+ A
+ Vectors can be decomposed into components- horizontal and vertical. See the diagram below. +
+
+ If the length of the horizontal leg is
+
+
+
+
+
+
+ Which option best describes the vertical component of the vector? +
+
+
+
+
+
+
+ Use your answers to (a) and (b) to write the velocity vector in component form. +
+
+ Use your answers to (a) and (b) to write the velocity vector using the standard unit vectors. Remember that the ball is thrown in the
+ Why would we consider the ball to be thrown in the
+ In order to fly from Oklahoma City to Dayton, OH, a plane must fly
+ What is the ground speed of the plane when the wind begins to blow? +
++ What is the new bearing of the plane? +
+
+ An object is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let
+Identify, create, combine, modify, and apply vectors in two- or three-dimensional space +
\ No newline at end of file diff --git a/source/calculus/source/00-VE/outcomes/02.ptx b/source/calculus/source/00-VE/outcomes/02.ptx new file mode 100644 index 000000000..bab0e056b --- /dev/null +++ b/source/calculus/source/00-VE/outcomes/02.ptx @@ -0,0 +1,4 @@ + ++Compute and apply the dot product and cross product of vectors. +
\ No newline at end of file diff --git a/source/calculus/source/00-VE/outcomes/03.ptx b/source/calculus/source/00-VE/outcomes/03.ptx new file mode 100644 index 000000000..d2e42fefb --- /dev/null +++ b/source/calculus/source/00-VE/outcomes/03.ptx @@ -0,0 +1,4 @@ + ++ Determine information about lines and planes in space, including: distance, angles between objects, and equations in various forms. +
\ No newline at end of file diff --git a/source/calculus/source/00-VE/readiness.ptx b/source/calculus/source/00-VE/readiness.ptx new file mode 100644 index 000000000..c0474b775 --- /dev/null +++ b/source/calculus/source/00-VE/readiness.ptx @@ -0,0 +1,106 @@ + + ++ Before beginning this chapter, you should be able to... +
+Use function notation and evaluate functions
+Review:
Practice:
+Find the domain of a function +
+Review:
Practice:
Determine vertical asymptotes, horizontal asymptotes, and holes (removable discontinuities) of rational functions
+Review:
+Practice:
+Perform basic operations with polynomials
+Review:
+Practice:
+Factor quadratic expressions
+Review:
Practice:
Represent intervals using number lines, inequalities, and interval notation +
+Review:
- A
- Vectors can be decomposed into components- horizontal and vertical. See the diagram below. -
-
- If the length of the horizontal leg is
-
-
-
-
-
-
- Which option best describes the vertical component of the vector? -
-
-
-
-
-
-
- Use your answers to (a) and (b) to write the velocity vector in component form. -
-
- Use your answers to (a) and (b) to write the velocity vector using the standard unit vectors. Remember that the ball is thrown in the
- Why would we consider the ball to be thrown in the
- In order to fly from Oklahoma City to Dayton, OH, a plane must fly
- What is the ground speed of the plane when the wind begins to blow? -
-- What is the new bearing of the plane? -
-
- An object is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let
+ A
+ Vectors can be decomposed into components- horizontal and vertical. See the diagram below. +
+
+ If the length of the horizontal leg is
+
+
+
+
+ Which expression best identifies the direction of the vector, denoted
+
+
+
+
+
+
+ Use trigonometry and/or part (a) to find expressions for the length of
+ A vector
+ Consider the points
+ Write the component form of the vector beginning at
+
+
+
+
+ What is the direction of this vector? +
+
+ Write the component form of the vector beginning at
+
+
+
+
+ What is the direction of this vector? +
+
+ What do you notice about the vectors
+ Now consider the points
+ Find the vector
+ Consider the additional point
+
+
+
+
+ Sketch
+ Now consider the point
+ Sketch
+ Vectors can be added and subtracted like regular numbers. Addition happens component-wise; for example, adding the vectors
+ Let
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ This activity will develop the parallelogram rule, a method for visualizing the addition or subtraction of two vectors. +
+
+ Consider the vectors
+ Find
+ On a coordinate grid, draw the vectors
+ Create a parallelogram by translating
+ Compute
+ Let
+ (Commutativity of Vectors)
+
+ (Associativity of Vectors)
+
+ (Additive Identity for Vectors)
+
+ (Additive Inverse for Vectors)
+
+ (Associativity of Scalar Muliplication)
