Question 1

Determine whether the speed of the object is increasing, decreasing, or constant at the given point.

Accepted Answer

A)  increasing 
B)  decreasing 
C)  constant 
A)  increasing 
B)  decreasing 
C)  constant

Question 2

Graph the curvature function   for   and find the limit of the curvature as   .

Accepted Answer

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

Question 3

Ted tells his incredulous nephews that he can turn a bucket of water upside down without the water pouring out! To illustrate, he puts a small amount of water in a bucket and swings it around in a circle so that the bucket travels from his knees, up over his head, and swings back down to his knees (see figure). If the water is 2.2 feet from the center of the circle of motion and is weightless at the top of the loop, what is the linear speed of the bucket? [Ignore the fact that the water in the bucket has depth; that is, pretend all the water is 2.2 feet from the loop's center.]

Accepted Answer

A)  8.4 ft/s 
B)  3.8 ft/s 
C)  4.2 ft/s 
D)  16.8 ft/s 
A)  8.4 ft/s 
B)  3.8 ft/s 
C)  4.2 ft/s 
D)  16.8 ft/s

Question 4

Which of the following is an arc length parameterization of the given two-dimensional curve? The line segment from   to

Accepted Answer

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

Question 5

Compute the arc length.

Accepted Answer

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

Question 6

Match the vector-valued function with its graph.

Accepted Answer

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

Question 7

Match the vector-valued function with its graph.

Accepted Answer

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

Question 8

Find the velocity function for the given position function.

Accepted Answer

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

Question 9

Determine whether the following is true or false.

Accepted Answer

Answers may vary.
False. Let @#IMG-DLM& be the ant

Question 10

Find the binormal vector   at   .

Accepted Answer

A)    
B)    
C)    
D)  The binormal vector is undefined at this point. 
A)    
B)    
C)    
D)  The binormal vector is undefined at this point.

Question 11

Find t such that   and   are perpendicular.

Accepted Answer

A)    
B)  t = 0 
C)    
D)  The vectors   and   are not perpendicular for any value of t. 
A)    
B)  t = 0 
C)    
D)  The vectors   and   are not perpendicular for any value of t.

Question 12

Find all values of t such that   is parallel to the xy-plane.

Accepted Answer

A)    
B)    
C)    
D)    is not parallel to the xy-plane for any real value of t 
A)    
B)    
C)    
D)    is not parallel to the xy-plane for any real value of t

Question 13

Find the position function from the given acceleration function.

Accepted Answer

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

Question 14

Find the unit normal vector at the given point.

Accepted Answer

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

Question 15

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 16

Find the tangential and normal components of acceleration for the given position function at the indicated point.

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

Determine whether the speed of the object is increasing, decreasing, or constant at the given point.

Accepted Answer

A)  decreasing 
B)  increasing 
C)  constant 
A)  decreasing 
B)  increasing 
C)  constant

Question 19

Use a CAS to estimate the arc length of the curve.

Accepted Answer

A)  21.4 
B)  84.8 
C)  33.8 
D)  7.6 
A)  21.4 
B)  84.8 
C)  33.8 
D)  7.6

Question 20

Identify and sketch a graph of the surface.

Accepted Answer

hyperbolic

Determine whether the speed of the object is increasing, decreasing, or constant at the given point.

Graph the curvature function for and find the limit of the curvature as .

Which of the following is an arc length parameterization of the given two-dimensional curve? The line segment from to

Compute the arc length.

Match the vector-valued function with its graph.

Match the vector-valued function with its graph.

Find the velocity function for the given position function.

Determine whether the following is true or false.

Find the binormal vector at .

Find t such that and are perpendicular.

Find all values of t such that is parallel to the xy-plane.

Find the position function from the given acceleration function.

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.]

Find the tangential and normal components of acceleration for the given position function at the indicated point.

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

Determine whether the speed of the object is increasing, decreasing, or constant at the given point.

Use a CAS to estimate the arc length of the curve.

Identify and sketch a graph of the surface.

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

Determine whether the speed of the object is increasing, decreasing, or constant at the given point.

Graph the curvature function for and find the limit of the curvature as .

Which of the following is an arc length parameterization of the given two-dimensional curve? The line segment from to

Compute the arc length.

Match the vector-valued function with its graph.

Match the vector-valued function with its graph.

Find the velocity function for the given position function.

Determine whether the following is true or false.

Find the binormal vector at .

Find t such that and are perpendicular.

Find all values of t such that is parallel to the xy-plane.

Find the position function from the given acceleration function.

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.]

Find the tangential and normal components of acceleration for the given position function at the indicated point.

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

Determine whether the speed of the object is increasing, decreasing, or constant at the given point.

Use a CAS to estimate the arc length of the curve.

Identify and sketch a graph of the surface.

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