp-growth-a-b-t.xml index 7ba4e75b..a8ea4254 100755 --- a/source/activities/act-exp-growth-a-b-t.xml +++ b/source/activities/act-exp-growth-a-b-t.xml @@ -1,4 +1,4 @@ - + @@ -12,63 +12,83 @@ - - - - -

+ + + +

In Desmos, define the function g(t) = ab^t and create sliders for both a and b when prompted. Click on the sliders to set the minimum value for each to 0.1 and the maximum value to 10. Note that for g to be an exponential function, we require b \ne 1, even though the slider for b will allow this value.

- -

-

    -
  1. -

    +

    + + + +

    What is the domain of g(t) = ab^t?

    -
    2. -
  2. -

    + + + + + + + +

    What is the range of g(t) = ab^t?

    -
    3. -
  3. -

    + + + + + + + +

    What is the y-intercept of g(t) = ab^t?

    -
    4. -
  4. -

    + + + + + + + +

    How does changing the value of b affect the shape and behavior of the graph of g(t) = ab^t? Write several sentences to explain.

    -
    5. -
  5. -

    + + + + + + + +

    For what values of the growth factor b is the corresponding growth rate positive? For which b-values is the growth rate negative?

    -
    6. -
  6. -

    + + + + + + + +

    Consider the graphs of the exponential functions p and q provided in Figure. If p(t) = ab^t and q(t) = cd^t, what can you say about the values a, b, c, and d (beyond the fact that all are positive and b \ne 1 and d \ne 1)? For instance, can you say a certain value is larger than another? Or that one of the values is less than 1?

    -
    - Graphs of exponential functions p and q. - -
    -
    7. -
-

- - -

- -

- - -

- -

- - - - +
+ Graphs of exponential functions p and q. + +
+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-growth-find-a-b.xml b/source/activities/act-exp-growth-find-a-b.xml index 3e61bf45..b776c78d 100755 --- a/source/activities/act-exp-growth-find-a-b.xml +++ b/source/activities/act-exp-growth-find-a-b.xml @@ -1,4 +1,4 @@ - + @@ -12,49 +12,59 @@ - - - - -

+ + + +

The value of an automobile is depreciating. When the car is 3 years old, its value is $12500; when the car is 7 years old, its value is $6500.

- -

-

    -
  1. -

    +

    + + + +

    Suppose the car's value t years after its purchase is given by the function V(t) and that V is exponential with form V(t) = ab^t, what are the values of a and b? Find a and b both exactly and approximately.

    -
    2. -
  2. -

    + + + + + + + +

    Using the exponential model determined in (a), determine the purchase value of the car and then use Desmosestimate when the car will be worth less than $1000.

    -
    3. -
  3. -

    + + + + + + + +

    Suppose instead that the car's value is modeled by a linear function L and satisfies the values stated at the outset of this activity. Find a formula for L(t) and determine both the purchase value of the car and when the car will be worth $1000.

    -
    4. -
  4. -

    + + + + + + + +

    Which model do you think is more realistic? Why?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-growth-rates.xml b/source/activities/act-exp-growth-rates.xml index f602d6af..d7661eab 100755 --- a/source/activities/act-exp-growth-rates.xml +++ b/source/activities/act-exp-growth-rates.xml @@ -1,4 +1,4 @@ - + @@ -12,54 +12,69 @@ - - - - -

+ + + +

For each of the following prompts, give an example of a function that satisfies the stated characteristics by both providing a formula and sketching a graph.

- -

-

    -
  1. -

    +

    + + + +

    A function p that is always decreasing and decreases at a constant rate.

    -
    2. -
  2. -

    + + + + + + + +

    A function q that is always increasing and increases at an increasing rate.

    -
    3. -
  3. -

    + + + + + + + +

    A function r that is always increasing for t \lt 2, always decreasing for t \gt 2, and is always changing at a decreasing rate.

    -
    4. -
  4. -

    + + + + + + + +

    A function s that is always increasing and increases at a decreasing rate. (Hint: to find a formula, think about how you might use a transformation of a familiar function.)

    -
    5. -
  5. -

    + + + + + + + +

    A function u that is always decreasing and decreases at a decreasing rate.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-log-base-10.xml b/source/activities/act-exp-log-base-10.xml index 462cdea9..5c1e3867 100755 --- a/source/activities/act-exp-log-base-10.xml +++ b/source/activities/act-exp-log-base-10.xml @@ -1,4 +1,4 @@ - + @@ -12,64 +12,89 @@ - - - - -

+ + + +

For each of the following equations, determine the exact value of the unknown variable. If the exact value involves a logarithm, use a computational device to also report an approximate value. For instance, if the exact value is y = \log_{10}(2), you can also note that y \approx 0.301.

- -

-

    -
  1. -

    - 10^t = 0.00001 -

    -
    2. -
  2. -

    - \log_{10}(1000000) = t -

    -
    3. -
  3. -

    - 10^t = 37 -

    -
    4. -
  4. -

    - \log_{10}(y) = 1.375 -

    -
    5. -
  5. -

    - 10^t = 0.04 -

    -
    6. -
  6. -

    - 3 \cdot 10^t + 11 = 147 -

    -
    7. -
  7. -

    - 2\log_{10}(y) + 5 = 1 -

    -
    8. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ 10^t = 0.00001 +

+ + + + + + + +

+ \log_{10}(1000000) = t +

+ + + + + + + +

+ 10^t = 37 +

+ + + + + + + +

+ \log_{10}(y) = 1.375 +

+ + + + + + + +

+ 10^t = 0.04 +

+ + + + + + + +

+ 3 \cdot 10^t + 11 = 147 +

+ + + + + + + +

+ 2\log_{10}(y) + 5 = 1 +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-log-equations.xml b/source/activities/act-exp-log-equations.xml index 76113404..4ae48412 100755 --- a/source/activities/act-exp-log-equations.xml +++ b/source/activities/act-exp-log-equations.xml @@ -1,4 +1,4 @@ - + @@ -12,69 +12,99 @@ - - - - -

+ + + +

Solve each of the following equations for the exact value of the unknown variable. If there is no solution to the equation, explain why not.

- -

-

    -
  1. -

    - e^t = \frac{1}{10} -

    -
    2. -
  2. -

    - 5e^{t}=7 -

    -
    3. -
  3. -

    - \ln(t) = -\frac{1}{3} -

    -
    4. -
  4. -

    - e^{1-3t} = 4 -

    -
    5. -
  5. -

    - 2\ln(t) + 1 = 4 -

    -
    6. -
  6. -

    - 4 - 3e^{2t} = 2 -

    -
    7. -
  7. -

    - 4 + 3e^{2t} = 2 -

    -
    8. -
  8. -

    - \ln(5 - 6t) = -2 -

    -
    9. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ e^t = \frac{1}{10} +

+ + + + + + + +

+ 5e^{t}=7 +

+ + + + + + + +

+ \ln(t) = -\frac{1}{3} +

+ + + + + + + +

+ e^{1-3t} = 4 +

+ + + + + + + +

+ 2\ln(t) + 1 = 4 +

+ + + + + + + +

+ 4 - 3e^{2t} = 2 +

+ + + + + + + +

+ 4 + 3e^{2t} = 2 +

+ + + + + + + +

+ \ln(5 - 6t) = -2 +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-log-exponential-equations.xml b/source/activities/act-exp-log-exponential-equations.xml index 05ef7c92..e4ddeb9e 100755 --- a/source/activities/act-exp-log-exponential-equations.xml +++ b/source/activities/act-exp-log-exponential-equations.xml @@ -1,4 +1,4 @@ - + @@ -12,65 +12,79 @@ - - - - -

+ + + +

Solve each of the following equations exactly and then find an estimate that is accurate to 5 decimal places.

- -

-

    -
  1. -

    - 3^t = 5 -

    -
    2. - -
  2. -

    - 4 \cdot 2^t - 2 = 3 -

    -
    3. - -
  3. -

    - 3.7 \cdot (0.9)^{0.3t} + 1.5 = 2.1 -

    -
    4. - -
  4. -

    - 72 - 30(0.7)^{0.05t} = 60 -

    -
    5. - -
  5. -

    - \ln(t) = -2 -

    -
    6. - -
  6. -

    - 3 + 2\log_{10}(t) = 3.5 -

    -
    7. - -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ 3^t = 5 +

+ + + + + + + +

+ 4 \cdot 2^t - 2 = 3 +

+ + + + + + + +

+ 3.7 \cdot (0.9)^{0.3t} + 1.5 = 2.1 +

+ + + + + + + +

+ 72 - 30(0.7)^{0.05t} = 60 +

+ + + + + + + +

+ \ln(t) = -2 +

+ + + + + + + +

+ 3 + 2\log_{10}(t) = 3.5 +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-log-natural.xml b/source/activities/act-exp-log-natural.xml index 3b623822..b40b0263 100755 --- a/source/activities/act-exp-log-natural.xml +++ b/source/activities/act-exp-log-natural.xml @@ -1,4 +1,4 @@ - + @@ -12,108 +12,149 @@ - - - - -

+ + + +

Let E(t) = e^t and N(y) = \ln(y) be the natural exponential function and the natural logarithm function, respectively.

- -

-

    -
  1. -

    +

    + + + +

    What are the domain and range of E?

    -
    2. -
  2. -

    + + + + + + + +

    What are the domain and range of N?

    -
    3. -
  3. -

    + + + + + + + +

    What can you say about \ln(e^t) for every real number t?

    -
    4. -
  4. -

    + + + + + + + +

    What can you say about e^{\ln(y)} for every positive real number y?

    -
    5. -
  5. -

    + + + + + + + +

    Complete Table and Table with both exact and approximate values of E and N. Then, plot the corresponding ordered pairs from each table on the axes provided in Figure and connect the points in an intuitive way. When you plot the ordered pairs on the axes, in both cases view the first line of the table as generating values on the horizontal axis and the second line of the table as producing values on the vertical axisNote that when we take this perspective for plotting the data in Table, we are viewing N as a function of t, writing N(t) = \ln(t) in order to plot the function on the t-y axes; label each ordered pair you plot appropriately.

    - - -

    - - Values of <m>y = E(t)</m>. - - - t - -2 - -1 - 0 - 1 - 2 - - - E(t)=e^t - e^{-2} \approx 0.135 - - - - - - -
    - - - Values of <m>t = N(y)</m>. - - - y - e^{-2} - e^{-1} - 1 - e^1 - e^2 - - - N(y)=\ln(y) - -2 - - - - - - -
    -

    - -
    - Axes for plotting data from Table and Table along with the graphs of the natural exponential and natural logarithm functions. - -
    - - -
    6. -
-

- - -

- -

- - -

- -

- - - - +

+ + Values of <m>y = E(t)</m>. + + + + t + + + -2 + + + -1 + + + 0 + + + 1 + + + 2 + + + + + E(t)=e^t + + + e^{-2} \approx 0.135 + + + + + + + +
+ + Values of <m>t = N(y)</m>. + + + + y + + + e^{-2} + + + e^{-1} + + + 1 + + + e^1 + + + e^2 + + + + + N(y)=\ln(y) + + + -2 + + + + + + + +
+

+
+ Axes for plotting data from Table and Table along with the graphs of the natural exponential and natural logarithm functions. + +
+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-log-properties-exp-or-log.xml b/source/activities/act-exp-log-properties-exp-or-log.xml index 35877dda..6bf579df 100755 --- a/source/activities/act-exp-log-properties-exp-or-log.xml +++ b/source/activities/act-exp-log-properties-exp-or-log.xml @@ -1,4 +1,4 @@ - + @@ -12,88 +12,100 @@ - - - - -

+ + + +

In the questions that follow, we compare and contrast the properties and behaviors of exponential and logarithmic functions.

- -

-

    -
  1. -

    +

    + + + +

    Let f(t) = 1 - e^{-(t-1)} and g(t) = \ln(t). Plot each function on the same set of coordinate axes. What properties do the two functions have in common? For what properties do the two functions differ? Consider each function's domain, range, t-intercept, y-intercept, increasing/decreasing behavior, concavity, and long-term behavior.

    -
    2. -
  2. -

    + + + + + + + +

    Let h(t) = a - be^{-k(t-c)}, where a, b, c, and k are positive constants. Describe h as a transformation of the function E(t) = e^t.

    -
    3. -
  3. -

    + + + + + + + +

    Let r(t) = a + b\ln(t-c), where a, b, and c are positive constants. Describe r as a transformation of the function L(t) = \ln(t).

    -
    4. -
  4. -

    + + + + + + + +

    Data for the height of a tree is given in the Table; time t is measured in years and height is given in feet. At http://gvsu.edu/s/0yy, you can find a Desmos worksheet with this data already input.

    - - - The height of a tree as a function of time <m>t</m> in years. - - - t - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - - - h(t) - 6 - 9.5 - 13 - 15 - 16.5 - 17.5 - 18.5 - 19 - 19.5 - 19.7 - 19.8 - - -
    - -

    + + The height of a tree as a function of time <m>t</m> in years. + + + + t + + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + + + + h(t) + + 6 + 9.5 + 13 + 15 + 16.5 + 17.5 + 18.5 + 19 + 19.5 + 19.7 + 19.8 + + +
    +

    Do you think this data is better modeled by a logarithmic function of form p(t) = a + b\ln(t-c) or by an exponential function of form q(t) = m + ne^{-rt}. Provide reasons based in how the data appears and how you think a tree grows, as well as by experimenting with sliders appropriately in Desmos. (Note: you may need to adjust the upper and lower bounds of several of the sliders in order to match the data well.)

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-log-properties-find-k.xml b/source/activities/act-exp-log-properties-find-k.xml index 03a07601..91d41002 100755 --- a/source/activities/act-exp-log-properties-find-k.xml +++ b/source/activities/act-exp-log-properties-find-k.xml @@ -1,4 +1,4 @@ - + @@ -12,49 +12,59 @@ - - - - -

+ + + +

Solve each of the following equations for the exact value of k.

- -

-

    -
  1. -

    - 41 = 50e^{-k \cdot 7} -

    -
    2. -
  2. -

    - 65 = 34 + 47e^{-k \cdot 45} -

    -
    3. -
  3. -

    - 7e^{2k-1} + 4 = 32 -

    -
    4. -
  4. -

    - \frac{5}{1+2e^{-10k}} = 4 -

    -
    5. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ 41 = 50e^{-k \cdot 7} +

+ + + + + + + +

+ 65 = 34 + 47e^{-k \cdot 45} +

+ + + + + + + +

+ 7e^{2k-1} + 4 = 32 +

+ + + + + + + +

+ \frac{5}{1+2e^{-10k}} = 4 +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-modeling-behavior.xml b/source/activities/act-exp-modeling-behavior.xml index 70e4e4da..9a46b35e 100755 --- a/source/activities/act-exp-modeling-behavior.xml +++ b/source/activities/act-exp-modeling-behavior.xml @@ -1,4 +1,4 @@ - + @@ -12,77 +12,111 @@ - - - - -

+ + + +

For each of the following functions, without using graphing technology, determine whether the function is -

    -
  1. -

    + + In addition, state the y-intercept and the range of the function. For each function, write a sentence that explains your thinking and sketch a rough graph of how the function appears. +

    +

    + + + +

    always increasing or always decreasing;

    -
    2. -
  2. -

    + + + + + + + +

    always concave up or always concave down; and

    -
    3. -
  3. -

    + + + + + + + +

    increasing without bound, decreasing without bound, or increasing/decreasing toward a finite value.

    -
    4. -
- In addition, state the y-intercept and the range of the function. For each function, write a sentence that explains your thinking and sketch a rough graph of how the function appears. -

- -

-

    -
  1. -

    - p(t) = 4372 (1.000235)^t + 92856 -

    -
    2. -
  2. -

    - q(t) = 27931 (0.97231)^t + 549786 -

    -
    3. -
  3. -

    - r(t) = -17398 (0.85234)^t -

    -
    4. -
  4. -

    - s(t) = -17398 (0.85234)^t + 19411 -

    -
    5. -
  5. -

    - u(t) = -7522 (1.03817)^t -

    -
    6. -
  6. -

    - v(t) = -7522 (1.03817)^t + 6731 -

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + + +

+ p(t) = 4372 (1.000235)^t + 92856 +

+ + + + + + + +

+ q(t) = 27931 (0.97231)^t + 549786 +

+ + + + + + + +

+ r(t) = -17398 (0.85234)^t +

+ + + + + + + +

+ s(t) = -17398 (0.85234)^t + 19411 +

+ + + + + + + +

+ u(t) = -7522 (1.03817)^t +

+ + + + + + + +

+ v(t) = -7522 (1.03817)^t + 6731 +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-modeling-potato.xml b/source/activities/act-exp-modeling-potato.xml index 7f663cea..8459e3ff 100755 --- a/source/activities/act-exp-modeling-potato.xml +++ b/source/activities/act-exp-modeling-potato.xml @@ -1,4 +1,4 @@ - + @@ -12,67 +12,82 @@ - - - - -

+ + + +

A potato initially at room temperature (68^\circ) is placed in an oven (at 350^\circ) at time t = 0. It is known that the potato's temperature at time t is given by the function F(t) = a - b(0.98)^t for some positive constants a and b, where F is measured in degrees Fahrenheit and t is time in minutes.

- -

-

    -
  1. -

    +

    + + + +

    What is the numerical value of F(0)? What does this tell you about the value of a - b?

    -
    2. -
  2. -

    + + + + + + + +

    Based on the context of the problem, what should be the long-range behavior of the function F(t)? Use this fact along with the behavior of (0.98)^t to determine the value of a. Write a sentence to explain your thinking.

    -
    3. -
  3. -

    + + + + + + + +

    What is the value of b? Why?

    -
    4. -
  4. -

    + + + + + + + +

    Check your work above by plotting the function F using graphing technology in an appropriate window. Record your results on the axes provided in Figure, labeling the scale on the axes. Then, use the graph to estimate the time at which the potato's temperature reaches 325 degrees.

    -
    - Axes for plotting F. - -
    -
    5. -
  5. -

    +

    + Axes for plotting F. + +
    + + + + + + + +

    How can we view the function F(t) = a - b(0.98)^t as a transformation of the parent function f(t) = (0.98)^t? Explain.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-modeling-soda.xml b/source/activities/act-exp-modeling-soda.xml index 3dee55d4..3a93332b 100755 --- a/source/activities/act-exp-modeling-soda.xml +++ b/source/activities/act-exp-modeling-soda.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

A can of soda (at room temperature) is placed in a refrigerator at time t = 0 (in minutes) and its temperature, F(t), in degrees Fahrenheit, is computed at regular intervals. Based on the data, a model is formulated for the object's temperature, given by @@ -24,63 +23,79 @@ F(t) = 42 + 30(0.95)^{t} .

- -

-

    -
  1. -

    +

    + + + +

    Consider the simpler (parent) function p(t) = (0.95)^t. How do you expect the graph of this function to appear? How will it behave as time increases? Without using graphing technology, sketch a rough graph of p and write a sentence of explanation.

    -
    2. -
  2. -

    + + + + + + + +

    For the slightly more complicated function r(t) = 30 (0.95)^{t}, how do you expect this function to look in comparison to p? What is the long-range behavior of this function as t increases? Without using graphing technology, sketch a rough graph of r and write a sentence of explanation.

    -
    3. -
  3. -

    + + + + + + + +

    Finally, how do you expect the graph of F(t) = 42 + 30(0.95)^{t} to appear? Why? First sketch a rough graph without graphing technology, and then use technology to check your thinking and report an accurate, labeled graph on the axes provided in Figure.

    -
    - Axes for plotting F. - -
    -
    4. -
  4. -

    +

    + Axes for plotting F. + +
    + + + + + + + +

    What is the temperature of the refrigerator? What is the room temperature of the surroundings outside the refrigerator? Why?

    -
    5. -
  5. -

    + + + + + + + +

    Determine the average rate of change of F on the intervals [10,20], [20,30], and [30,40]. Write at least two careful sentences that explain the meaning of the values you found, including units, and discuss any overall trend in how the average rate of change is changing.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-temp-pop-NLOC1.xml b/source/activities/act-exp-temp-pop-NLOC1.xml index 62db3d63..9bcaefbc 100755 --- a/source/activities/act-exp-temp-pop-NLOC1.xml +++ b/source/activities/act-exp-temp-pop-NLOC1.xml @@ -1,4 +1,4 @@ - + @@ -12,63 +12,69 @@ - - - - -

+ + + +

A can of soda is initially at room temperature, 72.3^\circ Fahrenheit, and at time t = 0 is placed in a refrigerator set at 37.7^\circ. In addition, we know that after 30 minutes, the soda's temperature has dropped to 59.5^\circ. Let F(t) represent the temperature of the soda in degrees Fahrenheit at time t in minutes.

- -

-

    -
  1. -

    +

    + + + +

    Use algebraic reasoning and your understanding of the physical situation to determine the exact values of a, c, and k in the model F(t) = ae^{-kt}+c. Write at least one careful sentence to explain your thinking.

    -
    2. - -
  2. -

    + + + + + + + +

    Determine the exact time the object's temperature is 42.4^\circ. Clearly show your algebraic work and thinking.

    -
    3. - -
  3. -

    + + + + + + + +

    In Desmos, enter the values you found for a, c, and k in order to define the function F. Then, use Desmos to find the average rate of change of F on the interval [25,30]. What is the meaning (with units) of this value?

    -
    4. - -
  4. -

    + + + + + + + +

    If everything stayed the same except the value of F(0), and instead F(0) = 65, would the value of k be larger or smaller? Why?

    -
    5. - -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-temp-pop-logistic-Desmos.xml b/source/activities/act-exp-temp-pop-logistic-Desmos.xml index ab386653..d847f750 100755 --- a/source/activities/act-exp-temp-pop-logistic-Desmos.xml +++ b/source/activities/act-exp-temp-pop-logistic-Desmos.xml @@ -1,4 +1,4 @@ - + @@ -12,54 +12,63 @@ - - - - -

+ + + +

In Desmos, define P(t) = \frac{A}{1 + Me^{-kt}} and accept sliders for A, M, and k. Set the slider ranges for these parameters as follows: 0.01 \le A \le 10; 0.01 \le M \le 10; 0.01 \le k \le 5.

- -

-

    -
  1. -

    +

    + + + +

    Sketch a typical graph of P(t) on the axes provided and write several sentences to explain the effects of A, M, and k on the graph of P.

    - -
    - Axes for plotting a typical logistic function P. - -
    -
    2. -
  2. -

    +

    + Axes for plotting a typical logistic function P. + +
    + + + + + + + +

    On a typical logistic graph, where does it appear that the population is growing most rapidly? How is this value connected to the carrying capacity, A?

    -
    3. -
  3. -

    + + + + + + + +

    How does the function 1 + Me^{-kt} behave as t decreases without bound? What is the algebraic reason that this occurs?

    -
    4. -
  4. -

    + + + + + + + +

    Use your Desmos worksheet to find a logistic function P that has the following properties: P(0) = 2, P(2) = 4, and P(t) approaches 9 as t increases without bound. What are the approximate values of A, M, and k that make the function P fit these criteria?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-exp-temp-pop-logistic-exact.xml b/source/activities/act-exp-temp-pop-logistic-exact.xml index 5236fb30..0879c1ac 100755 --- a/source/activities/act-exp-temp-pop-logistic-exact.xml +++ b/source/activities/act-exp-temp-pop-logistic-exact.xml @@ -1,4 +1,4 @@ - + @@ -12,51 +12,56 @@ - - - - -

+ + + +

Suppose that a population of animals (measured in thousands) that lives on an island is known to grow according to the logistic model, where t is measured in years. We know the following information: P(0) = 2.45, P(3) = 4.52, and as t increases without bound, P(t) approaches 11.7.

- -

-

    -
  1. -

    +

    + + + +

    Determine the exact values of A, M, and k in the logistic model P(t) = \frac{A}{1 + Me^{-kt}} . Clearly show your algebraic work and thinking.

    -
    2. -
  2. -

    + + + + + + + +

    Plot your model from (a) and check that its values match the desired characteristics. Then, compute the average rate of change of P on the intervals [0,2], [2,4], [4,6], and [6,8]. What is the meaning (with units) of the values you've found? How is the population growing on these intervals?

    -
    3. -
  3. -

    + + + + + + + +

    Find the exact time value when the population will be 10 (thousand). Show your algebraic work and thinking.

    -
    4. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-infty-limit.xml b/source/activities/act-poly-infty-limit.xml index 30e4fbda..227e1ee6 100755 --- a/source/activities/act-poly-infty-limit.xml +++ b/source/activities/act-poly-infty-limit.xml @@ -1,4 +1,4 @@ - + @@ -12,89 +12,147 @@ - - - - -

+ + + +

Complete the Table by entering - \infty, -\infty, - 0, or no limit + \infty,-\infty,0, or no limit to identify how the function behaves as either x increases or decreases without bound. As much as possible, work to decide the behavior without using a graphing utility.

- - - Some familiar functions and their limits as <m>x \to \infty</m> or <m>x \to -\infty</m>. - - - f(x) - \lim_{x \to \infty} f(x) - \lim_{x \to -\infty} f(x) - - - e^x - - - - - e^{-x} - - - - - \ln(x) - - - - - x - - - - - x^2 - - - - - x^3 - - - - - x^4 - - - - - \frac{1}{x} - - - - - \frac{1}{x^2} - - - - - \sin(x) - - - - -
- - -

- -

- - -

- -

- - - - + + Some familiar functions and their limits as <m>x \to \infty</m> or <m>x \to -\infty</m>. + + + + f(x) + + + \lim_{x \to \infty} f(x) + + + \lim_{x \to -\infty} f(x) + + + + + e^x + + + + + + + + + + + e^{-x} + + + + + + + + + + + \ln(x) + + + + + + + + + + + x + + + + + + + + + + + x^2 + + + + + + + + + + + x^3 + + + + + + + + + + + x^4 + + + + + + + + + + + \frac{1}{x} + + + + + + + + + + + \frac{1}{x^2} + + + + + + + + + + + \sin(x) + + + + + + + + + +
+ + +

+ + +

+ + + diff --git a/source/activities/act-poly-infty-natural-powers.xml b/source/activities/act-poly-infty-natural-powers.xml index 95bb8ec8..b825b2cb 100755 --- a/source/activities/act-poly-infty-natural-powers.xml +++ b/source/activities/act-poly-infty-natural-powers.xml @@ -1,4 +1,4 @@ - + @@ -12,69 +12,74 @@ - - - - -

+ + + +

Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zu. In what follows, we explore the behavior of power functions of the form y = x^n where n \ge 1.

- -

-

    -
  1. -

    +

    + + + +

    Press the play button next to the slider labeled n. Watch at least two loops of the animation and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

    -
    2. - -
  2. -

    + + + + + + + +

    Click the icons next to each of the following 8 functions so that you can see all of y = x, y = x^2, \ldots, y = x^8 graphed at once. On the interval 0 \lt x \lt 1, how do the graphs of x^a and x^b compare if a \lt b?

    -
    3. - -
  3. -

    + + + + + + + +

    Uncheck the icons on each of the 8 functions to hide their graphs. Click the settings icon to change the domain settings for the axes, and change them to -10 \le x \le 10 and -10,000 \le y \le 10,000. Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

    -
    4. - -
  4. -

    + + + + + + + +

    Click the icons next to each of the following 8 functions so that you can see all of y = x, y = x^2, \ldots, y = x^8 graphed at once. On the interval x \gt 1, how do the graphs of x^a and x^b compare if a \lt b?

    -
    5. - -
-

- - - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-infty-negative-powers.xml b/source/activities/act-poly-infty-negative-powers.xml index fae42d96..5d695160 100755 --- a/source/activities/act-poly-infty-negative-powers.xml +++ b/source/activities/act-poly-infty-negative-powers.xml @@ -1,4 +1,4 @@ - + @@ -12,28 +12,31 @@ - - - - -

+ + + +

Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zv. In what follows, we explore the behavior of power functions y = x^n where n \le -1.

- -

-

    -
  1. -

    +

    + + + +

    Press the play button next to the slider labeled n. Watch two loops of the animation and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

    -
    2. - -
  2. -

    + + + + + + + +

    Click the icons next to each of the following 8 functions so that you can see all of y = x^{-1}, y = x^{-2}, \ldots, y = x^{-8} graphed at once. @@ -41,43 +44,50 @@ x^a and x^b compare if a \lt b? (Be careful with negative numbers here: e.g., -3 \lt -2.)

    -
    3. - -
  3. -

    + + + + + + + +

    How do your answers change on the interval 0 \lt x \lt 1?

    -
    4. - -
  4. -

    + + + + + + + +

    Uncheck the icons on each of the 8 functions to hide their graphs. Click the settings icon to change the domain settings for the axes, and change them to -10 \le x \le 10 and -10,000 \le y \le 10,000. Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.

    -
    5. - -
  5. -

    + + + + + + + +

    Explain why \lim_{x \to \infty} \frac{1}{x^n} = 0 for any choice of n = 1, 2, \ldots.

    -
    6. -
-

- - - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-polynomial-applications-Taylor.xml b/source/activities/act-poly-polynomial-applications-Taylor.xml index 1c11e428..b80a6e9b 100755 --- a/source/activities/act-poly-polynomial-applications-Taylor.xml +++ b/source/activities/act-poly-polynomial-applications-Taylor.xml @@ -1,4 +1,4 @@ - + @@ -12,58 +12,72 @@ - - - - -

+ + + +

We understand the theoretical rule behind the function f(t) = \sin(t): given an angle t in radians, \sin(t) measures the value of the y-coordinate of the corresponding point on the unit circle. For special values of t, we have determined the exact value of \sin(t). For example, \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}. But note that we don't have a formula for \sin(t). Instead, we use a button on our calculator or command on our computer to find values like \sin(1.35). It turns out that a combination of calculus and polynomial functions explains how computers determine values of the sine function.

- -

+

At http://gvsu.edu/s/0zA, you'll find a Desmos worksheet that has the sine function already defined, along with a sequence of polynomials labeled T_1(x), T_3(x), T_5(x), T_7(x), \ldots. You can see these functions' graphs by clicking on their respective icons.

- -

-

    -
  1. -

    +

    + + + +

    For what values of x does it appear that \sin(x) \approx T_1(x)?

    -
    2. -
  2. -

    + + + + + + + +

    For what values of x does it appear that \sin(x) \approx T_3(x)?

    -
    3. -
  3. -

    + + + + + + + +

    For what values of x does it appear that \sin(x) \approx T_5(x)?

    -
    4. -
  4. -

    + + + + + + + +

    What overall trend do you observe? How good is the approximation generated by T_{19}(x)?

    -
    5. -
  5. -

    + + + + + + + +

    In a new Desmos worksheet, plot the function y = \cos(x) along with the following functions: P_2(x) = 1 - \frac{x^2}{2!} and P_4(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!}. Based on the patterns with the coefficients in the polynomials approximating \sin(x) and the polynomials P_2 and P_4 here, conjecture formulas for P_6, P_8, and P_{18} and plot them. How well can we approximate y = \cos(x) using polynomials?

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-polynomial-applications-postal.xml b/source/activities/act-poly-polynomial-applications-postal.xml index 8f5ef667..b605ed2a 100755 --- a/source/activities/act-poly-polynomial-applications-postal.xml +++ b/source/activities/act-poly-polynomial-applications-postal.xml @@ -1,4 +1,4 @@ - + @@ -12,63 +12,76 @@ - - - - -

+ + + +

According to a shipping company's regulations, the girth plus the length of a parcel they transport for their lowest rate may not exceed 120 inches, where by girth we mean the perimeter of one end.

- -
- A rectangular parcel with a square end. - -
- -

+

+ A rectangular parcel with a square end. + +
+

Suppose that we want to ship a parcel that has a square end of width x and an overall length of y, both measured in inches.

- -

-

    -
  1. -

    +

    + + + +

    Label the provided picture, using x for the length of each side of the square end, and y for the other edge of the package.

    -
    2. -
  2. -

    + + + + + + + +

    How does the length plus girth of 120 inches result in an equation (often called a constraint equation) that relates x and y? Explain, and state the equation.

    -
    3. -
  3. -

    + + + + + + + +

    Solve the equation you found in (b) for one of the variables present.

    -
    4. -
  4. -

    + + + + + + + +

    Hence determine the volume, V, of the package as a function of a single variable.

    -
    5. -
  5. -

    + + + + + + + +

    What is the domain of the function V in the context of the physical setting of this problem? (Hint: neither x nor y can equal 0.)

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-polynomial-applications-soup.xml b/source/activities/act-poly-polynomial-applications-soup.xml index c0ee5bc9..e4af14ad 100755 --- a/source/activities/act-poly-polynomial-applications-soup.xml +++ b/source/activities/act-poly-polynomial-applications-soup.xml @@ -1,4 +1,4 @@ - + @@ -12,53 +12,62 @@ - - - - -

+ + + +

Suppose that we want to construct a cylindrical can using 60 square inches of material for the surface of the can. In this context, how does the can's volume depend on the radius we choose?

- -

+

Let the cylindrical can have base radius r and height h.

- -

-

    -
  1. -

    +

    + + + +

    Use the formula for the surface area of a cylinder and the given constraint that the can's surface area is 60 square inches to write an equation that connects the radius r and height h.

    -
    2. -
  2. -

    + + + + + + + +

    Solve the equation you found in (a) for h in terms of r.

    -
    3. -
  3. -

    + + + + + + + +

    Recall that the volume of a cylinder is V = \pi r^2 h. Use your work in (b) to write V as a function of the single variable r; simplify the formula as much as possible.

    -
    4. -
  4. -

    + + + + + + + +

    What is the domain of the function V in the context of the physical setting of this problem? (Hint: how does the constraint on surface area provide an upper bound for the value of r? Think about the maximum area that can be allocated to the top and bottom of the can.)

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-polynomials-find.xml b/source/activities/act-poly-polynomials-find.xml index b15f7c04..5af2d4d3 100755 --- a/source/activities/act-poly-polynomials-find.xml +++ b/source/activities/act-poly-polynomials-find.xml @@ -1,4 +1,4 @@ - + @@ -12,49 +12,59 @@ - - - - -

+ + + +

By experimenting with coefficients in Desmos, find a formula for a polynomial function that has the stated properties, or explain why no such polynomial exists. (If you enter p(x)=a+bx+cx^2+dx^3+fx^4+gx^5 in DesmosWe skip using e as one of the constants since Desmos reserves e as the Euler constant., you'll get prompted to add sliders that make it easy to explore a degree 5 polynomial.)

- -

-

    -
  1. -

    +

    + + + +

    A polynomial p of degree 5 with exactly 3 real zeros, 4 turning points, and such that \lim_{x \to -\infty} p(x) = +\infty and \lim_{x \to \infty} p(x) = -\infty.

    -
    2. -
  2. -

    + + + + + + + +

    A polynomial p of degree 4 with exactly 4 real zeros, 3 turning points, and such that \lim_{x \to -\infty} p(x) = +\infty and \lim_{x \to \infty} p(x) = -\infty.

    -
    3. -
  3. -

    + + + + + + + +

    A polynomial p of degree 6 with exactly 2 real zeros, 3 turning points, and such that \lim_{x \to -\infty} p(x) = -\infty and \lim_{x \to \infty} p(x) = -\infty.

    -
    4. -
  4. -

    + + + + + + + +

    A polynomial p of degree 5 with exactly 5 real zeros, 3 turning points, and such that \lim_{x \to -\infty} p(x) = +\infty and \lim_{x \to \infty} p(x) = -\infty.

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-polynomials-multiple-zeros.xml b/source/activities/act-poly-polynomials-multiple-zeros.xml index 993c57f5..3a0f9a65 100755 --- a/source/activities/act-poly-polynomials-multiple-zeros.xml +++ b/source/activities/act-poly-polynomials-multiple-zeros.xml @@ -1,4 +1,4 @@ - + @@ -12,70 +12,81 @@ - - - - -

+ + + +

For each of the following prompts, try to determine a formula for a polynomial that satisfies the given criteria. If no such polynomial exists, explain why.

- -

-

    -
  1. -

    +

    + + + +

    A polynomial f of degree 10 whose zeros are x = -12 (multiplicity 3), x = -9 (multiplicity 2), x = 4 (multiplicity 4), and x = 10 (multiplicity 1), and f satisfies f(0) = 21. What can you say about the values of \lim_{x \to -\infty} f(x) and \lim_{x \to \infty} f(x)?

    -
    2. -
  2. -

    + + + + + + + +

    A polynomial p of degree 9 that satisfies p(0) = -2 and has the graph shown in Figure. Assume that all of the zeros of p are shown in the figure.

    - - -
    3. -
  3. -

    + + + + + + + +

    A polynomial q of degree 8 with 3 distinct real zeros (possibly of different multiplicities) such that q has the sign chart in Figure and satisfies q(0) = -10.

    - - -
    - + +
    + A polynomial p. - -
    -
    - A sign chart for the polynomial q. - -
    -     - -
    4. -
  4. -

    + + +

    + A sign chart for the polynomial q. + +
    + + + + + + + + +

    A polynomial q of degree 9 with 3 distinct real zeros (possibly of different multiplicities) such that q satisfies the sign chart in Figure and satisfies q(0) = -10.

    -
    5. -
  5. -

    + + + + + + + +

    A polynomial p of degree 11 that satisfies p(0) = -2 and p has the graph shown in Figure. Assume that all of the zeros of p are shown in the figure.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-polynomials-sign-chart.xml b/source/activities/act-poly-polynomials-sign-chart.xml index 99723387..2cc95dfb 100755 --- a/source/activities/act-poly-polynomials-sign-chart.xml +++ b/source/activities/act-poly-polynomials-sign-chart.xml @@ -1,4 +1,4 @@ - + @@ -12,62 +12,82 @@ - - - - -

+ + + +

Consider the polynomial function given by p(x) = 4692(x+1520)(x^2+10000)(x-3471)^2(x-9738) .

- -

-

    -
  1. -

    +

    + + + +

    What is the degree of p? How can you tell without fully expanding the factored form of the function?

    -
    2. -
  2. -

    + + + + + + + +

    What can you say about the sign of the factor (x^2 + 10000)?

    -
    3. -
  3. -

    + + + + + + + +

    What are the zeros of the polynomial p?

    -
    4. -
  4. -

    + + + + + + + +

    Construct a sign chart for p by using the zeros you identified in (c) and then analyzing the sign of each factor of p.

    -
    5. -
  5. -

    + + + + + + + +

    Without using a graphing utility, construct an approximate graph of p that has the zeros of p carefully labeled on the x-axis.

    -
    6. -
  6. -

    + + + + + + + +

    Use a graphing utility to check your earlier work. What is challenging or misleading when using technology to graph p?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-rational-application.xml b/source/activities/act-poly-rational-application.xml index cb1e0203..fb3d9c41 100755 --- a/source/activities/act-poly-rational-application.xml +++ b/source/activities/act-poly-rational-application.xml @@ -1,4 +1,4 @@ - + @@ -12,59 +12,79 @@ - - - - -

+ + + +

Suppose that we want to build an open rectangular box (that is, without a top) that holds 15 cubic feet of volume. If we want one side of the base to be twice as long as the other, how does the amount of material required depend on the shorter side of the base? We investigate this question through the following sequence of prompts.

- -

-

    -
  1. -

    +

    + + + +

    Draw a labeled picture of the box. Let x represent the shorter side of the base and h the height of the box. What is the length of the longer side of the base in terms of x?

    -
    2. -
  2. -

    + + + + + + + +

    Use the given volume constraint to write an equation that relates x and h, and solve the equation for h in terms of x.

    -
    3. -
  3. -

    + + + + + + + +

    Determine a formula for the surface area, S, of the box in terms of x and h.

