From 56b07c00420c664069d2ec156254646449fd709c Mon Sep 17 00:00:00 2001 From: mitchkeller Date: Wed, 10 Dec 2025 17:43:20 +0000 Subject: [PATCH] Add initial image descriptions --- .github/workflows/pretext-cli.yml | 2 +- .github/workflows/pretext-deploy.yml | 2 +- requirements.txt | 4 +- source/acs-solution-manual.ptx | 34 +++++------ source/activities/act-0-0-0.xml | 1 - .../activities/act-changing-aroc-graphs.xml | 13 ++-- .../act-changing-aroc-population.xml | 1 - .../activities/act-changing-aroc-trends.xml | 5 +- .../act-changing-combining-arithmetic.xml | 5 +- .../act-changing-combining-context.xml | 1 - .../act-changing-combining-piecewise.xml | 5 +- .../act-changing-composite-aroc.xml | 1 - ...ct-changing-composite-crickets-celsius.xml | 5 +- .../act-changing-composite-tables-graphs.xml | 3 +- .../act-changing-functions-is-it.xml | 5 +- ...ging-functions-spherical-tank-draining.xml | 1 - .../act-changing-functions-spherical-tank.xml | 1 - .../act-changing-inverse-Dolbear.xml | 1 - .../act-changing-inverse-does-it.xml | 5 +- .../act-changing-inverse-rainfall.xml | 1 - .../act-changing-linear-Kilimanjaro.xml | 1 - .../act-changing-linear-finding-eqs.xml | 3 +- .../act-changing-linear-in-context.xml | 1 - .../act-changing-quadratic-falling-ball.xml | 1 - .../act-changing-quadratic-parameters.xml | 1 - .../act-changing-quadratic-properties.xml | 1 - .../act-changing-tandem-conical-tank.xml | 7 +-- .../act-changing-tandem-spherical-tank.xml | 7 +-- .../act-changing-transformations-combined.xml | 7 +-- ...-changing-transformations-translations.xml | 5 +- ...-changing-transformations-vert-stretch.xml | 9 ++- .../act-circular-sine-cosine-computing.xml | 1 - .../act-circular-sine-cosine-incr-CCU.xml | 1 - .../activities/act-circular-sine-cosine.xml | 3 +- .../act-circular-sinusoidal-horiz-stretch.xml | 9 ++- .../act-circular-sinusoidal-model.xml | 1 - .../act-circular-sinusoidal-oscillator.xml | 1 - .../act-circular-sinusoidal-period.xml | 1 - .../act-circular-traversing-2nd-ex.xml | 5 +- ...ct-circular-traversing-oscillator-aroc.xml | 3 +- .../act-circular-traversing-oscillator.xml | 3 +- .../act-circular-unit-circle-non-unit.xml | 1 - ...t-circular-unit-circle-radians-degrees.xml | 1 - ...circular-unit-circle-special-triangles.xml | 11 ++-- source/activities/act-exp-e-aroc-e.xml | 1 - source/activities/act-exp-e-graph-solve.xml | 1 - source/activities/act-exp-growth-a-b-t.xml | 3 +- source/activities/act-exp-growth-find-a-b.xml | 1 - source/activities/act-exp-growth-rates.xml | 1 - source/activities/act-exp-log-base-10.xml | 1 - source/activities/act-exp-log-equations.xml | 1 - .../act-exp-log-exponential-equations.xml | 1 - source/activities/act-exp-log-natural.xml | 3 +- .../act-exp-log-properties-exp-or-log.xml | 1 - .../act-exp-log-properties-find-k.xml | 1 - .../activities/act-exp-modeling-behavior.xml | 1 - source/activities/act-exp-modeling-potato.xml | 3 +- source/activities/act-exp-modeling-soda.xml | 3 +- source/activities/act-exp-temp-pop-NLOC1.xml | 1 - .../act-exp-temp-pop-logistic-Desmos.xml | 3 +- .../act-exp-temp-pop-logistic-exact.xml | 1 - source/activities/act-poly-infty-limit.xml | 1 - .../act-poly-infty-natural-powers.xml | 1 - .../act-poly-infty-negative-powers.xml | 1 - ...ct-poly-polynomial-applications-Taylor.xml | 1 - ...ct-poly-polynomial-applications-postal.xml | 3 +- .../act-poly-polynomial-applications-soup.xml | 1 - .../activities/act-poly-polynomials-find.xml | 1 - .../act-poly-polynomials-multiple-zeros.xml | 5 +- .../act-poly-polynomials-sign-chart.xml | 1 - .../act-poly-rational-application.xml | 1 - .../activities/act-poly-rational-domain.xml | 1 - .../act-poly-rational-features-ZAH.xml | 1 - .../activities/act-poly-rational-formula.xml | 5 +- .../act-poly-rational-long-term-1.xml | 1 - .../act-poly-rational-long-term-2.xml | 1 - .../act-trig-finding-angles-baseball.xml | 1 - .../act-trig-finding-angles-exactly.xml | 1 - .../act-trig-finding-angles-rocket.xml | 1 - .../act-trig-finding-angles-roof.xml | 1 - source/activities/act-trig-inverse-arccos.xml | 1 - source/activities/act-trig-inverse-arcsin.xml | 3 +- source/activities/act-trig-inverse-arctan.xml | 3 +- .../activities/act-trig-other-aroc-sine.xml | 1 - source/activities/act-trig-other-cot.xml | 3 +- source/activities/act-trig-other-csc.xml | 3 +- source/activities/act-trig-other-special.xml | 1 - source/activities/act-trig-right-SOH-CAH.xml | 1 - source/activities/act-trig-right-similar.xml | 3 +- .../act-trig-right-sine-cosine-ratio.xml | 3 +- .../activities/act-trig-tangent-mountain.xml | 3 +- source/activities/act-trig-tangent-tower.xml | 1 - .../act-trig-tangent-two-towers.xml | 3 +- source/activities/z-act-inverse-rainfall.xml | 1 - .../z-act-transformations-translations.xml | 7 ++- source/apc-activity-workbook.ptx | 30 +++++---- source/apc-solution-manual.ptx | 26 ++++---- source/backmatter.xml | 9 +-- source/bibinfo.xml | 12 ++-- source/bookinfo.xml | 9 +-- source/chap-changing-wb.xml | 23 +++---- source/chap-changing.xml | 23 +++---- source/chap-circular-wb.xml | 13 ++-- source/chap-circular.xml | 13 ++-- source/chap-exp-wb.xml | 17 +++--- source/chap-exp.xml | 17 +++--- source/chap-poly-wb.xml | 15 ++--- source/chap-poly.xml | 15 ++--- source/chap-trig-wb.xml | 15 ++--- source/chap-trig.xml | 15 ++--- source/colophon.xml | 4 +- source/exercises/ez-0-0.xml | 2 - source/exercises/ez-changing-aroc.xml | 22 +++---- source/exercises/ez-changing-combining.xml | 23 +++---- source/exercises/ez-changing-composite.xml | 25 ++++---- source/exercises/ez-changing-functions.xml | 24 ++++---- source/exercises/ez-changing-inverse.xml | 22 ++++--- source/exercises/ez-changing-linear.xml | 17 +++--- source/exercises/ez-changing-quadratic.xml | 36 +++++------ source/exercises/ez-changing-tandem.xml | 22 ++++--- .../exercises/ez-changing-transformations.xml | 17 +++--- source/exercises/ez-circular-sine-cosine.xml | 9 ++- source/exercises/ez-circular-sinusoidal.xml | 14 ++--- source/exercises/ez-circular-traversing.xml | 54 ++++++++-------- source/exercises/ez-circular-unit-circle.xml | 15 +++-- source/exercises/ez-exp-e.xml | 11 ++-- source/exercises/ez-exp-growth.xml | 18 +++--- source/exercises/ez-exp-log-properties.xml | 12 ++-- source/exercises/ez-exp-log.xml | 12 ++-- source/exercises/ez-exp-modeling.xml | 15 +++-- source/exercises/ez-exp-temp-pop.xml | 10 ++- source/exercises/ez-poly-infty.xml | 14 ++--- .../ez-poly-polynomial-applications.xml | 9 ++- source/exercises/ez-poly-polynomials.xml | 27 ++++---- .../exercises/ez-poly-rational-features.xml | 15 +++-- source/exercises/ez-poly-rational.xml | 18 +++--- source/exercises/ez-trig-finding-angles.xml | 8 +-- source/exercises/ez-trig-inverse.xml | 16 ++--- source/exercises/ez-trig-other.xml | 8 +-- source/exercises/ez-trig-right.xml | 9 ++- source/exercises/ez-trig-tangent.xml | 14 ++--- source/frontmatter.xml | 29 ++++----- source/main.ptx | 18 +++--- source/previews/PA-0-0.xml | 1 - source/previews/PA-changing-aroc.xml | 3 +- source/previews/PA-changing-combining.xml | 3 +- source/previews/PA-changing-composite.xml | 1 - .../PA-changing-functions-crickets.xml | 1 - source/previews/PA-changing-inverse-F-C.xml | 1 - source/previews/PA-changing-linear-3-ex.xml | 3 +- source/previews/PA-changing-quadratic.xml | 1 - .../previews/PA-changing-tandem-aquarium.xml | 7 ++- .../PA-changing-transformations-quadratic.xml | 1 - source/previews/PA-circular-sine.xml | 3 +- source/previews/PA-circular-sinusoidal.xml | 1 - source/previews/PA-circular-traversing.xml | 1 - source/previews/PA-circular-unit-circle.xml | 5 +- source/previews/PA-exp-e.xml | 1 - source/previews/PA-exp-growth.xml | 3 +- source/previews/PA-exp-log-properties.xml | 1 - source/previews/PA-exp-log.xml | 1 - source/previews/PA-exp-modeling.xml | 1 - source/previews/PA-exp-temp-pop.xml | 1 - source/previews/PA-poly-infty.xml | 1 - .../PA-poly-polynomial-applications.xml | 1 - source/previews/PA-poly-polynomials.xml | 1 - source/previews/PA-poly-rational-features.xml | 1 - source/previews/PA-poly-rational.xml | 1 - source/previews/PA-trig-finding-angles.xml | 1 - source/previews/PA-trig-inverse.xml | 3 +- source/previews/PA-trig-other.xml | 1 - source/previews/PA-trig-right.xml | 1 - source/previews/PA-trig-tangent.xml | 1 - source/sec-0-0.xml | 5 +- source/sec-changing-aroc-wb.xml | 11 ++-- source/sec-changing-aroc.xml | 19 +++--- source/sec-changing-combining-wb.xml | 11 ++-- source/sec-changing-combining.xml | 16 +++-- source/sec-changing-composite-wb.xml | 11 ++-- source/sec-changing-composite.xml | 19 +++--- source/sec-changing-functions-models-wb.xml | 11 ++-- source/sec-changing-functions-models.xml | 22 +++---- source/sec-changing-in-tandem-wb.xml | 9 +-- source/sec-changing-in-tandem.xml | 28 +++++---- source/sec-changing-inverse-wb.xml | 11 ++-- source/sec-changing-inverse.xml | 36 ++++++----- source/sec-changing-linear-wb.xml | 11 ++-- source/sec-changing-linear.xml | 22 +++---- source/sec-changing-quadratic-wb.xml | 11 ++-- source/sec-changing-quadratic.xml | 34 ++++++----- source/sec-changing-transformations-wb.xml | 11 ++-- source/sec-changing-transformations.xml | 61 +++++++++++-------- source/sec-circular-sine-cosine-wb.xml | 11 ++-- source/sec-circular-sine-cosine.xml | 28 +++++---- source/sec-circular-sinusoidal-wb.xml | 13 ++-- source/sec-circular-sinusoidal.xml | 32 +++++----- source/sec-circular-traversing-wb.xml | 11 ++-- source/sec-circular-traversing.xml | 35 ++++++----- source/sec-circular-unit-circle-wb.xml | 11 ++-- source/sec-circular-unit-circle.xml | 22 +++---- source/sec-exp-e-wb.xml | 9 +-- source/sec-exp-e.xml | 46 +++++++------- source/sec-exp-growth-wb.xml | 11 ++-- source/sec-exp-growth.xml | 24 ++++---- source/sec-exp-log-properties-wb.xml | 11 ++-- source/sec-exp-log-properties.xml | 19 +++--- source/sec-exp-log-wb.xml | 11 ++-- source/sec-exp-log.xml | 13 ++-- source/sec-exp-modeling-wb.xml | 11 ++-- source/sec-exp-modeling.xml | 45 +++++++------- source/sec-exp-temp-pop-wb.xml | 11 ++-- source/sec-exp-temp-pop.xml | 13 ++-- source/sec-poly-infty-wb.xml | 11 ++-- source/sec-poly-infty.xml | 19 +++--- .../sec-poly-polynomial-applications-wb.xml | 11 ++-- source/sec-poly-polynomial-applications.xml | 25 ++++---- source/sec-poly-polynomials-wb.xml | 12 ++-- source/sec-poly-polynomials.xml | 48 ++++++++------- source/sec-poly-rational-features-wb.xml | 9 +-- source/sec-poly-rational-features.xml | 34 ++++++----- source/sec-poly-rational-wb.xml | 13 ++-- source/sec-poly-rational.xml | 15 ++--- source/sec-trig-finding-angles-wb.xml | 13 ++-- source/sec-trig-finding-angles.xml | 21 +++---- source/sec-trig-inverse-wb.xml | 11 ++-- source/sec-trig-inverse.xml | 28 ++++----- source/sec-trig-other-wb.xml | 13 ++-- source/sec-trig-other.xml | 26 ++++---- source/sec-trig-right-wb.xml | 11 ++-- source/sec-trig-right.xml | 30 ++++----- source/sec-trig-tangent-wb.xml | 11 ++-- source/sec-trig-tangent.xml | 28 +++++---- source/titlepage.xml | 4 +- 233 files changed, 1019 insertions(+), 1266 deletions(-) diff --git a/.github/workflows/pretext-cli.yml b/.github/workflows/pretext-cli.yml index e9a1feab..8713a695 100644 --- a/.github/workflows/pretext-cli.yml +++ b/.github/workflows/pretext-cli.yml @@ -1,4 +1,4 @@ -# This file was automatically generated with PreTeXt 2.28.2. +# This file was automatically generated with PreTeXt 2.33.0. # If you modify this file, PreTeXt will no longer automatically update it. # # This workflow file can be used to automatically build a project and create diff --git a/.github/workflows/pretext-deploy.yml b/.github/workflows/pretext-deploy.yml index aa794d2c..39615ff2 100644 --- a/.github/workflows/pretext-deploy.yml +++ b/.github/workflows/pretext-deploy.yml @@ -1,4 +1,4 @@ -# This file was automatically generated with PreTeXt 2.28.2. +# This file was automatically generated with PreTeXt 2.33.0. # If you modify this file, PreTeXt will no longer automatically update it. # diff --git a/requirements.txt b/requirements.txt index 23d0b63b..a85f8b33 100644 --- a/requirements.txt +++ b/requirements.txt @@ -1,2 +1,2 @@ -# This file was automatically generated with PreTeXt 2.28.2. -pretextbook == 2.28.2 +# This file was automatically generated with PreTeXt 2.33.0. +pretext == 2.33.0 diff --git a/source/acs-solution-manual.ptx b/source/acs-solution-manual.ptx index 21f6c09b..5650fb3d 100644 --- a/source/acs-solution-manual.ptx +++ b/source/acs-solution-manual.ptx @@ -1,4 +1,3 @@ - @@ -23,8 +22,8 @@ % This macro should only be used in ancillary production! \raggedbottom - - + + Preview Activity Motivating Questions @@ -36,7 +35,7 @@ - + Matthew Boelkins @@ -73,10 +72,10 @@ - + - +

@@ -87,7 +86,7 @@ so instructors can easily extract the solution for a single activity if they wish to print it for students or post on their course's learning management system site. After the activity solutions come the exercise solutions. Solutions for - exercises are not provided, + exercises are not provided, since those solutions are automatically available in this book's HTML version. Each exercise also starts at the top of its own page to make it easy to extract only the solutions to certain exercises.

@@ -107,19 +106,19 @@ - - - - - - - - + + + + + + + + - +

- This book was authored in . + This book was authored in .

@@ -127,4 +126,3 @@ - diff --git a/source/activities/act-0-0-0.xml b/source/activities/act-0-0-0.xml index bc459d30..9f6ba3c7 100755 --- a/source/activities/act-0-0-0.xml +++ b/source/activities/act-0-0-0.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-aroc-graphs.xml b/source/activities/act-changing-aroc-graphs.xml index 3a114823..cc8c872d 100755 --- a/source/activities/act-changing-aroc-graphs.xml +++ b/source/activities/act-changing-aroc-graphs.xml @@ -1,4 +1,3 @@ - @@ -25,8 +24,8 @@

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

- - +

ADD ALT TEXT TO THIS IMAGE

 +

ADD ALT TEXT TO THIS IMAGE

 @@ -38,8 +37,8 @@

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).

