Question 1

For   , show that each center of curvature lies on the curve traced out by   . Show your work.

Accepted Answer

Answers may vary.
Parameterize the curve   :   Curvature:   Unit normal vector:   Center trajectory:

Question 2

Evaluate the given integral.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)    A

Question 3

For the circle with radius c centered at   , show that the curvature at any point is equal to   . Show all of your work.

Accepted Answer

Answers may vary.

Question 4

Use the graph of   below to predict whether the curvature would be larger at   or   .

Accepted Answer

A)    
B)    
C)    
A)    
B)    
C)

Question 5

Use graphing technology to sketch the curve traced out by the given vector-valued function.   Use the viewpoint   and   (see figure below).

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 6

Find the unit normal vector at the given point.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 7

Show that there are no values of t such that   and   are parallel. Show your work.

Accepted Answer

Answers may vary.
The vectors

Question 8

Find the position function from the given velocity function.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 9

Find the magnitude of the net force on an object of mass 13 kg with the given position function (in units of meters and seconds).

Accepted Answer

A)  5070 Newtons 
B)  3900 Newtons 
C)  390 Newtons 
D)  1170 Newtons 
A)  5070 Newtons 
B)  3900 Newtons 
C)  390 Newtons 
D)  1170 Newtons

Question 10

Determine all values of t at which the given vector-valued function is continuous.

Accepted Answer

A)    
B)    
C)    
D)  all real numbers 
A)    
B)    
C)    
D)  all real numbers

Question 11

Which of the following is the graph of the given parametric surface?

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 12

Use the graph of   below to predict whether the curvature would be larger at   or   .

Accepted Answer

A)    
B)    
C)    
A)    
B)    
C)

Question 13

Find the unit normal vector at the given point.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 14

Label the statement as true (i.e., always true) or false and explain your answer. [Assume that T, N, and B are defined.]

Accepted Answer

A) True 
 B)False

Question 15

A stationary merry-go-round of radius 6 feet is started in motion by a push consisting of a force of 15 pounds on the outside edge, tangent to the circular edge of the merry-go-round, for 2 seconds. The moment of inertia of the merry-go-round is   . Find the resulting angular velocity of the merry-go-around.

Accepted Answer

A)  5 rad/sec 
B)  10 rad/sec 
C)  4 rad/sec 
D)  9 rad/sec 
A)  5 rad/sec 
B)  10 rad/sec 
C)  4 rad/sec 
D)  9 rad/sec

Question 16

Consider the claim that there exists a curve   for which   . Do you agree with this claim? Explain or justify your answer. [You need not give a rigorous proof.]

Accepted Answer

The claim cannot be true. A curve with a

Question 17

Sketch, by hand, the curve traced out by the given vector-valued function.

Accepted Answer

The answer of Sketch, by hand, the curve traced out...

Question 18

Which of the following is a parametric representation of the given surface?

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 19

Find the acceleration function for the given position function.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 20

Find an equation in x, y, and z for the plane defined below.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

For , show that each center of curvature lies on the curve traced out by . Show your work.

Evaluate the given integral.

For the circle with radius c centered at , show that the curvature at any point is equal to . Show all of your work.

Use the graph of below to predict whether the curvature would be larger at or .

Use graphing technology to sketch the curve traced out by the given vector-valued function. Use the viewpoint and (see figure below).

Find the unit normal vector at the given point.

Show that there are no values of t such that and are parallel. Show your work.

Find the position function from the given velocity function.

Find the magnitude of the net force on an object of mass 13 kg with the given position function (in units of meters and seconds).

Determine all values of t at which the given vector-valued function is continuous.

Which of the following is the graph of the given parametric surface?

Use the graph of below to predict whether the curvature would be larger at or .

Find the unit normal vector at the given point.

Label the statement as true (i.e., always true) or false and explain your answer. [Assume that T, N, and B are defined.]

Consider the claim that there exists a curve for which . Do you agree with this claim? Explain or justify your answer. [You need not give a rigorous proof.]

Sketch, by hand, the curve traced out by the given vector-valued function.

Which of the following is a parametric representation of the given surface?

Find the acceleration function for the given position function.

Find an equation in x, y, and z for the plane defined below.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Exam 12: Vector-Valued Functions

For , show that each center of curvature lies on the curve traced out by . Show your work.

Evaluate the given integral.

For the circle with radius c centered at , show that the curvature at any point is equal to . Show all of your work.

Use the graph of below to predict whether the curvature would be larger at or .

Use graphing technology to sketch the curve traced out by the given vector-valued function. Use the viewpoint and (see figure below).

Find the unit normal vector at the given point.

Show that there are no values of t such that and are parallel. Show your work.

Find the position function from the given velocity function.

Find the magnitude of the net force on an object of mass 13 kg with the given position function (in units of meters and seconds).

Determine all values of t at which the given vector-valued function is continuous.

Which of the following is the graph of the given parametric surface?

Use the graph of below to predict whether the curvature would be larger at or .

Find the unit normal vector at the given point.

Label the statement as true (i.e., always true) or false and explain your answer. [Assume that T, N, and B are defined.]

Consider the claim that there exists a curve for which . Do you agree with this claim? Explain or justify your answer. [You need not give a rigorous proof.]

Sketch, by hand, the curve traced out by the given vector-valued function.

Which of the following is a parametric representation of the given surface?

Find the acceleration function for the given position function.

Find an equation in x, y, and z for the plane defined below.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters