Question 1

The position vector of a particle is r(t). Find the requested vector.
-The velocity at t = 0 for r(t) = ln(   - 5   + 3)i -   j - 5cos(t)k

Accepted Answer

A)  v( 0) =   i -   j + 5k 
B)  v( 0) =   j 
C)  v( 0) =   i -   j 
D)  v( 0) = -   j 
A)  v( 0) =   i -   j + 5k 
B)  v( 0) =   j 
C)  v( 0) =   i -   j 
D)  v( 0) = -   j

Question 2

Evaluate the integral.
-

Accepted Answer

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

Question 3

Graph the curve described by the function, indicating the positive orientation.
-r(t) =   , for 0 $\le$ t $\le$ 2 $\pi$

Accepted Answer

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

Question 4

Find the unit tangent vector of the given curve. 
-r(t) = 3   i - 12   j + 4   k

Accepted Answer

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

Question 5

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

Accepted Answer

A)  a = (   sinh t)T + N 
B)  a = (-   sinh t)T + N 
C)  a = (sinh t)T + N 
D)  a = (-sinh t)T + N 
A)  a = (   sinh t)T + N 
B)  a = (-   sinh t)T + N 
C)  a = (sinh t)T + N 
D)  a = (-sinh t)T + N

Question 6

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

Accepted Answer

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

Question 7

The position vector of a particle is r(t). Find the requested vector.
-The velocity at t = 0 for r(t) = cos( 2t)i + 7ln(t - 3)j -   k

Accepted Answer

A)  v(0) =   j 
B)  v(0) = -2i -   j 
C)  v(0) = -   j 
D)  v(0) = 2i -   j 
A)  v(0) =   j 
B)  v(0) = -2i -   j 
C)  v(0) = -   j 
D)  v(0) = 2i -   j

Question 8

Evaluate the limit. 
-

Accepted Answer

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

Question 9

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
-Find the acceleration vector. r(t) = ( 6 cos t)i + ( 8 sin t)j

Accepted Answer

A)  a = (-6 sin t)i + (-8 cos t)j 
B)  a = ( 6 sin t)i + ( 8 cos t)j 
C)  a = ( 6 cos t)i + ( 8 sin t)j 
D)  a = (-6 cos t)i + (-8 sin t)j 
A)  a = (-6 sin t)i + (-8 cos t)j 
B)  a = ( 6 sin t)i + ( 8 cos t)j 
C)  a = ( 6 cos t)i + ( 8 sin t)j 
D)  a = (-6 cos t)i + (-8 sin t)j

Question 10

Find the unit tangent vector T and the principal unit normal vector N. 
-r(t) = (cosh t)i + (sinh t)j + tk

Accepted Answer

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

Question 11

The position vector of a particle is r(t). Find the requested vector.
-The acceleration at t = 0 for r(t) =   i + ( 10   - 2)j +   k

Accepted Answer

A)  a(0) = 2i - 2k 
B)  a(0) = 2i -   k 
C) a(0) = 2i -   k 
D)  a(0) = 2i +
 
   k 
A)  a(0) = 2i - 2k 
B)  a(0) = 2i -   k 
C) a(0) = 2i -   k 
D)  a(0) = 2i +
 
   k

Question 12

Find the unit tangent vector of the given curve. 
-r(t) =   i +   j - 12tk

Accepted Answer

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

Question 13

Find the curvature of the curve r(t).
-r(t) = ( 7 + cos 8t - sin 8t)i + ( 5 + sin 8t + cos 8t)j + 6k

Accepted Answer

A)    =   
B)    = 4 
C)   = 4   
D)    =   
A)    =   
B)    = 4 
C)   = 4   
D)    =

Question 14

FInd the tangential and normal components of the acceleration. 
-r(t) = (   - 3)i + ( 2t - 4)j + 9k

Accepted Answer

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

Question 15

Find the curvature of the curve r(t).
-r(t) = ( 10 + 6 cos 9t) i - ( 2 + 6 sin 9t)j + 2k

Accepted Answer

A)    = 6 
B)    =   
C)    =   
D)    =   
A)    = 6 
B)    =   
C)    =   
D)    =

Question 16

Compute r''(t).
-r(t) = (cos 2t)i + ( 3 sin t)j

Accepted Answer

A)  r''(t) = (-2 cos 2t)i + ( 3 sin t)j 
B)  r''(t) = (-4 cos 2t)i + (-3 sin t)j 
C)  r''(t) = (-4 cos 2t)i + (-9 sin t)j 
D)  r''(t) = ( 4 cos 2t)i + (-3 sin t)j 
A)  r''(t) = (-2 cos 2t)i + ( 3 sin t)j 
B)  r''(t) = (-4 cos 2t)i + (-3 sin t)j 
C)  r''(t) = (-4 cos 2t)i + (-9 sin t)j 
D)  r''(t) = ( 4 cos 2t)i + (-3 sin t)j