- - +

ADD ALT TEXT TO THIS IMAGE

 +

ADD ALT TEXT TO THIS IMAGE

 @@ -51,8 +50,8 @@

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

- - +

ADD ALT TEXT TO THIS IMAGE

 +

ADD ALT TEXT TO THIS IMAGE

 diff --git a/source/activities/act-changing-aroc-population.xml b/source/activities/act-changing-aroc-population.xml index e64dcd72..0b76d780 100755 --- a/source/activities/act-changing-aroc-population.xml +++ b/source/activities/act-changing-aroc-population.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-aroc-trends.xml b/source/activities/act-changing-aroc-trends.xml index b6c0638e..2a546740 100755 --- a/source/activities/act-changing-aroc-trends.xml +++ b/source/activities/act-changing-aroc-trends.xml @@ -1,4 +1,3 @@ - @@ -54,11 +53,11 @@ - +

ADD ALT TEXT TO THIS IMAGE

 - +

ADD ALT TEXT TO THIS IMAGE

 diff --git a/source/activities/act-changing-combining-arithmetic.xml b/source/activities/act-changing-combining-arithmetic.xml index 836124b6..fc4829ff 100755 --- a/source/activities/act-changing-combining-arithmetic.xml +++ b/source/activities/act-changing-combining-arithmetic.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,11 @@ - +

ADD ALT TEXT TO THIS IMAGE

 - +

ADD ALT TEXT TO THIS IMAGE

diff --git a/source/activities/act-changing-combining-context.xml b/source/activities/act-changing-combining-context.xml index 9d74b04c..f6370f69 100755 --- a/source/activities/act-changing-combining-context.xml +++ b/source/activities/act-changing-combining-context.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-combining-piecewise.xml b/source/activities/act-changing-combining-piecewise.xml index 1c93b637..6933598b 100755 --- a/source/activities/act-changing-combining-piecewise.xml +++ b/source/activities/act-changing-combining-piecewise.xml @@ -1,4 +1,3 @@ - @@ -68,11 +67,11 @@ - +

ADD ALT TEXT TO THIS IMAGE

- +

ADD ALT TEXT TO THIS IMAGE

 diff --git a/source/activities/act-changing-composite-aroc.xml b/source/activities/act-changing-composite-aroc.xml index c872a578..ac60b27f 100755 --- a/source/activities/act-changing-composite-aroc.xml +++ b/source/activities/act-changing-composite-aroc.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-composite-crickets-celsius.xml b/source/activities/act-changing-composite-crickets-celsius.xml index 46b895a6..ca7ada63 100755 --- a/source/activities/act-changing-composite-crickets-celsius.xml +++ b/source/activities/act-changing-composite-crickets-celsius.xml @@ -1,4 +1,3 @@ - @@ -48,11 +47,11 @@ - +

ADD ALT TEXT TO THIS IMAGE

 - +

ADD ALT TEXT TO THIS IMAGE

 diff --git a/source/activities/act-changing-composite-tables-graphs.xml b/source/activities/act-changing-composite-tables-graphs.xml index 3f39e02a..ecbddb72 100755 --- a/source/activities/act-changing-composite-tables-graphs.xml +++ b/source/activities/act-changing-composite-tables-graphs.xml @@ -1,4 +1,3 @@ - @@ -56,7 +55,7 @@ - +

ADD ALT TEXT TO THIS IMAGE

diff --git a/source/activities/act-changing-functions-is-it.xml b/source/activities/act-changing-functions-is-it.xml index 539a22f0..7b4ba376 100755 --- a/source/activities/act-changing-functions-is-it.xml +++ b/source/activities/act-changing-functions-is-it.xml @@ -1,4 +1,3 @@ - @@ -28,11 +27,11 @@ - +

ADD ALT TEXT TO THIS IMAGE

- +

ADD ALT TEXT TO THIS IMAGE

 diff --git a/source/activities/act-changing-functions-spherical-tank-draining.xml b/source/activities/act-changing-functions-spherical-tank-draining.xml index 36fd1331..e3b7e09b 100755 --- a/source/activities/act-changing-functions-spherical-tank-draining.xml +++ b/source/activities/act-changing-functions-spherical-tank-draining.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-functions-spherical-tank.xml b/source/activities/act-changing-functions-spherical-tank.xml index 73a53ab1..f32d3374 100755 --- a/source/activities/act-changing-functions-spherical-tank.xml +++ b/source/activities/act-changing-functions-spherical-tank.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-inverse-Dolbear.xml b/source/activities/act-changing-inverse-Dolbear.xml index e6e14ba1..65c5fe6e 100755 --- a/source/activities/act-changing-inverse-Dolbear.xml +++ b/source/activities/act-changing-inverse-Dolbear.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-inverse-does-it.xml b/source/activities/act-changing-inverse-does-it.xml index 643a1ee9..fd637c8c 100755 --- a/source/activities/act-changing-inverse-does-it.xml +++ b/source/activities/act-changing-inverse-does-it.xml @@ -1,4 +1,3 @@ - @@ -118,11 +117,11 @@ - +

ADD ALT TEXT TO THIS IMAGE

 - +

ADD ALT TEXT TO THIS IMAGE

 diff --git a/source/activities/act-changing-inverse-rainfall.xml b/source/activities/act-changing-inverse-rainfall.xml index 54303b3f..e9a1cb62 100755 --- a/source/activities/act-changing-inverse-rainfall.xml +++ b/source/activities/act-changing-inverse-rainfall.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-linear-Kilimanjaro.xml b/source/activities/act-changing-linear-Kilimanjaro.xml index e5635ac4..ebb32d6f 100755 --- a/source/activities/act-changing-linear-Kilimanjaro.xml +++ b/source/activities/act-changing-linear-Kilimanjaro.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-linear-finding-eqs.xml b/source/activities/act-changing-linear-finding-eqs.xml index bdf3b24e..75ad7cd6 100755 --- a/source/activities/act-changing-linear-finding-eqs.xml +++ b/source/activities/act-changing-linear-finding-eqs.xml @@ -1,4 +1,3 @@ - @@ -109,7 +108,7 @@

- +

ADD ALT TEXT TO THIS IMAGE

 diff --git a/source/activities/act-changing-linear-in-context.xml b/source/activities/act-changing-linear-in-context.xml index 1f5b7c26..f0466841 100755 --- a/source/activities/act-changing-linear-in-context.xml +++ b/source/activities/act-changing-linear-in-context.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-quadratic-falling-ball.xml b/source/activities/act-changing-quadratic-falling-ball.xml index f418cc07..f10d09cf 100755 --- a/source/activities/act-changing-quadratic-falling-ball.xml +++ b/source/activities/act-changing-quadratic-falling-ball.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-quadratic-parameters.xml b/source/activities/act-changing-quadratic-parameters.xml index cdebe58f..2aa55eed 100755 --- a/source/activities/act-changing-quadratic-parameters.xml +++ b/source/activities/act-changing-quadratic-parameters.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-quadratic-properties.xml b/source/activities/act-changing-quadratic-properties.xml index 2ab8d185..11df9e0b 100755 --- a/source/activities/act-changing-quadratic-properties.xml +++ b/source/activities/act-changing-quadratic-properties.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-changing-tandem-conical-tank.xml b/source/activities/act-changing-tandem-conical-tank.xml index 48e23301..840f7034 100755 --- a/source/activities/act-changing-tandem-conical-tank.xml +++ b/source/activities/act-changing-tandem-conical-tank.xml @@ -1,4 +1,3 @@ - @@ -111,7 +110,7 @@ - +

ADD ALT TEXT TO THIS IMAGE

 @@ -125,8 +124,8 @@ 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.

- - +

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 +

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 diff --git a/source/activities/act-changing-tandem-spherical-tank.xml b/source/activities/act-changing-tandem-spherical-tank.xml index 3f60c3ce..4baa5189 100755 --- a/source/activities/act-changing-tandem-spherical-tank.xml +++ b/source/activities/act-changing-tandem-spherical-tank.xml @@ -1,4 +1,3 @@ - @@ -117,7 +116,7 @@ - +

ADD ALT TEXT TO THIS IMAGE

 @@ -131,8 +130,8 @@ 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.

- - +

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 +

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 diff --git a/source/activities/act-changing-transformations-combined.xml b/source/activities/act-changing-transformations-combined.xml index 048bb3c9..f82caf17 100755 --- a/source/activities/act-changing-transformations-combined.xml +++ b/source/activities/act-changing-transformations-combined.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,11 @@ - +

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@@ -66,7 +65,7 @@ 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.

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 diff --git a/source/activities/act-changing-transformations-translations.xml b/source/activities/act-changing-transformations-translations.xml index 93698331..a8af2e3b 100755 --- a/source/activities/act-changing-transformations-translations.xml +++ b/source/activities/act-changing-transformations-translations.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,11 @@ - +

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diff --git a/source/activities/act-changing-transformations-vert-stretch.xml b/source/activities/act-changing-transformations-vert-stretch.xml index 45ddd800..d6f88d94 100755 --- a/source/activities/act-changing-transformations-vert-stretch.xml +++ b/source/activities/act-changing-transformations-vert-stretch.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,11 @@ - +

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@@ -56,8 +55,8 @@ 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.

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 diff --git a/source/activities/act-circular-sine-cosine-computing.xml b/source/activities/act-circular-sine-cosine-computing.xml index 805644a5..301ce1a1 100755 --- a/source/activities/act-circular-sine-cosine-computing.xml +++ b/source/activities/act-circular-sine-cosine-computing.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-circular-sine-cosine-incr-CCU.xml b/source/activities/act-circular-sine-cosine-incr-CCU.xml index 7721973b..9adce395 100755 --- a/source/activities/act-circular-sine-cosine-incr-CCU.xml +++ b/source/activities/act-circular-sine-cosine-incr-CCU.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-circular-sine-cosine.xml b/source/activities/act-circular-sine-cosine.xml index fc099959..f394cd66 100755 --- a/source/activities/act-circular-sine-cosine.xml +++ b/source/activities/act-circular-sine-cosine.xml @@ -1,4 +1,3 @@ - @@ -147,7 +146,7 @@

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 diff --git a/source/activities/act-circular-sinusoidal-horiz-stretch.xml b/source/activities/act-circular-sinusoidal-horiz-stretch.xml index 07e4303f..9ce430c8 100755 --- a/source/activities/act-circular-sinusoidal-horiz-stretch.xml +++ b/source/activities/act-circular-sinusoidal-horiz-stretch.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,11 @@ - +

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@@ -56,8 +55,8 @@ 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.

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 diff --git a/source/activities/act-circular-sinusoidal-model.xml b/source/activities/act-circular-sinusoidal-model.xml index 2b40df48..3ba927ef 100755 --- a/source/activities/act-circular-sinusoidal-model.xml +++ b/source/activities/act-circular-sinusoidal-model.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-circular-sinusoidal-oscillator.xml b/source/activities/act-circular-sinusoidal-oscillator.xml index 7792b318..d4ae90a6 100755 --- a/source/activities/act-circular-sinusoidal-oscillator.xml +++ b/source/activities/act-circular-sinusoidal-oscillator.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-circular-sinusoidal-period.xml b/source/activities/act-circular-sinusoidal-period.xml index 3e9aa84d..b5ed366a 100755 --- a/source/activities/act-circular-sinusoidal-period.xml +++ b/source/activities/act-circular-sinusoidal-period.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-circular-traversing-2nd-ex.xml b/source/activities/act-circular-traversing-2nd-ex.xml index 4b496524..0b6244b0 100755 --- a/source/activities/act-circular-traversing-2nd-ex.xml +++ b/source/activities/act-circular-traversing-2nd-ex.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,11 @@ - +

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diff --git a/source/activities/act-circular-traversing-oscillator-aroc.xml b/source/activities/act-circular-traversing-oscillator-aroc.xml index 93345253..43ab841b 100755 --- a/source/activities/act-circular-traversing-oscillator-aroc.xml +++ b/source/activities/act-circular-traversing-oscillator-aroc.xml @@ -1,4 +1,3 @@ - @@ -168,7 +167,7 @@ - +

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diff --git a/source/activities/act-circular-traversing-oscillator.xml b/source/activities/act-circular-traversing-oscillator.xml index 78da1b7e..8b02c781 100755 --- a/source/activities/act-circular-traversing-oscillator.xml +++ b/source/activities/act-circular-traversing-oscillator.xml @@ -1,4 +1,3 @@ - @@ -170,7 +169,7 @@ - +

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diff --git a/source/activities/act-circular-unit-circle-non-unit.xml b/source/activities/act-circular-unit-circle-non-unit.xml index d8523c9f..e60a79ee 100755 --- a/source/activities/act-circular-unit-circle-non-unit.xml +++ b/source/activities/act-circular-unit-circle-non-unit.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-circular-unit-circle-radians-degrees.xml b/source/activities/act-circular-unit-circle-radians-degrees.xml index 3010b879..e0a8565e 100755 --- a/source/activities/act-circular-unit-circle-radians-degrees.xml +++ b/source/activities/act-circular-unit-circle-radians-degrees.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-circular-unit-circle-special-triangles.xml b/source/activities/act-circular-unit-circle-special-triangles.xml index 9edda318..29b0ac94 100755 --- a/source/activities/act-circular-unit-circle-special-triangles.xml +++ b/source/activities/act-circular-unit-circle-special-triangles.xml @@ -1,4 +1,3 @@ - @@ -23,11 +22,11 @@ - +

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@@ -70,15 +69,15 @@ - +

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diff --git a/source/activities/act-exp-e-aroc-e.xml b/source/activities/act-exp-e-aroc-e.xml index 2b19c774..cfe8ee05 100755 --- a/source/activities/act-exp-e-aroc-e.xml +++ b/source/activities/act-exp-e-aroc-e.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-e-graph-solve.xml b/source/activities/act-exp-e-graph-solve.xml index 1fcd4736..ab0d0174 100755 --- a/source/activities/act-exp-e-graph-solve.xml +++ b/source/activities/act-exp-e-graph-solve.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-growth-a-b-t.xml b/source/activities/act-exp-growth-a-b-t.xml index 38e4979f..54c55a17 100755 --- a/source/activities/act-exp-growth-a-b-t.xml +++ b/source/activities/act-exp-growth-a-b-t.xml @@ -1,4 +1,3 @@ - @@ -77,7 +76,7 @@

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 diff --git a/source/activities/act-exp-growth-find-a-b.xml b/source/activities/act-exp-growth-find-a-b.xml index 8d50b862..4be850e7 100755 --- a/source/activities/act-exp-growth-find-a-b.xml +++ b/source/activities/act-exp-growth-find-a-b.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-growth-rates.xml b/source/activities/act-exp-growth-rates.xml index c6188a53..781d623c 100755 --- a/source/activities/act-exp-growth-rates.xml +++ b/source/activities/act-exp-growth-rates.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-log-base-10.xml b/source/activities/act-exp-log-base-10.xml index 821e6349..216b7c5a 100755 --- a/source/activities/act-exp-log-base-10.xml +++ b/source/activities/act-exp-log-base-10.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-log-equations.xml b/source/activities/act-exp-log-equations.xml index b568100a..bd907701 100755 --- a/source/activities/act-exp-log-equations.xml +++ b/source/activities/act-exp-log-equations.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-log-exponential-equations.xml b/source/activities/act-exp-log-exponential-equations.xml index f96a11c1..ee1c454f 100755 --- a/source/activities/act-exp-log-exponential-equations.xml +++ b/source/activities/act-exp-log-exponential-equations.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-log-natural.xml b/source/activities/act-exp-log-natural.xml index 8773eb36..83ccb9b0 100755 --- a/source/activities/act-exp-log-natural.xml +++ b/source/activities/act-exp-log-natural.xml @@ -1,4 +1,3 @@ - @@ -143,7 +142,7 @@

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 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 1b2ef3ea..8b0dd7ea 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 +1,3 @@ - diff --git a/source/activities/act-exp-log-properties-find-k.xml b/source/activities/act-exp-log-properties-find-k.xml index 3503a747..3994098f 100755 --- a/source/activities/act-exp-log-properties-find-k.xml +++ b/source/activities/act-exp-log-properties-find-k.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-modeling-behavior.xml b/source/activities/act-exp-modeling-behavior.xml index c0e777bc..8c09b602 100755 --- a/source/activities/act-exp-modeling-behavior.xml +++ b/source/activities/act-exp-modeling-behavior.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-modeling-potato.xml b/source/activities/act-exp-modeling-potato.xml index 629daf28..a2beb55a 100755 --- a/source/activities/act-exp-modeling-potato.xml +++ b/source/activities/act-exp-modeling-potato.xml @@ -1,4 +1,3 @@ - @@ -66,7 +65,7 @@

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 diff --git a/source/activities/act-exp-modeling-soda.xml b/source/activities/act-exp-modeling-soda.xml index 3e73bbb7..f057d5d4 100755 --- a/source/activities/act-exp-modeling-soda.xml +++ b/source/activities/act-exp-modeling-soda.xml @@ -1,4 +1,3 @@ - @@ -59,7 +58,7 @@

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 diff --git a/source/activities/act-exp-temp-pop-NLOC1.xml b/source/activities/act-exp-temp-pop-NLOC1.xml index 8901c4ac..9455a153 100755 --- a/source/activities/act-exp-temp-pop-NLOC1.xml +++ b/source/activities/act-exp-temp-pop-NLOC1.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-exp-temp-pop-logistic-Desmos.xml b/source/activities/act-exp-temp-pop-logistic-Desmos.xml index 629bf4b7..83a0a09d 100755 --- a/source/activities/act-exp-temp-pop-logistic-Desmos.xml +++ b/source/activities/act-exp-temp-pop-logistic-Desmos.xml @@ -1,4 +1,3 @@ - @@ -27,7 +26,7 @@

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 diff --git a/source/activities/act-exp-temp-pop-logistic-exact.xml b/source/activities/act-exp-temp-pop-logistic-exact.xml index 956c5a0d..7b542dab 100755 --- a/source/activities/act-exp-temp-pop-logistic-exact.xml +++ b/source/activities/act-exp-temp-pop-logistic-exact.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-infty-limit.xml b/source/activities/act-poly-infty-limit.xml index 51ff08b3..8c36c68a 100755 --- a/source/activities/act-poly-infty-limit.xml +++ b/source/activities/act-poly-infty-limit.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-infty-natural-powers.xml b/source/activities/act-poly-infty-natural-powers.xml index b825b2cb..56dc9c29 100755 --- a/source/activities/act-poly-infty-natural-powers.xml +++ b/source/activities/act-poly-infty-natural-powers.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-infty-negative-powers.xml b/source/activities/act-poly-infty-negative-powers.xml index f10fdb72..139a3e37 100755 --- a/source/activities/act-poly-infty-negative-powers.xml +++ b/source/activities/act-poly-infty-negative-powers.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-polynomial-applications-Taylor.xml b/source/activities/act-poly-polynomial-applications-Taylor.xml index 39c7b01b..6c9b7d00 100755 --- a/source/activities/act-poly-polynomial-applications-Taylor.xml +++ b/source/activities/act-poly-polynomial-applications-Taylor.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-polynomial-applications-postal.xml b/source/activities/act-poly-polynomial-applications-postal.xml index c51875be..80d2c3ab 100755 --- a/source/activities/act-poly-polynomial-applications-postal.xml +++ b/source/activities/act-poly-polynomial-applications-postal.xml @@ -1,4 +1,3 @@ - @@ -20,7 +19,7 @@

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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. diff --git a/source/activities/act-poly-polynomial-applications-soup.xml b/source/activities/act-poly-polynomial-applications-soup.xml index e4af14ad..4f40268b 100755 --- a/source/activities/act-poly-polynomial-applications-soup.xml +++ b/source/activities/act-poly-polynomial-applications-soup.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-polynomials-find.xml b/source/activities/act-poly-polynomials-find.xml index 4d3bd65e..1f34489a 100755 --- a/source/activities/act-poly-polynomials-find.xml +++ b/source/activities/act-poly-polynomials-find.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-polynomials-multiple-zeros.xml b/source/activities/act-poly-polynomials-multiple-zeros.xml index e108f064..e1dbd464 100755 --- a/source/activities/act-poly-polynomials-multiple-zeros.xml +++ b/source/activities/act-poly-polynomials-multiple-zeros.xml @@ -1,4 +1,3 @@ - @@ -36,7 +35,7 @@ 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.