    -
    4. -
  4. -

    + + + + + + + +

    Using the constraint equation from (b) together with your work in (c), write surface area, S, as a function of the single variable x.

    -
    5. -
  5. -

    + + + + + + + +

    What type of function is S? What is its domain?

    -
    6. -
  6. -

    + + + + + + + +

    Plot the function S using Desmos. What appears to be the least amount of material that can be used to construct the desired box that holds 15 cubic feet of volume?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-rational-domain.xml b/source/activities/act-poly-rational-domain.xml index ce2dff98..546b82e5 100755 --- a/source/activities/act-poly-rational-domain.xml +++ b/source/activities/act-poly-rational-domain.xml @@ -1,4 +1,4 @@ - + @@ -12,59 +12,79 @@ - - - - -

+ + + +

Determine the domain of each of the following functions. In each case, write a sentence to accurately describe the domain.

- -

-

    -
  1. -

    - \displaystyle f(x) = \frac{x^2-1}{x^2 + 1} -

    -
    2. -
  2. -

    - \displaystyle g(x) = \frac{x^2 - 1}{x^2 + 3x - 4} -

    -
    3. -
  3. -

    - \displaystyle h(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} -

    -
    4. -
  4. -

    - \displaystyle j(x) = \frac{(x+5)(x-3)(x+1)(x-4)}{(x+1)(x+3)(x-5)} -

    -
    5. -
  5. -

    - \displaystyle k(x) = \frac{2x^2 + 7}{3x^3 - 12x} -

    -
    6. -
  6. -

    - \displaystyle m(x) = \frac{5x^2 - 45}{7(x-2)(x-3)^2(x^2 + 9)(x+1)} -

    -
    7. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ \displaystyle f(x) = \frac{x^2-1}{x^2 + 1} +

+ + + + + + + +

+ \displaystyle g(x) = \frac{x^2 - 1}{x^2 + 3x - 4} +

+ + + + + + + +

+ \displaystyle h(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} +

+ + + + + + + +

+ \displaystyle j(x) = \frac{(x+5)(x-3)(x+1)(x-4)}{(x+1)(x+3)(x-5)} +

+ + + + + + + +

+ \displaystyle k(x) = \frac{2x^2 + 7}{3x^3 - 12x} +

+ + + + + + + +

+ \displaystyle m(x) = \frac{5x^2 - 45}{7(x-2)(x-3)^2(x^2 + 9)(x+1)} +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-rational-features-ZAH.xml b/source/activities/act-poly-rational-features-ZAH.xml index f218c26f..32bf9720 100755 --- a/source/activities/act-poly-rational-features-ZAH.xml +++ b/source/activities/act-poly-rational-features-ZAH.xml @@ -1,4 +1,4 @@ - + @@ -12,59 +12,79 @@ - - - - -

+ + + +

For each of the following rational functions, state the function's domain and determine the locations of all zeros, vertical asymptotes, and holes. Provide clear justification for your work by discussing the zeros of the numerator and denominator, as well as a table of values of the function near any point where you believe the function has a hole. In addition, state the value of the horizontal asymptote of the function or explain why the function has no such asymptote.

- -

-

    -
  1. -

    - \displaystyle f(x) = \frac{x^3 - 6x^2 + 5x}{x^2-1} -

    -
    2. -
  2. -

    - \displaystyle g(x) = \frac{11(x^2 + 1)(x-7)}{23(x-1)(x^2+4)} -

    -
    3. -
  3. -

    - \displaystyle h(x) = \frac{x^2 - 8x + 12}{x^2 - 3x - 18} -

    -
    4. -
  4. -

    - \displaystyle q(x) = \frac{(x-2)(x^2-9)}{(x-3)(x^2 + 4)} -

    -
    5. -
  5. -

    - \displaystyle r(x) = \frac{19(x-2) (x-3)^2 (x+1)}{17(x+1)(x-4)^2(x-5)} -

    -
    6. -
  6. -

    - \displaystyle s(x) = \frac{1}{x^2 + 1} -

    -
    7. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ \displaystyle f(x) = \frac{x^3 - 6x^2 + 5x}{x^2-1} +

+ + + + + + + +

+ \displaystyle g(x) = \frac{11(x^2 + 1)(x-7)}{23(x-1)(x^2+4)} +

+ + + + + + + +

+ \displaystyle h(x) = \frac{x^2 - 8x + 12}{x^2 - 3x - 18} +

+ + + + + + + +

+ \displaystyle q(x) = \frac{(x-2)(x^2-9)}{(x-3)(x^2 + 4)} +

+ + + + + + + +

+ \displaystyle r(x) = \frac{19(x-2) (x-3)^2 (x+1)}{17(x+1)(x-4)^2(x-5)} +

+ + + + + + + +

+ \displaystyle s(x) = \frac{1}{x^2 + 1} +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-rational-formula.xml b/source/activities/act-poly-rational-formula.xml index 877784e4..322f3bdc 100755 --- a/source/activities/act-poly-rational-formula.xml +++ b/source/activities/act-poly-rational-formula.xml @@ -1,4 +1,4 @@ - + @@ -12,65 +12,79 @@ - - - - -

+ + + +

Find a formula for a rational function that meets the stated criteria as given by words, a sign chart, or graph. Write several sentences to justify why your formula matches the specifications.

- -

-

    -
  1. -

    +

    + + + +

    A rational function r such that r has a vertical asymptote at x = -2, a zero at x = 1, a hole at x = 5, and a horizontal asymptote of y = -3.

    -
    2. -
  2. -

    + + + + + + + +

    A rational function u whose numerator has degree 3, denominator has degree 3, and that has exactly one vertical asymptote at x = -4 and a horizontal asymptote of y = \frac{3}{7}.

    -
    3. -
  3. -

    + + + + + + + +

    A rational function w whose formula generates a graph with all of the characteristics shown in Figure. Assume that w(5) = 0 but w(x) \gt 0 for all other x such that x \gt 3.

    -
    4. -
  4. -

    + + + + + + + +

    A rational function z whose formula satisfies the sign chart shown in Figure, and for which z has no horizontal asymptote and its only vertical asymptotes occur at the middle two values of x noted on the sign chart.

    - - -
    - Plot of the rational function w. - -
    - Sign chart for the rational function z. - -
    -     -
    5. - -
  5. -

    + +

    + Plot of the rational function w. + +
    +
    + Sign chart for the rational function z. + +
    + + + + + + + + +

    A rational function f that has exactly two holes, two vertical asymptotes, two zeros, and a horizontal asymptote.

    -
    6. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-rational-long-term-1.xml b/source/activities/act-poly-rational-long-term-1.xml index 04cddcca..be09d02d 100755 --- a/source/activities/act-poly-rational-long-term-1.xml +++ b/source/activities/act-poly-rational-long-term-1.xml @@ -1,4 +1,4 @@ - + @@ -12,27 +12,30 @@ - - - - -

+ + + +

Consider the rational function r(x) = \frac{3x^2 - 5x + 1}{7x^2 + 2x - 11}.

- -

+

Observe that the largest power of x that's present in r(x) is x^2. In addition, because of the dominant terms of 3x^2 in the numerator and 7x^2 in the denominator, both the numerator and denominator of r increase without bound as x increases without bound. In order to understand the long-range behavior of r, we choose to write the function in a different algebraic form.

- -

-

    -
  1. -

    +

    + + + +

    Note that we can multiply the formula for r by the form of 1 given by 1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}}. Do so, and distribute and simplify as much as possible in both the numerator and denominator to write r in a different algebraic form.

    -
    2. -
  2. -

    + + + + + + + +

    Having rewritten r, we are in a better position to evaluate \lim_{x \to \infty} r(x). Using our work from (a), we have @@ -40,33 +43,39 @@ . What is the exact value of this limit and why?

    -
    3. -
  3. -

    + + + + + + + +

    Next, determine \lim_{x \to -\infty} r(x) = \lim_{x \to -\infty} \frac{3 - \frac{5}{x} + \frac{1}{x^2}}{7 + \frac{2}{x} - \frac{11}{x^2}} .

    -
    4. -
  4. -

    + + + + + + + +

    Use Desmos to plot r on the interval [-10,10]. In addition, plot the horizontal line y = \frac{3}{7}. What is the meaning of the limits you found in (b) and (c)?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-poly-rational-long-term-2.xml b/source/activities/act-poly-rational-long-term-2.xml index d9d4d1ed..79e297cc 100755 --- a/source/activities/act-poly-rational-long-term-2.xml +++ b/source/activities/act-poly-rational-long-term-2.xml @@ -1,4 +1,4 @@ - + @@ -12,57 +12,67 @@ - - - - -

+ + + +

Let s(x) = \frac{3x - 5}{7x^2 + 2x - 11} and u(x) = \frac{3x^2 - 5x + 1}{7x + 2}. Note that both the numerator and denominator of each of these rational functions increases without bound as x \to \infty, and in addition that x^2 is the highest order term present in each of s and u.

- -

-

    -
  1. -

    +

    + + + +

    Using a similar algebraic approach to our work in Activity, multiply s(x) by 1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}} and hence evaluate \lim_{x \to \infty} \frac{3x - 5}{7x^2 + 2x - 11} . What value do you find?

    -
    2. -
  2. -

    + + + + + + + +

    Plot the function y = s(x) on the interval [-10,10]. What is the graphical meaning of the limit you found in (a)?

    -
    3. -
  3. -

    + + + + + + + +

    Next, use appropriate algebraic work to consider u(x) and evaluate \lim_{x \to \infty} \frac{3x^2 - 5x + 1}{7x + 2} . What do you find?

    -
    4. -
  4. -

    + + + + + + + +

    Plot the function y = u(x) on the interval [-10,10]. What is the graphical meaning of the limit you computed in (c)?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-finding-angles-baseball.xml b/source/activities/act-trig-finding-angles-baseball.xml index 3b062b5d..91f98a63 100755 --- a/source/activities/act-trig-finding-angles-baseball.xml +++ b/source/activities/act-trig-finding-angles-baseball.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

On a baseball diamond (which is a square with 90-foot sides), the third baseman fields the ball right on the line from third base to home plate and 10 feet away from third base @@ -24,22 +23,15 @@ (in degrees) does the line the ball travels make with the first base line? What angle does it make with the third base line? Draw a well-labeled diagram to support your thinking.

- -

+

What angles arise if he throws the ball to second base instead?

- - - -

- -

- - -

- -

- - - - + + +

+ + +

+ + + diff --git a/source/activities/act-trig-finding-angles-exactly.xml b/source/activities/act-trig-finding-angles-exactly.xml index f8544234..f40f75da 100755 --- a/source/activities/act-trig-finding-angles-exactly.xml +++ b/source/activities/act-trig-finding-angles-exactly.xml @@ -1,4 +1,4 @@ - + @@ -12,47 +12,52 @@ - - - - -

+ + + +

For each of the following different scenarios, draw a picture of the situation and use inverse trigonometric functions appropriately to determine the missing information both exactly and approximately.

- -

-

    -
  1. -

    +

    + + + +

    Consider a right triangle with legs of length 11 and 13. What are the measures (in radians) of the non-right angles and what is the length of the hypotenuse?

    -
    2. -
  2. -

    + + + + + + + +

    Consider an angle \alpha in standard position (vertex at the origin, one side on the positive x-axis) for which we know \cos(\alpha) = -\frac{1}{2} and \alpha lies in quadrant III. What is the measure of \alpha in radians? In addition, what is the value of \sin(\alpha)?

    -
    3. -
  3. -

    + + + + + + + +

    Consider an angle \beta in standard position for which we know \sin(\beta) = 0.1 and \beta lies in quadrant II. What is the measure of \beta in radians? In addition, what is the value of \cos(\beta)?

    -
    4. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-finding-angles-rocket.xml b/source/activities/act-trig-finding-angles-rocket.xml index b326c883..1c013514 100755 --- a/source/activities/act-trig-finding-angles-rocket.xml +++ b/source/activities/act-trig-finding-angles-rocket.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

A camera is tracking the launch of a SpaceX rocket. The camera is located 4000' from the rocket's launching pad, and the camera angle changes in order to keep the rocket in focus. @@ -25,8 +24,7 @@ is the camera tilted when the rocket is 3000' off the ground? Answer both exactly and approximately.

- -

+

Now, rather than considering the rocket at a fixed height of 3000', let its height vary and call the rocket's height h. Determine the camera's angle, \theta as a function of h, @@ -34,18 +32,12 @@ [5000,5500], and [7000,7500]. What do you observe about how the camera angle is changing?

- - - -

- -

- - -

- -

- - - - + + +

+ + +

+ + + diff --git a/source/activities/act-trig-finding-angles-roof.xml b/source/activities/act-trig-finding-angles-roof.xml index 90364d42..1387fdbf 100755 --- a/source/activities/act-trig-finding-angles-roof.xml +++ b/source/activities/act-trig-finding-angles-roof.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

A roof is being built with a 7-12 pitch. This means that the roof rises 7 inches vertically for every 12 inches of horizontal span; in other words, the slope of the roof is \frac{7}{12}. @@ -25,18 +24,12 @@ What is the approximate measure? What are the exact and approximate measures of the angle at the peak of the roof (made by the front and back portions of the roof that meet to form the ridge)?

- - - -

- -

- - -

- -

- - - - + + +

+ + +

+ + + diff --git a/source/activities/act-trig-inverse-arccos.xml b/source/activities/act-trig-inverse-arccos.xml index 37cee83a..36ebf3db 100755 --- a/source/activities/act-trig-inverse-arccos.xml +++ b/source/activities/act-trig-inverse-arccos.xml @@ -1,4 +1,4 @@ - + @@ -12,79 +12,119 @@ - - - - -

+ + + +

Use the special points on the unit circle (see, for instance, Figure) to determine the exact values of each of the following numerical expressions. Do so without using a computational device.

- -

-

    -
  1. -

    - \arccos(\frac{1}{2}) -

    -
    2. -
  2. -

    - \arccos(\frac{\sqrt{2}}{2}) -

    -
    3. -
  3. -

    - \arccos(\frac{\sqrt{3}}{2}) -

    -
    4. -
  4. -

    - \arccos(-\frac{1}{2}) -

    -
    5. -
  5. -

    - \arccos(-\frac{\sqrt{2}}{2}) -

    -
    6. -
  6. -

    - \arccos(-\frac{\sqrt{3}}{2}) -

    -
    7. -
  7. -

    - \arccos(-1) -

    -
    8. -
  8. -

    - \arccos(0) -

    -
    9. -
  9. -

    - \cos(\arccos(-\frac{1}{2})) -

    -
    10. -
  10. -

    - \arccos(\cos(\frac{7\pi}{6})) -

    -
    11. -
-

- - -

- -

- - -

- -

- - - - +

+ + + +

+ \arccos(\frac{1}{2}) +

+ + + + + + + +

+ \arccos(\frac{\sqrt{2}}{2}) +

+ + + + + + + +

+ \arccos(\frac{\sqrt{3}}{2}) +

+ + + + + + + +

+ \arccos(-\frac{1}{2}) +

+ + + + + + + +

+ \arccos(-\frac{\sqrt{2}}{2}) +

+ + + + + + + +

+ \arccos(-\frac{\sqrt{3}}{2}) +

+ + + + + + + +

+ \arccos(-1) +

+ + + + + + + +

+ \arccos(0) +

+ + + + + + + +

+ \cos(\arccos(-\frac{1}{2})) +

+ + + + + + + +

+ \arccos(\cos(\frac{7\pi}{6})) +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-inverse-arcsin.xml b/source/activities/act-trig-inverse-arcsin.xml index 5b61ed5e..94bec972 100755 --- a/source/activities/act-trig-inverse-arcsin.xml +++ b/source/activities/act-trig-inverse-arcsin.xml @@ -1,4 +1,4 @@ - + @@ -12,54 +12,63 @@ - - - - -

+ + + +

The goal of this activity is to understand key properties of the arcsine function in a way similar to our discussion of the arccosine function in Subsection.

- -

-

    -
  1. -

    +

    + + + +

    Using Definition, what are the domain and range of the arcsine function?

    -
    2. -
  2. -

    + + + + + + + +

    Determine the following values exactly: \arcsin(-1), \arcsin(-\frac{\sqrt{2}}{2}), \arcsin(0), \arcsin(\frac{1}{2}), and \arcsin(\frac{\sqrt{3}}{2}).

    -
    3. -
  3. -

    + + + + + + + +

    On the axes provided in Figure, sketch a careful plot of the restricted sine function on the interval [-\frac{\pi}{2},\frac{\pi}{2}] along with its corresponding inverse, the arcsine function. Label at least three points on each curve so that each point on the sine graph corresponds to a point on the arcsine graph. In addition, sketch the line y = t to demonstrate how the graphs are reflections of one another across this line.

    - -
    - Axes for plotting the restricted sine function and its inverse, the arcsine function. - -
    -
    4. -
  4. -

    +

    + Axes for plotting the restricted sine function and its inverse, the arcsine function. + +
    + + + + + + + +

    True or false: \arcsin(\sin(5\pi)) = 5\pi. Write a complete sentence to explain your reasoning.

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-inverse-arctan.xml b/source/activities/act-trig-inverse-arctan.xml index c7b1ec90..1e62e558 100755 --- a/source/activities/act-trig-inverse-arctan.xml +++ b/source/activities/act-trig-inverse-arctan.xml @@ -1,4 +1,4 @@ - + @@ -12,54 +12,63 @@ - - - - -

+ + + +

The goal of this activity is to understand key properties of the arctangent function.

- -

-

    -
  1. -

    +

    + + + +

    Using Definition, what are the domain and range of the arctangent function?

    -
    2. -
  2. -

    + + + + + + + +

    Determine the following values exactly: \arctan(-\sqrt{3}), \arctan(-1), \arctan(0), and \arctan(\frac{1}{\sqrt{3}}).

    -
    3. -
  3. -

    + + + + + + + +

    A plot of the restricted tangent function on the interval (-\frac{\pi}{2},\frac{\pi}{2}) is provided in Figure. Sketch its corresponding inverse function, the arctangent function, on the same axes. Label at least three points on each curve so that each point on the tangent graph corresponds to a point on the arctangent graph. In addition, sketch the line y = t to demonstrate how the graphs are reflections of one another across this line.

    - -
    - Axes for plotting the restricted tangent function and its inverse, the arctangent function. - -
    -
    4. -
  4. -

    - Complete the following sentence: as t increases without bound, \arctan(t) \ldots. +

    + Axes for plotting the restricted tangent function and its inverse, the arctangent function. + +
    + + + + + + + +

    + Complete the following sentence: as t increases without bound, \arctan(t)\ldots.

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-other-aroc-sine.xml b/source/activities/act-trig-other-aroc-sine.xml index a0706406..deed701d 100755 --- a/source/activities/act-trig-other-aroc-sine.xml +++ b/source/activities/act-trig-other-aroc-sine.xml @@ -1,4 +1,4 @@ - + @@ -12,63 +12,72 @@ - - - - -

+ + + +

In this activity, we investigate how a sum of two angles identity for the sine function helps us gain a different perspective on the average rate of change of the sine function.

- -

+

Recall that for any function f on an interval [a,a+h], its average rate of change is AV_{[a,a+h]} = \frac{f(a+h)-f(a)}{h} .

- -

-

    -
  1. -

    +

    + + + +

    Let f(x) = \sin(x). Use the definition of AV_{[a,a+h]} to write an expression for the average rate of change of the sine function on the interval [a,a+h].

    -
    2. -
  2. -

    + + + + + + + +

    Apply the sum of two angles identity for the sine function, \sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta), to the expression \sin(a+h).

    -
    3. -
  3. -

    + + + + + + + +

    Explain why your work in (a) and (b) together with some algebra shows that AV_{[a,a+h]} = \sin(a) \cdot \frac{\cos(h)-1}{h} - \cos(a) \cdot \frac{\sin(h)}{h} .

    -
    4. -
  4. -

    + + + + + + + +

    In calculus, we move from average rate of change to instantaneous rate of change by letting h approach 0 in the expression for average rate of change. Using a computational device in radian mode, investigate the behavior of \frac{\cos(h)-1}{h} as h gets close to 0. What happens? Similarly, how does \frac{\sin(h)}{h} behave for small values of h? What does this tell us about AV_{[a,a+h]} for the sine function as h approaches 0?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-other-cot.xml b/source/activities/act-trig-other-cot.xml index 41c62d24..4f7cacd5 100755 --- a/source/activities/act-trig-other-cot.xml +++ b/source/activities/act-trig-other-cot.xml @@ -1,4 +1,4 @@ - + @@ -12,200 +12,380 @@ - - - - -

+ + + +

In this activity, we develop the standard properties of the cotangent function, r(t) = \cot(t).

- - -

-

    -
  1. -

    +

    + + + +

    Complete Table and Table to determine the exact values of the cotangent function at the special points on the unit circle. Enter u for any value at which r(t) = \cot(t) is undefined. -

    - - Values of the sine, cosine, and tangent functions at special points on the unit circle. - - - t - 0 - \frac{\pi}{6} - \frac{\pi}{4} - \frac{\pi}{3} - \frac{\pi}{2} - \frac{2\pi}{3} - \frac{3\pi}{4} - \frac{5\pi}{6} - \pi - - - \sin(t) - 0 - \frac{1}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} - 1 - \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} - \frac{1}{2} - 0 - - - \cos(t) - 1 - \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} - \frac{1}{2} - 0 - -\frac{1}{2} - -\frac{\sqrt{2}}{2} - -\frac{\sqrt{3}}{2} - -1 - - - \tan(t) - 0 - \frac{1}{\sqrt{3}} - 1 - \frac{3}{\sqrt{3}} - u - -\frac{3}{\sqrt{3}} - -1 - -\frac{1}{\sqrt{3}} - 0 - - - \cot(t) - - - - - - - - - - - -
    - - Values of the sine, cosine, and tangent functions at special points on the unit circle. - - - t - \frac{7\pi}{6} - \frac{5\pi}{4} - \frac{4\pi}{3} - \frac{3\pi}{2} - \frac{5\pi}{3} - \frac{7\pi}{4} - \frac{11\pi}{6} - 2\pi - - - \sin(t) - -\frac{1}{2} - -\frac{\sqrt{2}}{2} - -\frac{\sqrt{3}}{2} - -1 - -\frac{\sqrt{3}}{2} - -\frac{\sqrt{2}}{2} - -\frac{1}{2} - 0 - - - \cos(t) - -\frac{\sqrt{3}}{2} - -\frac{\sqrt{2}}{2} - -\frac{1}{2} - 0 - \frac{1}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} - 1 - - - \tan(t) - \frac{1}{\sqrt{3}} - 1 - \frac{3}{\sqrt{3}} - u - -\frac{3}{\sqrt{3}} - -1 - -\frac{1}{\sqrt{3}} - 0 - - - \cot(t) - - - - - - - - - - -
    -
    2. -
  2. -

    +

    + + Values of the sine, cosine, and tangent functions at special points on the unit circle. + + + + t + + + 0 + + + \frac{\pi}{6} + + + \frac{\pi}{4} + + + \frac{\pi}{3} + + + \frac{\pi}{2} + + + \frac{2\pi}{3} + + + \frac{3\pi}{4} + + + \frac{5\pi}{6} + + + \pi + + + + + \sin(t) + + + 0 + + + \frac{1}{2} + + + \frac{\sqrt{2}}{2} + + + \frac{\sqrt{3}}{2} + + + 1 + + + \frac{\sqrt{3}}{2} + + + \frac{\sqrt{2}}{2} + + + \frac{1}{2} + + + 0 + + + + + \cos(t) + + + 1 + + + \frac{\sqrt{3}}{2} + + + \frac{\sqrt{2}}{2} + + + \frac{1}{2} + + + 0 + + + -\frac{1}{2} + + + -\frac{\sqrt{2}}{2} + + + -\frac{\sqrt{3}}{2} + + + -1 + + + + + \tan(t) + + + 0 + + + \frac{1}{\sqrt{3}} + + + 1 + + + \frac{3}{\sqrt{3}} + + u + + -\frac{3}{\sqrt{3}} + + + -1 + + + -\frac{1}{\sqrt{3}} + + + 0 + + + + + \cot(t) + + + + + + + + + + + + +
    + + Values of the sine, cosine, and tangent functions at special points on the unit circle. + + + + t + + + \frac{7\pi}{6} + + + \frac{5\pi}{4} + + + \frac{4\pi}{3} + + + \frac{3\pi}{2} + + + \frac{5\pi}{3} + + + \frac{7\pi}{4} + + + \frac{11\pi}{6} + + + 2\pi + + + + + \sin(t) + + + -\frac{1}{2} + + + -\frac{\sqrt{2}}{2} + + + -\frac{\sqrt{3}}{2} + + + -1 + + + -\frac{\sqrt{3}}{2} + + + -\frac{\sqrt{2}}{2} + + + -\frac{1}{2} + + + 0 + + + + + \cos(t) + + + -\frac{\sqrt{3}}{2} + + + -\frac{\sqrt{2}}{2} + + + -\frac{1}{2} + + + 0 + + + \frac{1}{2} + + + \frac{\sqrt{2}}{2} + + + \frac{\sqrt{3}}{2} + + + 1 + + + + + \tan(t) + + + \frac{1}{\sqrt{3}} + + + 1 + + + \frac{3}{\sqrt{3}} + + u + + -\frac{3}{\sqrt{3}} + + + -1 + + + -\frac{1}{\sqrt{3}} + + + 0 + + + + + \cot(t) + + + + + + + + + + + +
    + + + + + + + +

    In which quadrants is r(t) = \cot(t) positive? negative?

    -
    3. -
  3. -

    + + + + + + + +

    At what t-values does r(t) = \cot(t) have a vertical asymptote? Why?

    -
    4. -
  4. -

    + + + + + + + +

    What is the domain of the cotangent function? What is its range?

    -
    5. -
  5. -

    + + + + + + + +

    Sketch an accurate, labeled graph of r(t) = \cot(t) on the axes provided in Figure, including the special points that come from the unit circle.

    -
    - Axes for plotting r(t) = \cot(t). - -
    - -
    6. -
  6. -

    +

    + Axes for plotting r(t) = \cot(t). + +
    + + + + + + + +

    On intervals where the function is defined at every point in the interval, is r(t) = \cot(t) always increasing, always decreasing, or neither?

    -
    7. -
  7. -

    + + + + + + + +

    What is the period of the cotangent function?

    -
    8. -
  8. -

    + + + + + + + +

    How would you describe the relationship between the graphs of the tangent and cotangent functions?

    -
    9. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-other-csc.xml b/source/activities/act-trig-other-csc.xml index f98fa1d6..fd7e71a5 100755 --- a/source/activities/act-trig-other-csc.xml +++ b/source/activities/act-trig-other-csc.xml @@ -1,4 +1,4 @@ - + @@ -12,145 +12,276 @@ - - - - -

+ + + +

In this activity, we develop the standard properties of the cosecant function, q(t) = \csc(t).

- -
- Axes for plotting q(t) = \csc(t). - -
- -

-

    -
  1. -

    +

    + Axes for plotting q(t) = \csc(t). + +
    +

    + + + +

    Complete Table and Table to determine the exact values of the cosecant function at the special points on the unit circle. Enter u for any value at which q(t) = \csc(t) is undefined.

    - - - Values of the sine function at special points on the unit circle (Quadrants I and II). - - - t - 0 - \frac{\pi}{6} - \frac{\pi}{4} - \frac{\pi}{3} - \frac{\pi}{2} - \frac{2\pi}{3} - \frac{3\pi}{4} - \frac{5\pi}{6} - \pi - - - \sin(t) - 0 - \frac{1}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} - 1 - \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} - \frac{1}{2} - 0 - - - \csc(t) - - - - - - - - - - - -
    - - - Values of the sine function at special points on the unit circle (Quadrants III and IV). - - - t - \frac{7\pi}{6} - \frac{5\pi}{4} - \frac{4\pi}{3} - \frac{3\pi}{2} - \frac{5\pi}{3} - \frac{7\pi}{4} - \frac{11\pi}{6} - 2\pi - - - \sin(t) - -\frac{1}{2} - -\frac{\sqrt{2}}{2} - -\frac{\sqrt{3}}{2} - -1 - -\frac{\sqrt{3}}{2} - -\frac{\sqrt{2}}{2} - -\frac{1}{2} - 0 - - - \csc(t) - - - - - - - - - - -
    -
    2. -
  2. -

    + + Values of the sine function at special points on the unit circle (Quadrants I and II). + + + + t + + + 0 + + + \frac{\pi}{6} + + + \frac{\pi}{4} + + + \frac{\pi}{3} + + + \frac{\pi}{2} + + + \frac{2\pi}{3} + + + \frac{3\pi}{4} + + + \frac{5\pi}{6} + + + \pi + + + + + \sin(t) + + + 0 + + + \frac{1}{2} + + + \frac{\sqrt{2}}{2} + + + \frac{\sqrt{3}}{2} + + + 1 + + + \frac{\sqrt{3}}{2} + + + \frac{\sqrt{2}}{2} + + + \frac{1}{2} + + + 0 + + + + + \csc(t) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + Values of the sine function at special points on the unit circle (Quadrants III and IV). + + + + t + + + \frac{7\pi}{6} + + + \frac{5\pi}{4} + + + \frac{4\pi}{3} + + + \frac{3\pi}{2} + + + \frac{5\pi}{3} + + + \frac{7\pi}{4} + + + \frac{11\pi}{6} + + + 2\pi + + + + + \sin(t) + + + -\frac{1}{2} + + + -\frac{\sqrt{2}}{2} + + + -\frac{\sqrt{3}}{2} + + + -1 + + + -\frac{\sqrt{3}}{2} + + + -\frac{\sqrt{2}}{2} + + + -\frac{1}{2} + + + 0 + + + + + \csc(t) + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + + + + + + +

    In which quadrants is q(t) = \csc(t) positive? negative?

    -
    3. -
  3. -

    + + + + + + + +

    At what t-values does q(t) = \csc(t) have a vertical asymptote? Why?

    -
    4. -
  4. -

    + + + + + + + +

    What is the domain of the cosecant function? What is its range?

    -
    5. -
  5. -

    + + + + + + + +

    Sketch an accurate, labeled graph of q(t) = \csc(t) on the axes provided in Figure, including the special points that come from the unit circle.

    -
    6. -
  6. -

    + + + + + + + +

    What is the period of the cosecant function?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-other-special.xml b/source/activities/act-trig-other-special.xml index d6d04387..93d27f79 100755 --- a/source/activities/act-trig-other-special.xml +++ b/source/activities/act-trig-other-special.xml @@ -1,4 +1,4 @@ - + @@ -12,25 +12,18 @@ - - - - -

+ + + +

Suppose that \beta is an angle in standard position with its terminal side in quadrant II and you know that \sec(\beta) = -2. Without using a computational device in any way, determine the exact values of the other five trigonometric functions evaluated at \beta.

- - - -

- -

- - -

- -

- - - - + + +

+ + +

+ + + diff --git a/source/activities/act-trig-right-SOH-CAH.xml b/source/activities/act-trig-right-SOH-CAH.xml index 20cf1a39..bf92fa4a 100755 --- a/source/activities/act-trig-right-SOH-CAH.xml +++ b/source/activities/act-trig-right-SOH-CAH.xml @@ -1,4 +1,4 @@ - + @@ -12,59 +12,79 @@ - - - - -

+ + + +

In each of the following scenarios involving a right triangle, determine the exact values of as many of the remaining side lengths and angle measures (in radians) that you can. If there are quantities that you cannot determine, explain why. For every prompt, draw a labeled diagram of the situation.

- -

-

    -
  1. -

    +

    + + + +

    A right triangle with hypotenuse 7 and one non-right angle of measure \frac{\pi}{7}.

    -
    2. -
  2. -

    + + + + + + + +

    A right triangle with non-right angle \alpha that satisfies \sin(\alpha) = \frac{3}{5}.

    -
    3. -
  3. -

    + + + + + + + +

    A right triangle where one of the non-right angles has measure 1.2 and the hypotenuse has length 2.7.

    -
    4. -
  4. -

    + + + + + + + +

    A right triangle with hypotenuse 13 and one leg of length 6.5.

    -
    5. -
  5. -

    + + + + + + + +

    A right triangle with legs of length 5 and 12.

    -
    6. -
  6. -

    + + + + + + + +

    A right triangle where one of the non-right angles has measure \frac{\pi}{5} and the leg opposite this angle has length 4.

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-right-similar.xml b/source/activities/act-trig-right-similar.xml index a2387b14..ab626d3a 100755 --- a/source/activities/act-trig-right-similar.xml +++ b/source/activities/act-trig-right-similar.xml @@ -1,4 +1,4 @@ - + @@ -12,54 +12,63 @@ - - - - -

+ + + +

Consider right triangle OPQ given in Figure, and assume that the length of the hypotenuse is OP = r for some constant r \gt 1. Let point M lie on \overline{OP} (the line segment between O and P) in such a way that OM = 1, and let point N lie on \overline{OQ} so that \angle ONM is a right angle, as pictured. In addition, assume that point O corresponds to (0,0), point Q to (x,0), and point P to (x,y) so that OQ = x and PQ = y. Finally, let \theta be the measure of \angle POQ.

- -
- Two right triangles \triangle OPQ and \triangle OMN. - -
- -

-

    -
  1. -

    +

    + Two right triangles \triangle OPQ and \triangle OMN. + +
    +

    + + + +

    Explain why \triangle OPQ and \triangle OMN are similar triangles.

    -
    2. -
  2. -

    + + + + + + + +

    What is the value of the ratio \frac{OP}{OM}? What does this tell you about the ratios \frac{OQ}{ON} and \frac{PQ}{MN}?

    -
    3. -
  3. -

    + + + + + + + +

    What is the value of ON in terms of \theta? What is the value of MN in terms of \theta?

    -
    4. -
  4. -

    + + + + + + + +

    Use your conclusions in (b) and (c) to express the values of x and y in terms of r and \theta.

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-right-sine-cosine-ratio.xml b/source/activities/act-trig-right-sine-cosine-ratio.xml index fb7ba90a..31632f39 100755 --- a/source/activities/act-trig-right-sine-cosine-ratio.xml +++ b/source/activities/act-trig-right-sine-cosine-ratio.xml @@ -1,4 +1,4 @@ - + @@ -12,35 +12,27 @@ - - - - - -

+ + + + +

We want to determine the distance between two points A and B that are directly across from one another on opposite sides of a river, as pictured in Figure. We mark the locations of those points and walk 50 meters downstream from B to point P and use a sextant to measure \angle BPA. If the measure of \angle BPA is 56.4^{\circ}, how wide is the river? What other information about the situation can you determine?

- -
- Finding the width of the river. - -
- - - - -

- -

- - -

- -

- - - - +
+ Finding the width of the river. + +
+ + + +

+ + +

+ + + diff --git a/source/activities/act-trig-tangent-mountain.xml b/source/activities/act-trig-tangent-mountain.xml index 3f91fbe3..c2ff3fa5 100755 --- a/source/activities/act-trig-tangent-mountain.xml +++ b/source/activities/act-trig-tangent-mountain.xml @@ -1,4 +1,4 @@ - + @@ -12,64 +12,83 @@ - - - - -

+ + + +

Surveyors are trying to determine the height of a hill relative to sea level. First, they choose a point to take an initial measurement with a sextant that shows the angle of elevation from the ground to the peak of the hill is 19^\circ. Next, they move 1000 feet closer to the hill, staying at the same elevation relative to sea level, and find that the angle of elevation has increased to 25^\circ, as pictured in Figure. We let h represent the height of the hill relative to the two measurements, and x represent the distance from the second measurement location to the center of the hill that lies directly under the peak.

- -
- The surveyors' initial measurements. - -
- -

-

    -
  1. -

    +

    + The surveyors' initial measurements. + +
    +

    + + + +

    Using the right triangle with the 25^\circ angle, find an equation that relates x and h.

    -
    2. -
  2. -

    + + + + + + + +

    Using the right triangle with the 19^\circ angle, find a second equation that relates x and h.

    -
    3. -
  3. -

    + + + + + + + +

    Our work in (a) and (b) results in a system of two equations in the two unknowns x and h. Solve each of the two equations for h and then substitute appropriately in order to find a single equation in the variable x.

    -
    4. -
  4. -

    + + + + + + + +

    Solve the equation from (c) to find the exact value of x and determine an approximate value accurate to 3 decimal places.

    -
    5. -
  5. -

    + + + + + + + +

    Use your preceding work to solve for h exactly, plus determine an estimate accurate to 3 decimal places.

    -
    6. -
  6. -

    + + + + + + + +

    If the surveyors' initial measurements were taken from an elevation of 78 feet above sea level, how high above sea level is the peak of the hill?

    -
    7. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/activities/act-trig-tangent-tower.xml b/source/activities/act-trig-tangent-tower.xml index bb316493..0ea0c0b4 100755 --- a/source/activities/act-trig-tangent-tower.xml +++ b/source/activities/act-trig-tangent-tower.xml @@ -1,4 +1,4 @@ - + @@ -12,24 +12,18 @@ - - - - -

+ + + +

The top of a 225 foot tower is to be anchored by four cables that each make an angle of 32.5^{\circ} with the ground. How long do the cables have to be and how far from the base of the tower must they be anchored?

- - -

- -

- - -

- -

- - - - + + +

+ + +

+ + + diff --git a/source/activities/act-trig-tangent-two-towers.xml b/source/activities/act-trig-tangent-two-towers.xml index 2569fea1..6d1b2fda 100755 --- a/source/activities/act-trig-tangent-two-towers.xml +++ b/source/activities/act-trig-tangent-two-towers.xml @@ -1,4 +1,4 @@ - + @@ -12,34 +12,24 @@ - - - - - - -

+ + + + +

SupertallSee, for instance, this article high rises have changed the Manhattan skyline. These skyscrapers are known for their small footprint in proportion to their height, with their ratio of width to height at most 1:10, and some as extreme as 1:24. Suppose that a relatively short supertall has been built to a height of 635 feet, as pictured in Figure, and that a second supertall is built nearby. Given the two angles that are computed from the new building, how tall, s, is the new building, and how far apart, d, are the two towers? -

- -
- Two supertall skyscrapers. - -
- - - - - -

- -

- - -

- -

- - - - +

+
+ Two supertall skyscrapers. + +
+ + + +

+ + +

+ + + diff --git a/source/activities/z-act-inverse-rainfall.xml b/source/activities/z-act-inverse-rainfall.xml index e47e37a3..f0bea91a 100644 --- a/source/activities/z-act-inverse-rainfall.xml +++ b/source/activities/z-act-inverse-rainfall.xml @@ -1,4 +1,4 @@ - + @@ -12,34 +12,25 @@ - - - - -

- -

- -

-

    -
  1. -

    - -

    -
    2. -
-

- - -

- -

- - -

- -

- - - - + + + +

+

+ + + +

+ + + + + + +

+ + +

+ + + diff --git a/source/activities/z-act-transformations-translations.xml b/source/activities/z-act-transformations-translations.xml index 5aa9bac2..13085ce9 100644 --- a/source/activities/z-act-transformations-translations.xml +++ b/source/activities/z-act-transformations-translations.xml @@ -1,4 +1,4 @@ - + @@ -12,60 +12,69 @@ - - - - -

+ + + +

Consider the functions r and s given in Figure and Figure.

- - -
- A parent function r. - -
-
- A parent function s. - -
- - -

-

    -
  1. -

    + +

    + A parent function r. + +
    +
    + A parent function s. + +
    + +

    + + + +

    On the same axes as the plot of y = r(x), sketch the following graphs: y = g(x) = r(x) + 2, y = h(x) = r(x+1), and y = f(x) = r(x+1) + 2. Be sure to label the point on each of g, h, and f that corresponds to (-2,-1) on the original graph of r.