Question 17

Find the unit tangent vector T and the principal unit normal vector N. 
-r(t) = (   - 8)i + (2t - 6)j + 5k

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Find the curvature of the space curve.
-r(t) = -10i + ( 6 + 2t)j + (   + 2)k

Accepted Answer

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

Question 20

Find the domain of the vector-valued function.
-r(t) = sin 3t i +   j

Accepted Answer

A)  t $\ge$   
B)  t $\ge$ 0 
C)  t > 3 
D) All real numbers 
A)  t $\ge$   
B)  t $\ge$ 0 
C)  t > 3 
D) All real numbers

The position vector of a particle is r(t). Find the requested vector. -The velocity at t = 0 for r(t) = ln( - 5 + 3)i - j - 5cos(t)k

Evaluate the integral. -

Graph the curve described by the function, indicating the positive orientation. -r(t) = $Graph the curve described by the function, indicating the positive orientation. -r(t) = , for 0 \le t \le 2 \pi$ , for 0 $\le$ t $\le$ 2 $\pi$

Find the unit tangent vector of the given curve. -r(t) = 3 i - 12 j + 4 k

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

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

The position vector of a particle is r(t). Find the requested vector. -The velocity at t = 0 for r(t) = cos( 2t)i + 7ln(t - 3)j - k

Evaluate the limit. -

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the acceleration vector. r(t) = ( 6 cos t)i + ( 8 sin t)j

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = (cosh t)i + (sinh t)j + tk

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = 0 for r(t) = i + ( 10 - 2)j + k

Find the unit tangent vector of the given curve. -r(t) = i + j - 12tk

Find the curvature of the curve r(t). -r(t) = ( 7 + cos 8t - sin 8t)i + ( 5 + sin 8t + cos 8t)j + 6k

FInd the tangential and normal components of the acceleration. -r(t) = ( - 3)i + ( 2t - 4)j + 9k

Find the curvature of the curve r(t). -r(t) = ( 10 + 6 cos 9t) i - ( 2 + 6 sin 9t)j + 2k

Compute r''(t). -r(t) = (cos 2t)i + ( 3 sin t)j

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = ( - 8)i + (2t - 6)j + 5k

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

Find the curvature of the space curve. -r(t) = -10i + ( 6 + 2t)j + ( + 2)k

Find the domain of the vector-valued function. -r(t) = sin 3t i + 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

Exam 14: Vector-Valued Functions

The position vector of a particle is r(t). Find the requested vector. -The velocity at t = 0 for r(t) = ln( - 5 + 3)i - j - 5cos(t)k

Evaluate the integral. -

Graph the curve described by the function, indicating the positive orientation. -r(t) = , for 0 ≤\le≤ t ≤\le≤ 2 π\piπ

Find the unit tangent vector of the given curve. -r(t) = 3 i - 12 j + 4 k

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

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

The position vector of a particle is r(t). Find the requested vector. -The velocity at t = 0 for r(t) = cos( 2t)i + 7ln(t - 3)j - k

Evaluate the limit. -

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the acceleration vector. r(t) = ( 6 cos t)i + ( 8 sin t)j

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = (cosh t)i + (sinh t)j + tk

The position vector of a particle is r(t). Find the requested vector. -The acceleration at t = 0 for r(t) = i + ( 10 - 2)j + k

Find the unit tangent vector of the given curve. -r(t) = i + j - 12tk

Find the curvature of the curve r(t). -r(t) = ( 7 + cos 8t - sin 8t)i + ( 5 + sin 8t + cos 8t)j + 6k

FInd the tangential and normal components of the acceleration. -r(t) = ( - 3)i + ( 2t - 4)j + 9k

Find the curvature of the curve r(t). -r(t) = ( 10 + 6 cos 9t) i - ( 2 + 6 sin 9t)j + 2k

Compute r''(t). -r(t) = (cos 2t)i + ( 3 sin t)j

Find the unit tangent vector T and the principal unit normal vector N. -r(t) = ( - 8)i + (2t - 6)j + 5k

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

Find the curvature of the space curve. -r(t) = -10i + ( 6 + 2t)j + ( + 2)k

Find the domain of the vector-valued function. -r(t) = sin 3t i + 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

Graph the curve described by the function, indicating the positive orientation. -r(t) = $Graph the curve described by the function, indicating the positive orientation. -r(t) = , for 0 \le t \le 2 \pi$ , for 0 $\le$ t $\le$ 2 $\pi$