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 @@ -49,7 +48,7 @@ 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.

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 diff --git a/source/activities/act-poly-polynomials-sign-chart.xml b/source/activities/act-poly-polynomials-sign-chart.xml index 57f4f721..f4880773 100755 --- a/source/activities/act-poly-polynomials-sign-chart.xml +++ b/source/activities/act-poly-polynomials-sign-chart.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-rational-application.xml b/source/activities/act-poly-rational-application.xml index fb3d9c41..72e58458 100755 --- a/source/activities/act-poly-rational-application.xml +++ b/source/activities/act-poly-rational-application.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-rational-domain.xml b/source/activities/act-poly-rational-domain.xml index 55d68f81..7220cd35 100755 --- a/source/activities/act-poly-rational-domain.xml +++ b/source/activities/act-poly-rational-domain.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-rational-features-ZAH.xml b/source/activities/act-poly-rational-features-ZAH.xml index ac380887..989d0a4e 100755 --- a/source/activities/act-poly-rational-features-ZAH.xml +++ b/source/activities/act-poly-rational-features-ZAH.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-rational-formula.xml b/source/activities/act-poly-rational-formula.xml index a6b0d501..4dda2143 100755 --- a/source/activities/act-poly-rational-formula.xml +++ b/source/activities/act-poly-rational-formula.xml @@ -1,4 +1,3 @@ - @@ -46,7 +45,7 @@ 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.

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 @@ -58,7 +57,7 @@ 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.

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 diff --git a/source/activities/act-poly-rational-long-term-1.xml b/source/activities/act-poly-rational-long-term-1.xml index ff52c2ca..d8a763a9 100755 --- a/source/activities/act-poly-rational-long-term-1.xml +++ b/source/activities/act-poly-rational-long-term-1.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-poly-rational-long-term-2.xml b/source/activities/act-poly-rational-long-term-2.xml index 0989bc71..3d203fec 100755 --- a/source/activities/act-poly-rational-long-term-2.xml +++ b/source/activities/act-poly-rational-long-term-2.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-finding-angles-baseball.xml b/source/activities/act-trig-finding-angles-baseball.xml index 91f98a63..f688e09b 100755 --- a/source/activities/act-trig-finding-angles-baseball.xml +++ b/source/activities/act-trig-finding-angles-baseball.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-finding-angles-exactly.xml b/source/activities/act-trig-finding-angles-exactly.xml index 04372cb8..45ecace2 100755 --- a/source/activities/act-trig-finding-angles-exactly.xml +++ b/source/activities/act-trig-finding-angles-exactly.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-finding-angles-rocket.xml b/source/activities/act-trig-finding-angles-rocket.xml index 1c013514..216006f1 100755 --- a/source/activities/act-trig-finding-angles-rocket.xml +++ b/source/activities/act-trig-finding-angles-rocket.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-finding-angles-roof.xml b/source/activities/act-trig-finding-angles-roof.xml index 1387fdbf..906682fb 100755 --- a/source/activities/act-trig-finding-angles-roof.xml +++ b/source/activities/act-trig-finding-angles-roof.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-inverse-arccos.xml b/source/activities/act-trig-inverse-arccos.xml index 48786da5..65b66066 100755 --- a/source/activities/act-trig-inverse-arccos.xml +++ b/source/activities/act-trig-inverse-arccos.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-inverse-arcsin.xml b/source/activities/act-trig-inverse-arcsin.xml index b9bf4298..a0db87e0 100755 --- a/source/activities/act-trig-inverse-arcsin.xml +++ b/source/activities/act-trig-inverse-arcsin.xml @@ -1,4 +1,3 @@ - @@ -47,7 +46,7 @@

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 diff --git a/source/activities/act-trig-inverse-arctan.xml b/source/activities/act-trig-inverse-arctan.xml index acd4724f..13f4b05d 100755 --- a/source/activities/act-trig-inverse-arctan.xml +++ b/source/activities/act-trig-inverse-arctan.xml @@ -1,4 +1,3 @@ - @@ -47,7 +46,7 @@

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 diff --git a/source/activities/act-trig-other-aroc-sine.xml b/source/activities/act-trig-other-aroc-sine.xml index cb2378c9..fd09c6ae 100755 --- a/source/activities/act-trig-other-aroc-sine.xml +++ b/source/activities/act-trig-other-aroc-sine.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-other-cot.xml b/source/activities/act-trig-other-cot.xml index 377949ea..5d7f512d 100755 --- a/source/activities/act-trig-other-cot.xml +++ b/source/activities/act-trig-other-cot.xml @@ -1,4 +1,3 @@ - @@ -344,7 +343,7 @@

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 diff --git a/source/activities/act-trig-other-csc.xml b/source/activities/act-trig-other-csc.xml index d6f6d818..d75646a5 100755 --- a/source/activities/act-trig-other-csc.xml +++ b/source/activities/act-trig-other-csc.xml @@ -1,4 +1,3 @@ - @@ -20,7 +19,7 @@

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diff --git a/source/activities/act-trig-other-special.xml b/source/activities/act-trig-other-special.xml index 93d27f79..b0387044 100755 --- a/source/activities/act-trig-other-special.xml +++ b/source/activities/act-trig-other-special.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-right-SOH-CAH.xml b/source/activities/act-trig-right-SOH-CAH.xml index 8c584374..db263fa2 100755 --- a/source/activities/act-trig-right-SOH-CAH.xml +++ b/source/activities/act-trig-right-SOH-CAH.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-right-similar.xml b/source/activities/act-trig-right-similar.xml index 45ed00d3..fba4acd2 100755 --- a/source/activities/act-trig-right-similar.xml +++ b/source/activities/act-trig-right-similar.xml @@ -1,4 +1,3 @@ - @@ -20,7 +19,7 @@

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diff --git a/source/activities/act-trig-right-sine-cosine-ratio.xml b/source/activities/act-trig-right-sine-cosine-ratio.xml index dc9265e5..995db56f 100755 --- a/source/activities/act-trig-right-sine-cosine-ratio.xml +++ b/source/activities/act-trig-right-sine-cosine-ratio.xml @@ -1,4 +1,3 @@ - @@ -24,7 +23,7 @@

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 diff --git a/source/activities/act-trig-tangent-mountain.xml b/source/activities/act-trig-tangent-mountain.xml index 3960b40b..2f2d31b0 100755 --- a/source/activities/act-trig-tangent-mountain.xml +++ b/source/activities/act-trig-tangent-mountain.xml @@ -1,4 +1,3 @@ - @@ -20,7 +19,7 @@

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diff --git a/source/activities/act-trig-tangent-tower.xml b/source/activities/act-trig-tangent-tower.xml index 0ea0c0b4..1e0d50dd 100755 --- a/source/activities/act-trig-tangent-tower.xml +++ b/source/activities/act-trig-tangent-tower.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/act-trig-tangent-two-towers.xml b/source/activities/act-trig-tangent-two-towers.xml index 511dded8..f478bd96 100755 --- a/source/activities/act-trig-tangent-two-towers.xml +++ b/source/activities/act-trig-tangent-two-towers.xml @@ -1,4 +1,3 @@ - @@ -21,7 +20,7 @@

- +

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 diff --git a/source/activities/z-act-inverse-rainfall.xml b/source/activities/z-act-inverse-rainfall.xml index f0bea91a..47d5a590 100644 --- a/source/activities/z-act-inverse-rainfall.xml +++ b/source/activities/z-act-inverse-rainfall.xml @@ -1,4 +1,3 @@ - diff --git a/source/activities/z-act-transformations-translations.xml b/source/activities/z-act-transformations-translations.xml index 13085ce9..d70f14c6 100644 --- a/source/activities/z-act-transformations-translations.xml +++ b/source/activities/z-act-transformations-translations.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,13 @@
A parent function r. - +

A parent function r.

+
A parent function s. - +

A parent function s.

+

diff --git a/source/apc-activity-workbook.ptx b/source/apc-activity-workbook.ptx index 46d8525d..918134da 100644 --- a/source/apc-activity-workbook.ptx +++ b/source/apc-activity-workbook.ptx @@ -1,4 +1,3 @@ - @@ -14,7 +13,7 @@ - + - - + + APC @@ -37,9 +36,9 @@ Motivating Questions - ㅤ + ㅤ - + @@ -49,8 +48,8 @@ Active Prelude to Calculus - - + +

@@ -66,21 +65,20 @@ - - - - - + + + + + - + Back Matter

- This book was authored in . + This book was authored in .

- diff --git a/source/apc-solution-manual.ptx b/source/apc-solution-manual.ptx index a7453856..1af79c06 100644 --- a/source/apc-solution-manual.ptx +++ b/source/apc-solution-manual.ptx @@ -1,4 +1,3 @@ - @@ -15,7 +14,7 @@ - + @@ -27,7 +26,7 @@ - + Matthew Boelkins @@ -46,10 +45,10 @@ - + - +

@@ -60,18 +59,18 @@ so instructors can easily extract the solution for a single activity if they wish to print it for students or post on their course's learning management system site. After the activity solutions come the exercise solutions. Solutions for - exercises are not provided, + exercises are not provided, since those solutions are automatically available in this book's HTML version. Each exercise also starts at the top of its own page to make it easy to extract only the solutions to certain exercises.

-

+

Please do not post this solutions manual publicly on the internet nor in any electronic form where it is available in full to students. As much as possible, we aspire to keep this solutions manual as a resource for instructors only so that students get the full benefit of activities and exercises by having to struggle with them without looking at solutions.

-

+

By extracting individual pages from the PDF, it is fine to share solutions to individual activities or exercises via course management system software. Please do not post these individual pages publicly on the internet. @@ -79,12 +78,11 @@ - - - - - + + + + + - diff --git a/source/backmatter.xml b/source/backmatter.xml index 62b47648..e61d358d 100644 --- a/source/backmatter.xml +++ b/source/backmatter.xml @@ -1,4 +1,3 @@ - @@ -12,8 +11,7 @@ - - + Back Matter @@ -62,13 +60,12 @@ Index - +

- This book was authored in . + This book was authored in .

- diff --git a/source/bibinfo.xml b/source/bibinfo.xml index daf679cc..a2391548 100644 --- a/source/bibinfo.xml +++ b/source/bibinfo.xml @@ -1,4 +1,3 @@ - @@ -12,8 +11,7 @@ - - + Matthew Boelkins Department of Mathematics @@ -31,7 +29,7 @@ - + Cover Photo @@ -41,7 +39,7 @@ 2019 - + @@ -50,9 +48,9 @@ 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 + 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 - +

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 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 c6f90f6a..5df203bc 100644 --- a/source/bookinfo.xml +++ b/source/bookinfo.xml @@ -1,4 +1,3 @@ - @@ -12,9 +11,8 @@ - - + - + APC -   +   - diff --git a/source/chap-changing-wb.xml b/source/chap-changing-wb.xml index 11449219..42b788d9 100644 --- a/source/chap-changing-wb.xml +++ b/source/chap-changing-wb.xml @@ -1,4 +1,3 @@ - @@ -12,19 +11,17 @@ - - + Relating Changing Quantities - - - - - - - - - + + + + + + + + + - diff --git a/source/chap-changing.xml b/source/chap-changing.xml index ad458af8..1f97cae6 100755 --- a/source/chap-changing.xml +++ b/source/chap-changing.xml @@ -1,4 +1,3 @@ - @@ -12,19 +11,17 @@ - - + Relating Changing Quantities - - - - - - - - - + + + + + + + + + - diff --git a/source/chap-circular-wb.xml b/source/chap-circular-wb.xml index a38b5121..c4e3a872 100644 --- a/source/chap-circular-wb.xml +++ b/source/chap-circular-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ - - + Circular Functions - - - - + + + + - diff --git a/source/chap-circular.xml b/source/chap-circular.xml index fcf21bd3..395b5689 100755 --- a/source/chap-circular.xml +++ b/source/chap-circular.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ - - + Circular Functions - - - - + + + + - diff --git a/source/chap-exp-wb.xml b/source/chap-exp-wb.xml index c72879e3..c1f5130f 100644 --- a/source/chap-exp-wb.xml +++ b/source/chap-exp-wb.xml @@ -1,4 +1,3 @@ - @@ -12,16 +11,14 @@ - - + Exponential and Logarithmic Functions - - - - - - + + + + + + - diff --git a/source/chap-exp.xml b/source/chap-exp.xml index 9d5ef9eb..e73052c3 100755 --- a/source/chap-exp.xml +++ b/source/chap-exp.xml @@ -1,4 +1,3 @@ - @@ -12,16 +11,14 @@ - - + Exponential and Logarithmic Functions - - - - - - + + + + + + - diff --git a/source/chap-poly-wb.xml b/source/chap-poly-wb.xml index 5190e7f2..4f429701 100644 --- a/source/chap-poly-wb.xml +++ b/source/chap-poly-wb.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ - - + Polynomial and Rational Functions - - - - - + + + + + - diff --git a/source/chap-poly.xml b/source/chap-poly.xml index 8bfb1bcc..659499b3 100755 --- a/source/chap-poly.xml +++ b/source/chap-poly.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ - - + Polynomial and Rational Functions - - - - - + + + + + - diff --git a/source/chap-trig-wb.xml b/source/chap-trig-wb.xml index 08e8ecaf..05c29d99 100644 --- a/source/chap-trig-wb.xml +++ b/source/chap-trig-wb.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ - - + Trigonometry - - - - - + + + + + - diff --git a/source/chap-trig.xml b/source/chap-trig.xml index 197c673f..76a3eaf3 100755 --- a/source/chap-trig.xml +++ b/source/chap-trig.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ - - + Trigonometry - - - - - + + + + + - diff --git a/source/colophon.xml b/source/colophon.xml index b8c32c60..e256fcb7 100644 --- a/source/colophon.xml +++ b/source/colophon.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ - - + diff --git a/source/exercises/ez-0-0.xml b/source/exercises/ez-0-0.xml index e144cc03..cd03ea56 100755 --- a/source/exercises/ez-0-0.xml +++ b/source/exercises/ez-0-0.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ - @@ -15,28 +14,28 @@ - + - + - + - + - + - + - + - + @@ -125,7 +124,8 @@
The graph of y = s(t), the position of the car (measured in thousands of feet from its starting location) at time t in minutes. - +

The graph of y = s(t), the position of the car (measured in thousands of feet from its starting location) at time t in minutes.

+

@@ -221,4 +221,4 @@ - \ No newline at end of file + diff --git a/source/exercises/ez-changing-combining.xml b/source/exercises/ez-changing-combining.xml index 834e519c..89c0e815 100755 --- a/source/exercises/ez-changing-combining.xml +++ b/source/exercises/ez-changing-combining.xml @@ -1,4 +1,3 @@ - @@ -15,27 +14,27 @@ - + - + - + - + - + - + @@ -109,11 +108,13 @@

The graph of a piecewise function, s. - +

The graph of a piecewise function, s.

+
The graph of a piecewise function, g. - +

The graph of a piecewise function, g.

+
@@ -285,8 +286,8 @@

- - +

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 +

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 diff --git a/source/exercises/ez-changing-composite.xml b/source/exercises/ez-changing-composite.xml index 43aeca1f..6946671e 100755 --- a/source/exercises/ez-changing-composite.xml +++ b/source/exercises/ez-changing-composite.xml @@ -1,4 +1,3 @@ - @@ -15,13 +14,13 @@ - + - + - + - + - + - + - + @@ -69,11 +68,13 @@
Plot of y = f(x). - +

Plot of y = f(x).

+
Plot of y = g(x). - +

Plot of y = g(x).

+
@@ -302,8 +303,8 @@

- - +

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 +

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diff --git a/source/exercises/ez-changing-functions.xml b/source/exercises/ez-changing-functions.xml index f3098c4e..ec17e7ea 100644 --- a/source/exercises/ez-changing-functions.xml +++ b/source/exercises/ez-changing-functions.xml @@ -1,4 +1,3 @@ - @@ -22,22 +21,22 @@ --> - + - + - + - + - + - + @@ -59,8 +58,8 @@

- - +

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 +

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 @@ -77,8 +76,8 @@

- - +

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 +

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 @@ -107,7 +106,8 @@
The velocity graph for a person walking along a straight path. - +

The velocity graph for a person walking along a straight path.

+

diff --git a/source/exercises/ez-changing-inverse.xml b/source/exercises/ez-changing-inverse.xml index 4c3e5257..45ebef3b 100755 --- a/source/exercises/ez-changing-inverse.xml +++ b/source/exercises/ez-changing-inverse.xml @@ -1,4 +1,3 @@ - @@ -15,27 +14,27 @@ - + - + - + - + - + - + @@ -47,7 +46,8 @@

Plots of the graphs of p and q. - +

Plots of the graphs of p and q.

+

@@ -91,11 +91,13 @@

The conical tank. - +

The conical tank.

+
Axes to plot V = g(t). - +

Axes to plot V = g(t).

