Question 1

Find the curvature of the curve r(t).
-r(t) = ( 7t + 3)i - 7j + ( 4 -     )k

Accepted Answer

A)    =   
B)    = 7   
C)   = 7   
D)    =   
A)    =   
B)    = 7   
C)   = 7   
D)    =

Question 2

Evaluate the integral.
-

Accepted Answer

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

Question 3

Evaluate the integral.
-

Accepted Answer

A)  3i + 3j -   k 
B)  3i - 3j -   k 
C)  6i - 6j + 3k 
D)  6i + 6j - 3k 
A)  3i + 3j -   k 
B)  3i - 3j -   k 
C)  6i - 6j + 3k 
D)  6i + 6j - 3k

Question 4

FInd the tangential and normal components of the acceleration. 
-r(t) = 4   i + 4   j + 3tk

Accepted Answer

A)  a = T + 3   N 
B)  a = 3   N 
C)  a =     N 
D)  a = 3   T 
A)  a = T + 3   N 
B)  a = 3   N 
C)  a =     N 
D)  a = 3   T

Question 5

Compute r''(t).
-r(t) = ( 3 ln( 6t))i + ( 2   )j

Accepted Answer

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

Question 6

Find a function r(t) that describes the line or line segment.
-The line through P(4, 9, 3) and Q(1, 6, 7)

Accepted Answer

A)  r(t) =   
B)  r(t) =   
C)  r(t) =   
D)  r(t) =   
A)  r(t) =   
B)  r(t) =   
C)  r(t) =   
D)  r(t) =

Question 7

Find the length of the indicated portion of the trajectory.
-r(t) = (   cos t)i + (   sin t)j +   k, -ln 2 $\le$ t $\le$ 0

Accepted Answer

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

Question 8

Differentiate the function.  
-r(t) = (cot t)i + (csc t)j

Accepted Answer

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

Question 9

Find the curvature of the space curve.
-r(t) = -10i + (t + 5)j +(ln(cos t) + 4)k

Accepted Answer

A)    = sin t 
B)    = csc t 
C)    = sec t 
D)    = cos t 
A)    = sin t 
B)    = csc t 
C)    = sec t 
D)    = cos t

Question 10

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
-Find the velocity vector. r(t) = (cot t)i + (csc t)j

Accepted Answer

A)  v = (   t)i + (cot t csc t)j 
B)  v = (   t)i + (tan t sec t)j 
C)  v = (-   t)i - (cot t csc t)j 
D)  v = (-   t)i - (tan t sec t)j 
A)  v = (   t)i + (cot t csc t)j 
B)  v = (   t)i + (tan t sec t)j 
C)  v = (-   t)i - (cot t csc t)j 
D)  v = (-   t)i - (tan t sec t)j

Question 11

Evaluate the integral.
-

Accepted Answer

A)  11i + 4 ln 2j + (1 -   )k 
B)  5i + 4 ln 2j + (1 -   )k 
C)  11i + 4 ln 2j +   k 
D)  11i + 4 ln 2j + (1 +   )k 
A)  11i + 4 ln 2j + (1 -   )k 
B)  5i + 4 ln 2j + (1 -   )k 
C)  11i + 4 ln 2j +   k 
D)  11i + 4 ln 2j + (1 +   )k

Question 12

The position vector of a particle is r(t). Find the requested vector.
-The acceleration at t = 1 for r(t) =   i + 2ln   j +   k

Accepted Answer

A)  a(1) = 20i + 2j + 12k 
B)  a(1) = 20i +   j - 12k 
C)  a(1) = 20i - 2j - 12k 
D)  a(1) = 20i +   j + 12k 
A)  a(1) = 20i + 2j + 12k 
B)  a(1) = 20i +   j - 12k 
C)  a(1) = 20i - 2j - 12k 
D)  a(1) = 20i +   j + 12k

Question 13

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response
-Show that the arc length of one petal of the rose   is given by
 2   and use this formula to help make a conjecture about the limit of such arc lengths as

Accepted Answer

The answer of Write the word or phrase that best...

