Question 1

Find the unit tangent vector T and the principal unit normal vector N. 
-

Accepted Answer

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

Question 2

The position vector of a particle is r(t). Find the requested vector.
-The acceleration at t =   for r(t) = ( 9 sin 5t)i - ( 10 cos 5t)j + ( 2 csc 5t)k

Accepted Answer

A)  a   = -225i - 50k 
B)  a   = 225i + 50k 
C)  a   = -225i + 50k 
D)  a   = 250j + 50k 
A)  a   = -225i - 50k 
B)  a   = 225i + 50k 
C)  a   = -225i + 50k 
D)  a   = 250j + 50k C

Question 3

Find the unit tangent vector T and the principal unit normal vector N. 
-r(t) = (ln(cos t) + 9)i + 9j + ( 10 + t )k, -$\pi$/2 < t < $\pi$/2

Accepted Answer

A)  T = (sin t)i - (cos t)k; N = (cos t)i - (sin t)k; 
B)  T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k 
C)  T = (-sin t)i + (cos t)k; N = (-cos t)i + (sin t)k 
D)  T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k 
A)  T = (sin t)i - (cos t)k; N = (cos t)i - (sin t)k; 
B)  T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k 
C)  T = (-sin t)i + (cos t)k; N = (-cos t)i + (sin t)k 
D)  T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k D

Question 4

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 5

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility.
-r(t) = 3 cos t i + 2 sin t j + cos 5t k, for 0 $\le$ t $\le$ 2$\pi$

Accepted Answer

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

Question 6

The position vector of a particle is r(t). Find the requested vector.
-The acceleration at t = 2 for r(t) = ( 7t - 3   )i + ( 10 - t)j + ( 2   - 3t)k

Accepted Answer

A)  a( 2) = -144i - j + 4k 
B)  a( 2) = 144i + 4k 
C)  a( 2) = -144i + 4k 
D)  a( 2) = -36i + 4k 
A)  a( 2) = -144i - j + 4k 
B)  a( 2) = 144i + 4k 
C)  a( 2) = -144i + 4k 
D)  a( 2) = -36i + 4k

Question 7

FInd the tangential and normal components of the acceleration. 
-r(t) = (t + 6)i + (ln(sec t) - 1)j + 7k, -$\pi$/2 < t < $\pi$/2

Accepted Answer

A)  a = (sec t tan t)T + (sec t)N 
B)  a = (csc t)T + (sec t)N 
C)  a = (   t)T + (cos t)N 
D)  a = (cos t)T + (cos t)N 
A)  a = (sec t tan t)T + (sec t)N 
B)  a = (csc t)T + (sec t)N 
C)  a = (   t)T + (cos t)N 
D)  a = (cos t)T + (cos t)N

Question 8

Find the unit tangent vector T and the principal unit normal vector N. 
-r(t) = ( 7t sin t + 7cos t)i + ( 7t cos t - 7 sin t)j - 4k

Accepted Answer

A)  T = (cos t)i + (sin t)j; N = (-sin t)i + (cos t)j 
B)  T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j 
C)  T = (cos t)i - (sin t)j; N = (-sin t)i - (cos t)j 
D)  T = (-cos t)i - (sin t)j; N = (sin t)i + (cos t)j 
A)  T = (cos t)i + (sin t)j; N = (-sin t)i + (cos t)j 
B)  T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j 
C)  T = (cos t)i - (sin t)j; N = (-sin t)i - (cos t)j 
D)  T = (-cos t)i - (sin t)j; N = (sin t)i + (cos t)j

Question 9

Find the curvature of the space curve.
-r(t) = 12ti +   j +   k

Accepted Answer

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

Question 10

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

Accepted Answer

A)  B(t) =   ,   = 0 
B)  B(t) =   ,   = 0 
C)  B(t) =   ,   = 1 
D)  B(t) =   , = 1 
A)  B(t) =   ,   = 0 
B)  B(t) =   ,   = 0 
C)  B(t) =   ,   = 1 
D)  B(t) =   , = 1

Question 11

The position vector of a particle is r(t). Find the requested vector.
-The velocity at t = 3 for r(t) = ( 8 - 4   )i + ( 6t + 7)j -   k

Accepted Answer

A) v( 3) = -24i + 6j + 5   k 
B)  v( 3) = -24i + 6j - 5   k 
C)  v( 3) = -12i +6j + 5   k 
D)  v( 3) = 24i + 6j + 5   k 
A) v( 3) = -24i + 6j + 5   k 
B)  v( 3) = -24i + 6j - 5   k 
C)  v( 3) = -12i +6j + 5   k 
D)  v( 3) = 24i + 6j + 5   k