+
diff --git a/source/exercises/ez-changing-linear.xml b/source/exercises/ez-changing-linear.xml index 291dbc3b..9ed2dd19 100755 --- a/source/exercises/ez-changing-linear.xml +++ b/source/exercises/ez-changing-linear.xml @@ -1,4 +1,3 @@ - @@ -15,23 +14,23 @@ - + - + - + - + - + @@ -125,7 +124,8 @@
The velocity functions A = f(t) and D = g(t) for Alicia and Damon, respectively. - +

The velocity functions A = f(t) and D = g(t) for Alicia and Damon, respectively.

+

@@ -178,7 +178,8 @@

The inverted conical tank. - +

The inverted conical tank.

+

diff --git a/source/exercises/ez-changing-quadratic.xml b/source/exercises/ez-changing-quadratic.xml index 67c220d8..2805e79c 100755 --- a/source/exercises/ez-changing-quadratic.xml +++ b/source/exercises/ez-changing-quadratic.xml @@ -1,4 +1,3 @@ - @@ -15,19 +14,19 @@ - + - + - + - + - + @@ -54,7 +53,8 @@

Two quadratic functions, f and g. - +

Two quadratic functions, f and g.

+

@@ -127,27 +127,27 @@ 0 - + 1 - + 2 - + 3 - + 4 - + 5 - + @@ -160,23 +160,23 @@ [0,1] - + [1,2] - + [2,3] - + [3,4] - + [4,5] - + diff --git a/source/exercises/ez-changing-tandem.xml b/source/exercises/ez-changing-tandem.xml index 08c0f75f..c2a1a214 100755 --- a/source/exercises/ez-changing-tandem.xml +++ b/source/exercises/ez-changing-tandem.xml @@ -1,4 +1,3 @@ - @@ -16,7 +15,7 @@ - + @@ -32,9 +31,10 @@ Suppose we have an unusual tank whose base is a perfect sphere with radius 3 feet, and then atop the spherical base is a cylindrical chimney that is a circular cylinder of radius 1 foot and height 2 feet, as shown in Figure. The tank is initially empty, but then a spigot is turned on that pumps water into the tank at a constant rate of 1.25 cubic feet per minute.

-
+
A spherical tank with a cylindrical chimney. - +

A spherical tank with a cylindrical chimney.

+

@@ -54,8 +54,8 @@

- - +

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 +

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@@ -106,8 +106,8 @@

- - +

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 +

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@@ -169,8 +169,10 @@

A graph of the relationship between a car's position s and time t - - +

A graph of the relationship between a car's position s and time t

+ +

A graph of the relationship between a car's position s and time t

+
diff --git a/source/exercises/ez-changing-transformations.xml b/source/exercises/ez-changing-transformations.xml index f2c2f64b..62768814 100755 --- a/source/exercises/ez-changing-transformations.xml +++ b/source/exercises/ez-changing-transformations.xml @@ -1,4 +1,3 @@ - @@ -15,23 +14,23 @@ - + - + - + - + - + @@ -15,18 +14,18 @@ - + - + - + - + diff --git a/source/exercises/ez-circular-sinusoidal.xml b/source/exercises/ez-circular-sinusoidal.xml index a1fe829b..b82bd4c6 100755 --- a/source/exercises/ez-circular-sinusoidal.xml +++ b/source/exercises/ez-circular-sinusoidal.xml @@ -1,4 +1,3 @@ - @@ -12,27 +11,26 @@ - - + - + - + - + - + - + diff --git a/source/exercises/ez-circular-traversing.xml b/source/exercises/ez-circular-traversing.xml index 92192288..cf9a54c4 100755 --- a/source/exercises/ez-circular-traversing.xml +++ b/source/exercises/ez-circular-traversing.xml @@ -1,4 +1,3 @@ - @@ -12,23 +11,22 @@ - - + - + - + - + @@ -15,26 +14,26 @@ - + - + - + - + - + - + - + diff --git a/source/exercises/ez-exp-e.xml b/source/exercises/ez-exp-e.xml index ab4621ea..66c449eb 100755 --- a/source/exercises/ez-exp-e.xml +++ b/source/exercises/ez-exp-e.xml @@ -1,4 +1,3 @@ - @@ -24,19 +23,19 @@ --> - + - + - + - + - + @@ -15,28 +14,28 @@ - + - + - + - + - + - + - + @@ -12,14 +11,13 @@ - - + - + - + - + - + diff --git a/source/exercises/ez-exp-log.xml b/source/exercises/ez-exp-log.xml index 3616537a..64835447 100755 --- a/source/exercises/ez-exp-log.xml +++ b/source/exercises/ez-exp-log.xml @@ -1,4 +1,3 @@ - @@ -12,20 +11,19 @@ - - + - + - + - + - + diff --git a/source/exercises/ez-exp-modeling.xml b/source/exercises/ez-exp-modeling.xml index 1f35dbea..c6bf9bc2 100755 --- a/source/exercises/ez-exp-modeling.xml +++ b/source/exercises/ez-exp-modeling.xml @@ -1,4 +1,3 @@ - @@ -19,13 +18,13 @@ --> - + - + - + @@ -78,12 +77,12 @@

- - +

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 +

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 - - +

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 +

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diff --git a/source/exercises/ez-exp-temp-pop.xml b/source/exercises/ez-exp-temp-pop.xml index 979c0243..bdf1aa1b 100755 --- a/source/exercises/ez-exp-temp-pop.xml +++ b/source/exercises/ez-exp-temp-pop.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ - - + - +

In Exercise below, use the following structure/formula for N(t): N(t)=\frac{L}{1+Ab^{-kt}}. In particular, note that when the instructions say find A, this use of A is not in reference to carrying capacity.

- + @@ -112,7 +110,7 @@

-

    +

    1. Recall we know that S(0) = 5 and S(1) = 20. In addition, assume that 5000 is the number of people who will eventually get sick. Use this information determine the exact values of A, M, and k in the logistic model. diff --git a/source/exercises/ez-poly-infty.xml b/source/exercises/ez-poly-infty.xml index 539905d1..cd44f5ce 100755 --- a/source/exercises/ez-poly-infty.xml +++ b/source/exercises/ez-poly-infty.xml @@ -1,4 +1,3 @@ - @@ -12,18 +11,17 @@ - - + - + - + @@ -73,11 +71,13 @@

      A plot of y = f(x) = \frac{1}{x}. - +

      A plot of y = f(x) = \frac{1}{x}.

      +
      A plot of a related function y = g(x). - +

      A plot of a related function y = g(x).

      +
      diff --git a/source/exercises/ez-poly-polynomial-applications.xml b/source/exercises/ez-poly-polynomial-applications.xml index 6688354e..a060cb1d 100755 --- a/source/exercises/ez-poly-polynomial-applications.xml +++ b/source/exercises/ez-poly-polynomial-applications.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,14 @@ - - + - + @@ -12,23 +11,22 @@ - - + - + - + - + - + - + - + @@ -164,24 +162,24 @@

      - In each following question, find a formula for a polynomial with certain properties, generate a plot that demonstrates you’ve found a function with the given specifications, and write several sentences to explain your thinking. + In each following question, find a formula for a polynomial with certain properties, generate a plot that demonstrates you’ve found a function with the given specifications, and write several sentences to explain your thinking.

      1. - A quadratic function q has zeros at x = −7 and x = 11 and its y-value at its vertex is 42. + A quadratic function q has zeros at x = −7 and x = 11 and its y-value at its vertex is 42.

      2. - A polynomial r of degree 4 has zeros at x = −3 and x = 5, both of multiplicity 2, and the function has a y-intercept at the point (0, 28). + A polynomial r of degree 4 has zeros at x = −3 and x = 5, both of multiplicity 2, and the function has a y-intercept at the point (0, 28).

      3. - A polynomial f has degree 11 and the following zeros: zeros of multiplicity 1 at x = −3 and x = 5, zeros of multiplicity 2 at x = −2 and x = 3, and a zero of multiplicity 3 at x = 1. In addition, \lim_{x \to \infty} f(x) = -\infty. + A polynomial f has degree 11 and the following zeros: zeros of multiplicity 1 at x = −3 and x = 5, zeros of multiplicity 2 at x = −2 and x = 3, and a zero of multiplicity 3 at x = 1. In addition, \lim_{x \to \infty} f(x) = -\infty.

      4. @@ -191,7 +189,8 @@
        A polynomial function g. - +

        A polynomial function g.

        +
      diff --git a/source/exercises/ez-poly-rational-features.xml b/source/exercises/ez-poly-rational-features.xml index a06e3ad1..c9561f3e 100755 --- a/source/exercises/ez-poly-rational-features.xml +++ b/source/exercises/ez-poly-rational-features.xml @@ -1,4 +1,3 @@ - @@ -12,23 +11,22 @@ - - + - + - + - + - + @@ -12,26 +11,25 @@ - - + - + - + - + - + - + - + - + diff --git a/source/exercises/ez-trig-finding-angles.xml b/source/exercises/ez-trig-finding-angles.xml index 0f4f8bde..f38b3813 100755 --- a/source/exercises/ez-trig-finding-angles.xml +++ b/source/exercises/ez-trig-finding-angles.xml @@ -1,4 +1,3 @@ - @@ -12,17 +11,16 @@ - - + - + - + @@ -12,20 +11,19 @@ - - + - + - + - + @@ -176,12 +174,14 @@
      The right triangle that corresponds to the angle \theta = \arcsin(x). - +

      The right triangle that corresponds to the angle \theta = \arcsin(x).

      +
      The right triangle that corresponds to the angle \alpha = \arctan(x). - +

      The right triangle that corresponds to the angle \alpha = \arctan(x).

      +
      diff --git a/source/exercises/ez-trig-other.xml b/source/exercises/ez-trig-other.xml index b59ecfdc..dfa66d98 100755 --- a/source/exercises/ez-trig-other.xml +++ b/source/exercises/ez-trig-other.xml @@ -1,4 +1,3 @@ - @@ -12,18 +11,17 @@ - - + - + - + diff --git a/source/exercises/ez-trig-right.xml b/source/exercises/ez-trig-right.xml index 90962532..ef6e78ea 100755 --- a/source/exercises/ez-trig-right.xml +++ b/source/exercises/ez-trig-right.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,13 @@ - - + - + @@ -12,28 +11,27 @@ - - + - + - + - + - + - + @@ -12,11 +11,10 @@ - - + - - + +

      @@ -31,9 +29,9 @@

      The beautiful full-color .eps graphics, as well as the occasional interactive JavaScript graphics, use David Austin's Python library that employs Bill Casselman's PiScript. - The .html version of the text is the result Rob Beezer's amazing work to develop the publishing language (formerly known as Mathbook XML); learn more at pretextbook.org. + The .html version of the text is the result Rob Beezer's amazing work to develop the publishing language (formerly known as Mathbook XML); learn more at pretextbook.org. I'm grateful to the American Institute of Mathematics for hosting and funding - a weeklong workshop in San Jose, CA, in April 2016, which enabled me to get started in . The ongoing support of the user group is invaluable, and David Farmer of AIM is has also been a source of major support and advocacy. + a weeklong workshop in San Jose, CA, in April 2016, which enabled me to get started in . The ongoing support of the user group is invaluable, and David Farmer of AIM is has also been a source of major support and advocacy. Mitch Keller of the University of Wisconsin is the production editor of both Active Calculus: Single Variable and this text; his technical expertise is a gift.

      @@ -155,7 +153,7 @@ Motivating Questions

      At the start of each section, - we list 23 motivating questions + we list 23 motivating questions that provide motivation for why the following material is of interest to us. One goal of each section is to answer each of the motivating questions.

      @@ -189,13 +187,13 @@ There are dozens of college algebra and trignometry texts with (collectively) tens of thousands of exercises. Rather than repeat standard and routine exercises in this text, we recommend the use of - + with its access to the Open Problem Library (OPL) and many thousands of relevant problems. In this text, - each section includes a small collection of anonymous exercises that offer + each section includes a small collection of anonymous exercises that offer students immediate feedback without penalty, - as well as 34 additional challenging exercises per section. - Each of the non- exercises has multiple parts, + as well as 34 additional challenging exercises per section. + Each of the non- exercises has multiple parts, requires the student to connect several key ideas, and expects that the student will do at least a modest amount of writing to answer the questions and explain their findings.

      @@ -259,7 +257,7 @@

      - At the end of each section, you'll find two types of Exercises. First, there are several anonymous exercises. These are online, interactive exercises that allow you to submit answers for immediate feedback with unlimited attempts without penalty; to submit answers, you have to be using the HTML version of the text (see this short video on the HTML version that includes a demonstration). You should use these exercises as a way to test your understanding of basic ideas in the preceding section. If your institution uses , you may also need to log in to a server as directed by your instructor to complete assigned sets as part of your course grade. The exercises included in this text are ungraded and not connected to any individual account. Following the exercises there are 3-4 additional challenging exercises that are designed to encourage you to connect ideas, investigate new situations, and write about your understanding. + At the end of each section, you'll find two types of Exercises. First, there are several anonymous exercises. These are online, interactive exercises that allow you to submit answers for immediate feedback with unlimited attempts without penalty; to submit answers, you have to be using the HTML version of the text (see this short video on the HTML version that includes a demonstration). You should use these exercises as a way to test your understanding of basic ideas in the preceding section. If your institution uses , you may also need to log in to a server as directed by your instructor to complete assigned sets as part of your course grade. The exercises included in this text are ungraded and not connected to any individual account. Following the exercises there are 3-4 additional challenging exercises that are designed to encourage you to connect ideas, investigate new situations, and write about your understanding.

      - The source code for the text can be found on GitHub. If you find errors in the text or have other suggestions, you can file an issue on GitHub or email the author directly. To engage with instructors who use the text, we maintain a Google group and a blog; you can request to join the Google group via the link at the instructors page. Finally, if you're interested in a video presentation on using the similar Active Calculus text, you can see this online video presentation to the MIT Electronic Seminar on Mathematics Education; at about the 17-minute mark, the portion begins where we demonstrate features of and how to use the text. + The source code for the text can be found on GitHub. If you find errors in the text or have other suggestions, you can file an issue on GitHub or email the author directly. To engage with instructors who use the text, we maintain a Google group and a blog; you can request to join the Google group via the link at the instructors page. Finally, if you're interested in a video presentation on using the similar Active Calculus text, you can see this online video presentation to the MIT Electronic Seminar on Mathematics Education; at about the 17-minute mark, the portion begins where we demonstrate features of and how to use the text.

      @@ -346,4 +344,3 @@ - diff --git a/source/main.ptx b/source/main.ptx index 0200e129..8028ee31 100644 --- a/source/main.ptx +++ b/source/main.ptx @@ -1,4 +1,3 @@ - @@ -14,22 +13,21 @@ - + Active Prelude to Calculus - + - - - - - + + + + + - + - diff --git a/source/previews/PA-0-0.xml b/source/previews/PA-0-0.xml index f4291ada..3f3a7118 100755 --- a/source/previews/PA-0-0.xml +++ b/source/previews/PA-0-0.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-changing-aroc.xml b/source/previews/PA-changing-aroc.xml index 548e6a77..ba44e289 100755 --- a/source/previews/PA-changing-aroc.xml +++ b/source/previews/PA-changing-aroc.xml @@ -1,4 +1,3 @@ - @@ -56,7 +55,7 @@

      - +

      ADD ALT TEXT TO THIS IMAGE

             diff --git a/source/previews/PA-changing-combining.xml b/source/previews/PA-changing-combining.xml index 84513657..c72e14fb 100755 --- a/source/previews/PA-changing-combining.xml +++ b/source/previews/PA-changing-combining.xml @@ -1,4 +1,3 @@ - @@ -56,7 +55,7 @@ - +

      ADD ALT TEXT TO THIS IMAGE

      diff --git a/source/previews/PA-changing-composite.xml b/source/previews/PA-changing-composite.xml index fb1beb7f..be75c29b 100755 --- a/source/previews/PA-changing-composite.xml +++ b/source/previews/PA-changing-composite.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-changing-functions-crickets.xml b/source/previews/PA-changing-functions-crickets.xml index 5982d9cc..7b3953f5 100755 --- a/source/previews/PA-changing-functions-crickets.xml +++ b/source/previews/PA-changing-functions-crickets.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-changing-inverse-F-C.xml b/source/previews/PA-changing-inverse-F-C.xml index 868350c8..ec046ad7 100755 --- a/source/previews/PA-changing-inverse-F-C.xml +++ b/source/previews/PA-changing-inverse-F-C.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-changing-linear-3-ex.xml b/source/previews/PA-changing-linear-3-ex.xml index fd3679b6..4c76a343 100755 --- a/source/previews/PA-changing-linear-3-ex.xml +++ b/source/previews/PA-changing-linear-3-ex.xml @@ -1,4 +1,3 @@ - @@ -128,7 +127,7 @@

      - +

      ADD ALT TEXT TO THIS IMAGE

      Determine AV_{[-5,-2]}, AV_{[-1,1]}, and AV_{[0,4]} for the function h. diff --git a/source/previews/PA-changing-quadratic.xml b/source/previews/PA-changing-quadratic.xml index 1c2defde..b0e44115 100755 --- a/source/previews/PA-changing-quadratic.xml +++ b/source/previews/PA-changing-quadratic.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-changing-tandem-aquarium.xml b/source/previews/PA-changing-tandem-aquarium.xml index 7d1ea33d..09ce5bba 100755 --- a/source/previews/PA-changing-tandem-aquarium.xml +++ b/source/previews/PA-changing-tandem-aquarium.xml @@ -1,4 +1,3 @@ - @@ -21,11 +20,13 @@

      The empty aquarium. - +

      The empty aquarium.

      +
      The aquarium, partially filled. - +

      The aquarium, partially filled.

