Question 1

Find the point on the curve   at which the curvature K is a maximum.

Accepted Answer

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

Question 2

The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path   (x and y are measured in miles) can safely go 30 miles per hour at   . How fast can it go at   ? Round your answer to two decimal places.

Accepted Answer

A)    mi/h 
B)    mi/h 
C)    mi/h 
D)    mi/h 
E)    mi/h 
A)    mi/h 
B)    mi/h 
C)    mi/h 
D)    mi/h 
E)    mi/h

Question 3

Find the length of the plane curve given below.

Accepted Answer

A)  14 
B)  8 
C)  11 
D)  13 
E)  12 
A)  14 
B)  8 
C)  11 
D)  13 
E)  12

Question 4

Use the given acceleration function and initial conditions to find the position at time t = 4.

Accepted Answer

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

Question 5

Find   given the following vector function. ​   ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 6

Sketch the curve represented by the vector-valued function   and give the orientation of the curve.

Accepted Answer

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

Question 7

The quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 62 yards directly downfield at a height of 3 feet. The pass is released at an angle of   with the horizontal. Find the maximum height of the football. Round your answer to one decimal place. ​

Accepted Answer

A)  337.1 feet 
B)  16.3 feet 
C)  245.4 feet 
D)  85.8 feet 
E)  17.5 feet 
A)  337.1 feet 
B)  16.3 feet 
C)  245.4 feet 
D)  85.8 feet 
E)  17.5 feet

Question 8

Determine the interval on which the vector-valued function   is continuous.

Accepted Answer

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

Question 9

Find the arc length for   over the interval   . Round your answer to two decimal places. ​

Accepted Answer

A)  110.36 
B)  140.93 
C)  129.56 
D)  57.80 
E)  102.75 
A)  110.36 
B)  140.93 
C)  129.56 
D)  57.80 
E)  102.75

Question 10

Use the properties of the derivative to find   given the following vector-valued functions.

Accepted Answer

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

Question 11

Suppose the two particles travel along the space curves   and   . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.

Accepted Answer

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

Question 12

The graph below is most likely the graph of which of the following equations? ​   ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 13

Use the properties of the derivative to find   given the following vector-valued functions.

Accepted Answer

A)  1 
B)    
C)    
D)  6 
E)  0 
A)  1 
B)    
C)    
D)  6 
E)  0

Question 14

The position vector   describes the path of an object moving in space. Find the speed   of the object.

Accepted Answer

A)  17 
B)    
C)    
D)  115 
E)    
A)  17 
B)    
C)    
D)  115 
E)

Question 15

A baseball, hit 7 feet above the ground, leaves the bat at an angle of   and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer. ​

Accepted Answer

A)  180 feet 
B)  18 feet 
C)  397 feet 
D)  278 feet 
E)  50 feet 
A)  180 feet 
B)  18 feet 
C)  397 feet 
D)  278 feet 
E)  50 feet

Question 16

Find   at time   for the space curve   .

Accepted Answer

A)  9 
B)  0 
C)  8 
D)  64 
E)  3 
A)  9 
B)  0 
C)  8 
D)  64 
E)  3

Question 17

Determine the range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 500 feet per second and at an angle of   above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

Accepted Answer

A)  2,669.517 feet 
B)  1,764.116 feet 
C)  5,337.349 feet 
D)  6,767.555 feet 
E)  3,525.690 feet 
A)  2,669.517 feet 
B)  1,764.116 feet 
C)  5,337.349 feet 
D)  6,767.555 feet 
E)  3,525.690 feet

Question 18

Use the properties of the derivative to find   given the following vector-valued functions.

Accepted Answer

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

Question 19

Match the equation with the graph shown in red below.

Accepted Answer

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

Question 20

Use the given acceleration function and initial conditions to find the position at time t = 3.

Accepted Answer

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

Find the point on the curve at which the curvature K is a maximum.

Find the length of the plane curve given below.

Use the given acceleration function and initial conditions to find the position at time t = 4.

Find given the following vector function.

Sketch the curve represented by the vector-valued function and give the orientation of the curve.

Determine the interval on which the vector-valued function is continuous.

Find the arc length for over the interval . Round your answer to two decimal places.

Use the properties of the derivative to find given the following vector-valued functions.

Suppose the two particles travel along the space curves and . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.

The graph below is most likely the graph of which of the following equations?

Use the properties of the derivative to find given the following vector-valued functions.

The position vector describes the path of an object moving in space. Find the speed of the object.

A baseball, hit 7 feet above the ground, leaves the bat at an angle of and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer.

Find at time for the space curve .

Determine the range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 500 feet per second and at an angle of above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

Use the properties of the derivative to find given the following vector-valued functions.

Match the equation with the graph shown in red below.

Use the given acceleration function and initial conditions to find the position at time t = 3.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Exam 12: Vector-Valued Functions

Find the point on the curve at which the curvature K is a maximum.

Find the length of the plane curve given below.

Use the given acceleration function and initial conditions to find the position at time t = 4.

Find given the following vector function. ​ ​

Sketch the curve represented by the vector-valued function and give the orientation of the curve.

Determine the interval on which the vector-valued function is continuous.

Find the arc length for over the interval . Round your answer to two decimal places. ​

Use the properties of the derivative to find given the following vector-valued functions.

Suppose the two particles travel along the space curves and . A collision will occur at the point of intersection P if both particles are at P at the same time. Find the point of collision.

The graph below is most likely the graph of which of the following equations? ​ ​

Use the properties of the derivative to find given the following vector-valued functions.

The position vector describes the path of an object moving in space. Find the speed of the object.

A baseball, hit 7 feet above the ground, leaves the bat at an angle of and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer. ​

Find at time for the space curve .

Determine the range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 500 feet per second and at an angle of above the horizontal. Use the model for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

Use the properties of the derivative to find given the following vector-valued functions.

Match the equation with the graph shown in red below.

Use the given acceleration function and initial conditions to find the position at time t = 3.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Find given the following vector function.

Find the arc length for over the interval . Round your answer to two decimal places.

The graph below is most likely the graph of which of the following equations?

A baseball, hit 7 feet above the ground, leaves the bat at an angle of and its caught by an outfielder 7 feet above the ground and 400 feet from home plate. How high does the ball rise? Round your answer to the nearest integer.