    -
    2. -
  2. -

    + + + + + + + +

    Is it possible to view the function f in (a) as the result of composition of g and h? If so, in what order should g and h be composed in order to produce f?

    -
    3. -
  3. -

    + + + + + + + +

    On the same axes as the plot of y = s(x), sketch the following graphs: y = k(x) = s(x) - 1, y = j(x) = s(x-2), and y = m(x) = s(x-2) - 1. Be sure to label the point on each of k, j, and m that corresponds to (-2,-3) on the original graph of r.

    -
    4. -
  4. -

    + + + + + + + +

    Now consider the function q(x) = x^2. Determine a formula for the function that is given by p(x) = q(x+3) - 4. How is p a transformation of q?

    -
    5. -
-

- - -

- -

- - -

- -

- - - - + + + + + + +

+ + +

+ + + diff --git a/source/previews/PA-0-0.xml b/source/previews/PA-0-0.xml index 13dcdb02..f4291ada 100755 --- a/source/previews/PA-0-0.xml +++ b/source/previews/PA-0-0.xml @@ -1,4 +1,4 @@ - + @@ -12,13 +12,10 @@ - - - - -

- -

- - - \ No newline at end of file + + + +

+ + + diff --git a/source/previews/PA-changing-aroc.xml b/source/previews/PA-changing-aroc.xml index e0d4717e..f4211fe3 100755 --- a/source/previews/PA-changing-aroc.xml +++ b/source/previews/PA-changing-aroc.xml @@ -1,4 +1,4 @@ - + @@ -12,31 +12,39 @@ - - - -

+ + + +

Let the height function for a ball tossed vertically be given by s(t) = 64 - 16(t-1)^2, where t is measured in seconds and s is measured in feet above the ground.

- -

-

    -
  1. -

    +

    + + + +

    Compute the value of AV_{[1.5,2.5]}.

    -
    2. - -
  2. -

    + + + + + + + +

    What are the units on the quantity AV_{[1.5,2.5]}? What is the meaning of this number in the context of the rising/falling ball?

    -
    3. - -
  3. -

    + + + + + + + +

    In Desmos, plot the function s(t) = 64 - 16(t-1)^2 along with the points (1.5,s(1.5)) and (2.5, s(2.5)). @@ -46,36 +54,50 @@ The range? Why?

    -
    - Axes for plotting the position function. - -
    -
    4. - -
  4. -

    +

    + Axes for plotting the position function. + +
    + + + + + + + +

    Work by hand to find the equation of the line through the points (1.5,s(1.5)) and (2.5, s(2.5)). Write the line in the form y = mt + b and plot the line in Desmos, as well as on the axes above.

    -
    5. - -
  5. -

    + + + + + + + +

    What is a geometric interpretation of the value AV_{[1.5,2.5]} in light of your work in the preceding questions?

    -
    6. - -
  6. -

    + + + + + + + +

    How do your answers in the preceding questions change if we instead consider the interval [0.25, 0.75]? [0.5, 1.5]? [1,3]?

    -
    7. -
-

- - + + + + + + + diff --git a/source/previews/PA-changing-combining.xml b/source/previews/PA-changing-combining.xml index fee0ed09..db7d4cb7 100755 --- a/source/previews/PA-changing-combining.xml +++ b/source/previews/PA-changing-combining.xml @@ -1,4 +1,4 @@ - + @@ -12,85 +12,114 @@ - - - - -

+ + + +

Consider the functions f and g defined by Table and the piecewise linear functions p and q defined by Figure. Assume that the lines in the figure pass through whole number coordinates where they appear to do so; for example, (2,2) lies on the graph of q, and (3,-3) lies on the graph of p.

- - - - Table defining functions <m>f</m> and <m>g</m>. - - - x - 0 - 1 - 2 - 3 - 4 - - - f(x) - 5 - 10 - 15 - 20 - 25 - - - g(x) - 9 - 5 - 3 - 2 - 3 - - -
-
- Graphs defining functions p and q. - -
- - -

-

    -
  1. -

    + + + Table defining functions <m>f</m> and <m>g</m>. + + + + x + + 0 + 1 + 2 + 3 + 4 + + + + f(x) + + 5 + 10 + 15 + 20 + 25 + + + + g(x) + + 9 + 5 + 3 + 2 + 3 + + +
    +

    + Graphs defining functions p and q. + +
    + +

    + + + +

    Let h(x) = f(x) + g(x). Determine h(3).

    -
    2. -
  2. -

    + + + + + + + +

    Let r(x) = p(x) - q(x). Determine r(-1) exactly.

    -
    3. -
  3. -

    + + + + + + + +

    Are there any values of x for which r(x) = 0? If not, explain why; if so, determine all such values, with justification.

    -
    4. -
  4. -

    + + + + + + + +

    Let k(x) = f(x) \cdot g(x). Determine k(0).

    -
    5. -
  5. -

    + + + + + + + +

    Let s(x) = \frac{p(x)}{q(x)}. Determine s(1) exactly.

    -
    6. -
  6. -

    + + + + + + + +

    Are there any values of x in the interval -4 \le x \le 4 for which s(x) is not defined? If not, explain why; if so, determine all such values, with justification.

    -
    7. -
-

- - - - + + + + + + + diff --git a/source/previews/PA-changing-composite.xml b/source/previews/PA-changing-composite.xml index ea9ed5f6..8547b371 100755 --- a/source/previews/PA-changing-composite.xml +++ b/source/previews/PA-changing-composite.xml @@ -1,4 +1,4 @@ - + @@ -12,38 +12,53 @@ - - - - -

+ + + +

Let y = p(x) = 3x - 4 and x = q(t) = t^2 - 1.

- -

-

    -
  1. -

    +

    + + + +

    Let r(t) = p(q(t)). Determine a formula for r that depends only on t and not on p or q.

    -
    2. -
  2. -

    + + + + + + + +

    Recall Example, which involved functions similar to p and q. What is the biggest difference between your work in (a) above and in Example?

    -
    3. -
  3. -

    + + + + + + + +

    Let t = s(z) = \frac{1}{z+4} and recall that x = q(t) = t^2 - 1. Determine a formula for x = q(s(z)) that depends only on z.

    -
    4. -
  4. -

    + + + + + + + +

    Suppose that h(t) = \sqrt{2t^2 + 5}. Determine formulas for two related functions, y = f(x) and x = g(t), so that h(t) = f(g(t)).

    -
    5. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-changing-functions-crickets.xml b/source/previews/PA-changing-functions-crickets.xml index 5f088001..8b54a68c 100755 --- a/source/previews/PA-changing-functions-crickets.xml +++ b/source/previews/PA-changing-functions-crickets.xml @@ -1,4 +1,4 @@ - + @@ -12,43 +12,63 @@ - - - - -

+ + + +

Use Equation to respond to the questions below.

- -

-

    -
  1. -

    +

    + + + +

    If we hear snowy tree crickets chirping at a rate of 92 chirps per minute, what does Dolbear's model suggest should be the outside temperature?

    -
    2. -
  2. -

    + + + + + + + +

    If the outside temperature is 77^\circ F, how many chirps per minute should we expect to hear?

    -
    3. -
  3. -

    + + + + + + + +

    Is the model valid for determining the number of chirps one should hear when the outside temperature is 35^\circ F? Why or why not?

    -
    4. -
  4. -

    + + + + + + + +

    Suppose that in the morning an observer hears 65 chirps per minute, and several hours later hears 75 chirps per minute. How much has the temperature risen between observations?

    -
    5. -
  5. -

    + + + + + + + +

    Dolbear's Law is known to be accurate for temperatures from 50^\circ to 85^\circ. What is the fewest number of chirps per minute an observer could expect to hear? the greatest number of chirps per minute?

    -
    6. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-changing-inverse-F-C.xml b/source/previews/PA-changing-inverse-F-C.xml index 08514d74..f040d089 100755 --- a/source/previews/PA-changing-inverse-F-C.xml +++ b/source/previews/PA-changing-inverse-F-C.xml @@ -1,4 +1,4 @@ - + @@ -12,45 +12,56 @@ - - - - -

+ + + +

Recall that F = g(C) = \frac{9}{5}C + 32 is the function that takes Celsius temperature inputs and produces the corresponding Fahrenheit temperature outputs.

- -

-

    -
  1. -

    +

    + + + +

    Show that it is possible to solve the equation F = \frac{9}{5}C + 32 for C in terms of F and that doing so results in the equation C = \frac{5}{9}(F-32).

    -
    2. - -
  2. -

    + + + + + + + +

    Note that the equation C = \frac{5}{9}(F-32) expresses C as a function of F. Call this function h so that C = h(F) = \frac{5}{9}(F-32).

    - -

    +

    Find the simplest expression that you can for the composite function j(C) = h(g(C)).

    -
    3. - -
  3. -

    + + + + + + + +

    Find the simplest expression that you can for the composite function k(F) = g(h(F)).

    -
    4. - -
  4. -

    + + + + + + + +

    Why are the functions j and k so simple? Explain by discussing how the functions g and h process inputs to generate outputs and what happens when we first execute one followed by the other.

    -
    5. -
-

- - - + + + + + + + diff --git a/source/previews/PA-changing-linear-3-ex.xml b/source/previews/PA-changing-linear-3-ex.xml index 0c72dd02..7937ee1d 100755 --- a/source/previews/PA-changing-linear-3-ex.xml +++ b/source/previews/PA-changing-linear-3-ex.xml @@ -1,4 +1,4 @@ - + @@ -12,91 +12,155 @@ - - - - - -

-

    -
  1. -

    + + + +

    + + + +

    Let y = f(x) = 7 - 3x. Determine AV_{[-3,-1]}, AV_{[2,5]}, and AV_{[4,10]} for the function f.

    -
    2. -
  2. -

    + + + + + + + +

    Let y = g(x) be given by the data in Table.

    - - - A table that defines the function <m>y = g(x)</m>. - - - x - -5 - -4 - -3 - -2 - -1 - 0 - 1 - 2 - 3 - 4 - 5 - - - g(x) - -2.75 - -2.25 - -1.75 - -1.25 - -0.75 - -0.25 - 0.25 - 0.75 - 1.25 - 1.75 - 2.25 - - -
    - -

    + + A table that defines the function <m>y = g(x)</m>. + + + + x + + + -5 + + + -4 + + + -3 + + + -2 + + + -1 + + + 0 + + + 1 + + + 2 + + + 3 + + + 4 + + + 5 + + + + + g(x) + + + -2.75 + + + -2.25 + + + -1.75 + + + -1.25 + + + -0.75 + + + -0.25 + + + 0.25 + + + 0.75 + + + 1.25 + + + 1.75 + + + 2.25 + + + +
    +

    Determine AV_{[-5,-2]}, AV_{[-1,1]}, and AV_{[0,4]} for the function g.

    -
    3. -
  3. -

    + + + + + + + +

    Consider the function y = h(x) defined by the graph in Figure.

    - -
    - The graph of y = h(x). - -
    - -

    +

    + The graph of y = h(x). + +
    +

    Determine AV_{[-5,-2]}, AV_{[-1,1]}, and AV_{[0,4]} for the function h.

    -
    4. -
  4. -

    + + + + + + + +

    What do all three examples above have in common? How do they differ?

    -
    5. -
  5. -

    + + + + + + + +

    For the function y = f(x) = 7 - 3x from (a), find the simplest expression you can for AV_{[a,b]} = \frac{f(b)-f(a)}{b-a} where a \ne b.

    -
    6. -
-

- - - + + + + + + + diff --git a/source/previews/PA-changing-quadratic.xml b/source/previews/PA-changing-quadratic.xml index 60cd0178..45248cfc 100755 --- a/source/previews/PA-changing-quadratic.xml +++ b/source/previews/PA-changing-quadratic.xml @@ -1,4 +1,4 @@ - + @@ -12,108 +12,155 @@ - - - - -

+ + + +

A water balloon is tossed vertically from a fifth story window. Its height, h, in meters, at time t, in seconds, is modeled by the function h = q(t) = -5t^2 + 20t + 25 .

- -

-

    -
  1. -

    +

    + + + +

    Execute appropriate computations to complete both of the following tables.

    - - - Function values for <m>h</m> at select inputs. - - - t - h = q(t) - - - 0 - q(0) = 25 - - - 1 - - - - 2 - - - - 3 - - - - 4 - - - - 5 - - - -
    - - Average rates of change for <m>h</m> on select intervals. - - - [a,b] - AV_{[a,b]} - - - [0,1] - AV_{[0,1]} = 15 m/s - - - [1,2] - - - - [2,3] - - - - [3,4] - - - - [4,5] - - - - - - -
    -     -
    2. -
  2. -

    + + + Function values for <m>h</m> at select inputs. + + + + t + + + h = q(t) + + + + + 0 + + + q(0) = 25 + + + + + 1 + + + + + + 2 + + + + + + 3 + + + + + + 4 + + + + + + 5 + + + + +
    + + Average rates of change for <m>h</m> on select intervals. + + + + [a,b] + + + AV_{[a,b]} + + + + + [0,1] + + AV_{[0,1]} = 15 m/s + + + + [1,2] + + + + + + [2,3] + + + + + + [3,4] + + + + + + [4,5] + + + + + + + +
    +     + + + + + + + +

    What pattern(s) do you observe in Tables and ?

    -
    3. -
  3. -

    + + + + + + + +

    Explain why h = q(t) is not a linear function. Use Definition in your response.

    -
    4. -
  4. -

    + + + + + + + +

    What is the average velocity of the water balloon in the final second before it lands? How does this value compare to the average velocity on the time interval [4.9, 5]?

    -
    5. -
-

- - - + + + + + + + diff --git a/source/previews/PA-changing-tandem-aquarium.xml b/source/previews/PA-changing-tandem-aquarium.xml index c817b9f7..31c01ed8 100755 --- a/source/previews/PA-changing-tandem-aquarium.xml +++ b/source/previews/PA-changing-tandem-aquarium.xml @@ -1,4 +1,4 @@ - + @@ -12,46 +12,63 @@ - - - -

+ + + +

Suppose that a rectangular aquarium is being filled with water. The tank is 4 feet long by 2 feet wide by 3 feet high, and the hose that is filling the tank is delivering water at a rate of 0.5 cubic feet per minute.

- - -
- The empty aquarium. - -
-
- The aquarium, partially filled. - -
- -

-

    -
  1. -

    + +

    + The empty aquarium. + +
    +
    + The aquarium, partially filled. + +
    + +

    + + + +

    What are some different quantities that are changing in this scenario?

    -
    2. -
  2. -

    + + + + + + + +

    After 1 minute has elapsed, how much water is in the tank? At this moment, how deep is the water?

    -
    3. -
  3. -

    + + + + + + + +

    How much water is in the tank and how deep is the water after 2 minutes? After 3 minutes?

    -
    4. -
  4. -

    + + + + + + + +

    How long will it take for the tank to be completely full? Why?

    -
    5. -
-

- - + + + + + + + diff --git a/source/previews/PA-changing-transformations-quadratic.xml b/source/previews/PA-changing-transformations-quadratic.xml index 15af9dd7..23dbea53 100755 --- a/source/previews/PA-changing-transformations-quadratic.xml +++ b/source/previews/PA-changing-transformations-quadratic.xml @@ -1,4 +1,4 @@ - + @@ -12,60 +12,71 @@ - - - - -

+ + + +

Open a new Desmos graph and define the function f(x) = x^2. Adjust the window so that the range is for -4 \le x \le 4 and -10 \le y \le 10.

- -

-

    -
  1. -

    +

    + + + +

    In Desmos, define the function g(x) = f(x) + a. (That is, in Desmos on line 2, enter g(x) = f(x) + a.) You will get prompted to add a slider for a. Do so.

    - -

    +

    Explore by moving the slider for a and write at least one sentence to describe the effect that changing the value of a has on the graph of g.

    -
    2. -
  2. -

    + + + + + + + +

    Next, define the function h(x) = f(x-b). (That is, in Desmos on line 4, enter h(x) = f(x-b) and add the slider for b.)

    - -

    +

    Move the slider for b and write at least one sentence to describe the effect that changing the value of b has on the graph of h.

    -
    3. -
  3. -

    + + + + + + + +

    Now define the function p(x) = cf(x). (That is, in Desmos on line 6, enter p(x) = cf(x) and add the slider for c.)

    - -

    +

    Move the slider for c and write at least one sentence to describe the effect that changing the value of c has on the graph of p. In particular, when c = -1, how is the graph of p related to the graph of f?

    -
    4. -
  4. -

    + + + + + + + +

    Finally, click on the icons next to g, h, and p to temporarily hide them, and go back to Line 1 and change your formula for f. You can make it whatever you'd like, but try something like f(x) = x^2 + 2x + 3 or f(x) = x^3 - 1. @@ -73,9 +84,10 @@ and c to see the effects on g, h, and p (unhiding them appropriately). Write a couple of sentences to describe your observations of your explorations.

    -
    5. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-circular-sine.xml b/source/previews/PA-circular-sine.xml index 80fe8bcb..dfaf59cf 100755 --- a/source/previews/PA-circular-sine.xml +++ b/source/previews/PA-circular-sine.xml @@ -1,4 +1,4 @@ - + @@ -12,99 +12,157 @@ - - - - -

+ + + +

If we consider the unit circle in Figure, start at t = 0, and traverse the circle counterclockwise, we may view the height, h, of the traversing point as a function of the angle, t, in radians. From there, we can plot the resulting (t,h) ordered pairs and connect them to generate the circular function pictured in Figure.

- -
- Plot of the circular function that tracks the height of a point traversing the unit circle. - -
- -

-

    -
  1. -

    +

    + Plot of the circular function that tracks the height of a point traversing the unit circle. + +
    +

    + + + +

    What is the exact value of f( \frac{\pi}{4} )? of f( \frac{\pi}{3} )?

    -
    2. -
  2. -

    + + + + + + + +

    Complete the following table with the exact values of h that correspond to the stated inputs.

    - - Exact values of <m>h</m> as a function of <m>t</m>. - - - t - 0 - \frac{\pi}{6} - \frac{\pi}{4} - \frac{\pi}{3} - \frac{\pi}{2} - \frac{2\pi}{3} - \frac{3\pi}{4} - \frac{5\pi}{6} - \pi - - - h - - - - - - - - - - - - - - - t - \pi - \frac{7\pi}{6} - \frac{5\pi}{4} - \frac{4\pi}{3} - \frac{3\pi}{2} - \frac{5\pi}{3} - \frac{7\pi}{4} - \frac{11\pi}{6} - 2\pi - - - h - - - - - - - - - - - -
    -
    3. -
  3. -

    + + Exact values of <m>h</m> as a function of <m>t</m>. + + + + t + + + 0 + + + \frac{\pi}{6} + + + \frac{\pi}{4} + + + \frac{\pi}{3} + + + \frac{\pi}{2} + + + \frac{2\pi}{3} + + + \frac{3\pi}{4} + + + \frac{5\pi}{6} + + + \pi + + + + + h + + + + + + + + + + + + + + + + + t + + + \pi + + + \frac{7\pi}{6} + + + \frac{5\pi}{4} + + + \frac{4\pi}{3} + + + \frac{3\pi}{2} + + + \frac{5\pi}{3} + + + \frac{7\pi}{4} + + + \frac{11\pi}{6} + + + 2\pi + + + + + h + + + + + + + + + + + + +
    + + + + + + + +

    What is the exact value of f( \frac{11\pi}{4} )? of f( \frac{14\pi}{3} )?

    -
    4. -
  4. -

    + + + + + + + +

    Give four different values of t for which f(t) = -\frac{\sqrt{3}}{2}.

    -
    5. -
-

- - - + + + + + + + diff --git a/source/previews/PA-circular-sinusoidal.xml b/source/previews/PA-circular-sinusoidal.xml index 9c952766..c9b03d97 100755 --- a/source/previews/PA-circular-sinusoidal.xml +++ b/source/previews/PA-circular-sinusoidal.xml @@ -1,4 +1,4 @@ - + @@ -12,38 +12,46 @@ - - - - -

+ + + +

Let f(t) = \cos(t). First, answer all of the questions below without using Desmos; then use Desmos to confirm your conjectures. For each prompt, describe the graphs of g and h as transformations of f and, in addition, state the amplitude, midline, and period of both g and h.

-

-

    -
  1. -

    - g(t) = 3\cos(t) and h(t) = -\frac{1}{4}\cos(t) -

    -
    2. -
  2. -

    - g(t) = \cos(t-\pi) and h(t) = \cos\left(t+ \frac{\pi}{2}\right) -

    -
    3. -
  3. -

    - g(t) = \cos(t)+4 and h(t) = \cos\left(t\right)-2 -

    -
    4. -
  4. -

    - g(t) = 3\cos(t-\pi)+4 and h(t) = -\frac{1}{4}\cos\left(t+ \frac{\pi}{2}\right)-2 -

    -
    5. -
-

- - - +

+ + + +

g(t) = 3\cos(t) and h(t) = -\frac{1}{4}\cos(t)

+ + + + + + + +

g(t) = \cos(t-\pi) and h(t) = \cos\left(t+ \frac{\pi}{2}\right)

+ + + + + + + +

g(t) = \cos(t)+4 and h(t) = \cos\left(t\right)-2

+ + + + + + + +

g(t) = 3\cos(t-\pi)+4 and h(t) = -\frac{1}{4}\cos\left(t+ \frac{\pi}{2}\right)-2

+ + + + + + + diff --git a/source/previews/PA-circular-traversing.xml b/source/previews/PA-circular-traversing.xml index 5cb1ada9..c03e15b0 100755 --- a/source/previews/PA-circular-traversing.xml +++ b/source/previews/PA-circular-traversing.xml @@ -1,4 +1,4 @@ - + @@ -12,52 +12,76 @@ - - - - -

+ + + +

In the context of the ferris wheel pictured in Figure, assume that the height, h, of the moving point (the cab in which you are riding), and the distance, d, that the point has traveled around the circumference of the ferris wheel are both measured in meters.

- -

+

Further, assume that the circumference of the ferris wheel is 150 meters. In addition, suppose that after getting in your cab at the lowest point on the wheel, you traverse the full circle several times.

- -

-

    -
  1. -

    +

    + + + +

    Recall that the circumference, C, of a circle is connected to the circle's radius, r, by the formula C = 2\pi r. What is the radius of the ferris wheel? How high is the highest point on the ferris wheel?

    -
    2. -
  2. -

    + + + + + + + +

    How high is the cab after it has traveled 1/4 of the circumference of the circle?

    -
    3. -
  3. -

    + + + + + + + +

    How much distance along the circle has the cab traversed at the moment it first reaches a height of \frac{150}{\pi} \approx 47.75 meters?

    -
    4. -
  4. -

    + + + + + + + +

    Can h be thought of as a function of d? Why or why not?

    -
    5. -
  5. -

    + + + + + + + +

    Can d be thought of as a function of h? Why or why not?

    -
    6. -
  6. -

    + + + + + + + +

    Why do you think the curve shown at right in Figure has the shape that it does? Write several sentences to explain.

    -
    7. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-circular-unit-circle.xml b/source/previews/PA-circular-unit-circle.xml index a43e785c..68a58570 100755 --- a/source/previews/PA-circular-unit-circle.xml +++ b/source/previews/PA-circular-unit-circle.xml @@ -1,4 +1,4 @@ - + @@ -12,58 +12,70 @@ - - - - -

+ + + +

In Figure there are 24 equally spaced points on the unit circle. Since the circumference of the unit circle is 2\pi, each of the points is \frac{1}{24} \cdot 2\pi = \frac{\pi}{12} units apart (traveled along the circle). Thus, the first point counterclockwise from (1,0) corresponds to the distance t = \frac{\pi}{12} traveled along the unit circle. The second point is twice as far, and thus t = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6} units along the circle away from (1,0).

- -
- The unit circle with 24 equally-spaced points. - -
- -

-

    -
  1. -

    +

    + The unit circle with 24 equally-spaced points. + +
    +

    + + + +

    Label each of the subsequent points on the unit circle with the exact distance they lie counter-clockwise away from (1,0); write each fraction in lowest terms.

    -
    2. -
  2. -

    + + + + + + + +

    Which distance along the unit circle corresponds to \frac{1}{4} of a full rotation around? to \frac{5}{8} of a full rotation?

    -
    3. -
  3. - -

    - One way to measure angles is connected to the arc length along a circle. For an angle whose vertex is at (0,0) in the unit circle, we say the angle's measure is 1 radian radian provided that the angle intercepts an arc of the circle that is 1 unit in length, as pictured in Figure. Note particularly that an angle measuring 1 radian intercepts an arc of the same length as the circle's radius. + + + + + + + + +

    + One way to measure angles is connected to the arc length along a circle. For an angle whose vertex is at (0,0) in the unit circle, we say the angle's measure is 1 radianradian provided that the angle intercepts an arc of the circle that is 1 unit in length, as pictured in Figure. Note particularly that an angle measuring 1 radian intercepts an arc of the same length as the circle's radius.

    - -
    - +
    + An angle \theta of measure 1 radian. - -
    - - -

    + +

    +     +

    Suppose that \alpha and \beta are angles with respective radian measures \alpha = \frac{\pi}{3} and \beta = \frac{3\pi}{4}. Assuming that we view \alpha and \beta as having their vertex at (0,0) and one side along the positive x-axis, sketch the angles \alpha and \beta on the unit circle in Figure.

    -
    4. -
  4. -

    + + + + + + + +

    What is the radian measure that corresponds to a 90^\circ angle?

    -
    5. -
-

- - - + + + + + + + diff --git a/source/previews/PA-exp-e.xml b/source/previews/PA-exp-e.xml index 56ab8aa8..455dec28 100755 --- a/source/previews/PA-exp-e.xml +++ b/source/previews/PA-exp-e.xml @@ -1,4 +1,4 @@ - + @@ -12,43 +12,63 @@ - - - - -

+ + + +

Open a new Desmos worksheet and define the following functions: f(t) = 2^t, g(t) = 3^t, h(t) = (\frac{1}{3})^t, and p(t) = f(kt). After you define p, accept the slider for k, and set the range of the slider to be -2 \le k \le 2.

- -

-

    -
  1. -

    +

    + + + +

    By experimenting with the value of k, find a value of k so that the graph of p(t) = f(kt) = 2^{kt} appears to align with the graph of g(t) = 3^t. What is the value of k?

    -
    2. -
  2. -

    + + + + + + + +

    Similarly, experiment to find a value of k so that the graph of p(t) = f(kt) = 2^{kt} appears to align with the graph of h(t) = (\frac{1}{3})^t. What is the value of k?

    -
    3. -
  3. -

    + + + + + + + +

    For the value of k you determined in (a), compute 2^k. What do you observe?

    -
    4. -
  4. -

    + + + + + + + +

    For the value of k you determined in (b), compute 2^k. What do you observe?

    -
    5. -
  5. -

    + + + + + + + +

    Given any exponential function of the form b^t, do you think it's possible to find a value of k to that p(t) = f(kt) = 2^{kt} is the same function as b^t? Why or why not?

    -
    6. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-exp-growth.xml b/source/previews/PA-exp-growth.xml index 5b9c46c4..10370e2f 100755 --- a/source/previews/PA-exp-growth.xml +++ b/source/previews/PA-exp-growth.xml @@ -1,4 +1,4 @@ - + @@ -12,46 +12,60 @@ - - - - -

+ + + +

Suppose that at age 20 you have $20000 and you can choose between one of two ways to use the money: you can invest it in a mutual fund that will, on average, earn 8% interest annually, or you can purchase a new automobile that will, on average, depreciate 12% annually. Let's explore how the $20000 changes over time.

- -

+

Let I(t) denote the value of the $20000 after t years if it is invested in the mutual fund, and let V(t) denote the value of the automobile t years after it is purchased.

- -

-

    -
  1. -

    +

    + + + +

    Determine I(0), I(1), I(2), and I(3).

    -
    2. -
  2. -

    + + + + + + + +

    Note that if a quantity depreciates 12% annually, after a given year, 88% of the quantity remains. Compute V(0), V(1), V(2), and V(3).

    -
    3. -
  3. -

    + + + + + + + +

    Based on the patterns in your computations in (a) and (b), determine formulas for I(t) and V(t).

    -
    4. -
  4. -

    + + + + + + + +

    Use Desmos to define I(t) and V(t). Plot each function on the interval 0 \le t \le 20 and record your results on the axes in Figure, being sure to label the scale on the axes. What trends do you observe in the graphs? How do I(20) and V(20) compare? -

    -
    - Blank axes for plotting I and V. - -
    -
    5. -
-

- - - +

+
+ Blank axes for plotting I and V. + +
+ + + + + + + diff --git a/source/previews/PA-exp-log-properties.xml b/source/previews/PA-exp-log-properties.xml index d2011e49..c26d822a 100755 --- a/source/previews/PA-exp-log-properties.xml +++ b/source/previews/PA-exp-log-properties.xml @@ -1,4 +1,4 @@ - + @@ -12,45 +12,66 @@ - - - - -

+ + + +

In the following questions, we investigate how \log_{10}(a \cdot b) can be equivalently written in terms of \log_{10}(a) and \log_{10}(b).

- -

-

    -
  1. -

    +

    + + + +

    Write 10^x \cdot 10^y as 10 raised to a single power. That is, complete the equation 10^x \cdot 10^y = 10^{\Box} by filling in the box with an appropriate expression involving x and y.

    -
    2. -
  2. -

    + + + + + + + +

    What is the simplest possible way to write \log_{10}10^x? What about the simplest equivalent expression for \log_{10}10^y?

    -
    3. -
  3. -

    + + + + + + + +

    Explain why each of the following three equal signs is valid in the sequence of equalities: - - \log_{10}(10^x \cdot 10^y) &= \log_{10}(10^{x+y}) - &= x+y - &= \log_{10}(10^x) + \log_{10}(10^y) - . + \log_{10}(10^x \cdot 10^y) &= \log_{10}(10^{x+y}) &= x+y &= \log_{10}(10^x) + \log_{10}(10^y).

    -
    4. -
  4. - Suppose that a and b are positive real numbers, so we can think of a as 10^x for some real number x and b as 10^y for some real number y. That is, say that a = 10^x and b = 10^y. What does our work in (c) tell us about \log_{10}(ab)? -
    5. -
-

- - - \ No newline at end of file + + + + + + + + a + b + a + 10^x + x + b + 10^y + y + a = 10^x + b = 10^y + \log_{10}(ab) + + + + + + + diff --git a/source/previews/PA-exp-log.xml b/source/previews/PA-exp-log.xml index bfd751f4..779b306a 100755 --- a/source/previews/PA-exp-log.xml +++ b/source/previews/PA-exp-log.xml @@ -1,4 +1,4 @@ - + @@ -12,102 +12,161 @@ - - - - -

+ + + +

Let P(t) be the powers of 10 function, which is given by P(t) = 10^t.

- -

-

    -
  1. -

    +

    + + + +

    Complete Table to generate certain values of P.

    - - - Select values of the powers of <m>10</m> function. - - - t - -3 - -2 - -1 - 0 - 1 - 2 - 3 - - - y = P(t) = 10^t - - - - - - - - - -
    -
    2. -
  2. -

    + + Select values of the powers of <m>10</m> function. + + + + t + + -3 + -2 + -1 + 0 + 1 + 2 + 3 + + + + y = P(t) = 10^t + + + + + + + + + + + + + + + + + + + + + + +
    + + + + + + + +

    Why does P have an inverse function?

    -
    3. -
  3. -

    + + + + + + + +

    Since P has an inverse function, we know there exists some other function, say L, such that writing y = P(t) says the exact same thing as writing t = L(y). In words, where P produces the result of raising 10 to a given power, the function L reverses this process and instead tells us the power to which we need to raise 10, given a desired result. Complete Table to generate a collection of values of L.

    - - - Select values of the function <m>L</m> that is the inverse of <m>P</m>. - - - y - 10^{-3} - 10^{-2} - 10^{-1} - 10^{0} - 10^{1} - 10^{2} - 10^{3} - - - - - - - - - - - - - L(y) - - - - - - - - - -
    -
    4. -
  4. -

    + + Select values of the function <m>L</m> that is the inverse of <m>P</m>. + + + + y + + + 10^{-3} + + + 10^{-2} + + + 10^{-1} + + + 10^{0} + + + 10^{1} + + + 10^{2} + + + 10^{3} + + + + + + + + + + + + + + + L(y) + + + + + + + + + + + + + + + + + + + + + + +
    + + + + + + + +

    What are the domain and range of the function P? What are the domain and range of the function L?

    -
    5. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-exp-modeling.xml b/source/previews/PA-exp-modeling.xml index 6e7990f2..782f0855 100755 --- a/source/previews/PA-exp-modeling.xml +++ b/source/previews/PA-exp-modeling.xml @@ -1,4 +1,4 @@ - + @@ -12,11 +12,10 @@ - - - - -

+ + + +

In Desmos, define g(t) = ab^t+c and accept the prompt for sliders for a, b, and c. Edit the sliders so that a has values from a = 5 to a = 50, @@ -27,35 +26,50 @@ let P = (0, g(0)) and check the box to show the label. Finally, zoom out so that the window shows an interval of t-values from -30 \le t \le 30.

- -

-

    -
  1. -

    +

    + + + +

    Set b = 1.1 and explore the effects of changing the values of a and c. Write several sentences to summarize your observations.

    -
    2. -
  2. -

    - Follow the directions for (a) again, this time with b = 0.9 -

    -
    3. -
  3. -

    + + + + + + + +

    + Follow the directions for (a) again, this time with b = 0.9

    + + + + + + + +

    Set a = 5 and c = 4. Explore the effects of changing the value of b; be sure to include values of b both less than and greater than 1. Write several sentences to summarize your observations.

    -
    4. -
  4. -

    + + + + + + + +

    When 0 \lt b \lt 1, what happens to the graph of g when we consider positive t-values that get larger and larger?

    -
    5. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-exp-temp-pop.xml b/source/previews/PA-exp-temp-pop.xml index d41ec808..d3b8cf9d 100755 --- a/source/previews/PA-exp-temp-pop.xml +++ b/source/previews/PA-exp-temp-pop.xml @@ -1,4 +1,4 @@ - + @@ -12,29 +12,37 @@ - - - - -

+ + + +

In each of the following situations, determine the exact value of the unknown quantity that is identified.

- -

-

    -
  1. -

    +

    + + + +

    The temperature of a warming object in an oven is given by F(t) = 275 - 203e^{-kt}, and we know that the object's temperature after 20 minutes is F(20) = 101. Determine the exact value of k.

    -
    2. -
  2. -

    + + + + + + + +

    The temperature of a cooling object in a refrigerator is modeled by F(t) = a + 37.4e^{-0.05t}, and the temperature of the refrigerator is 39.8^\circ. By thinking about the long-term behavior of e^{-0.05t} and the long-term behavior of the object's temperature, determine the exact value of a.

    -
    3. - -
  3. -

    + + + + + + + +

    Later in this section, we'll learn that one model for how a population grows over time can be given by a function of the form P(t) = \frac{A}{1 + Me^{-kt}} @@ -44,15 +52,20 @@ Solve Equation for the exact value of x.

    -
    4. -
  4. -

    + + + + + + + +

    Suppose that y = a + be^{-kt}. Solve for t in terms of a, b, k, and y. What does this new equation represent?

    -
    5. -
-

- - - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-poly-infty.xml b/source/previews/PA-poly-infty.xml index 44ad7211..2b40fc31 100755 --- a/source/previews/PA-poly-infty.xml +++ b/source/previews/PA-poly-infty.xml @@ -1,4 +1,4 @@ - + @@ -12,53 +12,83 @@ - - - - -

+ + + +

Complete each of the following statements with an appropriate number or the symbols \infty or -\infty. Do your best to do so without using a graphing utility; instead use your understanding of the function's graph.

- -

-

    -
  1. -

    - As t \to \infty, e^{-t} \to . +

    + + + +

    + As t \to \infty, e^{-t} \to .

    -
    2. -
  2. -

    - As t \to \infty, \ln(t) \to . + + + + + + + +

    + As t \to \infty, \ln(t) \to .

    -
    3. -
  3. -

    - As t \to \infty, e^{t} \to . + + + + + + + +

    + As t \to \infty, e^{t} \to .

    -
    4. -
  4. -

    - As t \to 0^+, e^{-t} \to . (When we write t \to 0^+, this means that we are letting t get closer and closer to 0, but only allowing t to take on positive values.) + + + + + + + +

    + As t \to 0^+, e^{-t} \to . (When we write t \to 0^+, this means that we are letting t get closer and closer to 0, but only allowing t to take on positive values.)

    -
    5. -
  5. -

    - As t \to \infty, 35 + 53e^{-0.025t} \to . + + + + + + + +

    + As t \to \infty, 35 + 53e^{-0.025t} \to.

    -
    6. -
  6. -

    - As t \to \frac{\pi}{2}^-, \tan(t) \to . (When we write t \to \frac{\pi}{2}^-, this means that we are letting t get closer and closer to \frac{\pi}{2}^-, but only allowing t to take on values that lie to the left of \frac{\pi}{2}.) + + + + + + + +

    + As t \to \frac{\pi}{2}^-, \tan(t) \to . (When we write t \to \frac{\pi}{2}^-, this means that we are letting t get closer and closer to \frac{\pi}{2}^-, but only allowing t to take on values that lie to the left of \frac{\pi}{2}.)

    -
    7. -
  7. -

    - As t \to \frac{\pi}{2}^+, \tan(t) \to . (When we write t \to \frac{\pi}{2}^+, this means that we are letting t get closer and closer to \frac{\pi}{2}^+, but only allowing t to take on values that lie to the right of \frac{\pi}{2}.) + + + + + + + +

    + As t \to \frac{\pi}{2}^+, \tan(t) \to . (When we write t \to \frac{\pi}{2}^+, this means that we are letting t get closer and closer to \frac{\pi}{2}^+, but only allowing t to take on values that lie to the right of \frac{\pi}{2}.)

    -
    8. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-poly-polynomial-applications.xml b/source/previews/PA-poly-polynomial-applications.xml index a5a79dd0..60d5cbf5 100755 --- a/source/previews/PA-poly-polynomial-applications.xml +++ b/source/previews/PA-poly-polynomial-applications.xml @@ -1,4 +1,4 @@ - + @@ -12,43 +12,63 @@ - - - - -

+ + + +

A piece of cardboard that is 12 \times 18 (each measured in inches) is being made into a box without a top. To do so, squares are cut from each corner of the cardboard and the remaining sides are folded up.

- -

-

    -
  1. -

    +

    + + + +

    Let x be the side length of the squares being cut from the corners of the cardboard. Draw a labeled diagram that shows the given information and the variable being used.

    -
    2. -
  2. -

    + + + + + + + +

    Determine a formula for the function V whose output is the volume of the box that results from a square of size x \times x being cut from each corner of the cardboard.

    -
    3. -
  3. -

    + + + + + + + +

    What familiar kind of function is V?

    -
    4. -
  4. -

    + + + + + + + +

    If we start with a small positive value for x and let that value get larger and larger, what is the first value of xwe encounter that makes it impossible to remove x \times x squares from the cardboard and still form a box?

    -
    5. -
  5. -

    + + + + + + + +

    What are the zeros of V? What is the domain of the model V in the context of the rectangular box?

    -
    6. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-poly-polynomials.xml b/source/previews/PA-poly-polynomials.xml index fdac39a2..382a59bb 100755 --- a/source/previews/PA-poly-polynomials.xml +++ b/source/previews/PA-poly-polynomials.xml @@ -1,4 +1,4 @@ - + @@ -12,56 +12,79 @@ - - - - -

+ + + +

Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zy. There you'll find a degree 4 polynomial of the form p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4, where a_0, \ldots, a_4 are set up as sliders. In the questions that follow, you'll experiment with different values of a_0, \ldots, a_4 to investigate different possible behaviors in a degree 4 polynomial. Note that we require a_4 \ne 0 in order to ensure p is a degree 4 polynomial.

- -

-

    -
  1. -

    +

    + + + +

    What is the largest number of distinct points at which p(x) can cross the x-axis?

    - -

    +

    Recall from Definitionwhat we mean by a zero of the polynomial. Give examples of values for a_0, \ldots, a_4 that lead to that largest number of zeros for p(x).

    -
    2. -
  2. -

    + + + + + + + +

    What other numbers of zeros are possible for p(x)? Said differently, can you get each possible number of fewer zeros than the largest number that you found in (a)? Why or why not?

    -
    3. -
  3. -

    - We say that a function has a turning point turning point if the function changes from decreasing to increasing or increasing to decreasing at the point. For example, any quadratic function has a turning point at its vertex. + + + + + + + +

    + We say that a function has a turning pointturning point if the function changes from decreasing to increasing or increasing to decreasing at the point. For example, any quadratic function has a turning point at its vertex.

    - -

    +

    What is the largest number of turning points that p(x) (the function in the Desmos worksheet) can have? Experiment with the sliders, and give examples of values for a_0, \ldots, a_4 that lead to that largest number of turning points for p(x).

    -
    4. -
  4. -

    + + + + + + + +

    What other numbers of turning points are possible for p(x)? Can it have no turning points? Just one? Exactly two? Experiment and explain.

    -
    5. -
  5. -

    + + + + + + + +

    What long-range behavior is possible for p(x)? Said differently, what are the possible results for \displaystyle \lim_{x \to -\infty} p(x) and \displaystyle \lim_{x \to \infty} p(x)?

    -
    6. -
  6. -

    + + + + + + + +

    What happens when we plot y = a_4 x^4 in Desmos and compare p(x) and a_4 x^4? How do they look when we zoom out? (Experiment with different values of each of the sliders, too.)

    -
    7. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-poly-rational-features.xml b/source/previews/PA-poly-rational-features.xml index bfaa1a85..725899e7 100755 --- a/source/previews/PA-poly-rational-features.xml +++ b/source/previews/PA-poly-rational-features.xml @@ -1,4 +1,4 @@ - + @@ -12,155 +12,265 @@ - - - - -

+ + + +

Consider the rational function r(x) = \frac{x^2 - 1}{x^2 - 3x - 4}, and let p(x) = x^2 - 1 (the numerator of r(x)) and q(x) = x^2 - 3x - 4 (the denominator of r(x)).

- - - -

-

    -
  1. -

    +

    + + + +

    Reasoning algebraically, for what values of x is p(x) = 0?

    -
    2. -
  2. -

    + + + + + + + +

    Again reasoning algebraically, for what values of x is q(x) = 0?

    -
    3. -
  3. -

    + + + + + + + +

    Define r(x) in Desmos, and evaluate the function appropriately to find numerical values for the output of r and hence complete the following tables.

    - - - - - x - r(x) - - - 4.1 - - - - 4.01 - - - - 4.001 - - - - 3.9 - - - - 3.99 - - - - 3.999 - - - - - - - x - r(x) - - - 1.1 - - - - 1.01 - - - - 1.001 - - - - 0.9 - - - - 0.99 - - - - 0.999 - - - - - - - x - r(x) - - - -1.1 - - - - -1.01 - - - - -1.001 - - - - -0.9 - - - - -0.99 - - - - -0.999 - - - - -
    4. -
  4. -

    + + + + + x + + + r(x) + + + + + 4.1 + + + + + + + + 4.01 + + + + + + + + 4.001 + + + + + + + + 3.9 + + + + + + + + 3.99 + + + + + + + + 3.999 + + + + + + + + + + x + + + r(x) + + + + + 1.1 + + + + + + + + 1.01 + + + + + + + + 1.001 + + + + + + + + 0.9 + + + + + + + + 0.99 + + + + + + + + 0.999 + + + + + + + + + + x + + + r(x) + + + + + -1.1 + + + + + + + + -1.01 + + + + + + + + -1.001 + + + + + + + + -0.9 + + + + + + + + -0.99 + + + + + + + + -0.999 + + + + + + + + + + + + + + +

    Why does r behave the way it does near x = 4? Explain by describing the behavior of the numerator and denominator.

    -
    5. -
  5. -

    + + + + + + + +

    Why does r behave the way it does near x = 1? Explain by describing the behavior of the numerator and denominator.

    -
    6. -
  6. -

    + + + + + + + +

    Why does r behave the way it does near x = -1? Explain by describing the behavior of the numerator and denominator.

    -
    7. -
  7. -

    + + + + + + + +

    Plot r in Desmos. Is there anything surprising or misleading about the graph that Desmos generates?

    -
    8. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-poly-rational.xml b/source/previews/PA-poly-rational.xml index a39a3b94..8a7fe6dd 100755 --- a/source/previews/PA-poly-rational.xml +++ b/source/previews/PA-poly-rational.xml @@ -1,4 +1,4 @@ - + @@ -12,49 +12,74 @@ - - - - -

+ + + +

A drug companyThis activity is based on p. 457ff in Functions Modeling Change, by Connally et al. estimates that to produce a new drug, it will cost $5 million in startup resources, and that once they reach production, each gram of the drug will cost $2500 to make.

- -

-

    -
  1. -

    +

    + + + +

    Determine a formula for a function C(q) that models the cost of producing q grams of the drug. What familiar kind of function is C?

    -
    2. -
  2. -

    + + + + + + + +

    The drug company needs to sell the drug at a price of more than $2500 per gram in order to at least break even. To investigate how they might set prices, they first consider what their average cost per gram is. What is the total cost of producing 1000 grams? What is the average cost per gram to produce 1000 grams?

    -
    3. -
  3. -

    + + + + + + + +

    What is the total cost of producing 10000 grams? What is the average cost per gram to produce 10000 grams?

    -
    4. -
  4. -

    + + + + + + + +

    Our computations in (b) and (c) naturally lead us to define the average cost per gram function, A(q), whose output is the average cost of producing q grams of the drug. What is a formula for A(q)?

    -
    5. -
  5. -

    + + + + + + + +

    Explain why another formula for A is A(q) = 2500 + \frac{5000000}{q}.

    -
    6. -
  6. -

    + + + + + + + +

    What can you say about the long-range behavior of A? What does this behavior mean in the context of the problem?

    -
    7. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-trig-finding-angles.xml b/source/previews/PA-trig-finding-angles.xml index c360b4a3..68b9d73e 100755 --- a/source/previews/PA-trig-finding-angles.xml +++ b/source/previews/PA-trig-finding-angles.xml @@ -1,4 +1,4 @@ - + @@ -12,43 +12,63 @@ - - - - -

+ + + +

Consider a right triangle that has one leg of length 3 and another leg of length \sqrt{3}. Let \theta be the angle that lies opposite the shorter leg.

- -

-

    -
  1. -

    +

    + + + +

    Sketch a labeled picture of the triangle.

    -
    2. -
  2. -

    + + + + + + + +

    What is the exact length of the triangle's hypotenuse?

    -
    3. -
  3. -

    + + + + + + + +

    What is the exact value of \sin(\theta)?

    -
    4. -
  4. -

    + + + + + + + +

    Rewrite your equation from (c) using the arcsine function in the form \arcsin(\Box) = \Delta, where \Box and \Delta are numerical values.

    -
    5. -
  5. -

    + + + + + + + +

    What special angle from the unit circle is \theta?

    -
    6. -
-

- - - + + + + + + + diff --git a/source/previews/PA-trig-inverse.xml b/source/previews/PA-trig-inverse.xml index e299bedf..5b242bc8 100755 --- a/source/previews/PA-trig-inverse.xml +++ b/source/previews/PA-trig-inverse.xml @@ -1,4 +1,4 @@ - + @@ -12,69 +12,87 @@ - - - - -

+ + + +

Consider the plot of the standard cosine function in Figure along with the emphasized portion of the graph on [0,\pi].

- -
- The cosine function on [-\frac{5\pi}{2},\frac{5\pi}{2}] with the portion on [0,\pi] emphasized. - -
- -

+

+ The cosine function on [-\frac{5\pi}{2},\frac{5\pi}{2}] with the portion on [0,\pi] emphasized. + +
+

Let g be the function whose domain is 0 \le t \le \pi and whose outputs are determined by the rule g(t) = \cos(t). Note well: g is defined in terms of the cosine function, but because it has a different domain, it is not the cosine function.

- -

-

    -
  1. -

    +

    + + + +

    What is the domain of g?

    -
    2. - -
  2. -

    + + + + + + + +

    What is the range of g?

    -
    3. - -
  3. -

    + + + + + + + +

    Does g pass the horizontal line test? Why or why not?

    -
    4. - -
  4. -

    + + + + + + + +

    Explain why g has an inverse function, g^{-1}, and state the domain and range of g^{-1}.

    -
    5. - -
  5. -

    + + + + + + + +

    We know that g(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}. What is the exact value of g^{-1}(\frac{\sqrt{2}}{2})? How about the exact value of g^{-1}(-\frac{\sqrt{2}}{2})?

    -
    6. - -
  6. -

    + + + + + + + +

    Determine the exact values of g^{-1}(-\frac{1}{2}), g^{-1}(\frac{\sqrt{3}}{2}), g^{-1}(0), and g^{-1}(-1). Use proper notation to label your results.

    -
    7. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-trig-other.xml b/source/previews/PA-trig-other.xml index ec7f5f55..5e6f6c29 100755 --- a/source/previews/PA-trig-other.xml +++ b/source/previews/PA-trig-other.xml @@ -1,4 +1,4 @@ - + @@ -12,16 +12,15 @@ - - - - -

+ + + +

Consider a right triangle with hypotenuse of length 61 and one leg of length 11. Let \alpha be the angle opposite the side of length 11. Find the exact length of the other leg and then determine the value of each of the six trigonometric functions evaluated at \alpha. In addition, what are the exact and approximate measures of the two non-right angles in the triangle?

- - - \ No newline at end of file + + + diff --git a/source/previews/PA-trig-right.xml b/source/previews/PA-trig-right.xml index f7ee3e41..5979a4b1 100755 --- a/source/previews/PA-trig-right.xml +++ b/source/previews/PA-trig-right.xml @@ -1,4 +1,4 @@ - + @@ -12,48 +12,73 @@ - - - - -

+ + + +

For each of the following situations, sketch a right triangle that satisfies the given conditions, and then either determine the requested missing information in the triangle or explain why you don't have enough information to determine it. Assume that all angles are being considered in radian measure.

- -

-

    -
  1. -

    +

    + + + +

    The length of the other leg of a right triangle with hypotenuse of length 1 and one leg of length \frac{3}{5}.

    -
    2. -
  2. -

    + + + + + + + +

    The lengths of the two legs in a right triangle with hypotenuse of length 1 where one of the non-right angles measures \frac{\pi}{3}.

    -
    3. -
  3. -

    + + + + + + + +

    The length of the other leg of a right triangle with hypotenuse of length 7 and one leg of length 6.

    -
    4. -
  4. -

    + + + + + + + +

    The lengths of the two legs in a right triangle with hypotenuse 5 and where one of the non-right angles measures \frac{\pi}{4}.

    -
    5. -
  5. -

    + + + + + + + +

    The length of the other leg of a right triangle with hypotenuse of length 1 and one leg of length \cos(0.7).

    -
    6. -
  6. -

    + + + + + + + +

    The measures of the two angles in a right triangle with hypotenuse of length 1 where the two legs have lengths \cos(1.1) and \sin(1.1), respectively.

    -
    7. -
-

- - - \ No newline at end of file + + + + + + + diff --git a/source/previews/PA-trig-tangent.xml b/source/previews/PA-trig-tangent.xml index b2044439..6a25e911 100755 --- a/source/previews/PA-trig-tangent.xml +++ b/source/previews/PA-trig-tangent.xml @@ -1,4 +1,4 @@ - + @@ -12,18 +12,17 @@ - - - - -

+ + + +

Through the following questions, we work to understand the special values and overall behavior of the tangent function.

- -

-

    -
  1. -

    +

    + + + +

    Without using computational device, find the exact value of \tan(t) at the following values: t = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}. - -

    -
    2. - -
  2. -

    +-->

    + + + + + + + +

    Why is \tan \left( \frac{\pi}{2} \right) not defined? What are three other input values x for which \tan(x) is not defined?

    -
    3. - -
  3. -

    + + + + + + + +

    Point your browser to http://gvsu.edu/s/0yO (zero-y-Oh) to find a Desmos @@ -80,36 +85,27 @@ \sin(\frac{11\pi}{24}) and \cos(\frac{11\pi}{24})? Why is the value of \tan(\frac{11\pi}{24}) so large relative to the other values of \tan(x) in the table?

    -
    4. - -
  4. -

    + + + + + + + +

    At the top of the input lists on the left side of the Desmos worksheet, click the circle to highlight the function T(x) = \tan(x) and thus show its plot along with the data points in orange. Use the plot and your work above to answer the following important questions about the tangent function: -

      -
    • -

      +

      • What is the domain of y = \tan(x)? -

        -
        • - -
      • -

        +

      • What is the period of y = \tan(x)? -

        -
        • - -
      • -

        +

      • What is the range of y = \tan(x)? -

        -
        • - - -
      -

      -
      • -
-

- - - \ No newline at end of file +-->

+ + + + + + + diff --git a/xsl/activity-task.xsl b/xsl/activity-task.xsl new file mode 100644 index 00000000..4c2b1541 --- /dev/null +++ b/xsl/activity-task.xsl @@ -0,0 +1,107 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ws- + + + + + + + + + + + + + + + + + + + + + + + 0 + + + + + + + + + + + + + + + + + + + -task- + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file From a73f55043d7619211f177b77e1fb3066aa5af9f4 Mon Sep 17 00:00:00 2001 From: mitchkeller Date: Fri, 2 Aug 2024 17:56:06 -0500 Subject: [PATCH 02/10] Set up structure for workbook with worksheets --- project.ptx | 2 +- publication/publication-workbook.ptx | 21 +++ source/apc-activity-workbook.ptx | 18 +- source/chap-changing-wb.xml | 30 +++ source/chap-circular-wb.xml | 25 +++ source/chap-exp-wb.xml | 27 +++ source/chap-poly-wb.xml | 26 +++ source/chap-trig-wb.xml | 26 +++ source/sec-changing-aroc-wb.xml | 26 +++ source/sec-changing-in-tandem-wb.xml | 22 +++ xsl/apc-activity-workbook.xsl | 268 +++++++++++++++------------ 11 files changed, 365 insertions(+), 126 deletions(-) create mode 100644 publication/publication-workbook.ptx create mode 100644 source/chap-changing-wb.xml create mode 100644 source/chap-circular-wb.xml create mode 100644 source/chap-exp-wb.xml create mode 100644 source/chap-poly-wb.xml create mode 100644 source/chap-trig-wb.xml create mode 100644 source/sec-changing-aroc-wb.xml create mode 100644 source/sec-changing-in-tandem-wb.xml diff --git a/project.ptx b/project.ptx index 5fb18594..04fe8c1a 100644 --- a/project.ptx +++ b/project.ptx @@ -55,7 +55,7 @@ latex - source/apc-activity-workbook.xml + source/apc-activity-workbook.ptx publication/publication-workbook.ptx output/workbook-latex xsl/apc-activity-workbook.xsl diff --git a/publication/publication-workbook.ptx b/publication/publication-workbook.ptx new file mode 100644 index 00000000..50fae4d7 --- /dev/null +++ b/publication/publication-workbook.ptx @@ -0,0 +1,21 @@ + + + + + + letterpaper,tmargin=.5in,bmargin=.3in,hmargin=.75in,includeheadfoot + + + + + + + + + + + + + + diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index 92ae5d9d..c18dd10f 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -25,7 +25,7 @@ --> - + APC @@ -35,6 +35,12 @@ Preview Activity Motivating Questions + + + ㅤ + + + @@ -60,11 +66,11 @@ - - - - - + + Back Matter diff --git a/source/chap-changing-wb.xml b/source/chap-changing-wb.xml new file mode 100644 index 00000000..85b86e74 --- /dev/null +++ b/source/chap-changing-wb.xml @@ -0,0 +1,30 @@ + + + + + + + + + + + + + + + + + Relating Changing Quantities + + + + + + diff --git a/source/chap-circular-wb.xml b/source/chap-circular-wb.xml new file mode 100644 index 00000000..a38b5121 --- /dev/null +++ b/source/chap-circular-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + + + Circular Functions + + + + + + + + diff --git a/source/chap-exp-wb.xml b/source/chap-exp-wb.xml new file mode 100644 index 00000000..a8c89129 --- /dev/null +++ b/source/chap-exp-wb.xml @@ -0,0 +1,27 @@ + + + + + + + + + + + + + + + + + Exponential and Logarithmic Functions + + + + + + + + + + diff --git a/source/chap-poly-wb.xml b/source/chap-poly-wb.xml new file mode 100644 index 00000000..c77344eb --- /dev/null +++ b/source/chap-poly-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + + Polynomial and Rational Functions + + + + + + + + + diff --git a/source/chap-trig-wb.xml b/source/chap-trig-wb.xml new file mode 100644 index 00000000..f5a9bb33 --- /dev/null +++ b/source/chap-trig-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + + Trigonometry + + + + + + + + + diff --git a/source/sec-changing-aroc-wb.xml b/source/sec-changing-aroc-wb.xml new file mode 100644 index 00000000..95898399 --- /dev/null +++ b/source/sec-changing-aroc-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + +
+ The Average Rate of Change of a Function + + + + + + + +
+ diff --git a/source/sec-changing-in-tandem-wb.xml b/source/sec-changing-in-tandem-wb.xml new file mode 100644 index 00000000..fc90f81b --- /dev/null +++ b/source/sec-changing-in-tandem-wb.xml @@ -0,0 +1,22 @@ + + + + + + + + + + + + + + + +
+ Changing in Tandem + + + +
+ diff --git a/xsl/apc-activity-workbook.xsl b/xsl/apc-activity-workbook.xsl index aef198da..1129812c 100644 --- a/xsl/apc-activity-workbook.xsl +++ b/xsl/apc-activity-workbook.xsl @@ -1,9 +1,9 @@ - + - + @@ -12,126 +12,101 @@ + + + %entities; +]> + - + - - - - + - - - - - - - - - - - - - + - - - - - - - - - - - - - - - - - - - - - - - - - + + + enhanced,frame hidden,interior hidden, sharp corners, + boxrule=0pt,borderline west={3pt}{0pt}{ActiveBlue}, + runintitlestyle, blockspacingstyle, after title={.\space}, + colback=white, + coltitle=black,after={\cleardoublepage} - - - - - - - - - - - + + + - - - - \cleardoublepage - - + + + + + + + %% Customized to load Palatino fonts + \usepackage[T1]{fontenc} + \renewcommand{\rmdefault}{zpltlf} %Roman font for use in math mode + \usepackage[scaled=.85]{beramono}% used only by \mathtt + \usepackage[type1]{cabin}%used only by \mathsf + \usepackage{amsmath,amssymb}%load before newpxmath + \usepackage[varg,cmintegrals,bigdelims,varbb]{newpxmath} + \usepackage[scr=rsfso]{mathalfa} + \usepackage{bm} %load after all math to give access to bold math + % Now load the otf text fonts using fontspec--wont affect math + \usepackage[no-math]{fontspec} + \setmainfont{TeXGyrePagellaX} + \defaultfontfeatures{Ligatures=TeX,Scale=1,Mapping=tex-text} + \linespread{1.02} + - - - + + %% Used to get WeBWorK logo into margin next to WW exercises + \usepackage{marginnote} + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % CC icon at bottom of first page of each chapter + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \newpagestyle{chapopen}{ + \sethead[][][] % even + {}{}{} % odd + \setfoot[\includegraphics[height=1pc]{external/images/CC-BY-SA-license.pdf}][][] +{}{}{\includegraphics[height=1pc]{external/images/CC-BY-SA-license.pdf}}} + \assignpagestyle{\chapter}{chapopen} + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % Modified from Mitch Keller's chapter handling + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% This is from common + \definecolor{ActiveBlue}{cmyk}{1, 0.5, 0, 0.35} + \colorlet{chaptercolor}{ActiveBlue} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % Basic paragraph parameters + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \setlength{\parindent}{0mm} + \setlength{\parskip}{0.5pc} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % In print, trying to reduce color use + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \hypersetup{colorlinks=true,linkcolor=black,citecolor=black, + filecolor=black,urlcolor=black} + - - - - - - - \subcaption*{ - - - \captionof*{figure}{ - - - \captionof*{table}{ - - - \captionof*{listingcap}{ - - - \captionof*{namedlistcap}{ - - - \caption*{ - - - - \textbf{ - - - - :} - - - - - - - - - - - - - } + + + {\sethead[\textsl{\ifthechapter{\chaptertitlename{} + \thechapter }{} }][][] + {}{}{\textsl{\ifthesection{\thesection{} \sectiontitle} + {} }} + \setfoot[\thepage][][] + {}{}{\thepage}} @@ -140,19 +115,18 @@ \titleformat{\chapter}[display] {\raggedleft\normalfont\color{chaptercolor}\Large}{ \MakeUppercase{\divisionnameptx}\space - - \rlap{\enskip\resizebox{!}{0.95cm}{\thechapter} }}{10pt}{\normalfont\Huge\itshape#1} [{\Large\authorsptx}] \titleformat{name=\chapter,numberless}[display] {\raggedleft\normalfont\color{chaptercolor}\Huge\itshape}{}{0pt}{#1} [{\Large\authorsptx}] - \titlespacing*{\chapter}{0pt}{30pt}{20pt} + \titlespacing*{\chapter}{0pt}{0pt}{0pt} - + + \titleformat{\section}[block] -{\normalfont\Large\bfseries}{\thesection\space\titleptx}{1em}{} + {\normalfont\Large\bfseries}{\thesection\space\titleptx}{1em}{} [{\large\authorsptx}] \titleformat{name=\section,numberless}[block] {\normalfont\Large\bfseries}{}{0pt}{#1} @@ -174,10 +148,66 @@ \titlespacing*{\subsubsection}{0pt}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex} - - - - - - + + bwminimalstyle, runintitlestyle, blockspacingstyle, after title={\space}, + + + + bwminimalstyle, runintitlestyle, blockspacingstyle, after title={\space}, + + + + + + + enhanced,frame hidden,interior hidden,sharp corners, + blockspacingstyle,boxrule=0pt,left=0pt,right=0pt, + fonttitle=\large\bfseries, + borderline north={0.1ex}{0pt}{black}, + toptitle=0.5ex,top=2ex, bottom=0.5ex, + borderline south={0.1ex}{0pt}{black},coltitle=black, + + + + + + + skin=enhanced, arc=2ex, + colback=ActiveBlue!5,colframe=ActiveBlue!75!black, + colbacktitle=ActiveBlue!20, coltitle=black, + boxed title style={sharp corners, frame hidden}, + fonttitle=\bfseries, attach boxed title to top + left={xshift=4mm,yshift=-3mm}, top=3mm, + + + + + + + + + + (\nolinkurl{ + + + + + + }) + + + + + + \footnotetext[ + + ] + { + + + }% + + + + \ No newline at end of file From 53e796fb4a0aed376143073761e8f71fc7437624 Mon Sep 17 00:00:00 2001 From: Matt Boelkins Date: Sun, 4 Aug 2024 19:23:25 -0400 Subject: [PATCH 03/10] spacing work on the first four sections of chapter 1 --- .../activities/act-changing-aroc-graphs.xml | 4 +- .../act-changing-aroc-population.xml | 29 +++++----- .../activities/act-changing-aroc-trends.xml | 28 +++++----- .../act-changing-functions-is-it.xml | 28 +++++----- ...ging-functions-spherical-tank-draining.xml | 12 ++-- .../act-changing-functions-spherical-tank.xml | 10 ++-- .../act-changing-linear-Kilimanjaro.xml | 17 +++--- .../act-changing-linear-finding-eqs.xml | 55 +++++++++---------- .../act-changing-linear-in-context.xml | 10 ++-- .../act-changing-tandem-conical-tank.xml | 26 +++++---- .../act-changing-tandem-spherical-tank.xml | 24 ++++---- source/apc-activity-workbook.ptx | 2 + source/chap-changing-wb.xml | 6 +- source/previews/PA-changing-aroc.xml | 18 +++--- .../PA-changing-functions-crickets.xml | 10 ++-- source/previews/PA-changing-linear-3-ex.xml | 22 ++++---- .../previews/PA-changing-tandem-aquarium.xml | 8 +-- source/sec-changing-aroc-wb.xml | 5 +- source/sec-changing-functions-models-wb.xml | 24 ++++++++ source/sec-changing-linear-wb.xml | 25 +++++++++ source/sec-changing-quadratic-wb.xml | 25 +++++++++ 21 files changed, 232 insertions(+), 156 deletions(-) create mode 100755 source/sec-changing-functions-models-wb.xml create mode 100755 source/sec-changing-linear-wb.xml create mode 100755 source/sec-changing-quadratic-wb.xml diff --git a/source/activities/act-changing-aroc-graphs.xml b/source/activities/act-changing-aroc-graphs.xml index 690a6de6..3a114823 100755 --- a/source/activities/act-changing-aroc-graphs.xml +++ b/source/activities/act-changing-aroc-graphs.xml @@ -20,7 +20,7 @@

- +

f is a function defined on [-1,7] such that f(1) = 4 and AV_{[1,3]} = -2.

@@ -33,7 +33,7 @@ - +

g is a function defined on [-1,7] such that g(4) = 3, AV_{[0,4]} = 0.5, and g is not always increasing on (0,4).

diff --git a/source/activities/act-changing-aroc-population.xml b/source/activities/act-changing-aroc-population.xml index a1287d9a..e64dcd72 100755 --- a/source/activities/act-changing-aroc-population.xml +++ b/source/activities/act-changing-aroc-population.xml @@ -17,15 +17,15 @@

According to the US census, - the populations of Kent and Ottawa Counties in Michigan where GVSU is located - Grand Rapids is in Kent, Allendale in Ottawa. - + the populations of Kent and Ottawa Counties in Michigan where GVSU is located + (Grand Rapids is in Kent, Allendale in Ottawa) from 1960 to 2010 measured in 10-year intervals are given in the following tables.

- - Kent County population data. + + Year: 1960 1970 1980 @@ -34,6 +34,7 @@ 2010 + Kent County population: 363,187 411,044 444,506 @@ -42,11 +43,12 @@ 602,622 -
+ + Year: 1960 1970 1980 @@ -55,6 +57,7 @@ 2010 + Ottawa County Population: 98,719 128,181 157,174 @@ -63,13 +66,13 @@ 263,801 - +

Let K(Y) represent the population of Kent County in year Y and W(Y) the population of Ottawa County in year Y.

- +

Compute AV_{[1990,2010]} for both K and W. @@ -79,7 +82,7 @@ - +

What are the units on each of the quantities you computed in (a.)? @@ -89,7 +92,7 @@ - +

Write a careful sentence that explains the meaning of the average rate of change of the Ottawa county population on the time interval [1990,2010]. Your sentence should begin something like @@ -99,7 +102,7 @@ - +

Which county had a greater average rate of change during the time interval [2000,2010]? @@ -110,7 +113,7 @@ - +

Using the given data, what do you predict will be the population of Ottawa County in 2018? Why? diff --git a/source/activities/act-changing-aroc-trends.xml b/source/activities/act-changing-aroc-trends.xml index 557a4e8d..77cb897a 100755 --- a/source/activities/act-changing-aroc-trends.xml +++ b/source/activities/act-changing-aroc-trends.xml @@ -16,11 +16,11 @@

- Let's consider two different functions and see how different computations of their average rate of change tells us about their respective behavior. Plots of q and h are shown in Figures and . + Let's consider two different functions and see how different computations of their average rate of change tells us about their respective behavior. Plots of q and h are shown in the figures in part (c).

- +

Consider the function q(x) = 4-(x-2)^2. @@ -33,7 +33,7 @@ - +

Consider the function h(t) = 3 - 2(0.5)^t. @@ -46,27 +46,27 @@ - +

- On the graphs in Figures and , plot the line segments whose respective slopes are the average rates of change you computed in (a) and (b). + On the graphs that follow (q at left, h at right), plot the line segments whose respective slopes are the average rates of change you computed in (a) and (b).

-
- Plot of q from part (a). - -
+ + + + + - +

True or false: Since AV_{[0,3]} = 1, the function q is increasing on the interval (0,3). Justify your decision. @@ -76,7 +76,7 @@ - +

Give an example of a function that has the same average rate of change no matter what interval you choose. diff --git a/source/activities/act-changing-functions-is-it.xml b/source/activities/act-changing-functions-is-it.xml index 28d94c7b..7a6e48c5 100755 --- a/source/activities/act-changing-functions-is-it.xml +++ b/source/activities/act-changing-functions-is-it.xml @@ -20,20 +20,20 @@

- +

- The relationship between x and y in each of the graphs below (address each graph separately as a potential situation where y is a function of x). In Figure, any point on the circle relates x and y. For instance, the y-value \sqrt{7} is related to the x-value -3. In Figure, any point on the blue curve relates x and y. For instance, when x = -1, the corresponding y-value is y = 3. An unfilled circle indicates that there is not a point on the graph at that specific location. + The relationship between x and y in each of the graphs below (address each graph separately as a potential situation where y is a function of x). In the lefthand figure, any point on the circle relates x and y. For instance, the y-value \sqrt{7} is related to the x-value -3. In the righthand figure, any point on the blue curve relates x and y. For instance, when x = -1, the corresponding y-value is y = 3. An unfilled circle indicates that there is not a point on the graph at that specific location.

-
- A circle of radius 4 centered at (0,0). - -
+ + + + + @@ -50,7 +50,7 @@ - +

The relationship between a car's velocity and its odometer, where we attempt to view the car's odometer reading as a function of its velocity. @@ -60,13 +60,13 @@ - +

The relationship between x and y that is given in the following table where we attempt to view y as depending on x.

- - A table that relates <m>x</m> and <m>y</m> values. + @@ -109,7 +109,7 @@ -
+ diff --git a/source/activities/act-changing-functions-spherical-tank-draining.xml b/source/activities/act-changing-functions-spherical-tank-draining.xml index 6a51bc7c..36fd1331 100755 --- a/source/activities/act-changing-functions-spherical-tank-draining.xml +++ b/source/activities/act-changing-functions-spherical-tank-draining.xml @@ -27,7 +27,7 @@

- +

What is the height of the water when t = 0? When t = 1? When t = 2? How long will it take the tank to completely drain? Why? @@ -37,7 +37,7 @@ - +

What is the domain of the model h = q(t)? What is the domain of the model V = p(t)? @@ -47,7 +47,7 @@ - +

How much water is in the tank when the tank is full? What is the range of the model h = q(t)? What is the range of the model V = p(t)? @@ -57,10 +57,10 @@ - +

- We will frequently use a graphing utility to help us understand function behavior, and strongly recommend Desmos because it is intuitive, online, and free.To learn more about Desmos, see their outstanding online tutorials.

+ We will frequently use a graphing utility to help us understand function behavior, and strongly recommend Desmos because it is intuitive, online, and free. To learn more about Desmos, see their outstanding online tutorials.

In this prepared Desmos worksheet, you can see how we enter the (abstract) function V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t), as well as the corresponding graph the program generates. Make as many observations as you can about the model V = p(t). You should discuss its shape and overall behavior, its domain, its range, and more.

@@ -69,7 +69,7 @@ - +

How does the model V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t) differ from the abstract function y = r(x) = \frac{256\pi}{3} - \frac{\pi}{24} x^2(24-x)? In particular, how do the domain and range of the model differ from those of the abstract function, if at all? diff --git a/source/activities/act-changing-functions-spherical-tank.xml b/source/activities/act-changing-functions-spherical-tank.xml index 15bde161..73a53ab1 100755 --- a/source/activities/act-changing-functions-spherical-tank.xml +++ b/source/activities/act-changing-functions-spherical-tank.xml @@ -26,7 +26,7 @@

- +

What values of h make sense to consider in the context of this function? What values of V make sense in the same context? @@ -36,7 +36,7 @@ - +

What is the domain of the function f in the context of the spherical tank? Why? What is the corresponding codomain? Why? @@ -46,7 +46,7 @@ - +

Determine and interpret (with appropriate units) the values @@ -67,10 +67,10 @@ - +

- Can you determine a value of h for which f(h) = 300 cubic meters? + Can you determine a value of h for which f(h) = 300 cubic meters? Why or why not?

diff --git a/source/activities/act-changing-linear-Kilimanjaro.xml b/source/activities/act-changing-linear-Kilimanjaro.xml index 5c90cf2e..e5635ac4 100755 --- a/source/activities/act-changing-linear-Kilimanjaro.xml +++ b/source/activities/act-changing-linear-Kilimanjaro.xml @@ -17,16 +17,13 @@

The summit of Africa's largest peak, Mt. - Kilimanjaro - The main context of this problem comes from Exercise 30 on p.27 of Connally's - Functions Modeling Change, 5th ed. - , has two main ice fields and a glacier at its peak. + Kilimanjaro, has two main ice fields and a glacier at its peak. Geologists measured the ice cover in the year 2000 (t = 0) to be approximately 1951 m^2; in the year 2007, the ice cover measured 1555 m^2.

- +

Suppose that the amount of ice cover at the peak of Mt. @@ -41,7 +38,7 @@ - +

What do the slope and A-intercept mean in the model you found in (a)? In particular, what are the units on the slope? @@ -63,7 +60,7 @@ - +

If the model holds further into the future, @@ -74,7 +71,7 @@ - +

In light of your work above, @@ -92,5 +89,9 @@

+ +

The main context of the sequence of questions in this activity comes from Exercise 30 on p.27 of Connally's + Functions Modeling Change, 5th ed.

+ diff --git a/source/activities/act-changing-linear-finding-eqs.xml b/source/activities/act-changing-linear-finding-eqs.xml index ee31bbc3..bdf3b24e 100755 --- a/source/activities/act-changing-linear-finding-eqs.xml +++ b/source/activities/act-changing-linear-finding-eqs.xml @@ -21,7 +21,7 @@

- +

The line with slope \frac{3}{7} that passes through (-11, -17). @@ -31,7 +31,7 @@ - +

The line passing through the points (-2,5) and (3,-1). @@ -41,7 +41,7 @@ - +

The line passing through (4,9) that is parallel to the line 2x - 3y = 5. @@ -51,72 +51,67 @@ - +

- Explain why the function f given by Table appears to be linear and find a formula for f(x). + Explain why the function f given by the data in the following table appears to be linear and find a formula for f(x).

- + x - f(x) + 1 - - - 1 + 3 + + + 4 7 - + - 3 + f(x) - 3 + 7 - - - 4 + 3 1 - - - - 7 - -5 - -
- Plot of a linear function h. - -
- + + - +

- Find a formula for the linear function shown in Figure. + Find a formula for the linear function shown in the following figure.

+ + + diff --git a/source/activities/act-changing-linear-in-context.xml b/source/activities/act-changing-linear-in-context.xml index c55c28c2..1f5b7c26 100755 --- a/source/activities/act-changing-linear-in-context.xml +++ b/source/activities/act-changing-linear-in-context.xml @@ -20,7 +20,7 @@

- +

A town's population initially has 28750 people present and then grows at a constant rate of 825 people per year. @@ -31,7 +31,7 @@ - +

A different town's population Q is given by the function Q = g(t) = 42505 - 465t. What is the slope of this function and what is its meaning in the model? Write a complete sentence to explain. @@ -41,7 +41,7 @@ - +

A spherical tank is being drained with a pump. Initially the tank is full with \frac{32\pi}{3} cubic feet of water. Assume the tank is drained at a constant rate of 1.2 cubic feet per minute. Find a linear model V = p(t) for the total amount of water in the tank at time t. In addition, what is a reasonable approximate domain for the model? @@ -51,7 +51,7 @@ - +

A conical tank is being filled in such a way that the height of the water in the tank, h (in feet), at time t (in minutes) is given by the function h = q(t) = 0.65t. What can you say about how the water level is rising? Write at least one careful sentence to explain. @@ -61,7 +61,7 @@ - +

Suppose we know that a 5-year old car's value is $10200, and that after 10 years its value is $4600. Assuming that the car's value depreciates linearly, find a function C = L(t) whose output is the value of the car in year t. What is a reasonable domain for the model? What is the value and meaning of the slope of the line? Write at least one careful sentence to explain. diff --git a/source/activities/act-changing-tandem-conical-tank.xml b/source/activities/act-changing-tandem-conical-tank.xml index ae6f4d12..48e23301 100755 --- a/source/activities/act-changing-tandem-conical-tank.xml +++ b/source/activities/act-changing-tandem-conical-tank.xml @@ -20,7 +20,7 @@

- +

Sketch a labeled picture of the tank, including a snapshot of there being water in the tank prior to the tank being completely full. @@ -30,7 +30,7 @@ - +

What are some quantities that are changing in this scenario? What are some quantities that are not changing? @@ -40,14 +40,14 @@ - +

Fill in the following table of values to determine how much water, V, is in the tank at a given time in minutes, t, and thus generate a graph of the relationship between volume and time by plotting the data on the provided axes.

- - Table to record data on volume and time in the conical tank. + @@ -106,25 +106,27 @@ -
-
+ + + + + - +

Finally, think about how the height, h, of the water changes in tandem with time. Without attempting to determine specific values of h at particular values of t, how would you expect the data for the relationship between h and t to appear? Use the provided axes to sketch at least two possibilities; write at least one sentence to explain how you think the graph should appear.

- - + + diff --git a/source/activities/act-changing-tandem-spherical-tank.xml b/source/activities/act-changing-tandem-spherical-tank.xml index 28cfd104..3f60c3ce 100755 --- a/source/activities/act-changing-tandem-spherical-tank.xml +++ b/source/activities/act-changing-tandem-spherical-tank.xml @@ -20,7 +20,7 @@

- +

Sketch a labeled picture of the tank, including a snapshot of some water remaining in the tank prior to the tank being completely empty. @@ -40,7 +40,7 @@ - +

Recall that the volume of a sphere of radius r is V = \frac{4}{3} \pi r^3. When the tank is completely full at time t = 0 right before it starts being drained, how much water is present? @@ -60,14 +60,14 @@ - +

Fill in the following table of values to determine how much water, V, is in the tank at a given time in minutes, t, and thus generate a graph of the relationship between volume and time. Write a sentence to explain why the data's graph appears the way that it does.

- - Data for how volume and time change together. + @@ -114,25 +114,25 @@ -
+ + + - +

Finally, think about how the height of the water changes in tandem with time. What is the height of the water when t = 0? What is the height when the tank is empty? How would you expect the data for the relationship between h and t to appear? Use the provided axes to sketch at least two possibilities; write at least one sentence to explain how you think the graph should appear.

- - + + diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index c18dd10f..8d5776a5 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -67,10 +67,12 @@ + + Back Matter diff --git a/source/chap-changing-wb.xml b/source/chap-changing-wb.xml index 85b86e74..adeb2685 100644 --- a/source/chap-changing-wb.xml +++ b/source/chap-changing-wb.xml @@ -17,11 +17,11 @@ Relating Changing Quantities - diff --git a/source/previews/PA-changing-aroc.xml b/source/previews/PA-changing-aroc.xml index f4211fe3..548e6a77 100755 --- a/source/previews/PA-changing-aroc.xml +++ b/source/previews/PA-changing-aroc.xml @@ -31,7 +31,7 @@ - +

What are the units on the quantity AV_{[1.5,2.5]}? @@ -42,28 +42,28 @@ - +

In Desmos, plot the function s(t) = 64 - 16(t-1)^2 along with the points (1.5,s(1.5)) and (2.5, s(2.5)). - Make a copy of your plot on the axes in Figure, + Make a copy of your plot on the axes in the figure provided, labeling key points as well as the scale on your axes. What is the domain of the model? The range? Why?

-
- Axes for plotting the position function. + -
+ - +

Work by hand to find the equation of the line through the points @@ -77,7 +77,7 @@ - +

What is a geometric interpretation of the value @@ -88,7 +88,7 @@ - +

How do your answers in the preceding questions change if we instead consider the interval diff --git a/source/previews/PA-changing-functions-crickets.xml b/source/previews/PA-changing-functions-crickets.xml index 8b54a68c..5982d9cc 100755 --- a/source/previews/PA-changing-functions-crickets.xml +++ b/source/previews/PA-changing-functions-crickets.xml @@ -16,14 +16,14 @@

- Use Equation to respond to the questions below. + Use the equation T = 40 + 0.25N that relates the temperature, T, to the number of chirps per minute, N, to respond to the questions below. The equation is also called Dolbear's Law.

- If we hear snowy tree crickets chirping at a rate of 92 chirps per minute, what does Dolbear's model suggest should be the outside temperature? + If we hear snowy tree crickets chirping at a rate of 92 chirps per minute, what does Dolbear's Law suggest should be the outside temperature?

@@ -40,7 +40,7 @@ - +

Is the model valid for determining the number of chirps one should hear when the outside temperature is 35^\circ F? Why or why not? @@ -50,7 +50,7 @@ - +

Suppose that in the morning an observer hears 65 chirps per minute, and several hours later hears 75 chirps per minute. How much has the temperature risen between observations? @@ -60,7 +60,7 @@ - +

Dolbear's Law is known to be accurate for temperatures from 50^\circ to 85^\circ. What is the fewest number of chirps per minute an observer could expect to hear? the greatest number of chirps per minute? diff --git a/source/previews/PA-changing-linear-3-ex.xml b/source/previews/PA-changing-linear-3-ex.xml index 7937ee1d..fd3679b6 100755 --- a/source/previews/PA-changing-linear-3-ex.xml +++ b/source/previews/PA-changing-linear-3-ex.xml @@ -30,10 +30,10 @@

- Let y = g(x) be given by the data in Table. + Let y = g(x) be given by the data in the following table.

- - A table that defines the function <m>y = g(x)</m>. + @@ -112,7 +112,7 @@ -
+

Determine AV_{[-5,-2]}, AV_{[-1,1]}, and AV_{[0,4]} for the function g.

@@ -124,12 +124,12 @@

- Consider the function y = h(x) defined by the graph in Figure. + Consider the function y = h(x) defined by the graph in the following figure.

-
- The graph of y = h(x). - -
+ + +

Determine AV_{[-5,-2]}, AV_{[-1,1]}, and AV_{[0,4]} for the function h.

@@ -138,7 +138,7 @@ - +

What do all three examples above have in common? How do they differ? @@ -148,7 +148,7 @@ - +

For the function y = f(x) = 7 - 3x from (a), find the simplest expression you can for diff --git a/source/previews/PA-changing-tandem-aquarium.xml b/source/previews/PA-changing-tandem-aquarium.xml index 31c01ed8..7d1ea33d 100755 --- a/source/previews/PA-changing-tandem-aquarium.xml +++ b/source/previews/PA-changing-tandem-aquarium.xml @@ -30,7 +30,7 @@

- +

What are some different quantities that are changing in this scenario? @@ -40,7 +40,7 @@ - +

After 1 minute has elapsed, how much water is in the tank? At this moment, how deep is the water? @@ -50,7 +50,7 @@ - +

How much water is in the tank and how deep is the water after 2 minutes? After 3 minutes? @@ -60,7 +60,7 @@ - +

How long will it take for the tank to be completely full? Why? diff --git a/source/sec-changing-aroc-wb.xml b/source/sec-changing-aroc-wb.xml index 95898399..c91dc0fb 100644 --- a/source/sec-changing-aroc-wb.xml +++ b/source/sec-changing-aroc-wb.xml @@ -16,11 +16,10 @@

The Average Rate of Change of a Function - - + - +
diff --git a/source/sec-changing-functions-models-wb.xml b/source/sec-changing-functions-models-wb.xml new file mode 100755 index 00000000..50cf712c --- /dev/null +++ b/source/sec-changing-functions-models-wb.xml @@ -0,0 +1,24 @@ + + + + + + + + + + + + + + + +
+ Functions: Modeling Relationships + + + + + +
+ diff --git a/source/sec-changing-linear-wb.xml b/source/sec-changing-linear-wb.xml new file mode 100755 index 00000000..76f3cb86 --- /dev/null +++ b/source/sec-changing-linear-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Linear Functions + + + + + + +
+ diff --git a/source/sec-changing-quadratic-wb.xml b/source/sec-changing-quadratic-wb.xml new file mode 100755 index 00000000..dbf4e299 --- /dev/null +++ b/source/sec-changing-quadratic-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Quadratic Functions + + + + + + +
+ From a7ff8bb620b9eb0f49a84c51257854ef1114a305 Mon Sep 17 00:00:00 2001 From: Matt Boelkins Date: Tue, 6 Aug 2024 06:22:24 -0400 Subject: [PATCH 04/10] spacing edits to the rest of chapter 1 for the workbook, plus starting chapter 2 --- .../act-changing-combining-arithmetic.xml | 18 ++++----- .../act-changing-combining-context.xml | 8 ++-- .../act-changing-combining-piecewise.xml | 28 ++++++------- .../act-changing-composite-aroc.xml | 8 ++-- ...ct-changing-composite-crickets-celsius.xml | 18 ++++----- .../act-changing-composite-tables-graphs.xml | 30 +++++++------- .../act-changing-inverse-Dolbear.xml | 10 ++--- .../act-changing-inverse-does-it.xml | 40 +++++++++---------- .../act-changing-inverse-rainfall.xml | 6 +-- .../act-changing-quadratic-falling-ball.xml | 10 ++--- .../act-changing-quadratic-parameters.xml | 10 ++--- .../act-changing-quadratic-properties.xml | 14 +++---- .../act-changing-transformations-combined.xml | 26 ++++++------ ...-changing-transformations-translations.xml | 20 +++++----- ...-changing-transformations-vert-stretch.xml | 24 +++++------ source/apc-activity-workbook.ptx | 4 +- source/chap-changing-wb.xml | 4 +- source/chap-circular-wb.xml | 4 +- source/previews/PA-changing-combining.xml | 18 ++++----- source/previews/PA-changing-composite.xml | 8 ++-- source/previews/PA-changing-inverse-F-C.xml | 8 ++-- source/previews/PA-changing-quadratic.xml | 24 +++++------ .../PA-changing-transformations-quadratic.xml | 8 ++-- source/sec-changing-combining-wb.xml | 25 ++++++++++++ source/sec-changing-composite-wb.xml | 25 ++++++++++++ source/sec-changing-inverse-wb.xml | 25 ++++++++++++ source/sec-changing-transformations-wb.xml | 25 ++++++++++++ source/sec-circular-traversing-wb.xml | 25 ++++++++++++ 28 files changed, 298 insertions(+), 175 deletions(-) create mode 100755 source/sec-changing-combining-wb.xml create mode 100755 source/sec-changing-composite-wb.xml create mode 100755 source/sec-changing-inverse-wb.xml create mode 100755 source/sec-changing-transformations-wb.xml create mode 100755 source/sec-circular-traversing-wb.xml diff --git a/source/activities/act-changing-combining-arithmetic.xml b/source/activities/act-changing-combining-arithmetic.xml index aa642957..836124b6 100755 --- a/source/activities/act-changing-combining-arithmetic.xml +++ b/source/activities/act-changing-combining-arithmetic.xml @@ -16,17 +16,17 @@

- Consider the functions f and g defined by Figure and Figure. Assume that the given lines and curves pass through intersection points on the grid when it looks plausible. For instance, (0,2.5) and (3,-0.5) lie on the graph of f, and (-1,3) and (1.5, 1.5) lie on the graph of g. + Consider the functions f and g defined by the following figures. Assume that the given lines and curves pass through intersection points on the grid when it looks plausible. For instance, (0,2.5) and (3,-0.5) lie on the graph of f, and (-1,3) and (1.5, 1.5) lie on the graph of g.

- -
- The function f. + + -
+ -
+

@@ -40,7 +40,7 @@ - +

Determine the exact value of (g-f)(1). @@ -50,7 +50,7 @@ - +

Determine the exact value of (f \cdot g)(-1). diff --git a/source/activities/act-changing-combining-context.xml b/source/activities/act-changing-combining-context.xml index 268e8ae8..9d74b04c 100755 --- a/source/activities/act-changing-combining-context.xml +++ b/source/activities/act-changing-combining-context.xml @@ -37,7 +37,7 @@ - +

Consider the function g(v) = \frac{1}{f(v)}. @@ -50,7 +50,7 @@ - +

Consider the function h(v) = v \cdot f(v). @@ -63,7 +63,7 @@ - +

Do f(60), g(60), and h(60) tell us fundamentally different information, or are they all essentially saying the same thing? Explain. @@ -73,7 +73,7 @@ - +

Suppose we also know that f(70) = 0.045. diff --git a/source/activities/act-changing-combining-piecewise.xml b/source/activities/act-changing-combining-piecewise.xml index 2adff79d..1c93b637 100755 --- a/source/activities/act-changing-combining-piecewise.xml +++ b/source/activities/act-changing-combining-piecewise.xml @@ -21,7 +21,7 @@

- +

Consider the function p defined by the following rule: @@ -40,7 +40,7 @@ - +

What point is the vertex of the quadratic part of p that is valid for x \lt 0? What point is the vertex of the quadratic part of p that is valid for x \ge 0? @@ -50,7 +50,7 @@ - +

For what values of x is p(x) = 0? In addition, what is the y-intercept of p? @@ -60,30 +60,30 @@ - +

- Sketch an accurate, labeled graph of y = p(x) on the axes provided in Figure. + Sketch an accurate, labeled graph of y = p(x) on the axes provided at left in the following figure.

- -
- Axes to plot y = p(x). + + -
-
- Graph of y = f(x). + -
+ - +

- For the function f defined by Figure, determine a piecewise-defined formula for f that is expressed in bracket notation similar to the definition of y = p(x) above. + For the function f defined by the righthand figure in (d), determine a piecewise-defined formula for f that is expressed in bracket notation similar to the definition of y = p(x) above.

diff --git a/source/activities/act-changing-composite-aroc.xml b/source/activities/act-changing-composite-aroc.xml index e02d45d5..c872a578 100755 --- a/source/activities/act-changing-composite-aroc.xml +++ b/source/activities/act-changing-composite-aroc.xml @@ -20,7 +20,7 @@

- +

Compute f(1+h) and expand and simplify the result as much as possible by combining like terms. @@ -30,7 +30,7 @@ - +

Determine the most simplified expression you can for the average rate of change of f on the interval [1,1+h]. @@ -41,7 +41,7 @@ - +

Compute g(1+h). @@ -52,7 +52,7 @@ - +

Determine the most simplified expression you can for the average rate of change of g on the interval [1,1+h]. diff --git a/source/activities/act-changing-composite-crickets-celsius.xml b/source/activities/act-changing-composite-crickets-celsius.xml index 51534dc5..46b895a6 100755 --- a/source/activities/act-changing-composite-crickets-celsius.xml +++ b/source/activities/act-changing-composite-crickets-celsius.xml @@ -20,7 +20,7 @@

- +

Determine a formula for the new function H = (G \circ D) that depends only on the variable N. @@ -40,27 +40,27 @@ - +

How does a plot of the function H = (G \circ D) compare to that of Dolbear's function? Sketch a plot of y = H(N) = (G \circ D)(N) on the blank axes to the right of the plot of Dolbear's function, and discuss the similarities and differences between them. Be sure to label the vertical scale on your axes.

- -
- Dolbear's function. + + -
+ - + - +

What is the domain of the function H = G \circ D? What is its range? diff --git a/source/activities/act-changing-composite-tables-graphs.xml b/source/activities/act-changing-composite-tables-graphs.xml index 6ba36325..3f39e02a 100755 --- a/source/activities/act-changing-composite-tables-graphs.xml +++ b/source/activities/act-changing-composite-tables-graphs.xml @@ -16,13 +16,11 @@

- Let functions p and q be given by the graphs in Figure - (which are each piecewise linear - that is, parts that look like straight lines are straight lines) - and let f and g be given by Table. + Let functions p and q be given by the graphs in the figure below, which are each piecewise linear - that is, parts that look like straight lines are straight lines. In addition, let f and g be given by the tables below.

- - Table that defines <m>f</m> and <m>g</m>. + @@ -55,18 +53,18 @@ 2 -
+ - +

Compute each of the following quantities or explain why they are not defined.

- +

p(q(0)) @@ -76,7 +74,7 @@ - +

q(p(0)) @@ -86,7 +84,7 @@ - + (p \circ p)(-1) @@ -94,7 +92,7 @@ - +

(f \circ g)(2) @@ -104,7 +102,7 @@ - +

(g \circ f)(3) @@ -114,7 +112,7 @@ - +

g(f(0)) @@ -124,7 +122,7 @@ - +

For what value(s) of x is f(g(x)) = 4? @@ -134,7 +132,7 @@ - +

For what value(s) of x is q(p(x)) = 1? diff --git a/source/activities/act-changing-inverse-Dolbear.xml b/source/activities/act-changing-inverse-Dolbear.xml index 2b3a5ca8..e6e14ba1 100755 --- a/source/activities/act-changing-inverse-Dolbear.xml +++ b/source/activities/act-changing-inverse-Dolbear.xml @@ -16,11 +16,11 @@

- Recall Dolbear's function F = D(N) = 40 + \frac{1}{4}N that converts the number, N, of snowy tree cricket chirps per minute to a corresponding Fahrenheit temperature. We have earlier established that the domain of D is [40,180] and the range of D is [50,85], as seen in Figure. + Recall Dolbear's function F = D(N) = 40 + \frac{1}{4}N that converts the number, N, of snowy tree cricket chirps per minute to a corresponding Fahrenheit temperature. We have earlier established that the domain of D is [40,180] and the range of D is [50,85].

- +

Solve the equation F = 40 + \frac{1}{4}N for N in terms of F. Call the resulting function N = E(F). @@ -30,7 +30,7 @@ - +

Explain in words the process or effect of the function N = E(F). What does it take as input? What does it generate as output? @@ -40,7 +40,7 @@ - +

Use the function E that you found in (a.) to compute j(N) = E(D(N)). Simplify your result as much as possible. Do likewise for k(F) = D(E(F)). What do you notice about these two composite functions j and k? @@ -50,7 +50,7 @@ - +

Consider the equations F = 40 + \frac{1}{4}N and N = 4(F-40). Do these equations express different relationships between F and N, or do they express the same relationship in two different ways? Explain. diff --git a/source/activities/act-changing-inverse-does-it.xml b/source/activities/act-changing-inverse-does-it.xml index 45dbb577..643a1ee9 100755 --- a/source/activities/act-changing-inverse-does-it.xml +++ b/source/activities/act-changing-inverse-does-it.xml @@ -20,13 +20,13 @@

- +

- The function f : S \to S given by Table, where S = \{0, 1, 2, 3, 4 \}. + The function f : S \to S given by the table of values below, where S = \{0, 1, 2, 3, 4 \}.

- - Values of <m>y = f(x)</m>. + @@ -49,19 +49,19 @@ 2 -
+ - +

- The function g : S \to S given by Table, where S = \{0, 1, 2, 3, 4 \}. + The function g : S \to S given by the table of values below, where S = \{0, 1, 2, 3, 4 \}.

- - Values of <m>y = g(x)</m>. + @@ -84,13 +84,13 @@ 2 -
+ - +

The function p given by p(t) = 7 - \frac{3}{5}t. Assume that the domain and codomain of p are both all real numbers. @@ -100,7 +100,7 @@ - +

The function q given by q(t) = 7 - \frac{3}{5}t^4. Assume that the domain and codomain of q are both all real numbers. @@ -110,20 +110,20 @@ - +

- The functions r and s given by the graphs in Figure and Figure. Assume that the graphs show all of the important behavior of the functions and that the apparent trends continue beyond what is pictured. + The functions r and s given by the graphs in the figures below. Assume that the graphs show all of the important behavior of the functions and that the apparent trends continue beyond what is pictured.

- -
- The graph of y = r(t). + + -
+ - + diff --git a/source/activities/act-changing-inverse-rainfall.xml b/source/activities/act-changing-inverse-rainfall.xml index 7d7e0001..54303b3f 100755 --- a/source/activities/act-changing-inverse-rainfall.xml +++ b/source/activities/act-changing-inverse-rainfall.xml @@ -33,7 +33,7 @@ - +

Compute the average rate of change of g on the time interval [3,5] and write two careful complete sentences to explain the meaning of this value in the context of the problem, including units. Explicitly address what the value you compute tells you about how rain is falling over a certain time interval, and what you should expect as time goes on. @@ -53,7 +53,7 @@ - +

Determine g^{-1} \left( \frac{9}{5} \right) and write a complete sentence to explain its meaning in the given context. @@ -63,7 +63,7 @@ - +

According to the model g, is there ever a time during the storm that the rain falls at a rate of exactly 1 centimeter per hour? Why or why not? Provide an algebraic justification for your answer. diff --git a/source/activities/act-changing-quadratic-falling-ball.xml b/source/activities/act-changing-quadratic-falling-ball.xml index 685a47c0..f418cc07 100755 --- a/source/activities/act-changing-quadratic-falling-ball.xml +++ b/source/activities/act-changing-quadratic-falling-ball.xml @@ -31,11 +31,11 @@ - +

Plot the function in Desmos - in an appropriate window. + in an appropriate window. Sketch a copy of the graph here.

@@ -52,7 +52,7 @@ - +

Use algebra to find the exact @@ -63,7 +63,7 @@ - +

Determine the exact time the water balloon reaches its highest point and its height at that time. @@ -73,7 +73,7 @@ - +

Compute the average rate of change of s on the intervals [1.5, 2], diff --git a/source/activities/act-changing-quadratic-parameters.xml b/source/activities/act-changing-quadratic-parameters.xml index ec754bea..cdebe58f 100755 --- a/source/activities/act-changing-quadratic-parameters.xml +++ b/source/activities/act-changing-quadratic-parameters.xml @@ -24,7 +24,7 @@

- +

Describe how changing the value of a affects the graph of q. @@ -34,7 +34,7 @@ - +

Describe how changing the value of b affects the graph of q. @@ -44,7 +44,7 @@ - +

Describe how changing the value of c affects the graph of q. @@ -54,7 +54,7 @@ - +

Which parameter seems to have the simplest effect? @@ -66,7 +66,7 @@ - +

Is it possible to find a formula for a quadratic function that passes through the points (0,8), diff --git a/source/activities/act-changing-quadratic-properties.xml b/source/activities/act-changing-quadratic-properties.xml index 2ec28fc7..2ab8d185 100755 --- a/source/activities/act-changing-quadratic-properties.xml +++ b/source/activities/act-changing-quadratic-properties.xml @@ -20,7 +20,7 @@

- +

How many quadratic functions have x-intercepts at (-5,0) and (10,0) and a y-intercept at (0,-1)? Can you determine an exact formula for such a function? If yes, do so. If not, explain why. @@ -30,7 +30,7 @@ - +

Suppose that a quadratic function q has vertex (-3,-4) and opens upward. How many x-intercepts can you guarantee the function has? Why? @@ -40,7 +40,7 @@ - +

In addition to the information in (b), suppose you know that q(-1) = -3. Can you determine an exact formula for q? If yes, do so. If not, explain why. @@ -50,20 +50,20 @@ - +

- Does the quadratic function p(x) = -3(x+1)^2 + 9 have 0, 1, or 2x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist. + Does the quadratic function p(x) = -3(x+1)^2 + 9 have 0, 1, or 2 x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.

- +

- Does the quadratic function w(x) = -2x^2 + 10x - 20 have 0, 1, or 2x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist. + Does the quadratic function w(x) = -2x^2 + 10x - 20 have 0, 1, or 2 x-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.

diff --git a/source/activities/act-changing-transformations-combined.xml b/source/activities/act-changing-transformations-combined.xml index 600679b9..048bb3c9 100755 --- a/source/activities/act-changing-transformations-combined.xml +++ b/source/activities/act-changing-transformations-combined.xml @@ -16,21 +16,21 @@

- Consider the functions f and g given in Figure and Figure. + Consider the functions f and g given in the following figures.

- -
- A parent function f. + + -
+ - +

- +

Sketch an accurate graph of the transformation y = p(x) = -\frac{1}{2}f(x-1)+2. Write at least one sentence to explain how you developed the graph of p, and identify the point on p that corresponds to the original point (-2,2) on the graph of f. @@ -40,7 +40,7 @@ - +

Sketch an accurate graph of the transformation y = q(x) = 2g(x+0.5)-0.75. Write at least one sentence to explain how you developed the graph of q, and identify the point on q that corresponds to the original point (1.5,1.5) on the graph of g. @@ -50,22 +50,22 @@ - +

- Is the function y = r(x) = \frac{1}{2}(-f(x-1) - 4) the same function as p or different? Why? Explain in two different ways: discuss the algebraic similarities and differences between p and r, and also discuss how each is a transformation of f. + Is the function y = r(x) = \frac{1}{2}(-f(x-1) - 4) the same function as p in part (a) or different? Why? Explain in two different ways: discuss the algebraic similarities and differences between p and r, and also discuss how each is a transformation of f.

- +

Find a formula for a function y = s(x) (in terms of g) that represents this transformation of g: a horizontal shift of 1.25 units left, followed by a reflection across the x-axis and a vertical stretch by a factor of 2.5 units, followed by a vertical shift of 1.75 units. Sketch an accurate, labeled graph of s on the following axes along with the given parent function g.

- + diff --git a/source/activities/act-changing-transformations-translations.xml b/source/activities/act-changing-transformations-translations.xml index d3ccce80..93698331 100755 --- a/source/activities/act-changing-transformations-translations.xml +++ b/source/activities/act-changing-transformations-translations.xml @@ -16,21 +16,21 @@

- Consider the functions r and s given in Figure and Figure. + Consider the functions r and s given in the figures below.

- -
- A parent function r. + + -
+ - +

- +

On the same axes as the plot of y = r(x), sketch the following graphs: y = g(x) = r(x) + 2, y = h(x) = r(x+1), and y = f(x) = r(x+1) + 2. Be sure to label the point on each of g, h, and f that corresponds to (-2,-1) on the original graph of r. In addition, write one sentence to explain the overall transformations that have resulted in g, h, and f. @@ -40,7 +40,7 @@ - +

On the same axes as the plot of y = s(x), sketch the following graphs: y = k(x) = s(x) - 1, y = j(x) = s(x-2), and y = m(x) = s(x-2) - 1. Be sure to label the point on each of k, j, and m that corresponds to (-2,-3) on the original graph of s. In addition, write one sentence to explain the overall transformations that have resulted in k, j, and m. @@ -50,7 +50,7 @@ - +

Now consider the function q(x) = x^2. Determine a formula for the function that is given by p(x) = q(x+3) - 4. How is p a transformation of q? diff --git a/source/activities/act-changing-transformations-vert-stretch.xml b/source/activities/act-changing-transformations-vert-stretch.xml index e90de288..45ddd800 100755 --- a/source/activities/act-changing-transformations-vert-stretch.xml +++ b/source/activities/act-changing-transformations-vert-stretch.xml @@ -16,21 +16,21 @@

- Consider the functions r and s given in Figure and Figure. + Consider the functions r and s given in the figures below.

- -
- A parent function r. + + -
+ - +

- +

On the same axes as the plot of y = r(x), sketch the following graphs: y = g(x) = 3r(x) and y = h(x) = \frac{1}{3}r(x). Be sure to label the point on g and h that corresponds to the point (-2,-1) on the original graph of r. In addition, write one sentence to explain the overall transformations that have resulted in g and h from r. @@ -40,7 +40,7 @@ - +

On the same axes as the plot of y = s(x), sketch the following graphs: y = k(x) = -s(x) and y = j(x) = -\frac{1}{2}s(x). Be sure to label the point on k and j that corresponds to the point (-2,-3) on the original graph of s. In addition, write one sentence to explain the overall transformations that have resulted in k and j from s. @@ -50,12 +50,12 @@ - +

On the additional copies of the two figures below, sketch the graphs of the following transformed functions: y = m(x) = 2r(x+1)-1 (at left) and y = n(x) = \frac{1}{2}s(x-2)+2. As above, be sure to label a key point on each graph that corresonds to the labeled point on the original parent function.

- + @@ -64,7 +64,7 @@ - +

Describe in words how the function y = m(x) = 2r(x+1)-1 is the result of three elementary transformations of y = r(x). Does the order in which these transformations occur matter? Why or why not? diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index 8d5776a5..ee1f21e3 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -68,8 +68,8 @@ - diff --git a/source/chap-changing-wb.xml b/source/chap-changing-wb.xml index adeb2685..11449219 100644 --- a/source/chap-changing-wb.xml +++ b/source/chap-changing-wb.xml @@ -21,10 +21,10 @@ - + diff --git a/source/chap-circular-wb.xml b/source/chap-circular-wb.xml index a38b5121..f2a83ecc 100644 --- a/source/chap-circular-wb.xml +++ b/source/chap-circular-wb.xml @@ -17,9 +17,9 @@ Circular Functions - + diff --git a/source/previews/PA-changing-combining.xml b/source/previews/PA-changing-combining.xml index db7d4cb7..84513657 100755 --- a/source/previews/PA-changing-combining.xml +++ b/source/previews/PA-changing-combining.xml @@ -16,11 +16,11 @@

- Consider the functions f and g defined by Table and the piecewise linear functions p and q defined by Figure. Assume that the lines in the figure pass through whole number coordinates where they appear to do so; for example, (2,2) lies on the graph of q, and (3,-3) lies on the graph of p. + Consider the functions f and g defined by the following table and the piecewise linear functions p and q defined by the following figure. Assume that the lines in the figure pass through whole number coordinates where they appear to do so; for example, (2,2) lies on the graph of q, and (3,-3) lies on the graph of p.

- - - Table defining functions <m>f</m> and <m>g</m>. + + @@ -53,15 +53,15 @@ 3 -
+ - +

- +

Let h(x) = f(x) + g(x). Determine h(3). @@ -71,7 +71,7 @@ - +

Let r(x) = p(x) - q(x). Determine r(-1) exactly. diff --git a/source/previews/PA-changing-composite.xml b/source/previews/PA-changing-composite.xml index 8547b371..fb1beb7f 100755 --- a/source/previews/PA-changing-composite.xml +++ b/source/previews/PA-changing-composite.xml @@ -20,7 +20,7 @@

- +

Let r(t) = p(q(t)). Determine a formula for r that depends only on t and not on p or q. @@ -33,14 +33,14 @@

- Recall Example, which involved functions similar to p and q. What is the biggest difference between your work in (a) above and in Example? + Review the introductory example with f(x) = x^2 - 1 and g(t) = 3t - 4, which involved functions similar to p and q in part (a). What is the biggest difference between your work in (a) above and in the introductory example?

- +

Let t = s(z) = \frac{1}{z+4} and recall that x = q(t) = t^2 - 1. Determine a formula for x = q(s(z)) that depends only on z. @@ -50,7 +50,7 @@ - +

Suppose that h(t) = \sqrt{2t^2 + 5}. Determine formulas for two related functions, y = f(x) and x = g(t), so that h(t) = f(g(t)). diff --git a/source/previews/PA-changing-inverse-F-C.xml b/source/previews/PA-changing-inverse-F-C.xml index f040d089..868350c8 100755 --- a/source/previews/PA-changing-inverse-F-C.xml +++ b/source/previews/PA-changing-inverse-F-C.xml @@ -20,7 +20,7 @@

- +

Show that it is possible to solve the equation F = \frac{9}{5}C + 32 for C in terms of F and that doing so results in the equation C = \frac{5}{9}(F-32). @@ -30,7 +30,7 @@ - +

Note that the equation C = \frac{5}{9}(F-32) expresses C as a function of F. Call this function h so that C = h(F) = \frac{5}{9}(F-32). @@ -43,7 +43,7 @@ - +

Find the simplest expression that you can for the composite function k(F) = g(h(F)). @@ -53,7 +53,7 @@ - +

Why are the functions j and k so simple? Explain by discussing how the functions g and h process inputs to generate outputs and what happens when we first execute one followed by the other. diff --git a/source/previews/PA-changing-quadratic.xml b/source/previews/PA-changing-quadratic.xml index 45248cfc..1c2defde 100755 --- a/source/previews/PA-changing-quadratic.xml +++ b/source/previews/PA-changing-quadratic.xml @@ -23,14 +23,14 @@

- +

- Execute appropriate computations to complete both of the following tables. + Execute appropriate computations to complete both of the following tables: values of the function h on the left, average rates of change for h on the right.

- - Function values for <m>h</m> at select inputs. + @@ -79,9 +79,9 @@ -
+ @@ -125,34 +125,34 @@ - + - +

- What pattern(s) do you observe in Tables and ? + What pattern(s) do you observe in the table of function values and in the table of average rates of change?

- +

- Explain why h = q(t) is not a linear function. Use Definition in your response. + Explain why h = q(t) is not a linear function. Use the definition of a linear function (that is, referencing average rate of change) in your response.

- +

What is the average velocity of the water balloon in the final second before it lands? How does this value compare to the average velocity on the time interval [4.9, 5]? diff --git a/source/previews/PA-changing-transformations-quadratic.xml b/source/previews/PA-changing-transformations-quadratic.xml index 23dbea53..c692a1d7 100755 --- a/source/previews/PA-changing-transformations-quadratic.xml +++ b/source/previews/PA-changing-transformations-quadratic.xml @@ -21,7 +21,7 @@

- +

In Desmos, @@ -39,7 +39,7 @@ - +

Next, @@ -56,7 +56,7 @@ - +

Now define the function p(x) = cf(x). (That is, @@ -74,7 +74,7 @@ - +

Finally, click on the icons next to g, h, and p to temporarily hide them, and go back to Line 1 and change your formula for f. diff --git a/source/sec-changing-combining-wb.xml b/source/sec-changing-combining-wb.xml new file mode 100755 index 00000000..b013169d --- /dev/null +++ b/source/sec-changing-combining-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +

+ Combining Functions + + + + + + +
+ diff --git a/source/sec-changing-composite-wb.xml b/source/sec-changing-composite-wb.xml new file mode 100755 index 00000000..541236d9 --- /dev/null +++ b/source/sec-changing-composite-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Composite Functions + + + + + + +
+ diff --git a/source/sec-changing-inverse-wb.xml b/source/sec-changing-inverse-wb.xml new file mode 100755 index 00000000..ee1308e3 --- /dev/null +++ b/source/sec-changing-inverse-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Inverse Functions + + + + + + +
+ diff --git a/source/sec-changing-transformations-wb.xml b/source/sec-changing-transformations-wb.xml new file mode 100755 index 00000000..ad367cf6 --- /dev/null +++ b/source/sec-changing-transformations-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Transformations of Functions + + + + + + +
+ diff --git a/source/sec-circular-traversing-wb.xml b/source/sec-circular-traversing-wb.xml new file mode 100755 index 00000000..5e69f73f --- /dev/null +++ b/source/sec-circular-traversing-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Traversing Circles + + + + + + +
+ From 6fd186377b2f27b63f392080a29784578fdc46e7 Mon Sep 17 00:00:00 2001 From: Matt Boelkins Date: Tue, 6 Aug 2024 07:40:07 -0400 Subject: [PATCH 05/10] spacing edits to chapter 2 activities for workbook, plus setup for chapter 3 --- .../act-circular-sine-cosine-computing.xml | 14 ++++---- .../act-circular-sine-cosine-incr-CCU.xml | 8 ++--- .../activities/act-circular-sine-cosine.xml | 24 ++++++------- .../act-circular-sinusoidal-horiz-stretch.xml | 26 +++++++------- .../act-circular-sinusoidal-period.xml | 10 +++--- .../act-circular-traversing-2nd-ex.xml | 30 ++++++++-------- ...ct-circular-traversing-oscillator-aroc.xml | 14 ++++---- .../act-circular-traversing-oscillator.xml | 8 ++--- .../act-circular-unit-circle-non-unit.xml | 8 ++--- ...t-circular-unit-circle-radians-degrees.xml | 12 +++---- ...circular-unit-circle-special-triangles.xml | 34 +++++++++---------- source/apc-activity-workbook.ptx | 8 +++-- source/chap-circular-wb.xml | 4 +-- source/chap-exp-wb.xml | 4 +-- source/previews/PA-circular-sine.xml | 22 ++++++------ source/previews/PA-circular-sinusoidal.xml | 8 ++--- source/previews/PA-circular-traversing.xml | 14 ++++---- source/previews/PA-circular-unit-circle.xml | 28 +++++++-------- source/sec-circular-sine-cosine-wb.xml | 25 ++++++++++++++ source/sec-circular-sinusoidal-wb.xml | 26 ++++++++++++++ source/sec-circular-unit-circle-wb.xml | 25 ++++++++++++++ source/sec-exp-growth-wb.xml | 25 ++++++++++++++ source/sec-exp-modeling-wb.xml | 25 ++++++++++++++ 23 files changed, 265 insertions(+), 137 deletions(-) create mode 100755 source/sec-circular-sine-cosine-wb.xml create mode 100755 source/sec-circular-sinusoidal-wb.xml create mode 100755 source/sec-circular-unit-circle-wb.xml create mode 100755 source/sec-exp-growth-wb.xml create mode 100755 source/sec-exp-modeling-wb.xml diff --git a/source/activities/act-circular-sine-cosine-computing.xml b/source/activities/act-circular-sine-cosine-computing.xml index 0bc4d3f6..805644a5 100755 --- a/source/activities/act-circular-sine-cosine-computing.xml +++ b/source/activities/act-circular-sine-cosine-computing.xml @@ -20,7 +20,7 @@

- +

The x coordinate of the point on the unit circle that lies in the third quadrant and whose y-coordinate is y = -\frac{3}{4}. @@ -30,7 +30,7 @@ - +

The y-coordinate of the point on the unit circle generated by a central angle opening counterclockwise with one side on the positive x-axis that measures t = 2 radians. @@ -40,7 +40,7 @@ - +

The x-coordinate of the point on the unit circle generated by a central angle with one side on the positive x-axis that measures t = -3.05 radians. (With the negative radian measure, we view the angle as opening counterclockwise from its initial side on the positive x-axis.) @@ -50,7 +50,7 @@ - +

The value of \cos(t) where t is an angle in Quadrant II that satisfies \sin(t) = \frac{1}{2}. @@ -60,7 +60,7 @@ - +

The value of \sin(t) where t is an angle in Quadrant III for which \cos(t) = -0.7. @@ -70,7 +70,7 @@ - +

The average rate of change of f(t) = \sin(t) on the intervals [0.1,0.2] and [0.8,0.9]. @@ -80,7 +80,7 @@ - +

The average rate of change of g(t) = \cos(t) on the intervals [0.1,0.2] and [0.8,0.9]. diff --git a/source/activities/act-circular-sine-cosine-incr-CCU.xml b/source/activities/act-circular-sine-cosine-incr-CCU.xml index 6c9fcd61..7721973b 100755 --- a/source/activities/act-circular-sine-cosine-incr-CCU.xml +++ b/source/activities/act-circular-sine-cosine-incr-CCU.xml @@ -16,7 +16,7 @@

- Use Figure to assist in answering the following questions. + Use Figure 2.3.12 in the text (which plots the sine and cosine functions ont the same axes) to assist in answering the following questions.

@@ -64,7 +64,7 @@ - +

Without doing any computation, on which interval is the average rate of change of g(t) = \cos(t) greater: [\pi, \pi+0.1] or [\frac{3\pi}{2}, \frac{3\pi}{2} + 0.1]? Why? @@ -74,7 +74,7 @@ - +

In general, how would you characterize the locations on the sine and cosine graphs where the functions are increasing or decreasingly most rapidly? @@ -84,7 +84,7 @@ - +

Thinking from the perspective of the unit circle, for which quadrants of the x-y plane is \cos(t) negative for an angle t that lies in that quadrant? diff --git a/source/activities/act-circular-sine-cosine.xml b/source/activities/act-circular-sine-cosine.xml index 87761cb3..fc099959 100755 --- a/source/activities/act-circular-sine-cosine.xml +++ b/source/activities/act-circular-sine-cosine.xml @@ -16,11 +16,11 @@

- Let k = g(t) be the function that tracks the x-coordinate of a point traversing the unit circle counterclockwise from (1,0). That is, g(t) = \cos(t). Use the information we know about the unit circle that is summarized in Figure to respond to the following questions. + Let k = g(t) be the function that tracks the x-coordinate of a point traversing the unit circle counterclockwise from (1,0). That is, g(t) = \cos(t). Use the information we know about the unit circle that is summarized in Figure 2.3.1 (with 16 labeled special points) to respond to the following questions.

- +

What is the exact value of \cos(\frac{\pi}{6})? of \cos(\frac{5\pi}{6})? \cos(-\frac{\pi}{3})? @@ -30,13 +30,13 @@ - +

Complete the following table with the exact values of k that correspond to the stated inputs.

- - Exact values of <m>k = g(t) = \cos(t)</m>. + @@ -134,21 +134,21 @@ -
+ - +

- On the axes provided in Figure, sketch an accurate graph of k = \cos(t). Label the exact location of several key points on the curve. + On the axes provided in the following figure, sketch an accurate graph of k = \cos(t). Label the exact location of several key points on the curve.

-
- Axes for plotting k = \cos(t). + -
+ @@ -174,7 +174,7 @@ - +

How is the graph of k = \cos(t) different from the graph of h = \sin(t)? How are the graphs similar? diff --git a/source/activities/act-circular-sinusoidal-horiz-stretch.xml b/source/activities/act-circular-sinusoidal-horiz-stretch.xml index 7f76a6d5..9827d767 100755 --- a/source/activities/act-circular-sinusoidal-horiz-stretch.xml +++ b/source/activities/act-circular-sinusoidal-horiz-stretch.xml @@ -16,21 +16,21 @@

- Consider the functions f and g given in Figure and Figure. + Consider the functions f and g given in the following figures.

- -
- A parent function f. + + -
-
- A parent function g. + -
+

- +

On the same axes as the plot of y = f(t), sketch the following graphs: y = h(t) = f(\frac{1}{3}t) and y = j(t) = f(4t). Be sure to label several points on each of f, h, and j with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in h and j from f. @@ -40,7 +40,7 @@ - +

On the same axes as the plot of y = g(t), sketch the following graphs: y = k(t) = g(2t) and y = m(t) = g(\frac{1}{2}t). Be sure to label several points on each of g, k, and m with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in k and m from g. @@ -50,12 +50,12 @@ - +

On the additional copies of the two figures below, sketch the graphs of the following transformed functions: y = r(t) = 2f(\frac{1}{2}t) (at left) and y = s(t) = \frac{1}{2}g(2t). As above, be sure to label several points on each graph and indicate their correspondence to points on the original parent function.

