Question 1

Find t such that   and   are perpendicular.

Accepted Answer

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

Question 2

Label as true or false. Explain your answer or give a counterexample. [Hint - it may be useful to think about circular motion about the origin.]

Accepted Answer

A) True 
 B)False

Question 3

Find the curvature of the polar curve at the indicated point.

Accepted Answer

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

Question 4

At Erik's sixth-year birthday party, the children are playing a game where each contestant tries to throw a tennis ball into a cardboard box located 16 feet away. The box is 18 inches tall and 24 inches wide (see figure). A child tosses a ball with an initial height of 16 inches and an angle of 45o from the horizontal. If the initial speed of the ball is 30 feet per second, will the ball land in the box? Show all of your work. [Ignore the radius of the ball. Assume that the only force acting on the ball is gravity.Assume that the ball is "on-target" and that there is nothing behind the box to cause a rebound. Note that g = 32 ft/s2 = 384 in/s2.]   [The figure not necessarily drawn to scale.]

Accepted Answer

For convenience, convert all distances t

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

A)  16.1 
B)  13.6 
C)  14.1 
D)  18.2 
A)  16.1 
B)  13.6 
C)  14.1 
D)  18.2

Question 7

Evaluate the given integral.

Accepted Answer

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

Question 8

Find the position function from the given acceleration function.

Accepted Answer

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

Question 9

Find the velocity function for the given position function.

Accepted Answer

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

Question 10

Show that the curve   lies on the elliptical paraboloid   . Show all your work.

Accepted Answer

The answer of Show that the curve   lies...

Question 11

Find the unit tangent vector to the curve at the indicated point.

Accepted Answer

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

Question 12

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 13

Evaluate the given integral.

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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 16

Find the binormal vector   at t = 2.

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 17

Find a parametric representation of the surface.

Accepted Answer

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

Question 18

Find the curvature of the polar curve at the indicated point.

Accepted Answer

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

Question 19

Phillip claims that the trajectory   for   is identical to the trajectory   for   , where t represents time. Sharon suggests that Phillip's claim is not necessarily true. Who is correct and why?

Accepted Answer

Sharon is correct. While the curves trac

Question 20

A projectile is fired with initial speed   feet per second from a height of h feet at an angle of   above the horizontal. Assuming that the only force acting on the object is gravity, find the speed at impact. Round to the nearest tenth of a foot.

Accepted Answer

A)  263.1 ft/s 
B)  725.9 ft/s 
C)  181.8 ft/s 
D)  1022.4 ft/s 
A)  263.1 ft/s 
B)  725.9 ft/s 
C)  181.8 ft/s 
D)  1022.4 ft/s

Find t such that and are perpendicular.

Label as true or false. Explain your answer or give a counterexample. [Hint - it may be useful to think about circular motion about the origin.]

Find the curvature of the polar curve at the indicated point.

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

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

Evaluate the given integral.

Find the position function from the given acceleration function.

Find the velocity function for the given position function.

Show that the curve lies on the elliptical paraboloid . Show all your work.

Find the unit tangent vector to the curve at the indicated point.

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

Evaluate the given integral.

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

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

Find the binormal vector at t = 2.

Find a parametric representation of the surface.

Find the curvature of the polar curve at the indicated point.

Phillip claims that the trajectory for is identical to the trajectory for , where t represents time. Sharon suggests that Phillip's claim is not necessarily true. Who is correct and why?

A projectile is fired with initial speed feet per second from a height of h feet at an angle of above the horizontal. Assuming that the only force acting on the object is gravity, find the speed at impact. Round to the nearest tenth of a foot.

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

Find t such that and are perpendicular.

Label as true or false. Explain your answer or give a counterexample. [Hint - it may be useful to think about circular motion about the origin.]

Find the curvature of the polar curve at the indicated point.

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

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

Evaluate the given integral.

Find the position function from the given acceleration function.

Find the velocity function for the given position function.

Show that the curve lies on the elliptical paraboloid . Show all your work.

Find the unit tangent vector to the curve at the indicated point.

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

Evaluate the given integral.

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

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

Find the binormal vector at t = 2.

Find a parametric representation of the surface.

Find the curvature of the polar curve at the indicated point.

Phillip claims that the trajectory for is identical to the trajectory for , where t represents time. Sharon suggests that Phillip's claim is not necessarily true. Who is correct and why?

A projectile is fired with initial speed feet per second from a height of h feet at an angle of above the horizontal. Assuming that the only force acting on the object is gravity, find the speed at impact. Round to the nearest tenth of a foot.

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