+
+ (Distributivity)
+
+ (Distributivity)
+
+ (Multiplicative Identity for Vectors)
+
+ (Multiplicative Zero for Vectors)
+
+ A convenient thing about working with vectors is that the same ideas that work in two dimensions work in any number of dimensions. The number of dimensions is equal to the number of components in the vector. For example, the vector
+ For the vectors
+
+
+
+
+
+
+
+ In order to avoid confusion, we talk about a three-dimensional coordinate system using the
+ Recall from
+ For a three-dimensional vector
+ Find the length of the vector
+ What is the length of the vector
+ A
+ For each vector, find its associated unit vector. +
+
+
+
+
+
+
+ The unit vector
+ Write each vector as a sum of the standard unit vectors
+
+
+
+
+
+
+
+
+
+
+
+
+
+ The speed of an object is the magnitude of its associated velocity vector. A football is thrown by a player with an initial speed of 82 miles per hour and an angle of elevation of
+ Which of the following best expresses the horizontal component of the vector? +
+
+
+
+
+
+
+ Which option best describes the vertical component of the vector? +
+
+
+
+
+
+
+ Use your answers to (a) and (b) to write the velocity vector in component form. +
+
+ Use your answers to (a) and (b) to write the velocity vector using the standard unit vectors. Remember that the ball is thrown in the
+ Why would we consider the ball to be thrown in the
+ In order to fly from Oklahoma City to Dayton, OH, a plane must fly
+ What is the ground speed of the plane when the wind begins to blow? +
++ What is the new bearing of the plane? +
+
+ An object is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let
+What are vectors and how can we use them? +
++By the end of this chapter, you should be able to... +
+
-Explain how to find the following for the function
-The open intervals where
-The function increases on
-The function decreases on
-The local extrema of
-The point
-The point
+ We have that
+ We have that
+ We have that
+ We have that
+ We have that
-For each of the following functions, describe the open intervals
-where it is concave up or concave down, and any inflection points.
+ For
-
-
+ We have that
+ We have that
+ We have that
+ We have that
- Consider the region bounded by
+ Sketch the region
+
+ Consider the solid of revolution generated by rotating
+ The volume is expressible as
Find an integral or sum of integrals which computes the volume of the solid formed by rotating this region about the
Using the washer/disk method, we obtain
+ Consider the solid of revolution generated by rotating
+ The volume is expressible as
Find an integral or sum of integrals which computes the volume of the solid formed by rotating this region about the
Using the washer/disk method, we obtain
-
+ Consider the solid of revolution generated by rotating
+ The volume is expressible as
+ Consider the solid of revolution generated by rotating
+ The volume is expressible as
There is a jump or infinite discontinuity at
There is a jump or infinite discontinuity at
There is a jump or infinite discontinuity at
There is a jump or infinite discontinuity at
There is a jump discontinuity at
-Use trigonometric identities and
-{{#hint_2}}
-
Explain and demonstrate how to use double-angle identities to
-rewrite
-
-Substitute
-Explain and demonstrate how to write each of the following improper integrals
-as a limit, and why this limit converges or diverges.
+For each of the following integrals, identify if the integral is proper or improper. If improper, then compute the integral by finding the antiderivative, using the fundamental theorem of calculus, and computing the limit.
Converges. Diverges. This integral is proper.
Imagine moving along the graph toward the missing portion from the left, so that you are climbing up and to the right toward the obscured area of the graph. What 2 2.5
+ B. 1
+
+
Think of the same process, but this time from the right. You're falling down and to the left this time as you come close to the missing portion. What 2 2.5
+ B. 1
+
@@ -75,11 +92,19 @@
What is the value of
+ Does not exist.
+
What is the 2 2.5
+ B. 1
+
What is the 2 2.5
+ B. 1
+
When studying functions in algebra, we often focused on the value of a function given a specific
What is the limit as The limit is 0 The limit is not defined
+ D. The limit is not defined
Which of the following best matches the definition of right and left limits? (Note that DNE is short for "does not exist.")