      +

      diff --git a/source/previews/PA-changing-transformations-quadratic.xml b/source/previews/PA-changing-transformations-quadratic.xml index c692a1d7..ac5b3a9c 100755 --- a/source/previews/PA-changing-transformations-quadratic.xml +++ b/source/previews/PA-changing-transformations-quadratic.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-circular-sine.xml b/source/previews/PA-circular-sine.xml index 0c81efbf..fecd8feb 100755 --- a/source/previews/PA-circular-sine.xml +++ b/source/previews/PA-circular-sine.xml @@ -1,4 +1,3 @@ - @@ -20,7 +19,7 @@

      - +

      ADD ALT TEXT TO THIS IMAGE

      diff --git a/source/previews/PA-circular-sinusoidal.xml b/source/previews/PA-circular-sinusoidal.xml index 67cfe129..0ee6b49e 100755 --- a/source/previews/PA-circular-sinusoidal.xml +++ b/source/previews/PA-circular-sinusoidal.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-circular-traversing.xml b/source/previews/PA-circular-traversing.xml index 15549e47..7096a361 100755 --- a/source/previews/PA-circular-traversing.xml +++ b/source/previews/PA-circular-traversing.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-circular-unit-circle.xml b/source/previews/PA-circular-unit-circle.xml index f3ea646b..9788e94d 100755 --- a/source/previews/PA-circular-unit-circle.xml +++ b/source/previews/PA-circular-unit-circle.xml @@ -1,4 +1,3 @@ - @@ -22,7 +21,7 @@

      - +

      ADD ALT TEXT TO THIS IMAGE

      @@ -56,7 +55,7 @@ An angle \theta of measure 1 radian. --> - +

      ADD ALT TEXT TO THIS IMAGE

      diff --git a/source/previews/PA-exp-e.xml b/source/previews/PA-exp-e.xml index edc090d9..66d2bf34 100755 --- a/source/previews/PA-exp-e.xml +++ b/source/previews/PA-exp-e.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-exp-growth.xml b/source/previews/PA-exp-growth.xml index 0db6c804..d8e24c98 100755 --- a/source/previews/PA-exp-growth.xml +++ b/source/previews/PA-exp-growth.xml @@ -1,4 +1,3 @@ - @@ -60,7 +59,7 @@

      - +

      ADD ALT TEXT TO THIS IMAGE

       diff --git a/source/previews/PA-exp-log-properties.xml b/source/previews/PA-exp-log-properties.xml index 24d806b0..c3aa13c1 100755 --- a/source/previews/PA-exp-log-properties.xml +++ b/source/previews/PA-exp-log-properties.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-exp-log.xml b/source/previews/PA-exp-log.xml index 1d023e7d..34758aff 100755 --- a/source/previews/PA-exp-log.xml +++ b/source/previews/PA-exp-log.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-exp-modeling.xml b/source/previews/PA-exp-modeling.xml index 782f0855..9a301a6a 100755 --- a/source/previews/PA-exp-modeling.xml +++ b/source/previews/PA-exp-modeling.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-exp-temp-pop.xml b/source/previews/PA-exp-temp-pop.xml index f3cafe0e..6cf8249d 100755 --- a/source/previews/PA-exp-temp-pop.xml +++ b/source/previews/PA-exp-temp-pop.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-poly-infty.xml b/source/previews/PA-poly-infty.xml index 9d7acee1..5b543e1f 100755 --- a/source/previews/PA-poly-infty.xml +++ b/source/previews/PA-poly-infty.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-poly-polynomial-applications.xml b/source/previews/PA-poly-polynomial-applications.xml index feaa362a..53e1c6f3 100755 --- a/source/previews/PA-poly-polynomial-applications.xml +++ b/source/previews/PA-poly-polynomial-applications.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-poly-polynomials.xml b/source/previews/PA-poly-polynomials.xml index 2fed84b6..ff8bbad7 100755 --- a/source/previews/PA-poly-polynomials.xml +++ b/source/previews/PA-poly-polynomials.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-poly-rational-features.xml b/source/previews/PA-poly-rational-features.xml index 725899e7..74c4626e 100755 --- a/source/previews/PA-poly-rational-features.xml +++ b/source/previews/PA-poly-rational-features.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-poly-rational.xml b/source/previews/PA-poly-rational.xml index 825cb2d7..8563b5a8 100755 --- a/source/previews/PA-poly-rational.xml +++ b/source/previews/PA-poly-rational.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-trig-finding-angles.xml b/source/previews/PA-trig-finding-angles.xml index 4ecfc3be..deb3b245 100755 --- a/source/previews/PA-trig-finding-angles.xml +++ b/source/previews/PA-trig-finding-angles.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-trig-inverse.xml b/source/previews/PA-trig-inverse.xml index 766f08b4..52420621 100755 --- a/source/previews/PA-trig-inverse.xml +++ b/source/previews/PA-trig-inverse.xml @@ -1,4 +1,3 @@ - @@ -20,7 +19,7 @@

      - +

      ADD ALT TEXT TO THIS IMAGE

      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. diff --git a/source/previews/PA-trig-other.xml b/source/previews/PA-trig-other.xml index 5e6f6c29..4fd4921f 100755 --- a/source/previews/PA-trig-other.xml +++ b/source/previews/PA-trig-other.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-trig-right.xml b/source/previews/PA-trig-right.xml index a999551b..20f39bde 100755 --- a/source/previews/PA-trig-right.xml +++ b/source/previews/PA-trig-right.xml @@ -1,4 +1,3 @@ - diff --git a/source/previews/PA-trig-tangent.xml b/source/previews/PA-trig-tangent.xml index fa59708d..8bb2ebc8 100755 --- a/source/previews/PA-trig-tangent.xml +++ b/source/previews/PA-trig-tangent.xml @@ -1,4 +1,3 @@ - diff --git a/source/sec-0-0.xml b/source/sec-0-0.xml index 235224cf..fcf629b9 100755 --- a/source/sec-0-0.xml +++ b/source/sec-0-0.xml @@ -1,4 +1,3 @@ - @@ -12,10 +11,8 @@ -

      - + <!-- <xi:include href="./activities/act-1-1-1.xml" /> --> <!-- <xi:include href="./exercises/ez-1-1.xml" /> --> </section> - diff --git a/source/sec-changing-aroc-wb.xml b/source/sec-changing-aroc-wb.xml index c91dc0fb..d2c16dd4 100644 --- a/source/sec-changing-aroc-wb.xml +++ b/source/sec-changing-aroc-wb.xml @@ -1,4 +1,3 @@ -<?xml version="1.0" encoding="UTF-8" ?> <!-- **********************************************************************--> <!-- Copyright 2019 --> <!-- Matthew Boelkins --> @@ -12,14 +11,12 @@ <!-- "share and share alike". All trademarks are the registered marks of --> <!-- their respective owners. --> <!-- **********************************************************************--> - <section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec-changing-aroc"> <title>The Average Rate of Change of a Function - - - - + + + +
      - diff --git a/source/sec-changing-aroc.xml b/source/sec-changing-aroc.xml index 333ef963..87e2d129 100755 --- a/source/sec-changing-aroc.xml +++ b/source/sec-changing-aroc.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
      The Average Rate of Change of a Function @@ -57,7 +55,7 @@ change in position divided by the change in time.

      - + @@ -72,11 +70,13 @@
      The average rate of change of s on [1.5,2.5] for the function in Preview Activity. - +

      The average rate of change of s on [1.5,2.5] for the function in Preview Activity.

      +
      The average rate of change of an abstract function f on the interval [a,b]. - +

      The average rate of change of an abstract function f on the interval [a,b].

      +
      @@ -102,7 +102,7 @@ and those units are always units of output per unit of input. average rate of changeunits Moreover, the average rate of change of f on [a,b] always corresponds to the slope of the line between the points (a,f(a)) and (b,f(b)), as seen in Figure.

      - +

      The average rate of change of a function on an interval gives us an excellent way to describe how the function behaves, on average. For instance, if we compute AV_{[1970,2000]} for Kent County, we find that @@ -153,7 +153,7 @@ If we compute the average rate of change of a function on an interval, we can decide if the function is increasing or decreasing on average on the interval, but it takes more workCalculus offers one way to justify that a function is always increasing or always decreasing on an interval. to decide if the function is increasing or decreasing always on the interval.

      - +

      It is helpful be able to connect information about a function's average rate of change and its graph. For instance, if we have determined that AV_{[-3,2]} = 1.75 for some function f, this tells us that, on average, the function rises between the points x = -3 and x = 2 and does so at an average rate of 1.75 vertical units for every horizontal unit. Moreover, we can even determine that the difference between f(2) and f(-3) is @@ -163,7 +163,7 @@ since \frac{f(2)-f(-3)}{2-(-3)} = 1.75.

      - + @@ -191,9 +191,8 @@

      - +
      - diff --git a/source/sec-changing-combining-wb.xml b/source/sec-changing-combining-wb.xml index b013169d..074834e6 100755 --- a/source/sec-changing-combining-wb.xml +++ b/source/sec-changing-combining-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
      Combining Functions - - - - + + + +
      - diff --git a/source/sec-changing-combining.xml b/source/sec-changing-combining.xml index a5d1b7a9..d0d0c3d5 100755 --- a/source/sec-changing-combining.xml +++ b/source/sec-changing-combining.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
      Combining Functions @@ -45,7 +43,7 @@ Just as we can add, subtract, multiply, and divide numbers, we can also add, subtract, multiply, and divide functions to create a new function from two or more given functions.

      - + @@ -130,7 +128,7 @@ - + @@ -172,7 +170,7 @@ - + @@ -211,7 +209,8 @@
      A plot of the absolute value function, A(x) = |x|. - +

      A plot of the absolute value function, A(x) = |x|.

      +
      @@ -223,7 +222,7 @@ As long as we are careful to make sure that each potential input has one and only one corresponding output, we can define a piecewise function using as many different functions on different intervals as we desire.

      - + @@ -253,9 +252,8 @@

      - +
      - diff --git a/source/sec-changing-composite-wb.xml b/source/sec-changing-composite-wb.xml index 541236d9..feaf5d18 100755 --- a/source/sec-changing-composite-wb.xml +++ b/source/sec-changing-composite-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
      Composite Functions - - - - + + + +
      - diff --git a/source/sec-changing-composite.xml b/source/sec-changing-composite.xml index 2036bcd2..9871a64e 100755 --- a/source/sec-changing-composite.xml +++ b/source/sec-changing-composite.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
      Composite Functions @@ -77,7 +75,7 @@ When we have a situation such as in Example where we use the output of one function as the input of another, we often say that we have composed two functions. In addition, we use the notation h(t) = f(g(t)) to denote that a new function, h, results from composing the two functions f and g.

      - + @@ -110,7 +108,7 @@ and the following activities prompt us to consider functions given in this way.

      - + @@ -127,7 +125,7 @@ For instance, a Fahrenheit temperature of 32 degrees corresponds to C = G(32) = 0 degrees Celsius.

      - +       @@ -145,11 +143,13 @@
      AV_{[a,b]} is the slope of the line joining the points (a,f(a)) and (b,f(b)) on the graph of f. - +

      ADD ALT TEXT TO THIS IMAGEAV_{[a,b]} is the slope of the line joining the points (a,f(a)) and (b,f(b)) on the graph of f.

      +
      AV_{[a,a+h]} is the slope of the line joining the points (a,f(a)) and (a,f(a+h)) on the graph of f. - +

      ADD ALT TEXT TO THIS IMAGEAV_{[a,a+h]} is the slope of the line joining the points (a,f(a)) and (a,f(a+h)) on the graph of f.

      +
      @@ -198,7 +198,7 @@ - +

      In Activity, we see an important setting where algebraic simplification plays a crucial role in calculus. Because the expresssion @@ -241,9 +241,8 @@

      - +
      - diff --git a/source/sec-changing-functions-models-wb.xml b/source/sec-changing-functions-models-wb.xml index 50cf712c..b28a8ab9 100755 --- a/source/sec-changing-functions-models-wb.xml +++ b/source/sec-changing-functions-models-wb.xml @@ -1,4 +1,3 @@ - @@ -12,13 +11,11 @@ -
      Functions: Modeling Relationships - - - - + + + +
      - diff --git a/source/sec-changing-functions-models.xml b/source/sec-changing-functions-models.xml index 291b6a58..f13fdc4d 100755 --- a/source/sec-changing-functions-models.xml +++ b/source/sec-changing-functions-models.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
      Functions: Modeling Relationships @@ -62,7 +60,7 @@ true. For instance, 70 = 40 + 0.25(120). Indeed, scientists who made many additional cricket chirp observations following Dolbear's initial counts found that the formula in Equation holds with remarkable accuracy for the snowy tree cricket in temperatures ranging from about 50^\circ F to 85^\circ F.

      - + @@ -125,7 +123,8 @@
      Graph of data from the function T = D(N) = 40 + 0.25N and the underlying curve. - +

      Graph of data from the function T = D(N) = 40 + 0.25N and the underlying curve.

      +
      @@ -180,7 +179,7 @@ The range of any function is always a subset of the codomain. It is possible for the range to equal the codomain.

      - + @@ -246,7 +245,8 @@
      Graph of the function y = f(x) and some data from the table. - +

      Graph of the function y = f(x) and some data from the table.

      +
      @@ -286,14 +286,15 @@
      Graph of the model h = g(t) and some data from the table. - +

      Graph of the model h = g(t) and some data from the table.

      +
      - + @@ -349,7 +350,7 @@ - +

      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 the left graph in , we can use the vertical line test to determine whether or not the graph represents a function. @@ -404,9 +405,8 @@

      - +
      - diff --git a/source/sec-changing-in-tandem-wb.xml b/source/sec-changing-in-tandem-wb.xml index fc90f81b..6fe1d2ce 100644 --- a/source/sec-changing-in-tandem-wb.xml +++ b/source/sec-changing-in-tandem-wb.xml @@ -1,4 +1,3 @@ - @@ -12,11 +11,9 @@ -
      Changing in Tandem - - - + + +
      - diff --git a/source/sec-changing-in-tandem.xml b/source/sec-changing-in-tandem.xml index a72e982c..376737dc 100755 --- a/source/sec-changing-in-tandem.xml +++ b/source/sec-changing-in-tandem.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
      Changing in Tandem @@ -35,7 +33,7 @@ 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.

      - + @@ -81,9 +79,10 @@ -
      +
      A visual representation of the data in Table. - +

      A visual representation of the data in Table.

      +
      @@ -103,7 +102,7 @@ Depending on which variable we solve for, we can either see how V depends on h through the equation V = 8h, or how h depends on V via the equation h = \frac{1}{8}V. From either perspective, we observe that as depth or volume increases, so must volume or depth correspondingly increase.

      - + @@ -116,11 +115,13 @@
      The empty conical tank. - +

      The empty conical tank.

      +
      The conical tank, partially filled. - +

      The conical tank, partially filled.

      +
      @@ -216,11 +217,13 @@
      Plotting V versus t. - +

      Plotting V versus t.

      +
      Plotting h versus t. - +

      Plotting h versus t.

      +
      @@ -228,7 +231,7 @@ These different behaviors make sense because of the shape of the tank. Since at first there is less volume relative to depth near the cone's point, as water flows in at a constant rate, the water's height will rise quickly. But as time goes on and more water is added at the same rate, there is more space for the water to fill in order to make the water level rise, and thus the water's height rises more and more slowly as time passes.

      - + @@ -250,9 +253,8 @@

      - +
      - diff --git a/source/sec-changing-inverse-wb.xml b/source/sec-changing-inverse-wb.xml index ee1308e3..2f0d35f8 100755 --- a/source/sec-changing-inverse-wb.xml +++ b/source/sec-changing-inverse-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
      Inverse Functions - - - - + + + +
      - diff --git a/source/sec-changing-inverse.xml b/source/sec-changing-inverse.xml index 4a137ee9..2e500a82 100755 --- a/source/sec-changing-inverse.xml +++ b/source/sec-changing-inverse.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
      Inverse Functions @@ -42,7 +40,7 @@ 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?

      - + @@ -77,7 +75,7 @@ When a given function f has a corresponding inverse function g, we usually rename g as f^{-1}, which we read aloud as f-inverse. inverse functionnotation The equation g(f(a))=a now reads f^{-1}(f(a)) = a, which we interpret as saying f-inverse converts f(a) back to a. We similarly write that f(f^{-1}(b)) = b.

      - +

      When a given function has an inverse function, it allows us to express the same relationship from two different points of view. For instance, if y = f(t) = 2t+1, we can showObserve that g(f(t)) = g(2t+1) = \frac{(2t+1)-1}{2} = \frac{2t}{2} = t. Similarly, f(g(y)) = f\left(\frac{y-1}{2}\right) = 2\left(\frac{y-1}{2} \right) + 1 = y-1 + 1 = y. that the function t = g(y) = \frac{y-1}{2} reverses the effect of f (and vice versa), and thus g = f^{-1}. We observe that @@ -191,13 +189,19 @@ The graph that defines function p. - +

      + The graph that defines function p. +

      +
The graph that defines function p. - +

+ The graph that defines function p. +

+
@@ -293,15 +297,17 @@
A plot of y = r(t) = 3 - \frac{1}{5}(t-1)^3. - +

A plot of y = r(t) = 3 - \frac{1}{5}(t-1)^3.

+
A plot of y = s(t) = 3 - \frac{1}{5}(t-1)^2. - +

A plot of y = s(t) = 3 - \frac{1}{5}(t-1)^2.

+
- + @@ -351,10 +357,11 @@
The graph of a function f along with its inverse, f^{-1}. - +

The graph of a function f along with its inverse, f^{-1}.

+
- + @@ -369,21 +376,20 @@

  • - We determine whether or not a given function f has a corresponding inverse function by determining if the process that defines f can be reversed so that we can also think of the outputs as a function of the inputs. If we have a graph of the function f, we know f has an inverse function if the graph passes the . If we have a formula for the function f, say y = f(t), we know f has an inverse function if we can solve for t and write t = f^{-1}(y). + We determine whether or not a given function f has a corresponding inverse function by determining if the process that defines f can be reversed so that we can also think of the outputs as a function of the inputs. If we have a graph of the function f, we know f has an inverse function if the graph passes the . If we have a formula for the function f, say y = f(t), we know f has an inverse function if we can solve for t and write t = f^{-1}(y).

  • - A good summary of the properties of an inverse function is provided in the . + A good summary of the properties of an inverse function is provided in the .