Question 12

Find a function r(t) that describes the curve where the surfaces intersect.
-z = 16; z =   +

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 13

Find the length of the indicated portion of the trajectory.
-r(t) = ( 2cos t)i + ( 2sin t)j + 5tk, 0 $\le$ t $\le$  $\pi$/2

Accepted Answer

A)     $\pi$ 
B)     $\pi$ 
C)     $\pi$ 
D)     $\pi$ 
A)     $\pi$ 
B)     $\pi$ 
C)     $\pi$ 
D)     $\pi$

Question 14

Find the curvature of the curve r(t).
-r(t) = ( 8 + ln(sec t))i + ( 3 + t)k, -$\pi$/2 < t < $\pi$/2

Accepted Answer

A)    = cos t 
B)   = sin t 
C)    = 1-cos t 
D)   = -cos t 
A)    = cos t 
B)   = sin t 
C)    = 1-cos t 
D)   = -cos t

Question 15

Verify that the curve r(t) lies on the surface. Give the name of the surface.  
-r(t) =   ; z=   +

Accepted Answer

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

Question 16

Find the length of the indicated portion of the trajectory.
-r(t) = (1 + 5t)i + (1 + 8t)j + ( 2 - 2t)k, -1 $\le$ t $\le$ 0

Accepted Answer

A)    
B)  2   
C)    
D)    
A)    
B)  2   
C)    
D)

Question 17

Find the length of the indicated portion of the trajectory.
-r(t) = ( 2 + 2t)i + ( 3 + 3t)j + ( 3 - 6t)k, -1 $\le$ t $\le$ 0

Accepted Answer

A)  8 
B)  5 
C)  9 
D)  7 
A)  8 
B)  5 
C)  9 
D)  7

Question 18

Find the curvature of the space curve.
-r(t) = ( 9 t sin t + 9 cos t)i + 9j + ( 9t cos t - 9 sin t)k

Accepted Answer

A)    = -   
B)    = 9t 
C)    =   
D)    =   
A)    = -   
B)    = 9t 
C)    =   
D)    =

Question 19

Differentiate the function.  
-r(t) = ( -7   - 6)i +   j

Accepted Answer

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

Question 20

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

Accepted Answer

A)  T = ( 6 sin 2t)i - ( 6 cos 2t)j 
B)  T = ( 36 sin 2t)i -( 36 cos 2t)j 
C)  T = (sin 2t)i - (cos 2t)j 
D)  T = ( 6 cos 2t)i - ( 6 sin 2t)j 
A)  T = ( 6 sin 2t)i - ( 6 cos 2t)j 
B)  T = ( 36 sin 2t)i -( 36 cos 2t)j 
C)  T = (sin 2t)i - (cos 2t)j 
D)  T = ( 6 cos 2t)i - ( 6 sin 2t)j

Find the unit tangent vector T and the principal unit normal vector N. -

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = for r(t) = ( 9 sin 5t)i - ( 10 cos 5t)j + ( 2 csc 5t)k

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = (ln(cos t) + 9)i + 9j + ( 10 + t )k, - $\pi$ /2 < t < $\pi$ /2

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

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility. -r(t) = 3 cos t i + 2 sin t j + cos 5t k, for 0 $\le$ t $\le$ 2 $\pi$

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = 2 for r(t) = ( 7t - 3 )i + ( 10 - t)j + ( 2 - 3t)k

FInd the tangential and normal components of the acceleration. -r(t) = (t + 6)i + (ln(sec t) - 1)j + 7k, - $\pi$ /2 < t < $\pi$ /2

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = ( 7t sin t + 7cos t)i + ( 7t cos t - 7 sin t)j - 4k

Find the curvature of the space curve. -r(t) = 12ti + j + k

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

The position vector of a particle is r(t). Find the requested vector. -The velocity at t = 3 for r(t) = ( 8 - 4 )i + ( 6t + 7)j - k

Find a function r(t) that describes the curve where the surfaces intersect. -z = 16; z = +

Find the length of the indicated portion of the trajectory. -r(t) = ( 2cos t)i + ( 2sin t)j + 5tk, 0 $\le$ t $\le$ $\pi$ /2