- + @@ -64,7 +64,7 @@ - +

Describe in words how the function y = r(t) = 2f(\frac{1}{2}t) is the result of composing two elementary transformations of y = f(t). Does the order in which these transformations are composed matter? Why or why not? diff --git a/source/activities/act-circular-sinusoidal-period.xml b/source/activities/act-circular-sinusoidal-period.xml index d1bac6b8..3e9aa84d 100755 --- a/source/activities/act-circular-sinusoidal-period.xml +++ b/source/activities/act-circular-sinusoidal-period.xml @@ -20,7 +20,7 @@

- +

p(x) = \sin(10x) + 2 @@ -30,7 +30,7 @@ - +

q(x) = -3\cos(0.25x) - 4 @@ -40,7 +40,7 @@ - +

r(x) = 2\sin\left( \frac{\pi}{4} x\right) + 5 @@ -50,7 +50,7 @@ - +

w(x) = 2\cos\left( \frac{\pi}{2} (x-3) \right) + 5 @@ -60,7 +60,7 @@ - +

u(x) = -0.25\sin\left(3x-6\right) + 5 diff --git a/source/activities/act-circular-traversing-2nd-ex.xml b/source/activities/act-circular-traversing-2nd-ex.xml index 6f41a075..4b496524 100755 --- a/source/activities/act-circular-traversing-2nd-ex.xml +++ b/source/activities/act-circular-traversing-2nd-ex.xml @@ -16,17 +16,17 @@

- Consider the circle pictured in Figure that is centered at the point (2,2) and that has circumference 8. Assume that we track the y-coordinate (that is, the height, h) of a point that is traversing the circle counterclockwise and that it starts at P_0 as pictured. + Consider the circle pictured in the following figure that is centered at the point (2,2) and that has circumference 8. Assume that we track the y-coordinate (that is, the height, h) of a point that is traversing the circle counterclockwise and that it starts at P_0 as pictured.

-
- A point traversing the circle. + -
-
- Axes for plotting h as a function of d. + -
+

@@ -50,13 +50,13 @@ - +

- Determine the y-coordinates of the remaining points on the circle (exactly where possible, otherwise approximately) and hence complete the entries in Table that track the height, h, of the point traversing the circle as a function of distance traveled, d. Note that the d-values in the table correspond to the point traversing the circle more than once. + Determine the y-coordinates of the remaining points on the circle (exactly where possible, otherwise approximately) and hence complete the entries in the following table that track the height, h, of the point traversing the circle as a function of distance traveled, d. Note that the d-values in the table correspond to the point traversing the circle more than once.

- - Data for <m>h</m> as a function of <m>d</m>. + @@ -171,16 +171,16 @@ -
+ - +

- By plotting the points in Table and connecting them in an intuitive way, sketch a graph of h as a function of d on the axes provided in Figure over the interval 0 \le d \le 16. Clearly label the scale of your axes and the coordinates of several important points on the curve. + By plotting the points in the table in part (c) and connecting them in an intuitive way, sketch a graph of h as a function of d over the interval 0 \le d \le 16 on the axes provided in the figure in part (a). Clearly label the scale of your axes and the coordinates of several important points on the curve.

@@ -190,7 +190,7 @@

- What is similar about your graph in comparison to the one in Figure? What is different? + What is similar about your graph in comparison to the one in Figure 2.1.5 in the text? What is different?

diff --git a/source/activities/act-circular-traversing-oscillator-aroc.xml b/source/activities/act-circular-traversing-oscillator-aroc.xml index 9485d850..93345253 100755 --- a/source/activities/act-circular-traversing-oscillator-aroc.xml +++ b/source/activities/act-circular-traversing-oscillator-aroc.xml @@ -172,7 +172,7 @@

- +

Determine AV_{[2,2.25]}, AV_{[2.25,2.5]}, AV_{[2.5,2.75]}, and AV_{[2.75,3]}. What do these four values tell us about how the weight is moving on the interval [2,3]? @@ -182,7 +182,7 @@ - +

Give an example of an interval of length 0.25 units on which f has its most negative average rate of change. Justify your choice. @@ -192,7 +192,7 @@ - +

Give an example of the longest interval you can find on which f is decreasing. @@ -202,16 +202,16 @@ - +

- Give an example of an interval on which f is concave up.Recall that a function is concave up on an interval provided that throughout the interval, the curve bends upward, similar to a parabola that opens up.

+ Give an example of an interval on which f is concave up. (Recall that a function is concave up on an interval provided that throughout the interval, the curve bends upward, similar to a parabola that opens up.)

 - +

On an interval where f is both decreasing and concave down, what does this tell us about how the weight is moving on that interval? For instance, is the weight moving toward or away from the wall? is it speeding up or slowing down? @@ -221,7 +221,7 @@ - +

What general conclusions can you make about the average rate of change of a circular function on intervals near its highest or lowest points? about its average rate of change on intervals near the function's midline? diff --git a/source/activities/act-circular-traversing-oscillator.xml b/source/activities/act-circular-traversing-oscillator.xml index b90479f6..78da1b7e 100755 --- a/source/activities/act-circular-traversing-oscillator.xml +++ b/source/activities/act-circular-traversing-oscillator.xml @@ -174,7 +174,7 @@

- +

Determine the period p, midline y = m, and amplitude a of the function f. @@ -184,7 +184,7 @@ - +

What is the greatest distance the weight is displaced from the wall? What is the least distance the weight is displaced from the wall? What is the range of f? @@ -194,7 +194,7 @@ - +

Determine the average rate of change of f on the intervals [4,4.25] and [4.75,5]. Write one careful sentence to explain the meaning of each (including units). In addition, write a sentence to compare the two different values you find and what they together say about the motion of the weight. @@ -204,7 +204,7 @@ - +

Based on the periodicity of the function, what is the value of f(6.75)? of f(11.25)? diff --git a/source/activities/act-circular-unit-circle-non-unit.xml b/source/activities/act-circular-unit-circle-non-unit.xml index b67e9c13..d8523c9f 100755 --- a/source/activities/act-circular-unit-circle-non-unit.xml +++ b/source/activities/act-circular-unit-circle-non-unit.xml @@ -20,7 +20,7 @@

- +

In a circle of radius 11, the arc length intercepted by a central angle of \frac{5\pi}{3}. @@ -30,7 +30,7 @@ - +

In a circle of radius 3, the central angle measure that intercepts an arc of length \frac{\pi}{4}. @@ -40,7 +40,7 @@ - +

The radius of the circle in which an angle of @@ -51,7 +51,7 @@ - +

The exact coordinates of the point on the circle of radius 5 that lies \frac{25\pi}{6} units counterclockwise along the circle from (5,0). diff --git a/source/activities/act-circular-unit-circle-radians-degrees.xml b/source/activities/act-circular-unit-circle-radians-degrees.xml index 5ffb8880..3010b879 100755 --- a/source/activities/act-circular-unit-circle-radians-degrees.xml +++ b/source/activities/act-circular-unit-circle-radians-degrees.xml @@ -20,7 +20,7 @@

- +

30^\circ @@ -30,7 +30,7 @@ - +

\frac{2\pi}{3} radians

@@ -39,7 +39,7 @@ - +

\frac{5\pi}{4} radians

@@ -48,7 +48,7 @@ - +

240^\circ @@ -58,7 +58,7 @@ - +

17^\circ @@ -68,7 +68,7 @@ - +

2 radians

diff --git a/source/activities/act-circular-unit-circle-special-triangles.xml b/source/activities/act-circular-unit-circle-special-triangles.xml index 3e106279..9edda318 100755 --- a/source/activities/act-circular-unit-circle-special-triangles.xml +++ b/source/activities/act-circular-unit-circle-special-triangles.xml @@ -21,18 +21,18 @@ 30^\circ-60^\circ-90^\circ triangles.

-
- A right triangle with two 45^\circ angles. + -
+ - +

- +

For the 45^\circ-45^\circ-90^\circ triangle with legs of length x and y and hypotenuse of length 1, what does the fact that the triangle is isosceles tell us about the relationship between x and y? What are their exact values? @@ -42,7 +42,7 @@ - +

Now consider the 30^\circ-60^\circ-90^\circ triangle with hypotenuse of length 1 and the longer leg (of length x) lying along the positive x-axis. What special kind of triangle is formed when we reflect this triangle across the x-axis? How can we use this perspective to determine the exact values of x and y? @@ -52,7 +52,7 @@ - +

Suppose we consider the related 30^\circ-60^\circ-90^\circ triangle with hypotenuse of length 1 and the shorter leg (of length x) lying along the positive x-axis. What are the exact values of x and y in this triangle? @@ -62,27 +62,27 @@ - +

We know from the conversion factor from degrees to radians that an angle of 30^\circ corresponds to an angle measuring \frac{\pi}{6} radians, an angle of 45^\circ corresponds to \frac{\pi}{4} radians, and 60^\circ corresponds to \frac{\pi}{3} radians.

-
- An angle measuring \frac{\pi}{6} radians. + -
+ - + - +

- Use your work in (a), (b), and (c) to label the noted point in each of Figure, Figure, and Figure, respectively, with its exact coordinates. + Use your work in (a), (b), and (c) to label the noted point in each of the three respective figures with its exact coordinates.

diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index ee1f21e3..eafd2f10 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -66,10 +66,12 @@ - + - - diff --git a/source/chap-circular-wb.xml b/source/chap-circular-wb.xml index f2a83ecc..a38b5121 100644 --- a/source/chap-circular-wb.xml +++ b/source/chap-circular-wb.xml @@ -17,9 +17,9 @@ Circular Functions - + diff --git a/source/chap-exp-wb.xml b/source/chap-exp-wb.xml index a8c89129..a6080770 100644 --- a/source/chap-exp-wb.xml +++ b/source/chap-exp-wb.xml @@ -17,11 +17,11 @@ Exponential and Logarithmic Functions - + diff --git a/source/previews/PA-circular-sine.xml b/source/previews/PA-circular-sine.xml index dfaf59cf..0c81efbf 100755 --- a/source/previews/PA-circular-sine.xml +++ b/source/previews/PA-circular-sine.xml @@ -16,15 +16,15 @@

- If we consider the unit circle in Figure, start at t = 0, and traverse the circle counterclockwise, we may view the height, h, of the traversing point as a function of the angle, t, in radians. From there, we can plot the resulting (t,h) ordered pairs and connect them to generate the circular function pictured in Figure. + If we consider the unit circle with 16 labeled special points in Figure 2.3.1, start at t = 0, and traverse the circle counterclockwise, we may view the height, h, of the traversing point as a function of the angle, t, in radians. From there, we can plot the resulting (t,h) ordered pairs and connect them to generate the circular function pictured in the following figure, which tracks the height of a point traversing the unit circle.

-
- Plot of the circular function that tracks the height of a point traversing the unit circle. + -
+

- +

What is the exact value of f( \frac{\pi}{4} )? of f( \frac{\pi}{3} )? @@ -34,13 +34,13 @@ - +

Complete the following table with the exact values of h that correspond to the stated inputs.

- - Exact values of <m>h</m> as a function of <m>t</m>. + @@ -138,13 +138,13 @@ -
+ - +

What is the exact value of f( \frac{11\pi}{4} )? of f( \frac{14\pi}{3} )? @@ -154,7 +154,7 @@ - +

Give four different values of t for which f(t) = -\frac{\sqrt{3}}{2}. diff --git a/source/previews/PA-circular-sinusoidal.xml b/source/previews/PA-circular-sinusoidal.xml index c9b03d97..67cfe129 100755 --- a/source/previews/PA-circular-sinusoidal.xml +++ b/source/previews/PA-circular-sinusoidal.xml @@ -21,7 +21,7 @@

- +

g(t) = 3\cos(t) and h(t) = -\frac{1}{4}\cos(t)

 @@ -29,7 +29,7 @@ - +

g(t) = \cos(t-\pi) and h(t) = \cos\left(t+ \frac{\pi}{2}\right)

 @@ -37,7 +37,7 @@ - +

g(t) = \cos(t)+4 and h(t) = \cos\left(t\right)-2

 @@ -45,7 +45,7 @@ - +

g(t) = 3\cos(t-\pi)+4 and h(t) = -\frac{1}{4}\cos\left(t+ \frac{\pi}{2}\right)-2

 diff --git a/source/previews/PA-circular-traversing.xml b/source/previews/PA-circular-traversing.xml index c03e15b0..15549e47 100755 --- a/source/previews/PA-circular-traversing.xml +++ b/source/previews/PA-circular-traversing.xml @@ -16,7 +16,7 @@

- In the context of the ferris wheel pictured in Figure, assume that the height, h, of the moving point (the cab in which you are riding), and the distance, d, that the point has traveled around the circumference of the ferris wheel are both measured in meters. + In the context of the ferris wheel pictured in Figure 2.1.1 in the text, assume that the height, h, of the moving point (the cab in which you are riding), and the distance, d, that the point has traveled around the circumference of the ferris wheel are both measured in meters.

Further, assume that the circumference of the ferris wheel is 150 meters. In addition, suppose that after getting in your cab at the lowest point on the wheel, you traverse the full circle several times. @@ -33,7 +33,7 @@ - +

How high is the cab after it has traveled 1/4 of the circumference of the circle? @@ -43,7 +43,7 @@ - +

How much distance along the circle has the cab traversed at the moment it first reaches a height of \frac{150}{\pi} \approx 47.75 meters? @@ -53,7 +53,7 @@ - +

Can h be thought of as a function of d? Why or why not? @@ -63,7 +63,7 @@ - +

Can d be thought of as a function of h? Why or why not? @@ -73,10 +73,10 @@ - +

- Why do you think the curve shown at right in Figure has the shape that it does? Write several sentences to explain. + Why do you think the curve shown at right in Figure 2.1.1 has the shape that it does? Write several sentences to explain.

diff --git a/source/previews/PA-circular-unit-circle.xml b/source/previews/PA-circular-unit-circle.xml index 68a58570..112338e3 100755 --- a/source/previews/PA-circular-unit-circle.xml +++ b/source/previews/PA-circular-unit-circle.xml @@ -16,17 +16,17 @@

- In Figure there are 24 equally spaced points on the unit circle. Since the circumference of the unit circle is 2\pi, each of the points is \frac{1}{24} \cdot 2\pi = \frac{\pi}{12} units apart (traveled along the circle). Thus, + In the following figure there are 24 equally spaced points on the unit circle. Since the circumference of the unit circle is 2\pi, each of the points is \frac{1}{24} \cdot 2\pi = \frac{\pi}{12} units apart (traveled along the circle). Thus, the first point counterclockwise from (1,0) corresponds to the distance t = \frac{\pi}{12} traveled along the unit circle. The second point is twice as far, and thus t = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6} units along the circle away from (1,0).

-
- The unit circle with 24 equally-spaced points. + -
+

- +

Label each of the subsequent points on the unit circle with the exact distance they lie counter-clockwise away from (1,0); write each fraction in lowest terms. @@ -46,21 +46,21 @@ - + - -

- One way to measure angles is connected to the arc length along a circle. For an angle whose vertex is at (0,0) in the unit circle, we say the angle's measure is 1 radianradian provided that the angle intercepts an arc of the circle that is 1 unit in length, as pictured in Figure. Note particularly that an angle measuring 1 radian intercepts an arc of the same length as the circle's radius. -

-
+

+ One way to measure angles is connected to the arc length along a circle. For an angle whose vertex is at (0,0) in the unit circle, we say the angle's measure is 1 radianradian provided that the angle intercepts an arc of the circle that is 1 unit in length, as pictured in the following figure. Note particularly that an angle measuring 1 radian intercepts an arc of the same length as the circle's radius. +

+ + -
+

- Suppose that \alpha and \beta are angles with respective radian measures \alpha = \frac{\pi}{3} and \beta = \frac{3\pi}{4}. Assuming that we view \alpha and \beta as having their vertex at (0,0) and one side along the positive x-axis, sketch the angles \alpha and \beta on the unit circle in Figure. + Suppose that \alpha and \beta are angles with respective radian measures \alpha = \frac{\pi}{3} and \beta = \frac{3\pi}{4}. Assuming that we view \alpha and \beta as having their vertex at (0,0) and one side along the positive x-axis, sketch the angles \alpha and \beta on the unit circle in part (a).

diff --git a/source/sec-circular-sine-cosine-wb.xml b/source/sec-circular-sine-cosine-wb.xml new file mode 100755 index 00000000..d4abfb46 --- /dev/null +++ b/source/sec-circular-sine-cosine-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ The Sine and Cosine Functions + + + + + + +
+ diff --git a/source/sec-circular-sinusoidal-wb.xml b/source/sec-circular-sinusoidal-wb.xml new file mode 100755 index 00000000..9a3b86ba --- /dev/null +++ b/source/sec-circular-sinusoidal-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + +
+ Sinusoidal Functions + + + + + + + +
+ diff --git a/source/sec-circular-unit-circle-wb.xml b/source/sec-circular-unit-circle-wb.xml new file mode 100755 index 00000000..89dedb52 --- /dev/null +++ b/source/sec-circular-unit-circle-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ The Unit Circle + + + + + + +
+ diff --git a/source/sec-exp-growth-wb.xml b/source/sec-exp-growth-wb.xml new file mode 100755 index 00000000..2ad7800e --- /dev/null +++ b/source/sec-exp-growth-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Exponential Growth and Decay + + + + + + +
+ diff --git a/source/sec-exp-modeling-wb.xml b/source/sec-exp-modeling-wb.xml new file mode 100755 index 00000000..cd3b4816 --- /dev/null +++ b/source/sec-exp-modeling-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Modeling with exponential functions + + + + + + +
+ From 2b769556b4572180e179b28fe1401d5c57f65d90 Mon Sep 17 00:00:00 2001 From: Matt Boelkins Date: Tue, 6 Aug 2024 11:58:47 -0400 Subject: [PATCH 06/10] spacing for chapter 3 activities for workbook, plus setting up chapter 4 --- source/activities/act-exp-e-aroc-e.xml | 2 +- source/activities/act-exp-e-graph-solve.xml | 12 ++-- source/activities/act-exp-growth-a-b-t.xml | 8 +-- source/activities/act-exp-growth-find-a-b.xml | 6 +- source/activities/act-exp-growth-rates.xml | 10 +-- source/activities/act-exp-log-base-10.xml | 14 ++-- source/activities/act-exp-log-equations.xml | 16 ++--- .../act-exp-log-exponential-equations.xml | 12 ++-- source/activities/act-exp-log-natural.xml | 37 +++++----- .../act-exp-log-properties-exp-or-log.xml | 16 ++--- .../act-exp-log-properties-find-k.xml | 8 +-- .../activities/act-exp-modeling-behavior.xml | 67 +++++++------------ source/activities/act-exp-modeling-potato.xml | 16 ++--- source/activities/act-exp-modeling-soda.xml | 12 ++-- source/activities/act-exp-temp-pop-NLOC1.xml | 8 +-- .../act-exp-temp-pop-logistic-Desmos.xml | 12 ++-- .../act-exp-temp-pop-logistic-exact.xml | 6 +- source/apc-activity-workbook.ptx | 7 +- source/chap-exp-wb.xml | 10 +-- source/chap-trig-wb.xml | 4 +- source/previews/PA-exp-e.xml | 6 +- source/previews/PA-exp-growth.xml | 16 ++--- source/previews/PA-exp-log-properties.xml | 18 ++--- source/previews/PA-exp-log.xml | 22 +++--- source/previews/PA-exp-temp-pop.xml | 14 ++-- source/sec-exp-e-wb.xml | 24 +++++++ source/sec-exp-log-properties-wb.xml | 25 +++++++ source/sec-exp-log-wb.xml | 25 +++++++ source/sec-exp-temp-pop-wb.xml | 25 +++++++ source/sec-trig-right-wb.xml | 25 +++++++ 30 files changed, 291 insertions(+), 192 deletions(-) create mode 100755 source/sec-exp-e-wb.xml create mode 100755 source/sec-exp-log-properties-wb.xml create mode 100755 source/sec-exp-log-wb.xml create mode 100755 source/sec-exp-temp-pop-wb.xml create mode 100755 source/sec-trig-right-wb.xml diff --git a/source/activities/act-exp-e-aroc-e.xml b/source/activities/act-exp-e-aroc-e.xml index 9f7d5599..2b19c774 100755 --- a/source/activities/act-exp-e-aroc-e.xml +++ b/source/activities/act-exp-e-aroc-e.xml @@ -73,7 +73,7 @@ - +

Adjust your definition of A in Desmos by changing 1 to 2 so that diff --git a/source/activities/act-exp-e-graph-solve.xml b/source/activities/act-exp-e-graph-solve.xml index a7b9aa51..1fcd4736 100755 --- a/source/activities/act-exp-e-graph-solve.xml +++ b/source/activities/act-exp-e-graph-solve.xml @@ -20,7 +20,7 @@

- +

e^t = 2 @@ -30,7 +30,7 @@ - +

e^{3t} = 5 @@ -40,7 +40,7 @@ - +

2e^t - 4 = 7 @@ -50,7 +50,7 @@ - +

3e^{0.25t} + 2 = 6 @@ -60,7 +60,7 @@ - +

4 - 2e^{-0.7t} = 3 @@ -70,7 +70,7 @@ - +

2e^{1.2t} = 1.5e^{1.6t} diff --git a/source/activities/act-exp-growth-a-b-t.xml b/source/activities/act-exp-growth-a-b-t.xml index a8ea4254..38e4979f 100755 --- a/source/activities/act-exp-growth-a-b-t.xml +++ b/source/activities/act-exp-growth-a-b-t.xml @@ -73,12 +73,12 @@

- Consider the graphs of the exponential functions p and q provided in Figure. If p(t) = ab^t and q(t) = cd^t, what can you say about the values a, b, c, and d (beyond the fact that all are positive and b \ne 1 and d \ne 1)? For instance, can you say a certain value is larger than another? Or that one of the values is less than 1? + Consider the graphs of the exponential functions p and q provided in the following figure. If p(t) = ab^t and q(t) = cd^t, what can you say about the values a, b, c, and d (beyond the fact that all are positive and b \ne 1 and d \ne 1)? For instance, can you say a certain value is larger than another? Or that one of the values is less than 1?

-
- Graphs of exponential functions p and q. + -
+ diff --git a/source/activities/act-exp-growth-find-a-b.xml b/source/activities/act-exp-growth-find-a-b.xml index b776c78d..8d50b862 100755 --- a/source/activities/act-exp-growth-find-a-b.xml +++ b/source/activities/act-exp-growth-find-a-b.xml @@ -20,7 +20,7 @@

- +

Suppose the car's value t years after its purchase is given by the function V(t) and that V is exponential with form V(t) = ab^t, what are the values of a and b? Find a and b both exactly and approximately. @@ -33,14 +33,14 @@

- Using the exponential model determined in (a), determine the purchase value of the car and then use Desmosestimate when the car will be worth less than $1000. + Using the exponential model determined in (a), determine the purchase value of the car and then use Desmos to estimate when the car will be worth less than $1000.

- +

Suppose instead that the car's value is modeled by a linear function L and satisfies the values stated at the outset of this activity. Find a formula for L(t) and determine both the purchase value of the car and when the car will be worth $1000. diff --git a/source/activities/act-exp-growth-rates.xml b/source/activities/act-exp-growth-rates.xml index d7661eab..c6188a53 100755 --- a/source/activities/act-exp-growth-rates.xml +++ b/source/activities/act-exp-growth-rates.xml @@ -20,7 +20,7 @@

- +

A function p that is always decreasing and decreases at a constant rate. @@ -30,7 +30,7 @@ - +

A function q that is always increasing and increases at an increasing rate. @@ -40,7 +40,7 @@ - +

A function r that is always increasing for t \lt 2, always decreasing for t \gt 2, and is always changing at a decreasing rate. @@ -50,7 +50,7 @@ - +

A function s that is always increasing and increases at a decreasing rate. (Hint: to find a formula, think about how you might use a transformation of a familiar function.) @@ -60,7 +60,7 @@ - +

A function u that is always decreasing and decreases at a decreasing rate. diff --git a/source/activities/act-exp-log-base-10.xml b/source/activities/act-exp-log-base-10.xml index 5c1e3867..821e6349 100755 --- a/source/activities/act-exp-log-base-10.xml +++ b/source/activities/act-exp-log-base-10.xml @@ -20,7 +20,7 @@

- +

10^t = 0.00001 @@ -30,7 +30,7 @@ - +

\log_{10}(1000000) = t @@ -40,7 +40,7 @@ - +

10^t = 37 @@ -50,7 +50,7 @@ - +

\log_{10}(y) = 1.375 @@ -60,7 +60,7 @@ - +

10^t = 0.04 @@ -70,7 +70,7 @@ - +

3 \cdot 10^t + 11 = 147 @@ -80,7 +80,7 @@ - +

2\log_{10}(y) + 5 = 1 diff --git a/source/activities/act-exp-log-equations.xml b/source/activities/act-exp-log-equations.xml index 4ae48412..b568100a 100755 --- a/source/activities/act-exp-log-equations.xml +++ b/source/activities/act-exp-log-equations.xml @@ -20,7 +20,7 @@

- +

e^t = \frac{1}{10} @@ -30,7 +30,7 @@ - +

5e^{t}=7 @@ -40,7 +40,7 @@ - +

\ln(t) = -\frac{1}{3} @@ -50,7 +50,7 @@ - +

e^{1-3t} = 4 @@ -60,7 +60,7 @@ - +

2\ln(t) + 1 = 4 @@ -70,7 +70,7 @@ - +

4 - 3e^{2t} = 2 @@ -80,7 +80,7 @@ - +

4 + 3e^{2t} = 2 @@ -90,7 +90,7 @@ - +

\ln(5 - 6t) = -2 diff --git a/source/activities/act-exp-log-exponential-equations.xml b/source/activities/act-exp-log-exponential-equations.xml index e4ddeb9e..f96a11c1 100755 --- a/source/activities/act-exp-log-exponential-equations.xml +++ b/source/activities/act-exp-log-exponential-equations.xml @@ -20,7 +20,7 @@

- +

3^t = 5 @@ -30,7 +30,7 @@ - +

4 \cdot 2^t - 2 = 3 @@ -40,7 +40,7 @@ - +

3.7 \cdot (0.9)^{0.3t} + 1.5 = 2.1 @@ -50,7 +50,7 @@ - +

72 - 30(0.7)^{0.05t} = 60 @@ -60,7 +60,7 @@ - +

\ln(t) = -2 @@ -70,7 +70,7 @@ - +

3 + 2\log_{10}(t) = 3.5 diff --git a/source/activities/act-exp-log-natural.xml b/source/activities/act-exp-log-natural.xml index b40b0263..8773eb36 100755 --- a/source/activities/act-exp-log-natural.xml +++ b/source/activities/act-exp-log-natural.xml @@ -20,7 +20,7 @@

- +

What are the domain and range of E? @@ -30,7 +30,7 @@ - +

What are the domain and range of N? @@ -40,7 +40,7 @@ - +

What can you say about \ln(e^t) for every real number t? @@ -50,7 +50,7 @@ - +

What can you say about e^{\ln(y)} for every positive real number y? @@ -60,14 +60,14 @@ - +

- Complete Table and Table with both exact and approximate values of E and N. Then, plot the corresponding ordered pairs from each table on the axes provided in Figure and connect the points in an intuitive way. When you plot the ordered pairs on the axes, in both cases view the first line of the table as generating values on the horizontal axis and the second line of the table as producing values on the vertical axisNote that when we take this perspective for plotting the data in Table, we are viewing N as a function of t, writing N(t) = \ln(t) in order to plot the function on the t-y axes; label each ordered pair you plot appropriately. + Complete the following tables with both exact and approximate values of E and N. Then, plot the corresponding ordered pairs from each table on the axes provided and connect the points in an intuitive way. When you plot the ordered pairs on the axes, in both cases view the first line of the table as generating values on the horizontal axis and the second line of the table as producing values on the vertical axis. Note that when we take this perspective for plotting the data in the table for N, we are viewing N as a function of t, writing N(t) = \ln(t) in order to plot the function on the t-y axes; label each ordered pair you plot appropriately.

- - Values of <m>y = E(t)</m>. + @@ -102,9 +102,9 @@ -
+ @@ -139,22 +139,17 @@ - +

-
- Axes for plotting data from Table and Table along with the graphs of the natural exponential and natural logarithm functions. - -
+ + + - -

- - -

- + diff --git a/source/activities/act-exp-log-properties-exp-or-log.xml b/source/activities/act-exp-log-properties-exp-or-log.xml index 6bf579df..1b2ef3ea 100755 --- a/source/activities/act-exp-log-properties-exp-or-log.xml +++ b/source/activities/act-exp-log-properties-exp-or-log.xml @@ -20,7 +20,7 @@

- +

Let f(t) = 1 - e^{-(t-1)} and g(t) = \ln(t). Plot each function on the same set of coordinate axes. What properties do the two functions have in common? For what properties do the two functions differ? Consider each function's domain, range, t-intercept, y-intercept, increasing/decreasing behavior, concavity, and long-term behavior. @@ -30,7 +30,7 @@ - +

Let h(t) = a - be^{-k(t-c)}, where a, b, c, and k are positive constants. Describe h as a transformation of the function E(t) = e^t. @@ -40,7 +40,7 @@ - +

Let r(t) = a + b\ln(t-c), where a, b, and c are positive constants. Describe r as a transformation of the function L(t) = \ln(t). @@ -50,14 +50,14 @@ - +

- Data for the height of a tree is given in the Table; + Data for the height of a tree is given in the following table; time t is measured in years and height is given in feet. At http://gvsu.edu/s/0yy, you can find a Desmos worksheet with this data already input.

- - The height of a tree as a function of time <m>t</m> in years. + @@ -92,7 +92,7 @@ 19.8 -
+

Do you think this data is better modeled by a logarithmic function of form p(t) = a + b\ln(t-c) or by an exponential function of form q(t) = m + ne^{-rt}. Provide reasons based in how the data appears and how you think a tree grows, as well as by experimenting with sliders appropriately in Desmos. (Note: you may need to adjust the upper and lower bounds of several of the sliders in order to match the data well.)

diff --git a/source/activities/act-exp-log-properties-find-k.xml b/source/activities/act-exp-log-properties-find-k.xml index 91d41002..3503a747 100755 --- a/source/activities/act-exp-log-properties-find-k.xml +++ b/source/activities/act-exp-log-properties-find-k.xml @@ -20,7 +20,7 @@

- +

41 = 50e^{-k \cdot 7} @@ -30,7 +30,7 @@ - +

65 = 34 + 47e^{-k \cdot 45} @@ -40,7 +40,7 @@ - +

7e^{2k-1} + 4 = 32 @@ -50,7 +50,7 @@ - +

\frac{5}{1+2e^{-10k}} = 4 diff --git a/source/activities/act-exp-modeling-behavior.xml b/source/activities/act-exp-modeling-behavior.xml index 9a46b35e..c0e777bc 100755 --- a/source/activities/act-exp-modeling-behavior.xml +++ b/source/activities/act-exp-modeling-behavior.xml @@ -17,42 +17,27 @@

For each of the following functions, without using graphing technology, determine whether the function is - - In addition, state the y-intercept and the range of the function. For each function, write a sentence that explains your thinking and sketch a rough graph of how the function appears. -

-

- - - -

- always increasing or always decreasing; -

- - - - - - - -

- always concave up or always concave down; and -

- - - - - - - -

- increasing without bound, decreasing without bound, or increasing/decreasing toward a finite value. -

- - - - - - +
    +
  1. +

    + always increasing or always decreasing; +

    +
    2. +
  2. +

    + always concave up or always concave down; and +

    +
    3. +
  3. +

    + increasing without bound, decreasing without bound, or increasing/decreasing toward a finite value. +

    +
    4. +
+ In addition, state the y-intercept and the range of the function. For each function, write a sentence that explains your thinking and sketch a rough graph of how the function appears. +

+ +

p(t) = 4372 (1.000235)^t + 92856 @@ -62,7 +47,7 @@ - +

q(t) = 27931 (0.97231)^t + 549786 @@ -72,7 +57,7 @@ - +

r(t) = -17398 (0.85234)^t @@ -82,7 +67,7 @@ - +

s(t) = -17398 (0.85234)^t + 19411 @@ -92,7 +77,7 @@ - +

u(t) = -7522 (1.03817)^t @@ -102,7 +87,7 @@ - +

v(t) = -7522 (1.03817)^t + 6731 diff --git a/source/activities/act-exp-modeling-potato.xml b/source/activities/act-exp-modeling-potato.xml index 8459e3ff..629daf28 100755 --- a/source/activities/act-exp-modeling-potato.xml +++ b/source/activities/act-exp-modeling-potato.xml @@ -23,7 +23,7 @@

- +

What is the numerical value of F(0)? @@ -34,7 +34,7 @@ - +

Based on the context of the problem, @@ -47,7 +47,7 @@ - +

What is the value of b? Why? @@ -61,19 +61,19 @@

Check your work above by plotting the function F using graphing technology in an appropriate window. - Record your results on the axes provided in Figure, labeling the scale on the axes. + Record your results on the axes provided below, labeling the scale on the axes. Then, use the graph to estimate the time at which the potato's temperature reaches 325 degrees.

-
- Axes for plotting F. + -
+ - +

How can we view the function F(t) = a - b(0.98)^t as a transformation of the parent function f(t) = (0.98)^t? Explain. diff --git a/source/activities/act-exp-modeling-soda.xml b/source/activities/act-exp-modeling-soda.xml index 3a93332b..3e73bbb7 100755 --- a/source/activities/act-exp-modeling-soda.xml +++ b/source/activities/act-exp-modeling-soda.xml @@ -50,17 +50,17 @@ - +

Finally, how do you expect the graph of F(t) = 42 + 30(0.95)^{t} to appear? Why? First sketch a rough graph without graphing technology, and then use technology to check your thinking - and report an accurate, labeled graph on the axes provided in Figure. + and report an accurate, labeled graph on the axes provided below.

-
- Axes for plotting F. + -
+ @@ -78,7 +78,7 @@ - +

Determine the average rate of change of F on the intervals [10,20], diff --git a/source/activities/act-exp-temp-pop-NLOC1.xml b/source/activities/act-exp-temp-pop-NLOC1.xml index 9bcaefbc..8901c4ac 100755 --- a/source/activities/act-exp-temp-pop-NLOC1.xml +++ b/source/activities/act-exp-temp-pop-NLOC1.xml @@ -20,7 +20,7 @@

- +

Use algebraic reasoning and your understanding of the physical situation to determine the exact values of a, @@ -33,7 +33,7 @@ - +

Determine the exact time the object's temperature is 42.4^\circ. @@ -44,7 +44,7 @@ - +

In Desmos, enter the values you found for a, c, and k in order to define the function F. Then, use Desmos to find the average rate of change of F on the interval [25,30]. @@ -57,7 +57,7 @@ - +

If everything stayed the same except the value of F(0), diff --git a/source/activities/act-exp-temp-pop-logistic-Desmos.xml b/source/activities/act-exp-temp-pop-logistic-Desmos.xml index d847f750..629bf4b7 100755 --- a/source/activities/act-exp-temp-pop-logistic-Desmos.xml +++ b/source/activities/act-exp-temp-pop-logistic-Desmos.xml @@ -25,16 +25,16 @@

Sketch a typical graph of P(t) on the axes provided and write several sentences to explain the effects of A, M, and k on the graph of P.

-
- Axes for plotting a typical logistic function P. + -
+ - +

On a typical logistic graph, where does it appear that the population is growing most rapidly? How is this value connected to the carrying capacity, A? @@ -44,7 +44,7 @@ - +

How does the function 1 + Me^{-kt} behave as t decreases without bound? What is the algebraic reason that this occurs? @@ -54,7 +54,7 @@ - +

Use your Desmos worksheet to find a logistic function P that has the following properties: P(0) = 2, P(2) = 4, and P(t) approaches 9 as t increases without bound. What are the approximate values of A, M, and k that make the function P fit these criteria? diff --git a/source/activities/act-exp-temp-pop-logistic-exact.xml b/source/activities/act-exp-temp-pop-logistic-exact.xml index 0879c1ac..956c5a0d 100755 --- a/source/activities/act-exp-temp-pop-logistic-exact.xml +++ b/source/activities/act-exp-temp-pop-logistic-exact.xml @@ -20,7 +20,7 @@

- +

Determine the exact values of A, M, and k in the logistic model @@ -34,7 +34,7 @@ - +

Plot your model from (a) and check that its values match the desired characteristics. @@ -47,7 +47,7 @@ - +

Find the exact time value when the population will be 10 (thousand). Show your algebraic work and thinking. diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index eafd2f10..152371b7 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -67,12 +67,13 @@ + + --> - + diff --git a/source/chap-exp-wb.xml b/source/chap-exp-wb.xml index a6080770..c72879e3 100644 --- a/source/chap-exp-wb.xml +++ b/source/chap-exp-wb.xml @@ -17,11 +17,11 @@ Exponential and Logarithmic Functions - + + + + + diff --git a/source/chap-trig-wb.xml b/source/chap-trig-wb.xml index f5a9bb33..f5c5dea8 100644 --- a/source/chap-trig-wb.xml +++ b/source/chap-trig-wb.xml @@ -17,10 +17,10 @@ Trigonometry - + diff --git a/source/previews/PA-exp-e.xml b/source/previews/PA-exp-e.xml index 455dec28..edc090d9 100755 --- a/source/previews/PA-exp-e.xml +++ b/source/previews/PA-exp-e.xml @@ -40,7 +40,7 @@ - +

For the value of k you determined in (a), compute 2^k. What do you observe? @@ -50,7 +50,7 @@ - +

For the value of k you determined in (b), compute 2^k. What do you observe? @@ -60,7 +60,7 @@ - +

Given any exponential function of the form b^t, do you think it's possible to find a value of k to that p(t) = f(kt) = 2^{kt} is the same function as b^t? Why or why not? diff --git a/source/previews/PA-exp-growth.xml b/source/previews/PA-exp-growth.xml index 10370e2f..0db6c804 100755 --- a/source/previews/PA-exp-growth.xml +++ b/source/previews/PA-exp-growth.xml @@ -23,7 +23,7 @@

- +

Determine I(0), I(1), I(2), and I(3). @@ -33,7 +33,7 @@ - +

Note that if a quantity depreciates 12% annually, after a given year, 88% of the quantity remains. Compute V(0), V(1), V(2), and V(3). @@ -43,7 +43,7 @@ - +

Based on the patterns in your computations in (a) and (b), determine formulas for I(t) and V(t). @@ -56,12 +56,12 @@

- Use Desmos to define I(t) and V(t). Plot each function on the interval 0 \le t \le 20 and record your results on the axes in Figure, being sure to label the scale on the axes. What trends do you observe in the graphs? How do I(20) and V(20) compare? + Use Desmos to define I(t) and V(t). Plot each function on the interval 0 \le t \le 20 and record your results on the axes inthe figure below, being sure to label the scale on the axes. What trends do you observe in the graphs? How do I(20) and V(20) compare?

-
- Blank axes for plotting I and V. - -
+ + + diff --git a/source/previews/PA-exp-log-properties.xml b/source/previews/PA-exp-log-properties.xml index c26d822a..24d806b0 100755 --- a/source/previews/PA-exp-log-properties.xml +++ b/source/previews/PA-exp-log-properties.xml @@ -20,7 +20,7 @@

- +

Write 10^x \cdot 10^y as 10 raised to a single power. That is, complete the equation @@ -57,17 +57,11 @@ - a - b - a - 10^x - x - b - 10^y - y - a = 10^x - b = 10^y - \log_{10}(ab) +

+ Suppose that a and b are positive real numbers so we can think of a as 10^x for some real number x and + b as 10^y for some real number y. That is, say that a = 10^x and b = 10^y. What does our work in (c) tell us about + \log_{10}(ab)? +

diff --git a/source/previews/PA-exp-log.xml b/source/previews/PA-exp-log.xml index 779b306a..1d023e7d 100755 --- a/source/previews/PA-exp-log.xml +++ b/source/previews/PA-exp-log.xml @@ -20,13 +20,13 @@

- +

- Complete Table to generate certain values of P. + Complete the followion table to generate certain values of P.