    • - + - diff --git a/source/sec-changing-linear-wb.xml b/source/sec-changing-linear-wb.xml index 76f3cb86..f8aab523 100755 --- a/source/sec-changing-linear-wb.xml +++ b/source/sec-changing-linear-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Linear Functions - - - - + + + +
    - diff --git a/source/sec-changing-linear.xml b/source/sec-changing-linear.xml index 30b1133c..c210475b 100755 --- a/source/sec-changing-linear.xml +++ b/source/sec-changing-linear.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Linear Functions @@ -45,7 +43,7 @@ Whether a function's graph is a straight line or not is connected directly to its average rate of change.

    - + @@ -130,7 +128,7 @@

    - +

    Visualizing the various components of point-slope form is important. For a line through (x_0,y_0) with slope m, we know its equation is y = y_0 + m(x-x_0). In Figure, we see that the line passes through (x_0,y_0) along with an arbitary point (x,y), which makes the vertical change between the two points given by y - y_0 and the horizontal change between the points x - x_0. This is consistent with the fact that @@ -147,11 +145,13 @@

    The point-slope form of a line's equation. - +

    The point-slope form of a line's equation.

    +
    The slope-intercept form of a line's equation. - +

    The slope-intercept form of a line's equation.

    +
    @@ -235,7 +235,8 @@
    The linear Dolbear function with slope m = 0.25 degrees Fahrenheit per chirp per minute. - +

    The linear Dolbear function with slope m = 0.25 degrees Fahrenheit per chirp per minute.

    +
    @@ -275,9 +276,9 @@

    - + - + @@ -304,9 +305,8 @@

    - +
    - diff --git a/source/sec-changing-quadratic-wb.xml b/source/sec-changing-quadratic-wb.xml index dbf4e299..5079673c 100755 --- a/source/sec-changing-quadratic-wb.xml +++ b/source/sec-changing-quadratic-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Quadratic Functions - - - - + + + +
    - diff --git a/source/sec-changing-quadratic.xml b/source/sec-changing-quadratic.xml index 65b89fd9..bcfbeb88 100755 --- a/source/sec-changing-quadratic.xml +++ b/source/sec-changing-quadratic.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Quadratic Functions @@ -49,7 +47,7 @@ b, and c are real numbers with a \ne 0. One of the reasons that quadratic functions are especially important is that they model the height of an object falling under the force of gravity.

    - + @@ -61,7 +59,7 @@ Quadratic functions are likely familiar to you from experience in previous courses. Throughout, we let y = q(x) = ax^2 + bx + c where a, b, and c are real numbers with a \ne 0. From the outset, it is important to note that when we write q(x) = ax^2 + bx + c we are thinking of an infinite family of functions where each member depends on the three paramaters a, b, and c.

    - +

    Because quadratic functions are familiar to us, we will quickly restate some of their important known properties. @@ -86,11 +84,13 @@

    Three examples of quadratic functions that open up. - +

    Three examples of quadratic functions that open up.

    +
    One example of a quadratic function that opens down. - +

    One example of a quadratic function that opens down.

    +
    @@ -119,11 +119,13 @@
    The vertex of a quadratic function that opens up. - +

    The vertex of a quadratic function that opens up.

    +
    The vertex of a quadratic function that opens down. - +

    The vertex of a quadratic function that opens down.

    +
    @@ -179,7 +181,7 @@

    - + @@ -195,9 +197,9 @@

    - One of the fantastic consequences of calculus which, + One of the fantastic consequences of calculus which, like the realization that acceleration due to gravity is constant, - is largely due to Sir Isaac Newton in the late 1600s is that the height of a falling object at time t is modeled by a quadratic function. + is largely due to Sir Isaac Newton in the late 1600s is that the height of a falling object at time t is modeled by a quadratic function.

    @@ -216,7 +218,7 @@ If height is measured instead in meters and velocity in meters per second, the gravitational constant is g = 9.8 and the function h has form h(t) = -4.9t^2 + v_0t + s_0. gravitygravitational constant (When height is measured in feet, the gravitational constant is g = 32.)

    - +     @@ -343,7 +345,8 @@
    Plot of h(t) = -5t^2 + 20t + 25 along with line segments whose slopes correspond to average rates of change. - +

    Plot of h(t) = -5t^2 + 20t + 25 along with line segments whose slopes correspond to average rates of change.

    +
    @@ -401,7 +404,7 @@ A summary of the information that can be read from the various algebraic forms of a quadratic function - + standard vertex factoredProvided q has 1 or 2 x-intercepts. In the case of just one, we take r = s. @@ -438,9 +441,8 @@

    - +
    - diff --git a/source/sec-changing-transformations-wb.xml b/source/sec-changing-transformations-wb.xml index ad367cf6..df46ddf5 100755 --- a/source/sec-changing-transformations-wb.xml +++ b/source/sec-changing-transformations-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Transformations of Functions - - - - + + + +
    - diff --git a/source/sec-changing-transformations.xml b/source/sec-changing-transformations.xml index bd48b1af..c631d0b9 100755 --- a/source/sec-changing-transformations.xml +++ b/source/sec-changing-transformations.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Transformations of Functions @@ -44,7 +42,7 @@ In Preview Activity, we investigate the effects of the constants a, b, and c in generating the function g(x) = af(x-b) + c in the context of already knowing the function f.

    - + @@ -69,9 +67,11 @@
    Interactive vertical translations demonstration (in the HTML version only). - - - + +

    Interactive vertical translations demonstration (in the HTML version only).

    + +     +

    Move the sliderHuge thanks to the amazing David Austin for making these interactive javascript graphics for the text. by clicking and dragging on the red point to see how changing a affects the graph of y = f(x) + a, which appears in blue. The graph of y = f(x) will appear in grey and remain fixed. @@ -87,11 +87,13 @@

    A vertical translation, g, of the function y = f(x) = |x|. - +

    A vertical translation, g, of the function y = f(x) = |x|.

    +
    A horizontal translation, h, of a different function y = f(x). - +

    A horizontal translation, h, of a different function y = f(x).

    +
    @@ -114,9 +116,11 @@
    Interactive horizontal translations demonstration (in the HTML version only). - - - + +

    Interactive horizontal translations demonstration (in the HTML version only).

    + +     +

    Move the slider by clicking and dragging on the red point to see how changing b affects the graph of y = f(x-b), which appears in blue. The graph of y = f(x) will appear in grey and remain fixed. @@ -141,7 +145,7 @@ We emphasize that in the horizontal translation h(x) = f(x-b), if b \gt 0 the graph of h lies b units to the right of f, while if b \lt 0, h lies b units to the left of f.

    - + @@ -167,11 +171,13 @@
    The parent function y = f(x) along with two different vertical stretches, v and u. - +

    The parent function y = f(x) along with two different vertical stretches, v and u.

    +
    The parent function y = f(x) along with a vertical reflection, z, and a corresponding stretch, w. - +

    The parent function y = f(x) along with a vertical reflection, z, and a corresponding stretch, w.

    +
    @@ -197,9 +203,11 @@
    Interactive vertical scaling demonstration (in the HTML version only). - - - + +

    Interactive vertical scaling demonstration (in the HTML version only).

    + +     +

    Move the slider by clicking and dragging on the red point to see how changing c affects the graph of y = cf(x), which is shown in blue. The graph of y = f(x) will appear in grey and remain fixed. @@ -224,7 +232,7 @@

    - + @@ -263,11 +271,13 @@
    The parent function y = r(x). - +

    The parent function y = r(x).

    +
    The parent function y = r(x) along with the horizontal shift y = p(x) = r(x+1). - +

    The parent function y = r(x) along with the horizontal shift y = p(x) = r(x+1).

    +
    @@ -278,11 +288,13 @@
    The function y = q(x) = 2p(x) = 2r(x+1) along with graphs of p and r. - +

    The function y = q(x) = 2p(x) = 2r(x+1) along with graphs of p and r.

    +
    The function y = m(x) = q(x)-1 = 2r(x+1) - 1 along with graphs of q, p and r. - +

    The function y = m(x) = q(x)-1 = 2r(x+1) - 1 along with graphs of q, p and r.

    +
    @@ -298,7 +310,7 @@ The quantity 2r(x+1) - 1 multiplies the function r(x+1) by 2 first (the stretch) and then the vertical shift follows; the quantity 2[r(x+1) - 1] shifts the function r(x+1) down 1 unit first, and then executes a vertical stretch by a factor of 2. In the latter scenario, the point (1,-1) that lies on r(x+1) gets transformed first to (1,-2) and then to (1,-4), which is not the same as the point (1,-3) that lies on m(x) = 2r(x+1) - 1.

    - + @@ -320,9 +332,8 @@

    - +
    - diff --git a/source/sec-circular-sine-cosine-wb.xml b/source/sec-circular-sine-cosine-wb.xml index d4abfb46..c04e6f7a 100755 --- a/source/sec-circular-sine-cosine-wb.xml +++ b/source/sec-circular-sine-cosine-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    The Sine and Cosine Functions - - - - + + + +
    - diff --git a/source/sec-circular-sine-cosine.xml b/source/sec-circular-sine-cosine.xml index 02607df9..5d0ce7c7 100755 --- a/source/sec-circular-sine-cosine.xml +++ b/source/sec-circular-sine-cosine.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    The Sine and Cosine Functions @@ -43,14 +41,15 @@
    The unit circle with 16 labeled special points. - +

    The unit circle with 16 labeled special points.

    +

    You can also use the Desmos file at http://gvsu.edu/s/0xt to review and study the special points on the unit circle.

    - + @@ -76,7 +75,8 @@

    The definition of the sine of an angle t. - +

    The definition of the sine of an angle t.

    +
    @@ -153,7 +153,8 @@
    Plot of the sine function on the interval [-\frac{\pi}{4}, \frac{7\pi}{4}]. - +

    Plot of the sine function on the interval [-\frac{\pi}{4}, \frac{7\pi}{4}].

    +
    @@ -180,7 +181,8 @@

    The definition of the cosine of an angle t. - +

    The definition of the cosine of an angle t.

    +
    @@ -189,7 +191,7 @@ Again because of the correspondence between the radian measure of an angle and arc length along the unit circle, we can view the value of \cos(t) as tracking the x-coordinate of a point traversing the unit circle clockwise a distance of t units along the circle from (1,0). We now use the data and information we have developed about the unit circle to build a table of values of \cos(t) as well as a graph of the curve it generates.

    - +

    As we work with the sine and cosine functions, it's always helpful to remember their definitions in terms of the unit circle and the motion of a point traversing the circle. At http://gvsu.edu/s/0xe you can explore and investigate a helpful Desmos animation that shows how this motion around the circle generates each of the respective graphs. @@ -248,7 +250,8 @@

    Graphs of the sine and cosine functions. - +

    Graphs of the sine and cosine functions.

    +

    @@ -281,7 +284,7 @@ There are additional trends and patterns in the two functions' graphs that we explore further in the following activity.

    - + @@ -299,7 +302,7 @@ It takes substantial and sophisticated mathematics to enable a computational device to evaluate the sine and cosine functions at any value we want; the algorithms involve an idea from calculus known as an infinite series. While your computational device is powerful, it's both helpful and important to understand the meaning of these values on the unit circle and to remember the special points for which we know the outputs of the sine and cosine functions exactly.

    - + @@ -326,9 +329,8 @@

    - +
    - diff --git a/source/sec-circular-sinusoidal-wb.xml b/source/sec-circular-sinusoidal-wb.xml index 9a3b86ba..89e4c442 100755 --- a/source/sec-circular-sinusoidal-wb.xml +++ b/source/sec-circular-sinusoidal-wb.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ -
    Sinusoidal Functions - - - - - + + + + +
    - diff --git a/source/sec-circular-sinusoidal.xml b/source/sec-circular-sinusoidal.xml index 9e44caec..e55806db 100755 --- a/source/sec-circular-sinusoidal.xml +++ b/source/sec-circular-sinusoidal.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Sinusoidal Functions @@ -39,7 +37,7 @@ 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.

    - + @@ -65,7 +63,8 @@
    A sequence of transformations of y = \cos(t). - +

    A sequence of transformations of y = \cos(t).

    +

    @@ -80,7 +79,7 @@ For example, in Figure, the anchor point (b,a+c) on y = a\cos(t-b)+c corresponds to the starting point (0,1) on y = \cos(t).

    - +     @@ -98,9 +97,11 @@
    Interactive horizontal scaling demonstration (in the HTML version only). - - - + +

    Interactive horizontal scaling demonstration (in the HTML version only).

    + +     +

    Move the slider by clicking and dragging on the red point to see how changing k affects the graph of y = f(kt). The graph of y = f(t) will appear in grey and remain fixed. @@ -137,7 +138,7 @@ While we will soon focus on horizontal stretches of the sine and cosine functions for the remainder of this section, it's important to note that horizontal scaling follows the same principles for any function we choose.

    - + @@ -153,7 +154,8 @@
    A plot of the parent function, f(t) = \sin(t) (dashed, in gray), and the transformed function g(t) = f(2t) = \sin(2t) (in blue). - +

    A plot of the parent function, f(t) = \sin(t) (dashed, in gray), and the transformed function g(t) = f(2t) = \sin(2t) (in blue).

    +

    @@ -162,7 +164,8 @@

    A plot of the parent function, f(t) = \sin(t) (dashed, in gray), and the transformed function h(t) = f(\frac{1}{2}t) = \sin(\frac{1}{2}t) (in blue). - +

    A plot of the parent function, f(t) = \sin(t) (dashed, in gray), and the transformed function h(t) = f(\frac{1}{2}t) = \sin(\frac{1}{2}t) (in blue).

    +

    @@ -187,9 +190,9 @@ Thus, if we know the k-value from the given function, we can deduce the period. If instead we know the desired period, we can determine k by the rule k = \frac{2\pi}{P}.

    - + - + @@ -224,9 +227,8 @@

    - +
    - diff --git a/source/sec-circular-traversing-wb.xml b/source/sec-circular-traversing-wb.xml index 5e69f73f..6625ff6b 100755 --- a/source/sec-circular-traversing-wb.xml +++ b/source/sec-circular-traversing-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Traversing Circles - - - - + + + +
    - diff --git a/source/sec-circular-traversing.xml b/source/sec-circular-traversing.xml index 46d55a0c..47c234ab 100755 --- a/source/sec-circular-traversing.xml +++ b/source/sec-circular-traversing.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Traversing Circles @@ -44,14 +42,16 @@
    A snapshot of the motion of a cab moving around a ferris wheel. Reprinted with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved. - +

    A snapshot of the motion of a cab moving around a ferris wheel. + Reprinted with permission from Illuminations by the National Council of Teachers of Mathematics. All rights reserved.

    +

    Because we have two quantities changing in tandem, it is natural to wonder if it is possible to represent one as a function of the other.

    - + @@ -64,7 +64,8 @@
    A point traversing a circle with circumference C = 24. - +

    A point traversing a circle with circumference C = 24.

    +

    @@ -145,7 +146,8 @@

    The height, h, of a point traversing a circle of circumference 24 as a function of distance, d, traversed around the circle. - +

    The height, h, of a point traversing a circle of circumference 24 as a function of distance, d, traversed around the circle.

    +

    @@ -175,7 +177,7 @@ the height of the point is a function of distance traversed and the resulting graph will have the same basic shape as the curve shown in Figure. It also turns out that if we track the location of the x-coordinate of the point on the circle, the x-coordinate is also a function of distance traversed and its curve has a similar shape to the graph of the height of the point (the y-coordinate). Both of these functions are circular functions because they are generated by motion around a circle.

    - + @@ -189,11 +191,13 @@
    A point traversing the circle. - +

    A point traversing the circle.

    +
    Plotting h as a function of d. - +

    Plotting h as a function of d.

    +
    @@ -233,7 +237,7 @@ Circular functions arise as models for important phenomena in the world around us, such as in a harmonic oscillator. harmonic oscillator Consider a mass attached to a spring where the mass sits on a frictionless surface. After setting the mass in motion by stretching or compressing the spring, the mass will oscillate indefinitely back and forth, and its distance from a fixed point on the surface turns out to be given by a circular function.

    - + @@ -265,11 +269,13 @@
    Comparing the average rate of change over 1/8 the circumference. - +

    Comparing the average rate of change over 1/8 the circumference.

    +
    Comparing the average rate of change over 1/20 the circumference. - +

    Comparing the average rate of change over 1/20 the circumference.

    +
    @@ -285,7 +291,7 @@ We can study the average rate of change not only on the circle itself, but also on a graph such as Figure, and thus make conclusions about where the function is increasing, decreasing, concave up, and concave down.

    - + @@ -312,9 +318,8 @@

    - +
    - diff --git a/source/sec-circular-unit-circle-wb.xml b/source/sec-circular-unit-circle-wb.xml index 89dedb52..a0d0295f 100755 --- a/source/sec-circular-unit-circle-wb.xml +++ b/source/sec-circular-unit-circle-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    The Unit Circle - - - - + + + +
    - diff --git a/source/sec-circular-unit-circle.xml b/source/sec-circular-unit-circle.xml index 683bd470..c85c0dac 100755 --- a/source/sec-circular-unit-circle.xml +++ b/source/sec-circular-unit-circle.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    The Unit Circle @@ -50,11 +48,13 @@
    Coordinates of a point on the unit circle. - +

    Coordinates of a point on the unit circle.

    +
    A point traversing the unit circle. - +

    A point traversing the unit circle.

    +
    @@ -66,7 +66,7 @@ As we work to better understand the unit circle, we will commonly use fractional multiples of \pi as these result in natural distances traveled along the unit circle.

    - + @@ -100,7 +100,7 @@

    - +

    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. @@ -115,7 +115,7 @@ Our in-depth study of the unit circle is motivated by our desire to better understand the behavior of circular functions. Recall that as we traverse a circle, the height of the point moving along the circle generates a function that depends on distance traveled along the circle. Wherever possible, we'd like to be able to identify the exact height of a given point on the unit circle. Two special right triangles enable us to locate exactly an important collection of points on the unit circle.