Find the curvature of the curve r(t). -r(t) = ( 8 + ln(sec t))i + ( 3 + t)k, - $\pi$ /2 < t < $\pi$ /2

Verify that the curve r(t) lies on the surface. Give the name of the surface. -r(t) = ; z= +

Find the length of the indicated portion of the trajectory. -r(t) = (1 + 5t)i + (1 + 8t)j + ( 2 - 2t)k, -1 $\le$ t $\le$ 0

Find the length of the indicated portion of the trajectory. -r(t) = ( 2 + 2t)i + ( 3 + 3t)j + ( 3 - 6t)k, -1 $\le$ t $\le$ 0

Find the curvature of the space curve. -r(t) = ( 9 t sin t + 9 cos t)i + 9j + ( 9t cos t - 9 sin t)k

Differentiate the function. -r(t) = ( -7 - 6)i + j

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

Functions

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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

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Exam 14: Vector-Valued Functions

Find the unit tangent vector T and the principal unit normal vector N. -

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = for r(t) = ( 9 sin 5t)i - ( 10 cos 5t)j + ( 2 csc 5t)k

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = (ln(cos t) + 9)i + 9j + ( 10 + t )k, - π\piπ /2 < t < π\piπ /2

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

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility. -r(t) = 3 cos t i + 2 sin t j + cos 5t k, for 0 ≤\le≤ t ≤\le≤ 2 π\piπ

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = 2 for r(t) = ( 7t - 3 )i + ( 10 - t)j + ( 2 - 3t)k

FInd the tangential and normal components of the acceleration. -r(t) = (t + 6)i + (ln(sec t) - 1)j + 7k, - π\piπ /2 < t < π\piπ /2

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = ( 7t sin t + 7cos t)i + ( 7t cos t - 7 sin t)j - 4k

Find the curvature of the space curve. -r(t) = 12ti + j + k

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

The position vector of a particle is r(t). Find the requested vector. -The velocity at t = 3 for r(t) = ( 8 - 4 )i + ( 6t + 7)j - k

Find a function r(t) that describes the curve where the surfaces intersect. -z = 16; z = +

Find the length of the indicated portion of the trajectory. -r(t) = ( 2cos t)i + ( 2sin t)j + 5tk, 0 ≤\le≤ t ≤\le≤ π\piπ /2

Find the curvature of the curve r(t). -r(t) = ( 8 + ln(sec t))i + ( 3 + t)k, - π\piπ /2 < t < π\piπ /2

Verify that the curve r(t) lies on the surface. Give the name of the surface. -r(t) = ; z= +

Find the length of the indicated portion of the trajectory. -r(t) = (1 + 5t)i + (1 + 8t)j + ( 2 - 2t)k, -1 ≤\le≤ t ≤\le≤ 0

Find the length of the indicated portion of the trajectory. -r(t) = ( 2 + 2t)i + ( 3 + 3t)j + ( 3 - 6t)k, -1 ≤\le≤ t ≤\le≤ 0

Find the curvature of the space curve. -r(t) = ( 9 t sin t + 9 cos t)i + 9j + ( 9t cos t - 9 sin t)k

Differentiate the function. -r(t) = ( -7 - 6)i + j

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

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

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = (ln(cos t) + 9)i + 9j + ( 10 + t )k, - $\pi$ /2 < t < $\pi$ /2

Graph the curve described by the function. Use analysis to anticipate the shape of the curve before using a graphing utility. -r(t) = 3 cos t i + 2 sin t j + cos 5t k, for 0 $\le$ t $\le$ 2 $\pi$

FInd the tangential and normal components of the acceleration. -r(t) = (t + 6)i + (ln(sec t) - 1)j + 7k, - $\pi$ /2 < t < $\pi$ /2

Find the length of the indicated portion of the trajectory. -r(t) = ( 2cos t)i + ( 2sin t)j + 5tk, 0 $\le$ t $\le$ $\pi$ /2

Find the curvature of the curve r(t). -r(t) = ( 8 + ln(sec t))i + ( 3 + t)k, - $\pi$ /2 < t < $\pi$ /2

Find the length of the indicated portion of the trajectory. -r(t) = (1 + 5t)i + (1 + 8t)j + ( 2 - 2t)k, -1 $\le$ t $\le$ 0

Find the length of the indicated portion of the trajectory. -r(t) = ( 2 + 2t)i + ( 3 + 3t)j + ( 3 - 6t)k, -1 $\le$ t $\le$ 0