Question 14

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility.
-r(t) = cos 2t sin t i + sin 2t sin t j +   k, for 0 $\le$ t $\le$ 16

Accepted Answer

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

Question 15

Evaluate the integral.
-

Accepted Answer

A)  7   i 
B)  0 
C)  7   i + 3   j 
D)  3   i 
A)  7   i 
B)  0 
C)  7   i + 3   j 
D)  3   i

Question 16

Find the curvature of the space curve.
-

Accepted Answer

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

Question 17

FInd the tangential and normal components of the acceleration. 
-r(t) =   i +   j + 12tk

Accepted Answer

A)  a =   N 
B)  a = 25T + 25N 
C)  a = T + 25N 
D)  a =   T 
A)  a =   N 
B)  a = 25T + 25N 
C)  a = T + 25N 
D)  a =   T

Question 18

Compute the unit binormal vector and torsion of the curve.
-r(t) =

Accepted Answer

A)  B(t) =   ,   =   
B) B(t) =   ,  =   
C)  B(t) =   ,   = -   
D)  B(t) =   ,  =   
A)  B(t) =   ,   =   
B) B(t) =   ,  =   
C)  B(t) =   ,   = -   
D)  B(t) =   ,  =

Question 19

Find the unit tangent vector of the given curve. 
-r(t) = ( 6 + 10   )i + ( 9 + 11   )j + ( 1 + 2   )k

Accepted Answer

A)  
  
B)  
  
C)  T = 10i + 11j + 2k 
D)  
  
A)  
  
B)  
  
C)  T = 10i + 11j + 2k 
D)

Question 20

Find the curvature of the space curve.
-r(t) = (t + 5)i + 8j + (ln(sec t) + 1)k

Accepted Answer

A)    = cos t 
B)    = sec t 
C)    = sin t 
D)   = csc t 
A)    = cos t 
B)    = sec t 
C)    = sin t 
D)   = csc t

Find the curvature of the curve r(t). -r(t) = ( 7t + 3)i - 7j + ( 4 - )k

Evaluate the integral. -

Evaluate the integral. -

FInd the tangential and normal components of the acceleration. -r(t) = 4 i + 4 j + 3tk

Compute r''(t). -r(t) = ( 3 ln( 6t))i + ( 2 )j

Find a function r(t) that describes the line or line segment. -The line through P(4, 9, 3) and Q(1, 6, 7)

Differentiate the function. -r(t) = (cot t)i + (csc t)j

Find the curvature of the space curve. -r(t) = -10i + (t + 5)j +(ln(cos t) + 4)k

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the velocity vector. r(t) = (cot t)i + (csc t)j

Evaluate the integral. -

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = 1 for r(t) = i + 2ln j + k

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response -Show that the arc length of one petal of the rose is given by 2 and use this formula to help make a conjecture about the limit of such arc lengths as

Evaluate the integral. -

Find the curvature of the space curve. -

FInd the tangential and normal components of the acceleration. -r(t) = i + j + 12tk

Compute the unit binormal vector and torsion of the curve. -r(t) =

Find the unit tangent vector of the given curve. -r(t) = ( 6 + 10 )i + ( 9 + 11 )j + ( 1 + 2 )k

Find the curvature of the space curve. -r(t) = (t + 5)i + 8j + (ln(sec t) + 1)k

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Functions of Several Variables

Multiple Integration

Vector Calculus

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Exam 14: Vector-Valued Functions

Find the curvature of the curve r(t). -r(t) = ( 7t + 3)i - 7j + ( 4 - )k

Evaluate the integral. -

Evaluate the integral. -

FInd the tangential and normal components of the acceleration. -r(t) = 4 i + 4 j + 3tk

Compute r''(t). -r(t) = ( 3 ln( 6t))i + ( 2 )j

Find a function r(t) that describes the line or line segment. -The line through P(4, 9, 3) and Q(1, 6, 7)

Find the length of the indicated portion of the trajectory. -r(t) = ( cos t)i + ( sin t)j + k, -ln 2 ≤\le≤ t ≤\le≤ 0

Differentiate the function. -r(t) = (cot t)i + (csc t)j

Find the curvature of the space curve. -r(t) = -10i + (t + 5)j +(ln(cos t) + 4)k

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the velocity vector. r(t) = (cot t)i + (csc t)j

Evaluate the integral. -

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = 1 for r(t) = i + 2ln j + k

Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response -Show that the arc length of one petal of the rose is given by 2 and use this formula to help make a conjecture about the limit of such arc lengths as

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility. -r(t) = cos 2t sin t i + sin 2t sin t j + k, for 0 ≤\le≤ t ≤\le≤ 16

Evaluate the integral. -

Find the curvature of the space curve. -

FInd the tangential and normal components of the acceleration. -r(t) = i + j + 12tk

Compute the unit binormal vector and torsion of the curve. -r(t) =

Find the unit tangent vector of the given curve. -r(t) = ( 6 + 10 )i + ( 9 + 11 )j + ( 1 + 2 )k

Find the curvature of the space curve. -r(t) = (t + 5)i + 8j + (ln(sec t) + 1)k

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Functions of Several Variables

Multiple Integration

Vector Calculus

Filters