    - +

    Our work in Activity enables us to identify exactly the location of 12 special points on the unit circle. In part (d) of the activity, we located the three noted points in Figure along with their respective radian measures. By symmetry across the coordinate axes and thinking about the signs of coordinates in the other three quadrants, we can now identify all of the coordinates of the remaining 9 points. @@ -123,7 +123,8 @@

    The unit circle with 16 special points whose location we can determine exactly. - +

    The unit circle with 16 special points whose location we can determine exactly.

    +

    @@ -161,7 +162,7 @@

    - + @@ -192,9 +193,8 @@

    - +
    - diff --git a/source/sec-exp-e-wb.xml b/source/sec-exp-e-wb.xml index cab5cd72..590ca36e 100755 --- a/source/sec-exp-e-wb.xml +++ b/source/sec-exp-e-wb.xml @@ -1,4 +1,3 @@ - @@ -12,13 +11,11 @@ -
    The special number <m>e</m> - - - + + +
    - diff --git a/source/sec-exp-e.xml b/source/sec-exp-e.xml index d4e52d27..2a73a8bc 100755 --- a/source/sec-exp-e.xml +++ b/source/sec-exp-e.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    The special number <m>e</m> @@ -37,14 +35,15 @@
    Plots of four different exponential functions of form b^t with b \gt 1. - +

    Plots of four different exponential functions of form b^t with b \gt 1.

    +

    Because the point (0,1) lies on the graph of each of the four functions in Figure, the functions cannot be vertical scalings of one another. However, it is possible that the functions are horizontal scalings of one another. This leads us to a natural question: might it be possible to find a single exponential function with a special base, say e, for which every other exponential function f(t) = b^t can be expressed as a horizontal scaling of E(t) = e^t?

    - + @@ -112,12 +111,12 @@ 4 - - - - - - + + + + + + t @@ -136,12 +135,12 @@ 7.389 - - - - - - + + + + + + t @@ -163,7 +162,8 @@
    Plot of e^t along with 2^t and 3^t. - +

    Plot of e^t along with 2^t and 3^t.

    +
    @@ -182,7 +182,7 @@ Like 2^t and 3^t, the function e^t passes through (0,1) is always increasing and always concave up, and its range is the set of all positive real numbers.

    - + @@ -202,12 +202,13 @@

    - Given b \gt 0, we can always find a corresponding value of k such that e^k = b because the function f(t) = e^t passes the , as seen in Figure. + Given b \gt 0, we can always find a corresponding value of k such that e^k = b because the function f(t) = e^t passes the , as seen in Figure.

    A plot of f(t) = e^t along with several choices of positive constants b viewed on the vertical axis. - +

    A plot of f(t) = e^t along with several choices of positive constants b viewed on the vertical axis.

    +

    @@ -218,7 +219,7 @@ It follows that the function f(t) = e^t has an inverse function, and hence there must be some other function g such that writing y = f(t) is the same as writing t = g(y). This important function g will be developed in Section and will enable us to find the value of k exactly for a given b. For now, we are content to work with these observations graphically and to hence find estimates for the value of k.

    - + @@ -238,9 +239,8 @@ - +
    - diff --git a/source/sec-exp-growth-wb.xml b/source/sec-exp-growth-wb.xml index 2ad7800e..ebd365aa 100755 --- a/source/sec-exp-growth-wb.xml +++ b/source/sec-exp-growth-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Exponential Growth and Decay - - - - + + + +
    - diff --git a/source/sec-exp-growth.xml b/source/sec-exp-growth.xml index 2256884a..eefb5ee3 100755 --- a/source/sec-exp-growth.xml +++ b/source/sec-exp-growth.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Exponential Growth and Decay @@ -69,7 +67,7 @@ Of course, in 3 years at 10%, the original investment P will have grown to 1.1^3 P. Here we see a new kind of pattern developing: annual growth of 10% is leading to powers of the base 1.1, where the power to which we raise 1.1 corresponds to the number of years the investment has grown. We often call this phenomenon exponential growth. exponential growthintroduction

    - + @@ -102,7 +100,7 @@ We explore the properties of functions of form f(t) = ab^t further in Activity.

    - +     @@ -205,12 +203,13 @@
    Plot of p(t) = ab^t that passes through (2,11) and (5,18). - +

    Plot of p(t) = ab^t that passes through (2,11) and (5,18).

    +
    - + @@ -224,16 +223,18 @@
    The exponential function f. - +

    The exponential function f.

    +
    The exponential function g. - +

    The exponential function g.

    +

    - If we consider an exponential function f with a growth factor b > 1, such as the function pictured in Figure, then the function is always increasing because higher powers of b are greater than lesser powers (for example, (1.2)^3 \gt (1.2)^2). On the other hand, if 0 \lt b \lt 1, then the exponential function will be decreasing because higher powers of positive numbers less than 1 get smaller (e.g., (0.9)^3 \lt (0.9)^2), as seen for the exponential function in Figure. + If we consider an exponential function f with a growth factor b > 1, such as the function pictured in Figure, then the function is always increasing because higher powers of b are greater than lesser powers (for example, (1.2)^3 \gt (1.2)^2). On the other hand, if 0 \lt b \lt 1, then the exponential function will be decreasing because higher powers of positive numbers less than 1 get smaller (e.g., (0.9)^3 \lt (0.9)^2), as seen for the exponential function in Figure.

    @@ -333,7 +334,7 @@ Observe how a function's average rate of change helps us classify the function's behavior on an interval: whether the average rate of change is always positive or always negative on the interval enables us to say if the function is always increasing or always decreasing, and then how the average rate of change itself changes enables us to potentially say how the function is increasing or decreasing through phrases such as decreasing at an increasing rate.

    - + @@ -391,9 +392,8 @@

    - +
    - diff --git a/source/sec-exp-log-properties-wb.xml b/source/sec-exp-log-properties-wb.xml index 3cd51d30..abc39f3b 100755 --- a/source/sec-exp-log-properties-wb.xml +++ b/source/sec-exp-log-properties-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Properties and applications of logarithmic functions - - - - + + + +
    - diff --git a/source/sec-exp-log-properties.xml b/source/sec-exp-log-properties.xml index 68932c77..a702ada1 100755 --- a/source/sec-exp-log-properties.xml +++ b/source/sec-exp-log-properties.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Properties and applications of logarithmic functions @@ -40,7 +38,7 @@ 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.

    - + @@ -167,7 +165,7 @@ The approach used in Example works in a wide range of settings: any time we have an exponential equation of the form p \cdot q^t + r = s, we can solve for t by first isolating the exponential expression q^t and then by taking the natural logarithm of both sides of the equation.

    - + @@ -180,11 +178,13 @@
    The natural exponential and natural logarithm functions on the interval [-3,3]. - +

    The natural exponential and natural logarithm functions on the interval [-3,3].

    +
    The natural exponential and natural logarithm functions on the interval [-15,15]. - +

    The natural exponential and natural logarithm functions on the interval [-15,15].

    +
    @@ -229,7 +229,7 @@ While the natural exponential function and the natural logarithm (and transformations of these functions) are connected and have certain similar properties, it's also important to be able to distinguish between behavior that is fundamentally exponential and fundamentally logarithmic.

    - + @@ -279,7 +279,7 @@ - + @@ -321,9 +321,8 @@ - +
    - diff --git a/source/sec-exp-log-wb.xml b/source/sec-exp-log-wb.xml index 0f806ad0..74ec9b05 100755 --- a/source/sec-exp-log-wb.xml +++ b/source/sec-exp-log-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    What a logarithm is - - - - + + + +
    - diff --git a/source/sec-exp-log.xml b/source/sec-exp-log.xml index dc65275a..9ede1c5f 100755 --- a/source/sec-exp-log.xml +++ b/source/sec-exp-log.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    What a logarithm is @@ -45,7 +43,7 @@ More formally, recall that a function y = f(x) (where f : A \to B) has an inverse function if and only if there exists another function g : B \to A such that g(f(x)) = x for every x in A, and f(g(y)) = y for every y in B. We know that given a function f, we can use the to determine whether or not f has an inverse function. Finally, whenever a function f has an inverse function, we call its inverse function f^{-1} and know that the two equations y = f(x) and x = f^{-1}(y) say the same thing from different perspectives.

    - + @@ -141,7 +139,7 @@ It's important to note that the logarithm function produces exact values. For instance, if we want to solve the equation 10^t = 5, then it follows that t = \log_{10}(5) is the exact solution to the equation. Like \sqrt{2} or \cos(1), \log_{10}(5) is a number that is an exact value. A computational device can give us a decimal approximation, and we normally want to distinguish between the exact value and the approximate one. For the three different numbers here, \sqrt{2} \approx 1.414, \cos(1) \approx 0.540, and \log_{10}(5) \approx 0.699.

    - + @@ -172,7 +170,7 @@ The former equation is true since the power to which we raise e to get e^{-1} is -1; the latter equation is true since when we raise e to the power to which we raise e to get 2, we get 2. The key relationships between the natural exponential and the natural logarithm function are investigated in Activity.

    - + @@ -202,7 +200,7 @@ In modeling important phenomena using exponential functions, we will frequently encounter equations where the variable is in the exponent, like in Example where we had to solve e^k = 3. It is in this context where logarithms find one of their most powerful applications. Activity provides some opportunities to practice solving equations involving the natural base, e, and the natural logarithm.

    - + @@ -240,9 +238,8 @@ - +
    - diff --git a/source/sec-exp-modeling-wb.xml b/source/sec-exp-modeling-wb.xml index cd3b4816..06da3d64 100755 --- a/source/sec-exp-modeling-wb.xml +++ b/source/sec-exp-modeling-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Modeling with exponential functions - - - - + + + +
    - diff --git a/source/sec-exp-modeling.xml b/source/sec-exp-modeling.xml index fdf2c683..a771a0f8 100755 --- a/source/sec-exp-modeling.xml +++ b/source/sec-exp-modeling.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Modeling with exponential functions @@ -71,13 +69,13 @@ 144 - - - - - - - + + + + + + + 18 @@ -102,7 +100,8 @@
    A plot of the data in Table. - +

    A plot of the data in Table.

    +
    @@ -110,7 +109,7 @@ In one sense, the data looks exponential: the points appear to lie on a curve that is always decreasing and decreasing at an increasing rate. However, we know that the function can't have the form f(t) = ab^t because such a function's range is the set of all positive real numbers, and it's impossible for the coffee's temperature to fall below room temperature (71^\circ). It is natural to wonder if a function of the form g(t) = ab^t + c will work. Thus, in order to find a function that fits the data in a situation such as Figure, we begin by investigating and understanding the roles of a, b, and c in the behavior of g(t) = ab^t + c.

    - + @@ -211,7 +210,8 @@
    Plots of p(t) = 2^t and q(t) = 2^{-t}. - +

    Plots of p(t) = 2^t and q(t) = 2^{-t}.

    +

    @@ -229,11 +229,13 @@

    Plot of f(t) = ab^t. - +

    Plot of f(t) = ab^t.

    +
    Plot of g(t) = ab^t+c. - +

    Plot of g(t) = ab^t+c.

    +
    @@ -265,7 +267,7 @@ It's an important skill to be able to look at an exponential function of the form g(t) = ab^t + c and form an accurate mental picture of the graph's main features in light of the values of a, b, and c.

    - + @@ -342,11 +344,13 @@
    Plot of f(t) = 103.503 (0.974)^t. - +

    Plot of f(t) = 103.503 (0.974)^t.

    +
    Plot of F(t) = 103.503 (0.974)^t + 71. - +

    Plot of F(t) = 103.503 (0.974)^t + 71.

    +
    @@ -358,9 +362,9 @@ Our preceding work with the coffee data can be done similarly with data for any cooling or warming object whose temperature initially differs from its surroundings. Indeed, it is possible to show that Newton's Law of Cooling implies that the object's temperature is given by a function of the form F(t) = ab^t + c.

    - + - + @@ -391,10 +395,9 @@

    - +
    - diff --git a/source/sec-exp-temp-pop-wb.xml b/source/sec-exp-temp-pop-wb.xml index 2ee203fb..92265974 100755 --- a/source/sec-exp-temp-pop-wb.xml +++ b/source/sec-exp-temp-pop-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Modeling temperature and population - - - - + + + +
    - diff --git a/source/sec-exp-temp-pop.xml b/source/sec-exp-temp-pop.xml index caae7d99..61807333 100755 --- a/source/sec-exp-temp-pop.xml +++ b/source/sec-exp-temp-pop.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Modeling temperature and population @@ -58,7 +56,7 @@ In Preview Activity, we revisit some key algebraic ideas with exponential and logarithmic equations in preparation for using these concepts in models for temperature and population.

    - + @@ -100,7 +98,7 @@

    - + @@ -124,9 +122,9 @@ Since k \gt 0, it follows that e^{-kt} \to 0 as t increases without bound, and thus the denominator of P approaches 1 as time goes on. Thus, we observe that P(t) tends to A as t increases without bound. We sometimes refer to A as the carrying capacity of the population.

    - + - + @@ -150,9 +148,8 @@ - +
    - diff --git a/source/sec-poly-infty-wb.xml b/source/sec-poly-infty-wb.xml index 2c78e2a6..ee028408 100755 --- a/source/sec-poly-infty-wb.xml +++ b/source/sec-poly-infty-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Infinity, limits, and power functions - - - - + + + +
    - diff --git a/source/sec-poly-infty.xml b/source/sec-poly-infty.xml index 4aa3c1bd..7c7f2915 100755 --- a/source/sec-poly-infty.xml +++ b/source/sec-poly-infty.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Infinity, limits, and power functions @@ -57,7 +55,7 @@ In Preview, we review some familiar functions and portions of their behavior that involve \infty.

    - + @@ -74,11 +72,13 @@
    Plots of y = e^t and y = e^{-t}. - +

    Plots of y = e^t and y = e^{-t}.

    +
    Plots of y = e^t and y = \ln(t). - +

    Plots of y = e^t and y = \ln(t).

    +
    @@ -127,7 +127,7 @@ For now, we are going to focus on the long-range behavior of certain basic, familiar functions and work to understand how they behave as the input increases or decreases without bound. Above we've used the input variable t in most of our previous work; going forward, we'll regularly use x as well.

    - + @@ -153,7 +153,7 @@ We first focus on the case where p is a natural number (that is, a positive whole number).

    - +

    In the situation where the power p is a negative integer (i.e., a negative whole number), power functions behave very differently. This is because of the property of exponents that states @@ -163,7 +163,7 @@ so for a power function such as p(x) = x^{-2}, we can equivalently consider p(x) = \frac{1}{x^2}. Note well that for these functions, their domain is the set of all real numbers except x = 0. Like with power functions with positive whole number powers, we want to know how power functions with negative whole number powers behave as x increases without bound, as well as how the functions behave near x = 0.

    - + @@ -322,9 +322,8 @@ - +
    - diff --git a/source/sec-poly-polynomial-applications-wb.xml b/source/sec-poly-polynomial-applications-wb.xml index 50e22d6f..629cb0ce 100755 --- a/source/sec-poly-polynomial-applications-wb.xml +++ b/source/sec-poly-polynomial-applications-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Modeling with polynomial functions - - - - + + + +
    - diff --git a/source/sec-poly-polynomial-applications.xml b/source/sec-poly-polynomial-applications.xml index c44ec26c..fd4e0e58 100755 --- a/source/sec-poly-polynomial-applications.xml +++ b/source/sec-poly-polynomial-applications.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Modeling with polynomial functions @@ -44,7 +42,7 @@ In other similar situations where we consider the volume of a box, tank, or other three-dimensional container, polynomial functions frequently arise. To develop a model function that represents a physical situation, we almost always begin by drawing one or more diagrams of the situation and then introduce one or more variables to represent quantities that are changing. From there, we explore relationships that are present and work to express one of the quantities in terms of the other(s).

    - + @@ -60,11 +58,13 @@
    A rectangular box. - +

    A rectangular box.

    +
    A circular cylinder. - +

    A circular cylinder.

    +
    @@ -91,9 +91,9 @@ Each of the volume and surface area equations (Equation, Equation, Equation, and Equation) involve only multiplication and addition, and thus have the potential to result in polynomial functions. At present, however, each of these equations involves at least two variables. The inclusion of additional constraints can enable us to use these formulas to generate polynomial functions of a single variable.

    - + - + @@ -115,11 +115,13 @@
    A cubic Bezier curve with control points in gray. - +

    A cubic Bezier curve with control points in gray.

    +
    The letter S in Palatino font, generated by Bezier curves. - +

    The letter S in Palatino font, generated by Bezier curves.

    +
    @@ -135,7 +137,7 @@ Another important application of polynomial functions is found in how they can be used to approximate the sine and cosine functions.

    - + @@ -163,9 +165,8 @@ - +
    - diff --git a/source/sec-poly-polynomials-wb.xml b/source/sec-poly-polynomials-wb.xml index 126dc197..b772da51 100755 --- a/source/sec-poly-polynomials-wb.xml +++ b/source/sec-poly-polynomials-wb.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,12 @@ -
    Polynomials - - - - + + + +
    - - diff --git a/source/sec-poly-polynomials.xml b/source/sec-poly-polynomials.xml index 83f4c4ec..ba112d9e 100755 --- a/source/sec-poly-polynomials.xml +++ b/source/sec-poly-polynomials.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Polynomials @@ -84,7 +82,7 @@ Since a polynomial is simply a sum of constant multiples of various power functions with positive integer powers, we often refer to those individual terms by referring to their individual degrees: the linear term, the quadratic term, and so on. In addition, since the domain of any power function of the form p(x) = x^n where n is a positive whole number is the set of all real numbers, it's also true the the domain of any polynomial function is the set of all real numbers.

    - + @@ -120,11 +118,13 @@
    Plot of a degree 7 polynomial function p. - +

    Plot of a degree 7 polynomial function p.

    +
    Plot of the same degree 7 polynomial function p, but zoomed out. - +

    Plot of the same degree 7 polynomial function p, but zoomed out.