The left limit is -1. The right limit DNE. The left limit DNE. The right limit DNE.
+ A. The left limit is -1. The right limit is 1.
@@ -53,8 +63,15 @@
+
What do you think the overall limit equals?
@@ -225,6 +276,12 @@The limit is 0
The limit is not defined
+ D.
The limit is not defined
+ +
Find
+
Find
+
Find
+
Find
+
For which
+ At A. -3 and D. 4, the limit exist. +
+Sketch the graph of a function
Suppose that:
@@ -320,12 +413,14 @@The part of the theorem that starts with “Suppose…” forms the assumptions of the theorem, while the part of the theorem that starts with “Then…” is the conclusion of the theorem. What are the assumptions of the Squeeze Theorem? What is the conclusion?
The assumptions of the Squeeze Theorem can be restated informally as “the function
Match the functions
+
Using trigonometry, one can show algebraically that
+
- Exercises available at
Based on the values of
the limit is approximately 0.1667
the limit is approximately 6.9999
+ D. The limit is approximately 0.1667 +
+The limit can be approximated to be 1 because the values appear to approach 1 and the graph appears to approach 1, but we should zoom in on the graph to be sure.
The limit cannot be approximated because the function might not exist at
+ C. The limit can be approximated to be 1 because the values appear to approach 1 and the graph appears to approach 1, but we should zoom in on the graph to be sure. +
+
+ B.
Consider the following function
Of the following options, at which values given would you evaluate
1.8, 1.9, 2.0, 2.1, 2.2
1.0, 1.5, 2.0, 2.5, 3.0
+ B. 1.98, 1.99, 2.0, 2.01, 2.02 +
+ Use the values that you chose in part (a) to calculate an approximation for
+ 41.22, 41.61, 42, 42.39, 42.78 +
+Which value best describes the limit that you obtained in part (b)?
@@ -254,7 +288,11 @@ the function asThe approximate value is 41.75
The approximate value is 42
+ D. The approximate value is 42. +
+ The limit does not exist because you are dividing by zero when
+ C. The limit does not exist because the function is oscillating between -1 and 1. +
+Would your conclusion that resulted from
+ No, both
Use technology to complete the following table of values.
@@ -364,7 +415,9 @@ Then explain how to use it to make an educated guess as to the value of
the limit
How would you use the language of limits to clarify the definition of continuity?
-
When removing
the discontinuity.
When the left and right limit exist separately, but are not equal, the discontinuity is not removable and is called a
- Exercises available at
- Exercises available at
If
Informally, we say that the limit of "
Informally, we say that the limit of "
When considering a ratio of functions
Recall the definition of a hole from
Recall the definition of a hole from
- Exercises available at
- Exercises available at
The first derivative encodes information about the rate of change of the original function. In particular,
+ The first derivative encodes information about the rate of change of the original function. In particular,
Consider the function
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
Sketch the tangent line
- Exercises available at
A spherical balloon is being inflated at a constant rate of 20 cubic inches per second.
How fast is the radius of the balloon changing at the instant the balloon's diameter is 12 inches?
Is the radius changing more rapidly when
@@ -346,7 +348,7 @@ recall that when
- Exercises available at
For each of the following figures, decide where the global extrema are located.
- Exercises available at
- Exercises available at
On the leftmost curve in
We must be extra careful with our language when dealing with negative numbers. @@ -93,7 +120,35 @@
Recall the terminology of concavity: when a curve bends upward, we say its shape is concave up. When a curve bends downwards, we say its shape is concave down.
Recall the terminology of concavity: when a curve bends upward, we say its shape is concave up. When a curve bends downwards, we say its shape is concave down.
- Exercises available at
Which of the following diagrams best illustrates how the box is created?
- Exercises available at
- Exercises available at
The graph of
Find
Find
- Exercises available at
Explain how to approximate the area under the curve
- Exercises available at
- Exercises available at
In
So far we have only been going from velocity to position of an object. Recall that to find the acceleration of an object, you can take the derivative of the velocity of an object. Let use say we have the acceleration of a falling object in m/s
What is the velocity of the falling object, if the initial velocity is given by
So far we have only been going from velocity to position of an object. Recall that to find the acceleration of an object, you can take the derivative of the velocity of an object. Let use say we have the acceleration of a falling object in m/s
What is the velocity of the falling object, if the initial velocity is given by
What is the position of the object, if the initial position is given by
What is the position of the object, if the initial position is given by
- Exercises available at
Approximate the area under the curve
What do you notice?