- - Select values of the powers of <m>10</m> function. + @@ -65,7 +65,7 @@ -
+ @@ -81,15 +81,15 @@ - +

Since P has an inverse function, we know there exists some other function, say L, such that writing y = P(t) - says the exact same thing as writing t = L(y). In words, where P produces the result of raising 10 to a given power, the function L reverses this process and instead tells us the power to which we need to raise 10, given a desired result. Complete Table to generate a collection of values of L. + says the exact same thing as writing t = L(y). In words, where P produces the result of raising 10 to a given power, the function L reverses this process and instead tells us the power to which we need to raise 10, given a desired result. Complete the followin table to generate a collection of values of L.

- - Select values of the function <m>L</m> that is the inverse of <m>P</m>. + @@ -152,13 +152,13 @@ -
+ - +

What are the domain and range of the function P? What are the domain and range of the function L? diff --git a/source/previews/PA-exp-temp-pop.xml b/source/previews/PA-exp-temp-pop.xml index d3b8cf9d..f3cafe0e 100755 --- a/source/previews/PA-exp-temp-pop.xml +++ b/source/previews/PA-exp-temp-pop.xml @@ -20,7 +20,7 @@

- +

The temperature of a warming object in an oven is given by F(t) = 275 - 203e^{-kt}, and we know that the object's temperature after 20 minutes is F(20) = 101. Determine the exact value of k. @@ -30,7 +30,7 @@ - +

The temperature of a cooling object in a refrigerator is modeled by F(t) = a + 37.4e^{-0.05t}, and the temperature of the refrigerator is 39.8^\circ. By thinking about the long-term behavior of e^{-0.05t} and the long-term behavior of the object's temperature, determine the exact value of a. @@ -40,24 +40,24 @@ - +

Later in this section, we'll learn that one model for how a population grows over time can be given by a function of the form P(t) = \frac{A}{1 + Me^{-kt}} . Models of this form lead naturally to equations that have structure like - + 3 = \frac{10}{1+x}. - - Solve Equation for the exact value of x. + + Solve the equation 3 = \frac{10}{1+x} for the exact value of x.

- +

Suppose that y = a + be^{-kt}. Solve for t in terms of a, b, k, and y. What does this new equation represent? diff --git a/source/sec-exp-e-wb.xml b/source/sec-exp-e-wb.xml new file mode 100755 index 00000000..cab5cd72 --- /dev/null +++ b/source/sec-exp-e-wb.xml @@ -0,0 +1,24 @@ + + + + + + + + + + + + + + + +

+ The special number <m>e</m> + + + + + +
+ diff --git a/source/sec-exp-log-properties-wb.xml b/source/sec-exp-log-properties-wb.xml new file mode 100755 index 00000000..3cd51d30 --- /dev/null +++ b/source/sec-exp-log-properties-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Properties and applications of logarithmic functions + + + + + + +
+ diff --git a/source/sec-exp-log-wb.xml b/source/sec-exp-log-wb.xml new file mode 100755 index 00000000..0f806ad0 --- /dev/null +++ b/source/sec-exp-log-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ What a logarithm is + + + + + + +
+ diff --git a/source/sec-exp-temp-pop-wb.xml b/source/sec-exp-temp-pop-wb.xml new file mode 100755 index 00000000..2ee203fb --- /dev/null +++ b/source/sec-exp-temp-pop-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Modeling temperature and population + + + + + + +
+ diff --git a/source/sec-trig-right-wb.xml b/source/sec-trig-right-wb.xml new file mode 100755 index 00000000..2ba9e42c --- /dev/null +++ b/source/sec-trig-right-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Right triangles + + + + + + +
+ From 3aca7f07892808f161f5640084c240a11bed3479 Mon Sep 17 00:00:00 2001 From: Matt Boelkins Date: Tue, 6 Aug 2024 13:29:47 -0400 Subject: [PATCH 07/10] spacing in activities in Chapter 4 for the workbook, plus starting chapter 5 --- .../act-trig-finding-angles-exactly.xml | 6 +-- source/activities/act-trig-inverse-arccos.xml | 6 +-- source/activities/act-trig-inverse-arcsin.xml | 16 ++++---- source/activities/act-trig-inverse-arctan.xml | 16 ++++---- .../activities/act-trig-other-aroc-sine.xml | 8 ++-- source/activities/act-trig-other-cot.xml | 38 +++++++++---------- source/activities/act-trig-other-csc.xml | 28 +++++++------- source/activities/act-trig-right-SOH-CAH.xml | 12 +++--- source/activities/act-trig-right-similar.xml | 16 ++++---- .../act-trig-right-sine-cosine-ratio.xml | 8 ++-- .../activities/act-trig-tangent-mountain.xml | 20 +++++----- .../act-trig-tangent-two-towers.xml | 10 ++--- source/apc-activity-workbook.ptx | 12 +++--- source/chap-poly-wb.xml | 5 ++- source/chap-trig-wb.xml | 6 +-- source/previews/PA-trig-finding-angles.xml | 6 +-- source/previews/PA-trig-inverse.xml | 14 +++---- source/previews/PA-trig-right.xml | 12 +++--- source/previews/PA-trig-tangent.xml | 8 ++-- source/sec-poly-infty-wb.xml | 25 ++++++++++++ source/sec-trig-finding-angles-wb.xml | 26 +++++++++++++ source/sec-trig-inverse-wb.xml | 25 ++++++++++++ source/sec-trig-other-wb.xml | 26 +++++++++++++ source/sec-trig-tangent-wb.xml | 25 ++++++++++++ 24 files changed, 250 insertions(+), 124 deletions(-) create mode 100755 source/sec-poly-infty-wb.xml create mode 100755 source/sec-trig-finding-angles-wb.xml create mode 100755 source/sec-trig-inverse-wb.xml create mode 100755 source/sec-trig-other-wb.xml create mode 100755 source/sec-trig-tangent-wb.xml diff --git a/source/activities/act-trig-finding-angles-exactly.xml b/source/activities/act-trig-finding-angles-exactly.xml index f40f75da..04372cb8 100755 --- a/source/activities/act-trig-finding-angles-exactly.xml +++ b/source/activities/act-trig-finding-angles-exactly.xml @@ -20,7 +20,7 @@

- +

Consider a right triangle with legs of length 11 and 13. What are the measures @@ -31,7 +31,7 @@ - +

Consider an angle \alpha in standard position (vertex at the origin, one side on the positive x-axis) for which we know \cos(\alpha) = -\frac{1}{2} and \alpha lies in quadrant III. What is the measure of \alpha in radians? In addition, what is the value of \sin(\alpha)? @@ -41,7 +41,7 @@ - +

Consider an angle \beta in standard position for which we know diff --git a/source/activities/act-trig-inverse-arccos.xml b/source/activities/act-trig-inverse-arccos.xml index 36ebf3db..48786da5 100755 --- a/source/activities/act-trig-inverse-arccos.xml +++ b/source/activities/act-trig-inverse-arccos.xml @@ -16,7 +16,7 @@

- Use the special points on the unit circle (see, for instance, Figure) to determine the exact values of each of the following numerical expressions. Do so without using a computational device. + Use the special points on the unit circle (see, for instance, the start of Section) to determine the exact values of each of the following numerical expressions. Do so without using a computational device.

@@ -50,7 +50,7 @@ - +

\arccos(-\frac{1}{2}) @@ -60,7 +60,7 @@ - +

\arccos(-\frac{\sqrt{2}}{2}) diff --git a/source/activities/act-trig-inverse-arcsin.xml b/source/activities/act-trig-inverse-arcsin.xml index 94bec972..b9bf4298 100755 --- a/source/activities/act-trig-inverse-arcsin.xml +++ b/source/activities/act-trig-inverse-arcsin.xml @@ -16,21 +16,21 @@

- The goal of this activity is to understand key properties of the arcsine function in a way similar to our discussion of the arccosine function in Subsection. + The goal of this activity is to understand key properties of the arcsine function in a way similar to our recent discussion of the arccosine function.

- Using Definition, what are the domain and range of the arcsine function? + Using the definition of the arcsine function, what are the domain and range of the arcsine function?

- +

Determine the following values exactly: \arcsin(-1), \arcsin(-\frac{\sqrt{2}}{2}), \arcsin(0), \arcsin(\frac{1}{2}), and \arcsin(\frac{\sqrt{3}}{2}). @@ -40,15 +40,15 @@ - +

- On the axes provided in Figure, sketch a careful plot of the restricted sine function on the interval [-\frac{\pi}{2},\frac{\pi}{2}] along with its corresponding inverse, the arcsine function. Label at least three points on each curve so that each point on the sine graph corresponds to a point on the arcsine graph. In addition, sketch the line y = t to demonstrate how the graphs are reflections of one another across this line. + On the axes provided, sketch a careful plot of the restricted sine function on the interval [-\frac{\pi}{2},\frac{\pi}{2}] along with its corresponding inverse, the arcsine function. Label at least three points on each curve so that each point on the sine graph corresponds to a point on the arcsine graph. In addition, sketch the line y = t to demonstrate how the graphs are reflections of one another across this line.

-
- Axes for plotting the restricted sine function and its inverse, the arcsine function. + -
+ diff --git a/source/activities/act-trig-inverse-arctan.xml b/source/activities/act-trig-inverse-arctan.xml index 1e62e558..acd4724f 100755 --- a/source/activities/act-trig-inverse-arctan.xml +++ b/source/activities/act-trig-inverse-arctan.xml @@ -23,14 +23,14 @@

- Using Definition, what are the domain and range of the arctangent function? + Using the definition of the arctangent function, what are the domain and range of the arctangent function?

- +

Determine the following values exactly: \arctan(-\sqrt{3}), \arctan(-1), \arctan(0), and \arctan(\frac{1}{\sqrt{3}}). @@ -40,15 +40,15 @@ - +

- A plot of the restricted tangent function on the interval (-\frac{\pi}{2},\frac{\pi}{2}) is provided in Figure. Sketch its corresponding inverse function, the arctangent function, on the same axes. Label at least three points on each curve so that each point on the tangent graph corresponds to a point on the arctangent graph. In addition, sketch the line y = t to demonstrate how the graphs are reflections of one another across this line. + A plot of the restricted tangent function on the interval (-\frac{\pi}{2},\frac{\pi}{2}) is provided in the following figure. Sketch its corresponding inverse function, the arctangent function, on the same axes. Label at least three points on each curve so that each point on the tangent graph corresponds to a point on the arctangent graph. In addition, sketch the line y = t to demonstrate how the graphs are reflections of one another across this line.

-
- Axes for plotting the restricted tangent function and its inverse, the arctangent function. + -
+ @@ -57,7 +57,7 @@

- Complete the following sentence: as t increases without bound, \arctan(t)\ldots. + Complete the following sentence: as t increases without bound, \arctan(t) \ldots.

diff --git a/source/activities/act-trig-other-aroc-sine.xml b/source/activities/act-trig-other-aroc-sine.xml index deed701d..cb2378c9 100755 --- a/source/activities/act-trig-other-aroc-sine.xml +++ b/source/activities/act-trig-other-aroc-sine.xml @@ -26,7 +26,7 @@

- +

Let f(x) = \sin(x). Use the definition of AV_{[a,a+h]} to write an expression for the average rate of change of the sine function on the interval [a,a+h]. @@ -36,7 +36,7 @@ - +

Apply the sum of two angles identity for the sine function, \sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta), to the expression \sin(a+h). @@ -46,7 +46,7 @@ - +

Explain why your work in (a) and (b) together with some algebra shows that @@ -59,7 +59,7 @@ - +

In calculus, we move from average rate of change to instantaneous rate of change by letting h approach 0 in the expression for average rate of change. Using a computational device in radian mode, investigate the behavior of diff --git a/source/activities/act-trig-other-cot.xml b/source/activities/act-trig-other-cot.xml index 4f7cacd5..377949ea 100755 --- a/source/activities/act-trig-other-cot.xml +++ b/source/activities/act-trig-other-cot.xml @@ -20,13 +20,13 @@

- +

- Complete Table and Table to determine the exact values of the cotangent function at the special points on the unit circle. Enter u for any value at which r(t) = \cot(t) is undefined. + Complete the following tables to determine the exact values of the cotangent function at the special points on the unit circle. Enter u for any value at which r(t) = \cot(t) is undefined.

- - Values of the sine, cosine, and tangent functions at special points on the unit circle. + @@ -169,9 +169,9 @@ -
+ @@ -301,13 +301,13 @@ - + - +

In which quadrants is r(t) = \cot(t) positive? negative? @@ -317,7 +317,7 @@ - +

At what t-values does r(t) = \cot(t) have a vertical asymptote? Why? @@ -327,7 +327,7 @@ - +

What is the domain of the cotangent function? What is its range? @@ -337,21 +337,21 @@ - +

- Sketch an accurate, labeled graph of r(t) = \cot(t) on the axes provided in Figure, including the special points that come from the unit circle. + Sketch an accurate, labeled graph of r(t) = \cot(t) on the axes provided below, including the special points that come from the unit circle.

-
- Axes for plotting r(t) = \cot(t). - -
+ + + - +

On intervals where the function is defined at every point in the interval, is r(t) = \cot(t) always increasing, always decreasing, or neither? @@ -361,7 +361,7 @@ - +

What is the period of the cotangent function? @@ -371,7 +371,7 @@ - +

How would you describe the relationship between the graphs of the tangent and cotangent functions? diff --git a/source/activities/act-trig-other-csc.xml b/source/activities/act-trig-other-csc.xml index fd7e71a5..d6f6d818 100755 --- a/source/activities/act-trig-other-csc.xml +++ b/source/activities/act-trig-other-csc.xml @@ -18,19 +18,19 @@

In this activity, we develop the standard properties of the cosecant function, q(t) = \csc(t).

-
- Axes for plotting q(t) = \csc(t). - -
+ + +

- +

- Complete Table and Table to determine the exact values of the cosecant function at the special points on the unit circle. Enter u for any value at which q(t) = \csc(t) is undefined. + Complete the tables below to determine the exact values of the cosecant function at the special points on the unit circle. Enter u for any value at which q(t) = \csc(t) is undefined.

- - Values of the sine function at special points on the unit circle (Quadrants I and II). + @@ -129,9 +129,9 @@ -
+ @@ -221,13 +221,13 @@ - + - +

In which quadrants is q(t) = \csc(t) positive? negative? @@ -257,10 +257,10 @@ - +

- Sketch an accurate, labeled graph of q(t) = \csc(t) on the axes provided in Figure, including the special points that come from the unit circle. + Sketch an accurate, labeled graph of q(t) = \csc(t) on the figure given at the start of this activity, including the special points that come from the unit circle.

diff --git a/source/activities/act-trig-right-SOH-CAH.xml b/source/activities/act-trig-right-SOH-CAH.xml index bf92fa4a..8c584374 100755 --- a/source/activities/act-trig-right-SOH-CAH.xml +++ b/source/activities/act-trig-right-SOH-CAH.xml @@ -20,7 +20,7 @@

- +

A right triangle with hypotenuse 7 and one non-right angle of measure \frac{\pi}{7}. @@ -30,7 +30,7 @@ - +

A right triangle with non-right angle \alpha that satisfies \sin(\alpha) = \frac{3}{5}. @@ -40,7 +40,7 @@ - +

A right triangle where one of the non-right angles has measure 1.2 and the hypotenuse has length 2.7. @@ -50,7 +50,7 @@ - +

A right triangle with hypotenuse 13 and one leg of length 6.5. @@ -60,7 +60,7 @@ - +

A right triangle with legs of length 5 and 12. @@ -70,7 +70,7 @@ - +

A right triangle where one of the non-right angles has measure \frac{\pi}{5} and the leg opposite this angle has length 4. diff --git a/source/activities/act-trig-right-similar.xml b/source/activities/act-trig-right-similar.xml index ab626d3a..45ed00d3 100755 --- a/source/activities/act-trig-right-similar.xml +++ b/source/activities/act-trig-right-similar.xml @@ -16,15 +16,15 @@

- Consider right triangle OPQ given in Figure, and assume that the length of the hypotenuse is OP = r for some constant r \gt 1. Let point M lie on \overline{OP} (the line segment between O and P) in such a way that OM = 1, and let point N lie on \overline{OQ} so that \angle ONM is a right angle, as pictured. In addition, assume that point O corresponds to (0,0), point Q to (x,0), and point P to (x,y) so that OQ = x and PQ = y. Finally, let \theta be the measure of \angle POQ. + Consider right triangle OPQ given in the following figure, and assume that the length of the hypotenuse is OP = r for some constant r \gt 1. Let point M lie on \overline{OP} (the line segment between O and P) in such a way that OM = 1, and let point N lie on \overline{OQ} so that \angle ONM is a right angle, as pictured. In addition, assume that point O corresponds to (0,0), point Q to (x,0), and point P to (x,y) so that OQ = x and PQ = y. Finally, let \theta be the measure of \angle POQ.

-
- Two right triangles \triangle OPQ and \triangle OMN. + -
+

- +

Explain why \triangle OPQ and \triangle OMN are similar triangles. @@ -34,7 +34,7 @@ - +

What is the value of the ratio \frac{OP}{OM}? What does this tell you about the ratios \frac{OQ}{ON} and \frac{PQ}{MN}? @@ -44,7 +44,7 @@ - +

What is the value of ON in terms of \theta? What is the value of MN in terms of \theta? @@ -54,7 +54,7 @@ - +

Use your conclusions in (b) and (c) to express the values of x and y in terms of r and \theta. diff --git a/source/activities/act-trig-right-sine-cosine-ratio.xml b/source/activities/act-trig-right-sine-cosine-ratio.xml index 31632f39..dc9265e5 100755 --- a/source/activities/act-trig-right-sine-cosine-ratio.xml +++ b/source/activities/act-trig-right-sine-cosine-ratio.xml @@ -17,15 +17,15 @@

- We want to determine the distance between two points A and B that are directly across from one another on opposite sides of a river, as pictured in Figure. + We want to determine the distance between two points A and B that are directly across from one another on opposite sides of a river, as pictured in the given diagram. We mark the locations of those points and walk 50 meters downstream from B to point P and use a sextant to measure \angle BPA. If the measure of \angle BPA is 56.4^{\circ}, how wide is the river? What other information about the situation can you determine?

-
- Finding the width of the river. + -
+ diff --git a/source/activities/act-trig-tangent-mountain.xml b/source/activities/act-trig-tangent-mountain.xml index c2ff3fa5..3960b40b 100755 --- a/source/activities/act-trig-tangent-mountain.xml +++ b/source/activities/act-trig-tangent-mountain.xml @@ -16,15 +16,15 @@

- Surveyors are trying to determine the height of a hill relative to sea level. First, they choose a point to take an initial measurement with a sextant that shows the angle of elevation from the ground to the peak of the hill is 19^\circ. Next, they move 1000 feet closer to the hill, staying at the same elevation relative to sea level, and find that the angle of elevation has increased to 25^\circ, as pictured in Figure. We let h represent the height of the hill relative to the two measurements, and x represent the distance from the second measurement location to the center of the hill that lies directly under the peak. + Surveyors are trying to determine the height of a hill relative to sea level. First, they choose a point to take an initial measurement with a sextant that shows the angle of elevation from the ground to the peak of the hill is 19^\circ. Next, they move 1000 feet closer to the hill, staying at the same elevation relative to sea level, and find that the angle of elevation has increased to 25^\circ, as pictured in the given diagram. We let h represent the height of the hill relative to the two measurements, and x represent the distance from the second measurement location to the center of the hill that lies directly under the peak.

-
- The surveyors' initial measurements. + -
+

- +

Using the right triangle with the 25^\circ angle, find an equation that relates x and h. @@ -34,7 +34,7 @@ - +

Using the right triangle with the 19^\circ angle, find a second equation that relates x and h. @@ -44,7 +44,7 @@ - +

Our work in (a) and (b) results in a system of two equations in the two unknowns x and h. Solve each of the two equations for h and then substitute appropriately in order to find a single equation in the variable x. @@ -54,7 +54,7 @@ - +

Solve the equation from (c) to find the exact value of x and determine an approximate value accurate to 3 decimal places. @@ -64,7 +64,7 @@ - +

Use your preceding work to solve for h exactly, plus determine an estimate accurate to 3 decimal places. @@ -74,7 +74,7 @@ - +

If the surveyors' initial measurements were taken from an elevation of 78 feet above sea level, how high above sea level is the peak of the hill? diff --git a/source/activities/act-trig-tangent-two-towers.xml b/source/activities/act-trig-tangent-two-towers.xml index 6d1b2fda..511dded8 100755 --- a/source/activities/act-trig-tangent-two-towers.xml +++ b/source/activities/act-trig-tangent-two-towers.xml @@ -15,14 +15,14 @@ - +

- SupertallSee, for instance, this article high rises have changed the Manhattan skyline. These skyscrapers are known for their small footprint in proportion to their height, with their ratio of width to height at most 1:10, and some as extreme as 1:24. Suppose that a relatively short supertall has been built to a height of 635 feet, as pictured in Figure, and that a second supertall is built nearby. Given the two angles that are computed from the new building, how tall, s, is the new building, and how far apart, d, are the two towers? + Supertall high rises have changed the Manhattan skyline. These skyscrapers are known for their small footprint in proportion to their height, with their ratio of width to height at most 1:10, and some as extreme as 1:24. Suppose that a relatively short supertall has been built to a height of 635 feet, as pictured in the given figure, and that a second supertall is built nearby. Given the two angles that are computed from the new building, how tall, s, is the new building, and how far apart, d, are the two towers?

-
- Two supertall skyscrapers. + -
+ diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index 152371b7..4c91820a 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -68,13 +68,11 @@ - - - - + + + --> + + Back Matter diff --git a/source/chap-poly-wb.xml b/source/chap-poly-wb.xml index c77344eb..f96a43cc 100644 --- a/source/chap-poly-wb.xml +++ b/source/chap-poly-wb.xml @@ -17,10 +17,11 @@ Polynomial and Rational Functions - + + diff --git a/source/chap-trig-wb.xml b/source/chap-trig-wb.xml index f5c5dea8..08e8ecaf 100644 --- a/source/chap-trig-wb.xml +++ b/source/chap-trig-wb.xml @@ -17,10 +17,10 @@ Trigonometry - + + diff --git a/source/previews/PA-trig-finding-angles.xml b/source/previews/PA-trig-finding-angles.xml index 68b9d73e..4ecfc3be 100755 --- a/source/previews/PA-trig-finding-angles.xml +++ b/source/previews/PA-trig-finding-angles.xml @@ -20,7 +20,7 @@

- +

Sketch a labeled picture of the triangle. @@ -30,7 +30,7 @@ - +

What is the exact length of the triangle's hypotenuse? @@ -40,7 +40,7 @@ - +

What is the exact value of \sin(\theta)? diff --git a/source/previews/PA-trig-inverse.xml b/source/previews/PA-trig-inverse.xml index 5b242bc8..766f08b4 100755 --- a/source/previews/PA-trig-inverse.xml +++ b/source/previews/PA-trig-inverse.xml @@ -16,18 +16,18 @@

- Consider the plot of the standard cosine function in Figure along with the emphasized portion of the graph on [0,\pi]. + Consider the plot of the standard cosine function in the following figure along with the emphasized portion of the graph on [0,\pi].

-
- The cosine function on [-\frac{5\pi}{2},\frac{5\pi}{2}] with the portion on [0,\pi] emphasized. + -
+

Let g be the function whose domain is 0 \le t \le \pi and whose outputs are determined by the rule g(t) = \cos(t). Note well: g is defined in terms of the cosine function, but because it has a different domain, it is not the cosine function.

- +

What is the domain of g? @@ -37,7 +37,7 @@ - +

What is the range of g? @@ -47,7 +47,7 @@ - +

Does g pass the horizontal line test? diff --git a/source/previews/PA-trig-right.xml b/source/previews/PA-trig-right.xml index 5979a4b1..a999551b 100755 --- a/source/previews/PA-trig-right.xml +++ b/source/previews/PA-trig-right.xml @@ -20,7 +20,7 @@

- +

The length of the other leg of a right triangle with hypotenuse of length 1 and one leg of length \frac{3}{5}. @@ -30,7 +30,7 @@ - +

The lengths of the two legs in a right triangle with hypotenuse of length 1 where one of the non-right angles measures \frac{\pi}{3}. @@ -40,7 +40,7 @@ - +

The length of the other leg of a right triangle with hypotenuse of length 7 and one leg of length 6. @@ -50,7 +50,7 @@ - +

The lengths of the two legs in a right triangle with hypotenuse 5 and where one of the non-right angles measures \frac{\pi}{4}. @@ -60,7 +60,7 @@ - +

The length of the other leg of a right triangle with hypotenuse of length 1 and one leg of length \cos(0.7). @@ -70,7 +70,7 @@ - +

The measures of the two angles in a right triangle with hypotenuse of length 1 where the two legs have lengths \cos(1.1) and \sin(1.1), respectively. diff --git a/source/previews/PA-trig-tangent.xml b/source/previews/PA-trig-tangent.xml index 6a25e911..fa59708d 100755 --- a/source/previews/PA-trig-tangent.xml +++ b/source/previews/PA-trig-tangent.xml @@ -20,7 +20,7 @@

- +

Without using computational device, @@ -57,7 +57,7 @@ - +

Why is \tan \left( \frac{\pi}{2} \right) not defined? @@ -68,7 +68,7 @@ - +

Point your browser to @@ -90,7 +90,7 @@ - +

At the top of the input lists on the left side of the diff --git a/source/sec-poly-infty-wb.xml b/source/sec-poly-infty-wb.xml new file mode 100755 index 00000000..2c78e2a6 --- /dev/null +++ b/source/sec-poly-infty-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +

+ Infinity, limits, and power functions + + + + + + +
+ diff --git a/source/sec-trig-finding-angles-wb.xml b/source/sec-trig-finding-angles-wb.xml new file mode 100755 index 00000000..bc81726d --- /dev/null +++ b/source/sec-trig-finding-angles-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + +
+ Finding Angles + + + + + + + +
+ diff --git a/source/sec-trig-inverse-wb.xml b/source/sec-trig-inverse-wb.xml new file mode 100755 index 00000000..ebb11271 --- /dev/null +++ b/source/sec-trig-inverse-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Inverses of trigonometric functions + + + + + + +
+ diff --git a/source/sec-trig-other-wb.xml b/source/sec-trig-other-wb.xml new file mode 100755 index 00000000..b474d4c6 --- /dev/null +++ b/source/sec-trig-other-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + +
+ Other Trigonometric Functions and Identities + + + + + + + +
+ diff --git a/source/sec-trig-tangent-wb.xml b/source/sec-trig-tangent-wb.xml new file mode 100755 index 00000000..93018d31 --- /dev/null +++ b/source/sec-trig-tangent-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ The Tangent Function + + + + + + +
+ From 964be54501b3a0cafa35933fa42f10ec08d572d9 Mon Sep 17 00:00:00 2001 From: Matt Boelkins Date: Tue, 6 Aug 2024 14:47:30 -0400 Subject: [PATCH 08/10] last spacing edits for workbook, mostly chapter 5 activities --- .../activities/act-changing-aroc-trends.xml | 2 +- .../act-changing-functions-is-it.xml | 2 +- .../act-circular-sinusoidal-horiz-stretch.xml | 2 +- source/activities/act-poly-infty-limit.xml | 8 ++--- .../act-poly-infty-negative-powers.xml | 6 ++-- ...ct-poly-polynomial-applications-Taylor.xml | 4 +-- ...ct-poly-polynomial-applications-postal.xml | 12 ++++---- .../activities/act-poly-polynomials-find.xml | 10 +++---- .../act-poly-polynomials-multiple-zeros.xml | 28 +++++++----------- .../act-poly-polynomials-sign-chart.xml | 6 ++-- .../activities/act-poly-rational-domain.xml | 12 ++++---- .../act-poly-rational-features-ZAH.xml | 12 ++++---- .../activities/act-poly-rational-formula.xml | 29 ++++++++----------- .../act-poly-rational-long-term-1.xml | 6 ++-- .../act-poly-rational-long-term-2.xml | 8 ++--- source/apc-activity-workbook.ptx | 6 ++-- source/chap-poly-wb.xml | 5 ++-- source/previews/PA-poly-infty.xml | 14 ++++----- .../PA-poly-polynomial-applications.xml | 4 +-- source/previews/PA-poly-polynomials.xml | 14 ++++----- source/previews/PA-poly-rational.xml | 13 ++++++--- .../sec-poly-polynomial-applications-wb.xml | 25 ++++++++++++++++ source/sec-poly-polynomials-wb.xml | 26 +++++++++++++++++ source/sec-poly-rational-features-wb.xml | 24 +++++++++++++++ source/sec-poly-rational-wb.xml | 26 +++++++++++++++++ 25 files changed, 198 insertions(+), 106 deletions(-) create mode 100755 source/sec-poly-polynomial-applications-wb.xml create mode 100755 source/sec-poly-polynomials-wb.xml create mode 100755 source/sec-poly-rational-features-wb.xml create mode 100755 source/sec-poly-rational-wb.xml diff --git a/source/activities/act-changing-aroc-trends.xml b/source/activities/act-changing-aroc-trends.xml index 77cb897a..b6c0638e 100755 --- a/source/activities/act-changing-aroc-trends.xml +++ b/source/activities/act-changing-aroc-trends.xml @@ -46,7 +46,7 @@ - +

On the graphs that follow (q at left, h at right), plot the line segments whose respective slopes are the average rates of change you computed in (a) and (b). diff --git a/source/activities/act-changing-functions-is-it.xml b/source/activities/act-changing-functions-is-it.xml index 7a6e48c5..539a22f0 100755 --- a/source/activities/act-changing-functions-is-it.xml +++ b/source/activities/act-changing-functions-is-it.xml @@ -40,7 +40,7 @@ - +

The relationship between the day of the year and the value of the S&P500 stock index (at the close of trading on a given day), where we attempt to consider the index's value (at the close of trading) as a function of the day of the year. diff --git a/source/activities/act-circular-sinusoidal-horiz-stretch.xml b/source/activities/act-circular-sinusoidal-horiz-stretch.xml index 9827d767..07e4303f 100755 --- a/source/activities/act-circular-sinusoidal-horiz-stretch.xml +++ b/source/activities/act-circular-sinusoidal-horiz-stretch.xml @@ -30,7 +30,7 @@

- +

On the same axes as the plot of y = f(t), sketch the following graphs: y = h(t) = f(\frac{1}{3}t) and y = j(t) = f(4t). Be sure to label several points on each of f, h, and j with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in h and j from f. diff --git a/source/activities/act-poly-infty-limit.xml b/source/activities/act-poly-infty-limit.xml index 227e1ee6..51ff08b3 100755 --- a/source/activities/act-poly-infty-limit.xml +++ b/source/activities/act-poly-infty-limit.xml @@ -16,13 +16,13 @@

- Complete the Table by entering + Complete the following table by entering \infty,-\infty,0, or no limit to identify how the function behaves as either x increases or decreases without bound. As much as possible, work to decide the behavior without using a graphing utility.

- - Some familiar functions and their limits as <m>x \to \infty</m> or <m>x \to -\infty</m>. + @@ -146,7 +146,7 @@ -
+

diff --git a/source/activities/act-poly-infty-negative-powers.xml b/source/activities/act-poly-infty-negative-powers.xml index 5d695160..f10fdb72 100755 --- a/source/activities/act-poly-infty-negative-powers.xml +++ b/source/activities/act-poly-infty-negative-powers.xml @@ -34,7 +34,7 @@ - +

Click the icons next to each of the following 8 functions so that you can see all of y = x^{-1}, @@ -59,7 +59,7 @@ - +

Uncheck the icons on each of the 8 functions to hide their graphs. @@ -73,7 +73,7 @@ - +

Explain why \lim_{x \to \infty} \frac{1}{x^n} = 0 for any choice of n = 1, 2, \ldots. diff --git a/source/activities/act-poly-polynomial-applications-Taylor.xml b/source/activities/act-poly-polynomial-applications-Taylor.xml index b80a6e9b..39c7b01b 100755 --- a/source/activities/act-poly-polynomial-applications-Taylor.xml +++ b/source/activities/act-poly-polynomial-applications-Taylor.xml @@ -53,7 +53,7 @@ - +

What overall trend do you observe? How good is the approximation generated by T_{19}(x)? @@ -63,7 +63,7 @@ - +

In a new Desmos worksheet, plot the function y = \cos(x) along with the following functions: P_2(x) = 1 - \frac{x^2}{2!} and P_4(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!}. Based on the patterns with the coefficients in the polynomials approximating \sin(x) and the polynomials P_2 and P_4 here, conjecture formulas for P_6, P_8, and P_{18} and plot them. How well can we approximate y = \cos(x) using polynomials? diff --git a/source/activities/act-poly-polynomial-applications-postal.xml b/source/activities/act-poly-polynomial-applications-postal.xml index b605ed2a..c51875be 100755 --- a/source/activities/act-poly-polynomial-applications-postal.xml +++ b/source/activities/act-poly-polynomial-applications-postal.xml @@ -18,16 +18,16 @@

According to a shipping company's regulations, the girth plus the length of a parcel they transport for their lowest rate may not exceed 120 inches, where by girth we mean the perimeter of one end.

-
- A rectangular parcel with a square end. - -
+ + +

Suppose that we want to ship a parcel that has a square end of width x and an overall length of y, both measured in inches.

- +

Label the provided picture, using x for the length of each side of the square end, and y for the other edge of the package. @@ -47,7 +47,7 @@ - +

Solve the equation you found in (b) for one of the variables present. diff --git a/source/activities/act-poly-polynomials-find.xml b/source/activities/act-poly-polynomials-find.xml index 5af2d4d3..4d3bd65e 100755 --- a/source/activities/act-poly-polynomials-find.xml +++ b/source/activities/act-poly-polynomials-find.xml @@ -16,11 +16,11 @@

- By experimenting with coefficients in Desmos, find a formula for a polynomial function that has the stated properties, or explain why no such polynomial exists. (If you enter p(x)=a+bx+cx^2+dx^3+fx^4+gx^5 in DesmosWe skip using e as one of the constants since Desmos reserves e as the Euler constant., you'll get prompted to add sliders that make it easy to explore a degree 5 polynomial.) + By experimenting with coefficients in Desmos, find a formula for a polynomial function that has the stated properties, or explain why no such polynomial exists. If you enter p(x)=a+bx+cx^2+dx^3+fx^4+gx^5 in Desmos, you'll get prompted to add sliders that make it easy to explore a degree 5 polynomial. (We skip using e as one of the constants since Desmos reserves e as the Euler constant.)

- +

A polynomial p of degree 5 with exactly 3 real zeros, 4 turning points, and such that \lim_{x \to -\infty} p(x) = +\infty and \lim_{x \to \infty} p(x) = -\infty. @@ -30,7 +30,7 @@ - +

A polynomial p of degree 4 with exactly 4 real zeros, 3 turning points, and such that \lim_{x \to -\infty} p(x) = +\infty and \lim_{x \to \infty} p(x) = -\infty. @@ -40,7 +40,7 @@ - +

A polynomial p of degree 6 with exactly 2 real zeros, 3 turning points, and such that \lim_{x \to -\infty} p(x) = -\infty and \lim_{x \to \infty} p(x) = -\infty. @@ -50,7 +50,7 @@ - +

A polynomial p of degree 5 with exactly 5 real zeros, 3 turning points, and such that \lim_{x \to -\infty} p(x) = +\infty and \lim_{x \to \infty} p(x) = -\infty. diff --git a/source/activities/act-poly-polynomials-multiple-zeros.xml b/source/activities/act-poly-polynomials-multiple-zeros.xml index 3a0f9a65..e108f064 100755 --- a/source/activities/act-poly-polynomials-multiple-zeros.xml +++ b/source/activities/act-poly-polynomials-multiple-zeros.xml @@ -30,33 +30,27 @@ - +

- A polynomial p of degree 9 that satisfies p(0) = -2 and has the graph shown in Figure. Assume that all of the zeros of p are shown in the figure. + A polynomial p of degree 9 that satisfies p(0) = -2 and has the graph shown in the following figure. Assume that all of the zeros of p are shown in the figure.

+ + + - +

- A polynomial q of degree 8 with 3 distinct real zeros (possibly of different multiplicities) such that q has the sign chart in Figure and satisfies q(0) = -10. + A polynomial q of degree 8 with 3 distinct real zeros (possibly of different multiplicities) such that q has the sign chart in the figure below and satisfies q(0) = -10.

- -
- - A polynomial p. - - -
-
- A sign chart for the polynomial q. + -
- + @@ -65,7 +59,7 @@

- A polynomial q of degree 9 with 3 distinct real zeros (possibly of different multiplicities) such that q satisfies the sign chart in Figure and satisfies q(0) = -10. + A polynomial q of degree 9 with 3 distinct real zeros (possibly of different multiplicities) such that q satisfies the sign chart in part (c) and satisfies q(0) = -10.

@@ -75,7 +69,7 @@

- A polynomial p of degree 11 that satisfies p(0) = -2 and p has the graph shown in Figure. Assume that all of the zeros of p are shown in the figure. + A polynomial p of degree 11 that satisfies p(0) = -2 and p has the graph shown in part (b). Assume that all of the zeros of p are shown in the figure.

diff --git a/source/activities/act-poly-polynomials-sign-chart.xml b/source/activities/act-poly-polynomials-sign-chart.xml index 2cc95dfb..57f4f721 100755 --- a/source/activities/act-poly-polynomials-sign-chart.xml +++ b/source/activities/act-poly-polynomials-sign-chart.xml @@ -53,7 +53,7 @@ - +

Construct a sign chart for p by using the zeros you identified in (c) and then analyzing the sign of each factor of p. @@ -63,7 +63,7 @@ - +

Without using a graphing utility, construct an approximate graph of p that has the zeros of p carefully labeled on the x-axis. @@ -73,7 +73,7 @@ - +

Use a graphing utility to check your earlier work. What is challenging or misleading when using technology to graph p? diff --git a/source/activities/act-poly-rational-domain.xml b/source/activities/act-poly-rational-domain.xml index 546b82e5..55d68f81 100755 --- a/source/activities/act-poly-rational-domain.xml +++ b/source/activities/act-poly-rational-domain.xml @@ -20,7 +20,7 @@

- +

\displaystyle f(x) = \frac{x^2-1}{x^2 + 1} @@ -30,7 +30,7 @@ - +

\displaystyle g(x) = \frac{x^2 - 1}{x^2 + 3x - 4} @@ -40,7 +40,7 @@ - +

\displaystyle h(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} @@ -50,7 +50,7 @@ - +

\displaystyle j(x) = \frac{(x+5)(x-3)(x+1)(x-4)}{(x+1)(x+3)(x-5)} @@ -60,7 +60,7 @@ - +

\displaystyle k(x) = \frac{2x^2 + 7}{3x^3 - 12x} @@ -70,7 +70,7 @@ - +

\displaystyle m(x) = \frac{5x^2 - 45}{7(x-2)(x-3)^2(x^2 + 9)(x+1)} diff --git a/source/activities/act-poly-rational-features-ZAH.xml b/source/activities/act-poly-rational-features-ZAH.xml index 32bf9720..ac380887 100755 --- a/source/activities/act-poly-rational-features-ZAH.xml +++ b/source/activities/act-poly-rational-features-ZAH.xml @@ -20,7 +20,7 @@

- +

\displaystyle f(x) = \frac{x^3 - 6x^2 + 5x}{x^2-1} @@ -30,7 +30,7 @@ - +

\displaystyle g(x) = \frac{11(x^2 + 1)(x-7)}{23(x-1)(x^2+4)} @@ -40,7 +40,7 @@ - +

\displaystyle h(x) = \frac{x^2 - 8x + 12}{x^2 - 3x - 18} @@ -50,7 +50,7 @@ - +

\displaystyle q(x) = \frac{(x-2)(x^2-9)}{(x-3)(x^2 + 4)} @@ -60,7 +60,7 @@ - +

\displaystyle r(x) = \frac{19(x-2) (x-3)^2 (x+1)}{17(x+1)(x-4)^2(x-5)} @@ -70,7 +70,7 @@ - +

\displaystyle s(x) = \frac{1}{x^2 + 1} diff --git a/source/activities/act-poly-rational-formula.xml b/source/activities/act-poly-rational-formula.xml index 322f3bdc..a6b0d501 100755 --- a/source/activities/act-poly-rational-formula.xml +++ b/source/activities/act-poly-rational-formula.xml @@ -20,7 +20,7 @@

- +

A rational function r such that r has a vertical asymptote at x = -2, a zero at x = 1, a hole at x = 5, and a horizontal asymptote of y = -3. @@ -30,7 +30,7 @@ - +

A rational function u whose numerator has degree 3, denominator has degree 3, and that has exactly one vertical asymptote at x = -4 and a horizontal asymptote of y = \frac{3}{7}. @@ -40,37 +40,32 @@ - +

- A rational function w whose formula generates a graph with all of the characteristics shown in Figure. Assume that w(5) = 0 but w(x) \gt 0 for all other x such that x \gt 3. + A rational function w whose formula generates a graph with all of the characteristics shown in the following figure. Assume that w(5) = 0 but w(x) \gt 0 for all other x such that x \gt 3.