    +
    @@ -140,7 +140,7 @@

    - + @@ -184,7 +184,8 @@
    A sign chart for the polynomial function p(x) = k(x-1)(x-a)(x-b). - +

    A sign chart for the polynomial function p(x) = k(x-1)(x-a)(x-b).

    +

    @@ -197,12 +198,13 @@

    The graph of the polynomial function p(x) = k(x-1)(x-a)(x-b). - +

    The graph of the polynomial function p(x) = k(x-1)(x-a)(x-b).

    +
    - + @@ -222,7 +224,8 @@
    A plot of g(x) = x^3(x-1) with zero x = 0 of multiplicity 3 and x = 1 of multiplicity 1. - +

    A plot of g(x) = x^3(x-1) with zero x = 0 of multiplicity 3 and x = 1 of multiplicity 1.

    +
    @@ -233,11 +236,13 @@
    A plot of g(x) = x^3(x-1) zoomed in on the zero x = 0 of multiplicity 3. - +

    A plot of g(x) = x^3(x-1) zoomed in on the zero x = 0 of multiplicity 3.

    +
    -
    +
    A plot of g(x) = x^3(x-1) zoomed in on the zero x = 1 of multiplicity 1. - +

    A plot of g(x) = x^3(x-1) zoomed in on the zero x = 1 of multiplicity 1.

    +
    @@ -248,11 +253,13 @@
    Plot of h(x) = x^2 (x-1)^2 with zeros x = 0 and x = 1 of multiplicity 2. - +

    Plot of h(x) = x^2 (x-1)^2 with zeros x = 0 and x = 1 of multiplicity 2.

    +
    Plot of k(x) = x^2(x-1)(x+1) with zeros x = 0 of multiplicity 2 and x = -1 and x = 1 of multiplicity 1. - +

    Plot of k(x) = x^2(x-1)(x+1) with zeros x = 0 of multiplicity 2 and x = -1 and x = 1 of multiplicity 1.

    +
    @@ -260,9 +267,10 @@ Finally, if we consider m(x) = (x+1)x(x-1)(x-2), which has 4 distinct real zeros each of multiplicity 1, we observe in Figure that zooming in on each zero individually, the function demonstrates approximately linear behavior as it passes through the x-axis.

    -
    +
    Plot of m(x) = (x+1)x(x-1)(x-2) with 4 distinct zeros of multiplicity 1. - +

    Plot of m(x) = (x+1)x(x-1)(x-2) with 4 distinct zeros of multiplicity 1.

    +

    @@ -277,7 +285,7 @@

    - + @@ -319,10 +327,8 @@

    - +
    - - diff --git a/source/sec-poly-rational-features-wb.xml b/source/sec-poly-rational-features-wb.xml index 27eeb3f0..b5894d02 100755 --- a/source/sec-poly-rational-features-wb.xml +++ b/source/sec-poly-rational-features-wb.xml @@ -1,4 +1,3 @@ - @@ -12,13 +11,11 @@ -
    Key features of rational functions - - - + + +
    - diff --git a/source/sec-poly-rational-features.xml b/source/sec-poly-rational-features.xml index 26a9eae9..86a60b55 100755 --- a/source/sec-poly-rational-features.xml +++ b/source/sec-poly-rational-features.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Key features of rational functions @@ -45,7 +43,7 @@ Connected to these questions, we want to understand both where a rational function's output value is zero, as well as where the function is undefined. In addition, from the behavior of simple rational power functions such as \frac{1}{x}, we expect that rational functions may not only have horizontal asymptotes (as investigated in Section), but also vertical asymptotes. At first glance, these questions about zeros and vertical asymptotes of rational functions may appear to be elementary ones whose answers simply depend on where the numerator and denominator of the rational function are zero. But in fact, rational functions often admit very subtle behavior that can escape the human eye and the graph generated by a computer.

    - + @@ -98,15 +96,18 @@
    A plot of r(x) = \frac{x^2 - 1}{x^2 - 3x - 4}. - +

    A plot of r(x) = \frac{x^2 - 1}{x^2 - 3x - 4}.

    +
    Zooming in on r(x) near x = -1. - +

    Zooming in on r(x) near x = -1.

    +
    How the graph of r(x) should actually appear near x = -1. - +

    How the graph of r(x) should actually appear near x = -1.

    +
    @@ -130,10 +131,10 @@ \frac{0.002001}{0.005001} \approx 0.4001 - - - - + + + + x @@ -215,7 +216,7 @@

    - + @@ -267,11 +268,13 @@
    The sign chart for q. - +

    The sign chart for q.

    +
    A plot of q. - +

    A plot of q.

    +
    @@ -285,7 +288,7 @@ To find a formula for a rational function with certain properties, we can reason in ways that are similar to our work with polynomials. Since the rational function must have a polynomial expression in both the numerator and denominator, by thinking about where the numerator and denominator must be zero, we can often generate a formula whose graph will satisfy the desired properties.

    - + @@ -314,9 +317,8 @@ - +
    - diff --git a/source/sec-poly-rational-wb.xml b/source/sec-poly-rational-wb.xml index f59f5001..f5363b32 100755 --- a/source/sec-poly-rational-wb.xml +++ b/source/sec-poly-rational-wb.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ -
    Rational Functions - - - - - + + + + +
    - diff --git a/source/sec-poly-rational.xml b/source/sec-poly-rational.xml index db01ad7a..90a15951 100755 --- a/source/sec-poly-rational.xml +++ b/source/sec-poly-rational.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Rational Functions @@ -64,7 +62,7 @@ Ratios of polynomial functions arise in several different important circumstances. Sometimes we are interested in what happens when the denominator approaches 0, which makes the overall ratio undefined. In other situations, we may want to know what happens in the long term and thus consider what happens when the input variable increases without bound.

    - + @@ -121,9 +119,9 @@ since 2500 times a quantity approaching 0 will still approach 0 as x increases.

    - + - +

    We summarize and generalize the results of Activity and Activity as follows. @@ -209,7 +207,7 @@ We note that when it comes to determining the domain of a rational function, the numerator is irrelevant: all that matters is where the denominator is 0.

    - + @@ -268,7 +266,7 @@ - + @@ -313,9 +311,8 @@ - +
    - diff --git a/source/sec-trig-finding-angles-wb.xml b/source/sec-trig-finding-angles-wb.xml index bc81726d..87c83edb 100755 --- a/source/sec-trig-finding-angles-wb.xml +++ b/source/sec-trig-finding-angles-wb.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ -
    Finding Angles - - - - - + + + + +
    - diff --git a/source/sec-trig-finding-angles.xml b/source/sec-trig-finding-angles.xml index 1213a462..eb6bf7c5 100755 --- a/source/sec-trig-finding-angles.xml +++ b/source/sec-trig-finding-angles.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Finding Angles @@ -43,7 +41,7 @@ say the exact same thing from two different perspectives, and that we read \arccos(x) as the angle whose cosine is x.

    - + @@ -67,7 +65,8 @@
    A right triangle with one known leg and known hypotenuse. - +

    A right triangle with one known leg and known hypotenuse.

    +
    @@ -88,7 +87,7 @@ - + @@ -98,11 +97,11 @@ Now that we have developed the (restricted) sine, cosine, and tangent functions and their respective inverses, in any setting in which we have a right triangle together with one side length and any one additional piece of information (another side length or a non-right angle measurement), we can determine all of the remaining pieces of the triangle. In the activities that follow, we explore these possibilities in a variety of different applied contexts.

    - + - + - + @@ -128,7 +127,8 @@
    Finding an angle from knowing the legs in a right triangle. - +

    Finding an angle from knowing the legs in a right triangle.

    +
    @@ -144,9 +144,8 @@ - +
    - diff --git a/source/sec-trig-inverse-wb.xml b/source/sec-trig-inverse-wb.xml index ebb11271..6ee9c42e 100755 --- a/source/sec-trig-inverse-wb.xml +++ b/source/sec-trig-inverse-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Inverses of trigonometric functions - - - - + + + +
    - diff --git a/source/sec-trig-inverse.xml b/source/sec-trig-inverse.xml index b4016bf4..e80ca146 100755 --- a/source/sec-trig-inverse.xml +++ b/source/sec-trig-inverse.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Inverses of trigonometric functions @@ -100,7 +98,7 @@ trigonometric functions so that we can discuss inverses in a meaningful way.

    - + @@ -163,7 +161,7 @@

    - +

    ADD ALT TEXT TO THIS IMAGE

     @@ -172,7 +170,7 @@ Just as the natural logarithm function allowed us to rewrite exponential equations in an equivalent way (for instance, y = e^t and t = \ln(y) say the exact same thing), the arccosine function allows us to do likewise for certain angles and cosine outputs. For instance, saying \cos(\frac{\pi}{2}) = 0 is the same as writing \frac{\pi}{2} = \arccos(0), which reads \frac{\pi}{2} is the angle whose cosine is 0. Indeed, these relationships are reflected in the plot above, where we see that any point (a,b) that lies on the graph of y = \cos(t) corresponds to the point (b,a) that lies on the graph of y = \arccos(t).

    - + @@ -182,7 +180,7 @@ We can develop an inverse function for a restricted version of the sine function in a similar way. As with the cosine function, we need to choose an interval on which the sine function is always increasing or always decreasing in order to have the function pass the horizontal line test. The standard choice is the domain [-\frac{\pi}{2}, \frac{\pi}{2}] on which f(t) = \sin(t) is increasing and attains all of the values in the range of the sine function. Thus, we consider f(t) = \sin(t) so that f : [-\frac{\pi}{2}, \frac{\pi}{2}] \to [-1,1] and hence define the corresponding arcsine function.

    - + inverse trigonometric functionsarcsine

    @@ -195,7 +193,7 @@ - + @@ -206,7 +204,7 @@ Finally, we develop an inverse function for a restricted version of the tangent function. We choose the domain (-\frac{\pi}{2}, \frac{\pi}{2}) on which h(t) = \tan(t) is increasing and attains all of the values in the range of the tangent function.

    - + inverse trigonometric functionsarctangent

    @@ -219,7 +217,7 @@ - + @@ -270,12 +268,14 @@

    The restricted cosine function (in light blue) and its inverse, y = g^{-1}(t) = \arccos(t) (in dark blue). - +

    The restricted cosine function (in light blue) and its inverse, y = g^{-1}(t) = \arccos(t) (in dark blue).

    +
    The restricted sine function (in light blue) and its inverse, y = f^{-1}(t) = \arcsin(t) (in dark blue). - +

    The restricted sine function (in light blue) and its inverse, y = f^{-1}(t) = \arcsin(t) (in dark blue).

    +
    @@ -290,7 +290,8 @@
    The restricted tangent function (in light blue) and its inverse, y = h^{-1}(t) = \arctan(t) (in dark blue). - +

    The restricted tangent function (in light blue) and its inverse, y = h^{-1}(t) = \arctan(t) (in dark blue).

    +
    @@ -298,9 +299,8 @@

    - +
    - diff --git a/source/sec-trig-other-wb.xml b/source/sec-trig-other-wb.xml index b474d4c6..425ac411 100755 --- a/source/sec-trig-other-wb.xml +++ b/source/sec-trig-other-wb.xml @@ -1,4 +1,3 @@ - @@ -12,15 +11,13 @@ -
    Other Trigonometric Functions and Identities - - - - - + + + + +
    - diff --git a/source/sec-trig-other.xml b/source/sec-trig-other.xml index 54987c12..8f3fc525 100755 --- a/source/sec-trig-other.xml +++ b/source/sec-trig-other.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Other Trigonometric Functions and Identities @@ -103,7 +101,8 @@

    A right triangle with angle \theta. - +

    A right triangle with angle \theta.

    +
    @@ -111,7 +110,7 @@ With these three additional trigonometric functions, we now have expressions that address all six possible combinations of two sides of a right triangle in a ratio.

    - + @@ -127,14 +126,15 @@
    A 3-4-5 right triangle. - +

    A 3-4-5 right triangle.

    +

    But we could also view \sin(\alpha) = \frac{3}{5} as \sin(\alpha) = \frac{\frac{3}{5}}{1}, and thus think of the right triangle has having hypotenuse 1 and vertical leg \frac{3}{5}. This triangle is similar to the originally considered 3-4-5 right triangle, but can be viewed as lying within the unit circle. The perspective of the unit circle is particularly valuable when ratios such as \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, and \frac{1}{2} arise in right triangles.

    - + @@ -243,7 +243,8 @@
    A plot of the secant function with special points that come from the unit circle, plus the cosine function (dotted, in light blue). - +

    A plot of the secant function with special points that come from the unit circle, plus the cosine function (dotted, in light blue).

    +

    @@ -274,9 +275,9 @@

    - + - + @@ -328,7 +329,7 @@

    - In calculus, it is also beneficial to know a couple of other standard identities for sums of angles or double angles. sum of two angles identity double angle identity We simply state these identities without justification. For more information about them, see Section 10.4 in College Trigonometry, by Stitz and ZeagerMore information on Stitz and Zeager's free texts can be found at .. + In calculus, it is also beneficial to know a couple of other standard identities for sums of angles or double angles. sum of two angles identity double angle identity We simply state these identities without justification. For more information about them, see Section 10.4 in College Trigonometry, by Stitz and ZeagerMore information on Stitz and Zeager's free texts can be found at ..

    @@ -356,7 +357,7 @@

    - + @@ -390,9 +391,8 @@

    - +
    - diff --git a/source/sec-trig-right-wb.xml b/source/sec-trig-right-wb.xml index 2ba9e42c..a1130f48 100755 --- a/source/sec-trig-right-wb.xml +++ b/source/sec-trig-right-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    Right triangles - - - - + + + +
    - diff --git a/source/sec-trig-right.xml b/source/sec-trig-right.xml index 148cfeda..745176d0 100755 --- a/source/sec-trig-right.xml +++ b/source/sec-trig-right.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    Right triangles @@ -49,11 +47,13 @@
    The values of \cos(t) and \sin(t) as coordinates on the unit circle. - +

    The values of \cos(t) and \sin(t) as coordinates on the unit circle.

    +
    The values of \cos(\theta) and \sin(\theta) as the lengths of the legs of a right triangle. - +

    The values of \cos(\theta) and \sin(\theta) as the lengths of the legs of a right triangle.

    +
    @@ -61,7 +61,7 @@ This right triangle perspective enables us to use the sine and cosine functions to determine missing information in certain right triangles. The field of mathematics that studies relationships among the angles and sides of triangles is called trigonometry. trigonometry In addition, it's important to recall both the Pythagorean Theorem and the Fundamental Trigonometric Identity. Pythagorean Theorem fundamental trigonometric identity The former states that in any right triangle with legs of length a and b and hypotenuse of length c, it follows a^2 + b^2 = c^2. The latter, which is a special case of the Pythagorean Theorem, says that for any angle \theta, \cos^2(\theta) + \sin^2(\theta) = 1.

    - + @@ -99,14 +99,15 @@
    The 5 potential unknowns in a right triangle. - +

    The 5 potential unknowns in a right triangle.

    +

    Because we know the values of the cosine and sine functions from the unit circle, right triangles with hypotentuse 1 are the easiest ones in which to determine missing information. In addition, we can relate any other right triangle to a right triangle with hypotenuse 1 through the concept of similarity. Recall that two triangles are similar similar triangles provided that one is a magnification of the other. More precisely, two triangles are similar whenever there is some constant k such that every side in one triangle is k times as long as the corresponding side in the other and the corresponding angles in the two triangles are equal. An important result from geometry tells us that if two triangles are known to have all three of their corresponding angles equal, then it follows that the two triangles are similar, and therefore their corresponding sides must be proportionate to one another.

    - + @@ -119,7 +120,8 @@
    The roles of r and \theta in a right triangle. - +

    The roles of r and \theta in a right triangle.

    +

    @@ -152,12 +154,12 @@ .

    - +

    ADD ALT TEXT TO THIS IMAGE

     - + @@ -219,7 +221,8 @@
    The given right triangle. - +

    The given right triangle.

    +
    @@ -259,7 +262,7 @@ Example demonstrates that a ratio of values of the sine and cosine function can be needed in order to determine the value of one of the missing sides of a right triangle, and also that we may need to work with two unknown quantities simultaneously in order to determine both of their values.

    - + @@ -289,9 +292,8 @@ - +
    - diff --git a/source/sec-trig-tangent-wb.xml b/source/sec-trig-tangent-wb.xml index 93018d31..06fc0c2b 100755 --- a/source/sec-trig-tangent-wb.xml +++ b/source/sec-trig-tangent-wb.xml @@ -1,4 +1,3 @@ - @@ -12,14 +11,12 @@ -
    The Tangent Function - - - - + + + +
    - diff --git a/source/sec-trig-tangent.xml b/source/sec-trig-tangent.xml index 2bccf567..26204162 100755 --- a/source/sec-trig-tangent.xml +++ b/source/sec-trig-tangent.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ -
    The Tangent Function @@ -48,7 +46,8 @@
    Finding the width of the river. - +

    Finding the width of the river.

    +
    @@ -70,7 +69,7 @@ - + @@ -80,11 +79,13 @@
    An angle t in standard position in the unit circle that intercepts an arc from (1,0) to (a,b). - +

    An angle t in standard position in the unit circle that intercepts an arc from (1,0) to (a,b).

    +
    A right triangle with legs adjacent and opposite angle \theta. - +

    A right triangle with legs adjacent and opposite angle \theta.

    +
    @@ -246,7 +247,8 @@
    A plot of the tangent function together with special points that come from the unit circle. - +

    A plot of the tangent function together with special points that come from the unit circle.

    +

    @@ -302,7 +304,8 @@

    A right triangle with one angle and one leg known. - +

    A right triangle with one angle and one leg known.

    +
    @@ -323,11 +326,11 @@ The tangent function finds a wide range of applications in finding missing information in right triangles where information about one or more legs of the triangle is known.

    - + - + - + @@ -362,9 +365,8 @@

    - +
    - diff --git a/source/titlepage.xml b/source/titlepage.xml index f22959fe..236f7b2d 100644 --- a/source/titlepage.xml +++ b/source/titlepage.xml @@ -1,4 +1,3 @@ - @@ -12,7 +11,6 @@ - - +