@@ -134,7 +179,14 @@Find the area bounded by the curves
- Exercises available at
- Exercises available at
A geometrical interpretation of
- Exercises available at
We now look for a general way of writing definite integrals for the area between two given curves,
- Exercises available at
- Exercises available at
Consider the integral
What is
Neither integral seems harder than the other one.
- Exercises available at
We might also use some other trigonometric identities to manipulate our integrands, listed in
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
Generally when solving problems without the aid of technology, it's useful to draw your region in
two dimensions, choose whether to use a horizontal or vertical line segment, and draw its rotation
@@ -195,7 +194,6 @@ When the shape is a cylindrical shell, this is called the
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
As seen in the previous activity, having too few terms may prevent us from finding a unique way to continue creating a sequence of numbers. In fact, we need sufficiently many terms to uniquely continue a sequence of numbers (and how many terms is sufficient depends on which sequence of numbers you are trying to generate). Sometimes, we do not want to write out all of the terms needed to allow for this. Therefore, we will want to find short-hand notation that allows us to do so.
-
- Exercises available at
- For each of the following sequences, determine which of the properties: monotonic, bounded and convergent, the sequence satisfies. If a sequence is convergent, determine to what it converges.
-
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
- Exercises available at
Consider a power series
Consider a power series
Then recall:
- Letting
- For what values of
- Letting
- For what values of
- Letting
- For what values of
- Given the power series
- The
If
- Exercises available at
- Exercises available at
- Exercises available at
This work includes materials used under license from the following works:
@@ -90,7 +93,7 @@ This work includes materials used under license from the following works: This book is a part of the
- For each of the following matrices, explain why it is not in reduced row - echelon form. + For each of the following matrices, mark the entries + where pivots should appear, and either label the matrix as + in reduced row echelon form, or explain what should be done + to put the matrix in reduced row echelon form and why.
-
Use technology to compute
Show step by step how to compute
Exercises available at
Exercises available at
+ In the following 4-part activity, students make decisions about which row operations are most helpful for them in producing the RREF. + The matrices appearing in parts (b) and (c) are obtained by applying specific choices of row operations to the matrix appearing previously; the matrix in part (d) is the one that we started with. +
-Consider the matrix
-
+ Consider the matrix:
+
+ Consider the matrix:
+
-Consider the matrix
-
-Consider the matrix
-
Free browser-based technologies for mathematical computation are available online.
-
- Go to
Type
+
+Go to
+In the dropdown on the right, you can select a number of different languages. +Select "Octave" for the Matlab-compatible syntax used by this text. +
+
+Type
Now try using whitespace to write out the matrix and compute
A = [1 3 2
- 2 5 7]
+ 2 5 7]
rref(A)
-
- -In the HTML version of this text, code cells are often embedded for your convenience -when RREFs need to be computed. -
-
-Try this out to compute
-
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Recall that in
Exercises available at
Exercises available at
Recall that in
Exercises available at
Exercises available at
+Recall that in
- Take the RREF of an appropriate matrix to determine
+ Use technology to find the RREF of an appropriate matrix and determine
if each of the following sets is a basis for
Exercises available at
Exercises available at
Exercises available at
Exercises available at
-Rewrite this solution space in the form
+ The set is linearly dependent. +
++ The set is linearly independent. +
++ The set spans the solution space. +
++ The set is a basis of the solution space. +
++ D. +
+
+Consider the homogeneous system of equations
+
+Find its solution set (a subspace of
+Rewrite this solution space in the form
Which of these choices best describes the set of two vectors
-
@@ -149,22 +215,32 @@ used in this span?
- The set spans all of
- The set fails to span the solution space. + The set is a basis for the solution space.
+ D. +
+
@@ -175,18 +251,24 @@ used in this span?
Thus if
-Consider the homogeneous system of equations
-
-Find a basis for its solution space. -
-
-Consider the homogeneous vector equation
-
-Find a basis for its solution space. -
-@@ -278,8 +312,14 @@ solution space?