+ + - +

- A rational function z whose formula satisfies the sign chart shown in Figure, and for which z has no horizontal asymptote and its only vertical asymptotes occur at the middle two values of x noted on the sign chart. + A rational function z whose formula satisfies the sign chart shown in the following figure, and for which z has no horizontal asymptote and its only vertical asymptotes occur at the middle two values of x noted on the sign chart.

- -
- Plot of the rational function w. - -
-
- Sign chart for the rational function z. - -
- + + + - +

A rational function f that has exactly two holes, two vertical asymptotes, two zeros, and a horizontal asymptote. diff --git a/source/activities/act-poly-rational-long-term-1.xml b/source/activities/act-poly-rational-long-term-1.xml index be09d02d..ff52c2ca 100755 --- a/source/activities/act-poly-rational-long-term-1.xml +++ b/source/activities/act-poly-rational-long-term-1.xml @@ -23,7 +23,7 @@

- +

Note that we can multiply the formula for r by the form of 1 given by 1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}}. Do so, and distribute and simplify as much as possible in both the numerator and denominator to write r in a different algebraic form. @@ -48,7 +48,7 @@ - +

Next, determine @@ -61,7 +61,7 @@ - +

Use Desmos to plot r on the interval [-10,10]. In addition, plot the horizontal line y = \frac{3}{7}. What is the meaning of the limits you found in (b) and (c)? diff --git a/source/activities/act-poly-rational-long-term-2.xml b/source/activities/act-poly-rational-long-term-2.xml index 79e297cc..0989bc71 100755 --- a/source/activities/act-poly-rational-long-term-2.xml +++ b/source/activities/act-poly-rational-long-term-2.xml @@ -20,7 +20,7 @@

- +

Using a similar algebraic approach to our work in Activity, multiply s(x) by 1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}} and hence evaluate @@ -34,7 +34,7 @@ - +

Plot the function y = s(x) on the interval [-10,10]. What is the graphical meaning of the limit you found in (a)? @@ -44,7 +44,7 @@ - +

Next, use appropriate algebraic work to consider u(x) and evaluate @@ -58,7 +58,7 @@ - +

Plot the function y = u(x) on the interval [-10,10]. What is the graphical meaning of the limit you computed in (c)? diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index 4c91820a..46d8525d 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -66,13 +66,11 @@ - - - + Back Matter diff --git a/source/chap-poly-wb.xml b/source/chap-poly-wb.xml index f96a43cc..5190e7f2 100644 --- a/source/chap-poly-wb.xml +++ b/source/chap-poly-wb.xml @@ -17,11 +17,10 @@ Polynomial and Rational Functions - - + diff --git a/source/previews/PA-poly-infty.xml b/source/previews/PA-poly-infty.xml index 2b40fc31..9d7acee1 100755 --- a/source/previews/PA-poly-infty.xml +++ b/source/previews/PA-poly-infty.xml @@ -20,7 +20,7 @@

- +

As t \to \infty, e^{-t} \to . @@ -30,7 +30,7 @@ - +

As t \to \infty, \ln(t) \to . @@ -40,7 +40,7 @@ - +

As t \to \infty, e^{t} \to . @@ -50,7 +50,7 @@ - +

As t \to 0^+, e^{-t} \to . (When we write t \to 0^+, this means that we are letting t get closer and closer to 0, but only allowing t to take on positive values.) @@ -60,7 +60,7 @@ - +

As t \to \infty, 35 + 53e^{-0.025t} \to. @@ -70,7 +70,7 @@ - +

As t \to \frac{\pi}{2}^-, \tan(t) \to . (When we write t \to \frac{\pi}{2}^-, this means that we are letting t get closer and closer to \frac{\pi}{2}^-, but only allowing t to take on values that lie to the left of \frac{\pi}{2}.) @@ -80,7 +80,7 @@ - +

As t \to \frac{\pi}{2}^+, \tan(t) \to . (When we write t \to \frac{\pi}{2}^+, this means that we are letting t get closer and closer to \frac{\pi}{2}^+, but only allowing t to take on values that lie to the right of \frac{\pi}{2}.) diff --git a/source/previews/PA-poly-polynomial-applications.xml b/source/previews/PA-poly-polynomial-applications.xml index 60d5cbf5..feaa362a 100755 --- a/source/previews/PA-poly-polynomial-applications.xml +++ b/source/previews/PA-poly-polynomial-applications.xml @@ -20,7 +20,7 @@

- +

Let x be the side length of the squares being cut from the corners of the cardboard. Draw a labeled diagram that shows the given information and the variable being used. @@ -60,7 +60,7 @@ - +

What are the zeros of V? What is the domain of the model V in the context of the rectangular box? diff --git a/source/previews/PA-poly-polynomials.xml b/source/previews/PA-poly-polynomials.xml index 382a59bb..2fed84b6 100755 --- a/source/previews/PA-poly-polynomials.xml +++ b/source/previews/PA-poly-polynomials.xml @@ -20,20 +20,20 @@

- +

What is the largest number of distinct points at which p(x) can cross the x-axis?

- Recall from Definitionwhat we mean by a zero of the polynomial. Give examples of values for a_0, \ldots, a_4 that lead to that largest number of zeros for p(x). + Recall from the definition of a polynoimal function what we mean by a zero of the polynomial. Give examples of values for a_0, \ldots, a_4 that lead to that largest number of zeros for p(x).

- +

What other numbers of zeros are possible for p(x)? Said differently, can you get each possible number of fewer zeros than the largest number that you found in (a)? Why or why not? @@ -43,7 +43,7 @@ - +

We say that a function has a turning pointturning point if the function changes from decreasing to increasing or increasing to decreasing at the point. For example, any quadratic function has a turning point at its vertex. @@ -56,7 +56,7 @@ - +

What other numbers of turning points are possible for p(x)? Can it have no turning points? Just one? Exactly two? Experiment and explain. @@ -66,7 +66,7 @@ - +

What long-range behavior is possible for p(x)? Said differently, what are the possible results for \displaystyle \lim_{x \to -\infty} p(x) and \displaystyle \lim_{x \to \infty} p(x)? @@ -76,7 +76,7 @@ - +

What happens when we plot y = a_4 x^4 in Desmos and compare p(x) and a_4 x^4? How do they look when we zoom out? (Experiment with different values of each of the sliders, too.) diff --git a/source/previews/PA-poly-rational.xml b/source/previews/PA-poly-rational.xml index 8a7fe6dd..825cb2d7 100755 --- a/source/previews/PA-poly-rational.xml +++ b/source/previews/PA-poly-rational.xml @@ -16,7 +16,7 @@

- A drug companyThis activity is based on p. 457ff in Functions Modeling Change, by Connally et al. estimates that to produce a new drug, + A drug company estimates that to produce a new drug, it will cost $5 million in startup resources, and that once they reach production, each gram of the drug will cost $2500 to make.

@@ -51,7 +51,7 @@ - +

Our computations in (b) and (c) naturally lead us to define the average cost per gram function, A(q), whose output is the average cost of producing q grams of the drug. What is a formula for A(q)? @@ -61,7 +61,7 @@ - +

Explain why another formula for A is A(q) = 2500 + \frac{5000000}{q}. @@ -71,7 +71,7 @@ - +

What can you say about the long-range behavior of A? What does this behavior mean in the context of the problem? @@ -81,5 +81,10 @@ + +

+ This activity is based on p. 457ff in Functions Modeling Change, by Connally et al. +

+ diff --git a/source/sec-poly-polynomial-applications-wb.xml b/source/sec-poly-polynomial-applications-wb.xml new file mode 100755 index 00000000..50e22d6f --- /dev/null +++ b/source/sec-poly-polynomial-applications-wb.xml @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +
+ Modeling with polynomial functions + + + + + + +
+ diff --git a/source/sec-poly-polynomials-wb.xml b/source/sec-poly-polynomials-wb.xml new file mode 100755 index 00000000..126dc197 --- /dev/null +++ b/source/sec-poly-polynomials-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + +
+ Polynomials + + + + + + +
+ + diff --git a/source/sec-poly-rational-features-wb.xml b/source/sec-poly-rational-features-wb.xml new file mode 100755 index 00000000..27eeb3f0 --- /dev/null +++ b/source/sec-poly-rational-features-wb.xml @@ -0,0 +1,24 @@ + + + + + + + + + + + + + + + +
+ Key features of rational functions + + + + + +
+ diff --git a/source/sec-poly-rational-wb.xml b/source/sec-poly-rational-wb.xml new file mode 100755 index 00000000..f59f5001 --- /dev/null +++ b/source/sec-poly-rational-wb.xml @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + +
+ Rational Functions + + + + + + + +
+ From 330ca88e9516f00bf5c3868b1c45efb5c0eb15be Mon Sep 17 00:00:00 2001 From: mitchkeller Date: Wed, 20 Aug 2025 19:43:08 +0000 Subject: [PATCH 09/10] Convert section/introduction to section/subsection with title Introduction and resolve deprecations --- publication/publication.ptx | 4 +- source/bibinfo.xml | 61 +++++++++++++++++++++ source/bookinfo.xml | 8 +-- source/colophon.xml | 25 +-------- source/exercises/ez-circular-traversing.xml | 2 +- source/frontmatter.xml | 1 + source/previews/PA-circular-unit-circle.xml | 2 +- source/sec-changing-aroc.xml | 4 +- source/sec-changing-combining.xml | 4 +- source/sec-changing-composite.xml | 4 +- source/sec-changing-functions-models.xml | 8 +-- source/sec-changing-in-tandem.xml | 4 +- source/sec-changing-inverse.xml | 4 +- source/sec-changing-linear.xml | 4 +- source/sec-changing-quadratic.xml | 4 +- source/sec-changing-transformations.xml | 4 +- source/sec-circular-sine-cosine.xml | 4 +- source/sec-circular-sinusoidal.xml | 4 +- source/sec-circular-traversing.xml | 4 +- source/sec-circular-unit-circle.xml | 8 +-- source/sec-exp-e.xml | 4 +- source/sec-exp-growth.xml | 4 +- source/sec-exp-log-properties.xml | 4 +- source/sec-exp-log.xml | 4 +- source/sec-exp-modeling.xml | 4 +- source/sec-exp-temp-pop.xml | 4 +- source/sec-poly-infty.xml | 4 +- source/sec-poly-polynomial-applications.xml | 4 +- source/sec-poly-polynomials.xml | 4 +- source/sec-poly-rational-features.xml | 4 +- source/sec-poly-rational.xml | 4 +- source/sec-trig-finding-angles.xml | 4 +- source/sec-trig-inverse.xml | 4 +- source/sec-trig-other.xml | 4 +- source/sec-trig-right.xml | 4 +- source/sec-trig-tangent.xml | 4 +- source/titlepage.xml | 19 +------ 37 files changed, 134 insertions(+), 112 deletions(-) create mode 100644 source/bibinfo.xml diff --git a/publication/publication.ptx b/publication/publication.ptx index 97461226..6fcbe811 100644 --- a/publication/publication.ptx +++ b/publication/publication.ptx @@ -4,8 +4,8 @@ /> - - + + diff --git a/source/bibinfo.xml b/source/bibinfo.xml new file mode 100644 index 00000000..daf679cc --- /dev/null +++ b/source/bibinfo.xml @@ -0,0 +1,61 @@ + + + + + + + + + + + + + + + + + + Matthew Boelkins + Department of Mathematics + Grand Valley State University + boelkinm@gvsu.edu + + + + Production Editor + + Mitchel T. Keller + Department of Mathematics + University of Wisconsin-Madison + mitch@rellek.net + + + + + + + Cover Photo + Noah Wyn Photography + + + 2019 + + + + + + + 2019 + Matthew Boelkins + CC BY-SA 4.0 License + + + Permission is granted to copy and (re)distribute this material in any format and/or adapt it (even commercially) under the terms of the Creative Commons Attribution-ShareAlike 4.0 International License. The work may be used for free in any way by any party so long as attribution is given to the author(s) and if the material is modified, the resulting contributions are distributed under the same license as this original. All trademarks are the registered marks of their respective owners. The graphic + + + + that may appear in other locations in the text shows that the work is licensed with the Creative Commons and that the work may be used for free by any party so long as attribution is given to the author(s) and if the material is modified, the resulting contributions are distributed under the same license as this original. Full details may be found by visiting https://creativecommons.org/licenses/by-sa/4.0/ or sending a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. + + + + diff --git a/source/bookinfo.xml b/source/bookinfo.xml index f3b88827..c6f90f6a 100644 --- a/source/bookinfo.xml +++ b/source/bookinfo.xml @@ -24,11 +24,8 @@ --> - + APC - - - + +   diff --git a/source/colophon.xml b/source/colophon.xml index 30d8b558..b8c32c60 100644 --- a/source/colophon.xml +++ b/source/colophon.xml @@ -14,28 +14,5 @@ - - Cover Photo - Noah Wyn Photography - - - 2019 - - - - - - - 2019 - Matthew Boelkins - CC BY-SA 4.0 License - - - Permission is granted to copy and (re)distribute this material in any format and/or adapt it (even commercially) under the terms of the Creative Commons Attribution-ShareAlike 4.0 International License. The work may be used for free in any way by any party so long as attribution is given to the author(s) and if the material is modified, the resulting contributions are distributed under the same license as this original. All trademarks are the registered marks of their respective owners. The graphic - - - - that may appear in other locations in the text shows that the work is licensed with the Creative Commons and that the work may be used for free by any party so long as attribution is given to the author(s) and if the material is modified, the resulting contributions are distributed under the same license as this original. Full details may be found by visiting https://creativecommons.org/licenses/by-sa/4.0/ or sending a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. - - + diff --git a/source/exercises/ez-circular-traversing.xml b/source/exercises/ez-circular-traversing.xml index b9272a9b..92192288 100755 --- a/source/exercises/ez-circular-traversing.xml +++ b/source/exercises/ez-circular-traversing.xml @@ -126,7 +126,7 @@

  • - What is similar about your graph in comparison to the one in Figure? What is different? + What is similar about your graph in comparison to the one in ? What is different?

  • diff --git a/source/frontmatter.xml b/source/frontmatter.xml index 1d0a399e..f5f48a81 100644 --- a/source/frontmatter.xml +++ b/source/frontmatter.xml @@ -14,6 +14,7 @@ + diff --git a/source/previews/PA-circular-unit-circle.xml b/source/previews/PA-circular-unit-circle.xml index 112338e3..f3ea646b 100755 --- a/source/previews/PA-circular-unit-circle.xml +++ b/source/previews/PA-circular-unit-circle.xml @@ -20,7 +20,7 @@ the first point counterclockwise from (1,0) corresponds to the distance t = \frac{\pi}{12} traveled along the unit circle. The second point is twice as far, and thus t = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6} units along the circle away from (1,0).

    - diff --git a/source/sec-changing-aroc.xml b/source/sec-changing-aroc.xml index 44686827..333ef963 100755 --- a/source/sec-changing-aroc.xml +++ b/source/sec-changing-aroc.xml @@ -36,7 +36,7 @@ - + Introduction

    Given a function that models a certain phenomenon, it's natural to ask such questions as @@ -59,7 +59,7 @@ - + Defining and interpreting the average rate of change of a function diff --git a/source/sec-changing-combining.xml b/source/sec-changing-combining.xml index 14bb1d11..a5d1b7a9 100755 --- a/source/sec-changing-combining.xml +++ b/source/sec-changing-combining.xml @@ -31,7 +31,7 @@ - + Introduction

    In arithmetic, we execute processes where we take two numbers to generate a new number. For example, 2 + 3 = 5: the number 5 results from adding 2 and 3. Similarly, we can multiply two numbers to generate a new one: 2 \cdot 3 = 6. @@ -47,7 +47,7 @@ - + Arithmetic with functions diff --git a/source/sec-changing-composite.xml b/source/sec-changing-composite.xml index ba0efcd5..2036bcd2 100755 --- a/source/sec-changing-composite.xml +++ b/source/sec-changing-composite.xml @@ -36,7 +36,7 @@ - + Introduction

    Recall that a function, by definition, is a process that takes a collection of inputs and produces a corresponding collection of outputs in such a way that the process produces one and only one output value for any single input value. Because every function is a process, it makes sense to think that it may be possible to take two function processes and do one of the processes first, and then apply the second process to the result.

    @@ -79,7 +79,7 @@ -     + Composing two functions diff --git a/source/sec-changing-functions-models.xml b/source/sec-changing-functions-models.xml index 632d63ee..291b6a58 100755 --- a/source/sec-changing-functions-models.xml +++ b/source/sec-changing-functions-models.xml @@ -30,7 +30,7 @@ - + Introduction

    A mathematical model is an abstract concept through which we use mathematical language and notation to describe a phenomenon in the world around us. One example of a mathematical model is found in Dolbear's LawYou can read more in the Wikipedia entry for Dolbear's Law, which has proven to be remarkably accurate for the behavior of snowy tree crickets. For even more of the story, including a reference to this phenomenon on the popular show The Big Bang Theory, see this article.. Dolbear's Law In the late 1800s, the physicist Amos Dolbear was listening to crickets chirp and noticed a pattern: how frequently the crickets chirped seemed to be connected to the outside temperature. If we let T represent the temperature in degrees Fahrenheit and N the number of chirps per minute, we can summarize Dolbear's observations in the following table.

    @@ -64,7 +64,7 @@ -     +     Functions @@ -352,7 +352,7 @@

    - For a relationship or process to be a function, each individual input must be associated with one and only one output. Thus, the usual way that we demonstrate a relationship or process is not a function is to find a particular input that is associated with two or more outputs. When the relationship is given graphically, such as in Figure, we can use the vertical line test to determine whether or not the graph represents a function. + For a relationship or process to be a function, each individual input must be associated with one and only one output. Thus, the usual way that we demonstrate a relationship or process is not a function is to find a particular input that is associated with two or more outputs. When the relationship is given graphically, such as in the left graph in , we can use the vertical line test to determine whether or not the graph represents a function.

    @@ -364,7 +364,7 @@

    - Since the vertical line x = -3 passes through the circle in Figure at both y = -\sqrt{7} and y = \sqrt{7}, the circle does not represent a relationship where y is a function of x. However, since any vertical line we draw in Figure intersects the blue curve at most one time, the graph indeed represents a function. + Since the vertical line x = -3 passes through the circle in the left graph in at both y = -\sqrt{7} and y = \sqrt{7}, the circle does not represent a relationship where y is a function of x. However, since any vertical line we draw in the right graph in intersects the blue curve at most one time, the graph indeed represents a function.

    diff --git a/source/sec-changing-in-tandem.xml b/source/sec-changing-in-tandem.xml index b33fba9e..a72e982c 100755 --- a/source/sec-changing-in-tandem.xml +++ b/source/sec-changing-in-tandem.xml @@ -30,14 +30,14 @@ - + Introduction

    Mathematics is the art of making sense of patterns. One way that patterns arise is when two quantities are changing in tandem. In this setting, we may make sense of the situation by expressing the relationship between the changing quantities through words, through images, through data, or through a formula.

    - +     Using Graphs to Represent Relationships diff --git a/source/sec-changing-inverse.xml b/source/sec-changing-inverse.xml index d93cac43..4a137ee9 100755 --- a/source/sec-changing-inverse.xml +++ b/source/sec-changing-inverse.xml @@ -36,7 +36,7 @@ - + Introduction

    Because every function is a process that converts a collection of inputs to a corresponding collection of outputs, a natural question is: for a particular function, can we change perspective and think of the original function's outputs as the inputs for a reverse process? If we phrase this question algebraically, it is analogous to asking: given an equation that defines y is a function of x, is it possible to find a corresponding equation where x is a function of y? @@ -44,7 +44,7 @@ - + When a function has an inverse function diff --git a/source/sec-changing-linear.xml b/source/sec-changing-linear.xml index 8e8d1ff2..30b1133c 100755 --- a/source/sec-changing-linear.xml +++ b/source/sec-changing-linear.xml @@ -36,7 +36,7 @@ - + Introduction

    Functions whose graphs are straight lines are both the simplest and the most important functions in mathematics. Lines often model important phenomena, @@ -47,7 +47,7 @@ - + diff --git a/source/sec-changing-quadratic.xml b/source/sec-changing-quadratic.xml index c3b8d60a..65b89fd9 100755 --- a/source/sec-changing-quadratic.xml +++ b/source/sec-changing-quadratic.xml @@ -36,7 +36,7 @@ - + Introduction

    After linear functions, @@ -52,7 +52,7 @@ - + Properties of Quadratic Functions diff --git a/source/sec-changing-transformations.xml b/source/sec-changing-transformations.xml index 9e914266..bd48b1af 100755 --- a/source/sec-changing-transformations.xml +++ b/source/sec-changing-transformations.xml @@ -31,7 +31,7 @@ - + Introduction

    In our preparation for calculus, we aspire to understand functions from a wide range of perspectives and to become familiar with a library of basic functions. So far, two basic families of functions we have considered are linear functions and quadratic functions, the simplest of which are L(x) = x and Q(x) = x^2. As we progress further, we will endeavor to understand a parent function as the most fundamental member of a family of functions, as well as how other similar but more complicated functions are the result of transforming the parent function.

    @@ -46,7 +46,7 @@ -     +     Translations of Functions diff --git a/source/sec-circular-sine-cosine.xml b/source/sec-circular-sine-cosine.xml index 5673fae3..02607df9 100755 --- a/source/sec-circular-sine-cosine.xml +++ b/source/sec-circular-sine-cosine.xml @@ -36,7 +36,7 @@ - + Introduction

    In Section, we saw how tracking the height of a point that is traversing a cirle generates a periodic function, such as in Figure. Then, in Section, we identified a collection of 16 special points on the unit circle, as seen in Figure.

    @@ -52,7 +52,7 @@ -     +     The definition of the sine function diff --git a/source/sec-circular-sinusoidal.xml b/source/sec-circular-sinusoidal.xml index bfa824c1..9e44caec 100755 --- a/source/sec-circular-sinusoidal.xml +++ b/source/sec-circular-sinusoidal.xml @@ -34,14 +34,14 @@
    • - + Introduction

    Recall our work in Section, where we studied how the graph of the function g defined by g(x) = af(x-b) + c is related to the graph of f, where a, b, and c are real numbers with a \ne 0. Because such transformations can shift and stretch a function, we are interested in understanding how we can use transformations of the sine and cosine functions to fit formulas to circular functions.

    -     + Shifts and vertical stretches of the sine and cosine functions

    diff --git a/source/sec-circular-traversing.xml b/source/sec-circular-traversing.xml index c2d4fadf..46d55a0c 100755 --- a/source/sec-circular-traversing.xml +++ b/source/sec-circular-traversing.xml @@ -36,7 +36,7 @@ - + Introduction

    Certain naturally occurring phenomena eventually repeat themselves, especially when the phenomenon is somehow connected to a circle. For example, suppose that you are taking a ride on a ferris wheel and we consider your height, h, above the ground and how your height changes in tandem with the distance, d, that you have traveled around the wheel. In Figure we see a snapshot of this situation, which is available as a full animationUsed with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved. at http://gvsu.edu/s/0Dt.

    @@ -53,7 +53,7 @@ - +     Circular Functions circular functions diff --git a/source/sec-circular-unit-circle.xml b/source/sec-circular-unit-circle.xml index 682e5047..683bd470 100755 --- a/source/sec-circular-unit-circle.xml +++ b/source/sec-circular-unit-circle.xml @@ -34,7 +34,7 @@ - + Introduction

    As demonstrated by several different examples in Section, certain periodic phenomena are closely linked to circles and circular motion. Rather than regularly work with circles of different center and radius, it turns out to be ideal to work with one standard circle and build all circular functions from it. The unit circle unit circleis the circle of radius 1 that is centered at the origin, (0,0).

    @@ -68,7 +68,7 @@ -     +     Radians and degrees @@ -86,7 +86,7 @@

    - As seen in Figure, in the unit circle this means that a central angle has measure 1 radian whenever it intercepts an arc of length 1 unit along the circumference. Because of this important correspondence between the unit circle and radian measure (one unit of arc length on the unit circle corresponds to 1 radian), we focus our discussion of radian measure within the unit circle. + As seen in , in the unit circle this means that a central angle has measure 1 radian whenever it intercepts an arc of length 1 unit along the circumference. Because of this important correspondence between the unit circle and radian measure (one unit of arc length on the unit circle corresponds to 1 radian), we focus our discussion of radian measure within the unit circle.

    @@ -103,7 +103,7 @@

    - Note that in Figure in the Preview Activity, we labeled 24 equally spaced points with their respective distances around the unit circle counterclockwise from (1,0). Because these distances are on the unit circle, they also correspond to the radian measure of the central angles that intercept them. In particular, each central angle with one of its sides on the positive x-axis generates a unique point on the unit circle, and with it, an associated length intercepted along the circumference of the circle. A good exercise at this point is to return to Figure and label each of the noted points with the degree measure that is intercepted by a central angle with one side on the positive x-axis, in addition to the arc lengths (radian measures) already identified. + Note that in , we labeled 24 equally spaced points with their respective distances around the unit circle counterclockwise from (1,0). Because these distances are on the unit circle, they also correspond to the radian measure of the central angles that intercept them. In particular, each central angle with one of its sides on the positive x-axis generates a unique point on the unit circle, and with it, an associated length intercepted along the circumference of the circle. A good exercise at this point is to return to and label each of the noted points with the degree measure that is intercepted by a central angle with one side on the positive x-axis, in addition to the arc lengths (radian measures) already identified.

    diff --git a/source/sec-exp-e.xml b/source/sec-exp-e.xml index c26653ce..d4e52d27 100755 --- a/source/sec-exp-e.xml +++ b/source/sec-exp-e.xml @@ -30,7 +30,7 @@ - + Introduction

    We have observed that the behavior of functions of the form f(t) = b^t is very consistent, where the only major differences depend on whether b \lt 1 or b \gt 1. Indeed, if we stipulate that b \gt 1, the graphs of functions with different bases b look nearly identical, as seen in the plots of p, q, r, and s in Figure.

    @@ -46,7 +46,7 @@ -     + The natural base <m>e</m> diff --git a/source/sec-exp-growth.xml b/source/sec-exp-growth.xml index c3207307..2256884a 100755 --- a/source/sec-exp-growth.xml +++ b/source/sec-exp-growth.xml @@ -36,7 +36,7 @@ - + Introduction

    Linear functions have constant average rate of change and model many important phenomena. In other settings, it is natural for a quantity to change at a rate that is proportional to the amount of the quantity present. For instance, whether you put $100 or $100000 or any other amount in a mutual fund, the investment's value changes at a rate proportional the amount present. We often measure that rate in terms of the annual percentage rate of return.

    @@ -70,7 +70,7 @@

    -     +     Exponential functions of form <m>f(t) = ab^t</m> diff --git a/source/sec-exp-log-properties.xml b/source/sec-exp-log-properties.xml index d8c1dec6..68932c77 100755 --- a/source/sec-exp-log-properties.xml +++ b/source/sec-exp-log-properties.xml @@ -35,14 +35,14 @@ - + Introduction

    Logarithms arise as inverses of exponential functions. In addition, we have motivated their development by our desire to solve exponential equations such as e^k = 3 for k. Because of the inverse relationship between exponential and logarithmic functions, there are several important properties logarithms have that are analogous to ones held by exponential functions. We will work to develop these properties and then show how they are useful in applied settings.

    -     +     Key properties of logarithms diff --git a/source/sec-exp-log.xml b/source/sec-exp-log.xml index b61084ee..dc65275a 100755 --- a/source/sec-exp-log.xml +++ b/source/sec-exp-log.xml @@ -36,7 +36,7 @@ - + Introduction

    In Section, we introduced the idea of an inverse function. The fundamental idea is that f has an inverse function if and only if there exists another function g such that f and g undo one another's respective processes. In other words, the process of the function f is reversible, and reversing f generates a related function g.

    @@ -47,7 +47,7 @@ -     +     The base-<m>10</m> logarithm diff --git a/source/sec-exp-modeling.xml b/source/sec-exp-modeling.xml index 67209294..fdf2c683 100755 --- a/source/sec-exp-modeling.xml +++ b/source/sec-exp-modeling.xml @@ -36,7 +36,7 @@ - + Introduction

    If a quantity changes so that its growth or decay occurs at a constant percentage rate with respect to time, the function is exponential. This is because if the growth or decay rate is r, the total amount of the quantity at time t is given by @@ -112,7 +112,7 @@ - + Long-term behavior of exponential functions diff --git a/source/sec-exp-temp-pop.xml b/source/sec-exp-temp-pop.xml index e8a9ca0d..caae7d99 100755 --- a/source/sec-exp-temp-pop.xml +++ b/source/sec-exp-temp-pop.xml @@ -31,7 +31,7 @@ - + Introduction

    We've seen that exponential functions can be used to model several different important phenomena, such as the growth of money due to continuously compounded interest, the decay of radioactive quanitities, and the temperature of an object that is cooling or warming due to its surroundings. From initial work with functions of the form f(t) = ab^t where b \gt 0 and b \ne 1, we found that shifted exponential functions of form g(t) = ab^t + c are also important. Moreover, the special base e allows us to represent all of these functions through horizontal scaling by writing @@ -60,7 +60,7 @@ - + Newton's Law of Cooling revisited diff --git a/source/sec-poly-infty.xml b/source/sec-poly-infty.xml index a20826e2..4aa3c1bd 100755 --- a/source/sec-poly-infty.xml +++ b/source/sec-poly-infty.xml @@ -36,7 +36,7 @@ - + Introduction

    In Section, we compared the behavior of the exponential functions p(t) = 2^t and q(t) = (\frac{1}{2})^t, and observed in Figure that as t increases without bound, p(t) also increases without bound, while q(t) approaches 0 (while having its value be always positive). We also introduced shorthand notation for describing these phenomena, writing @@ -59,7 +59,7 @@ - + Limit notation diff --git a/source/sec-poly-polynomial-applications.xml b/source/sec-poly-polynomial-applications.xml index 9dbb89e0..c44ec26c 100755 --- a/source/sec-poly-polynomial-applications.xml +++ b/source/sec-poly-polynomial-applications.xml @@ -31,7 +31,7 @@ - + Introduction

    Polynomial functions are the simplest of all functions in mathematics in part because they only involve multiplication and addition. In any applied setting where we can formulate key ideas using only those arithmetic operations, it's natural that polynomial functions model the corresponding phenomena. For example, in Activity, we saw that for a spherical tank of radius 4 m filling with water, the volume of water in the tank at a given instant, V, is a function of the depth, h, of the water in the tank at the same moment according to the formula @@ -46,7 +46,7 @@ - + Volume, surface area, and constraints diff --git a/source/sec-poly-polynomials.xml b/source/sec-poly-polynomials.xml index d8901cbb..83f4c4ec 100755 --- a/source/sec-poly-polynomials.xml +++ b/source/sec-poly-polynomials.xml @@ -36,7 +36,7 @@ - + Introduction

    We know that linear functions are the simplest of all functions we can consider: their graphs have the simplest shape, their average rate of change is always constant (regardless of the interval chosen), and their formula is elementary. Moreover, computing the value of a linear function only requires multiplication and addition.

    @@ -86,7 +86,7 @@ -     +     Key results about polynomial functions diff --git a/source/sec-poly-rational-features.xml b/source/sec-poly-rational-features.xml index a1dbe8bb..26a9eae9 100755 --- a/source/sec-poly-rational-features.xml +++ b/source/sec-poly-rational-features.xml @@ -36,7 +36,7 @@ - + Introduction

    Because any rational function is the ratio of two polynomial functions, it's natural to ask questions about rational functions similar to those we ask about polynomials. With polynomials, it is often helpful to know where the function's value is zero. In a rational function r(x) = \frac{p(x)}{q(x)}, we are curious to know where both p(x) = 0 and where q(x) = 0.

    @@ -47,7 +47,7 @@ -     +     When a rational function has a <q>hole</q> diff --git a/source/sec-poly-rational.xml b/source/sec-poly-rational.xml index 83d35cb5..db01ad7a 100755 --- a/source/sec-poly-rational.xml +++ b/source/sec-poly-rational.xml @@ -36,7 +36,7 @@ - + Introduction

    The average rate of change of a function on an interval always involves a ratio. Indeed, for a given function f that interests us near t = 2, we can investigate its average rate of change on intervals near this value by considering @@ -66,7 +66,7 @@ - + diff --git a/source/sec-trig-finding-angles.xml b/source/sec-trig-finding-angles.xml index a14c5e81..1213a462 100755 --- a/source/sec-trig-finding-angles.xml +++ b/source/sec-trig-finding-angles.xml @@ -30,7 +30,7 @@ </ul> </objectives> - <introduction> + <subsection><title>Introduction

    In our earlier work in Section and Section, we observed that in any right triangle, if we know the measure of one additional angle and the length of one additional side, we can determine all of the other parts of the triangle. With the inverse trigonometric functions that we developed in Section, we are now also able to determine the missing angles in any right triangle where we know the lengths of two sides.

    @@ -45,7 +45,7 @@ -     +     Evaluating inverse trigonometric functions diff --git a/source/sec-trig-inverse.xml b/source/sec-trig-inverse.xml index add6f72a..b4016bf4 100755 --- a/source/sec-trig-inverse.xml +++ b/source/sec-trig-inverse.xml @@ -36,7 +36,7 @@ - + Introduction

    In our prior work with inverse functions, we have seen several important principles, including

      @@ -102,7 +102,7 @@ - + The arccosine function diff --git a/source/sec-trig-other.xml b/source/sec-trig-other.xml index e5d6c5f0..54987c12 100755 --- a/source/sec-trig-other.xml +++ b/source/sec-trig-other.xml @@ -35,7 +35,7 @@
    - + Introduction

    The sine and cosine functions, originally defined in the context of a point traversing the unit circle, are also central in right triangle trigonometry. They enable us to find missing information in right triangles in a straightforward way when we know one of the non-right angles and one of the three sides of the triangle, or two of the sides where one is the hypotenuse. In addition, we defined the tangent function in terms of the sine and cosine functions, and the tangent function offers additional options for finding missing information in right triangles. We've also seen how the inverses of the restricted sine, cosine, and tangent functions enable us to find missing angles in a wide variety of settings involving right triangles.

    @@ -113,7 +113,7 @@ -     +     Ratios in right triangles diff --git a/source/sec-trig-right.xml b/source/sec-trig-right.xml index 4e18f9d2..148cfeda 100755 --- a/source/sec-trig-right.xml +++ b/source/sec-trig-right.xml @@ -36,7 +36,7 @@ - + Introduction

    In Section, we defined the cosine and sine functions as the functions that track the location of a point traversing the unit circle counterclockwise from (1,0). In particular, for a central angle of radian measure t that passes through the point (1,0), we define \cos(t) as the x-coordinate of the point where the other side of the angle intersects the unit circle, and \sin(t) as the y-coordinate of that same point, as pictured in Figure. @@ -63,7 +63,7 @@ - + The geometry of triangles diff --git a/source/sec-trig-tangent.xml b/source/sec-trig-tangent.xml index 29cbbb52..2bccf567 100755 --- a/source/sec-trig-tangent.xml +++ b/source/sec-trig-tangent.xml @@ -36,7 +36,7 @@ - + Introduction

    @@ -72,7 +72,7 @@ - + Two perspectives on the tangent function diff --git a/source/titlepage.xml b/source/titlepage.xml index 686217e4..f22959fe 100644 --- a/source/titlepage.xml +++ b/source/titlepage.xml @@ -14,22 +14,5 @@ - - Matthew Boelkins - Department of Mathematics - Grand Valley State University - boelkinm@gvsu.edu - - - - Production Editor - - Mitchel T. Keller - Department of Mathematics - University of Wisconsin-Madison - mitch@rellek.net - - - - + From a18b2b96c0d62bffd687a267e2e7ff69ec004104 Mon Sep 17 00:00:00 2001 From: mitchkeller Date: Wed, 20 Aug 2025 19:58:31 +0000 Subject: [PATCH 10/10] Convert project.ptx to v2 --- project.ptx | 118 ++++++++++++++-------------------------------------- 1 file changed, 31 insertions(+), 87 deletions(-) diff --git a/project.ptx b/project.ptx index 04fe8c1a..4caf98bb 100644 --- a/project.ptx +++ b/project.ptx @@ -4,93 +4,37 @@ project. To edit the content of your document, open `source/main.ptx` (default location). --> - + - - html - source/main.ptx - publication/publication.ptx - output/html - - - html - source/main.ptx - publication/publication.ptx - /home/activec2/public_html/prelude - - - latex - source/main.ptx - publication/publication.ptx - output/latex - xsl/apc-latex.xsl - - - pdf - source/main.ptx - publication/publication.ptx - output/pdf - xsl/apc-latex.xsl - - - pdf - source/apc-solution-manual.ptx - publication/publication-solutions.ptx - output/solutions-pdf - xsl/apc-solution-manual.xsl - - - latex - source/apc-solution-manual.ptx - publication/publication-solutions.ptx - output/solutions-latex - xsl/apc-solution-manual.xsl - - - pdf - source/apc-activity-workbook.ptx - publication/publication-workbook.ptx - output/workbook-pdf - xsl/apc-activity-workbook.xsl - - - latex - source/apc-activity-workbook.ptx - publication/publication-workbook.ptx - output/workbook-latex - xsl/apc-activity-workbook.xsl - - - html - source/main.ptx - publication/publication.ptx - output/subset - - - C-8 - xsl/acs-html.xsl - - - epub - source/main.ptx - publication/publication.ptx - output/epub - - + + + + + + + - - latex - pdflatex - xelatex - pdf2svg - asy - sage - convert - pdftops - pdf-crop-margins - pageres - node - file2brl -