+ A. +
+
-What homoegeneous linear system describes the positions
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
- What picture, or other study strategy would be helpful to you in conceptualizing how these defintions fit together? + What picture, or other study strategy would be helpful to you in conceptualizing how these definitions fit together?
-Which of the following properites of
- Consider the following applications of properites of the real numbers
+ Consider the following applications of properties of the real numbers
-Which of the following properites of
Consider the
- set
Let
@@ -687,14 +687,14 @@ for all
What are some objects that are important to you personally, academically, or otherwise that appear vector-like to you?
- What makes them feel vector-like? Which axiom for vector spaces does not hold for these objects, if any.
+ What makes them feel vector-like? Which axiom for vector spaces does not hold for these objects, if any?
- Our vector space axioms have eight properties. While these eight properties are enough to capture vectors, the objects that we study in the real-world often have additional structures not captured by these axioms. + Our vector space axioms have eight properties. While these eight properties are enough to capture vectors, the objects that we study in the real world often have additional structures not captured by these axioms. What are some structures that you have encountered in other classes, or in previous experiences, that are not captured by these eight axioms?
Exercises available at
Exercises available at
Describe the vector space involved in this problem, and an isomorphic Euclidean space and relevant Eucldean vectors that can be used to solve this problem.
+Describe the vector space involved in this problem, and an isomorphic Euclidean space, and relevant Euclidean vectors that can be used to solve this problem.
Show how to construct an appropriate Euclidean vector from an approriate set of Euclidean vectors.
+Show how to construct an appropriate Euclidean vector from an appropriate set of Euclidean vectors.
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
+
Which of the following matrices is invertible? Find the inverse for the one that is invertible.
+ Check your work using technology. Read the examples carefully, and + run them in the Octave cell below. +
++ Recall the syntax for entering and multiplying matrices. +
+
+ To invert a matrix
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
Exercises available at
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+ Calculate the change-of-basis matrix between two bases of
+ So far, when working with the Euclidean vector space
+ Let
+ Is
+
+ Yes. +
++ No. +
+
+ Since
+ Compute
+ Compute
+ Note that one way to compute
+ Given a basis
+ The vector
+ In particular, notation as above, we would have:
+
+ Let
+ Calculate
+ Use your result to calculate
+ Let
+ Let
+ Calculate the matrix
+ The matrix
+ Which of these two descriptions of
+ Let
+ Explain and demonstrate how to verify that
+ Explain and demonstrate how to use
+ We can accomplish both tasks by calculating the RREF of the following matrix:
+
+ By definition of the change of basis matrix, if
@@ -58,7 +58,7 @@ most important to least important.
@@ -187,7 +187,7 @@ the matrix equation
-Thus, our $978,000,000,000 problem is what kind of problem?
+Thus, our $2,110,000,000,000 problem is what kind of problem?
-Based upon this page rank vector,
+Based on this page rank vector,
here is a complete ranking of all seven pages from most important to least important:
-For example, at the bottom left node there are 3 forces acting.
+For example, at the bottom left node, 3 forces are acting.
-Since the angle of the normal forces for each anchor point are unknown,
+Since the angle of the normal forces for each anchor point is unknown,
two variables may be used to represent each.
The full augmented matrix given by the ten equations in this linear system
-is given below, where the eleven columns correspond to
In particular, the negative
This work includes materials used under license from the following works:
Recall that when solving a linear equation, you use addition, subtraction, multiplication and division to isolate the variable.
+
A linear equation with one unique solution is a
You can treat solving linear inequalities, just like solving an equation. The one exception is when you multiply or divide by a negative value, reverse the inequality symbol.
+
Solving a compound linear inequality, uses the same methods as a single linear inequality ensuring that you perform the same operations on all three parts. Alternatively, you can break the compound inquality up into two and solve separately.
+
Exercises available at
Exercises available at
Now that we have explored multiple methods for solving systems of linear equations, let's put those in to practice using some real-world application problems. +
Exercises available at
Exercises available at
Exercises available at
Exercises available at
This work includes materials used under license from the following works:
@@ -64,7 +67,7 @@ This work includes materials used under license from the following works:This work is supported in part by a grant from Affordable Learning Georgia.
+
+
+
+
+
+
+
+
+