Question 1

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 5 t - 2$ . What is the object's acceleration?

Accepted Answer

A)  2t + 5 
B)  2t 
C)  t 
D)  2 
A)  2t + 5 
B)  2t 
C)  t 
D)  2

Question 2

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 3$ . Find the velocity and acceleration of the object at time t. When is the object stationary?

Accepted Answer

a(t) = 6t-

Question 3

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 x ^ { 2 } - 5 x + 8 } { 6 - 5 x ^ { 3 } }$ at the point (1, 7).

Accepted Answer

The answer of Find the equation of the tangent line...

Question 4

For f (x) = 1 - x2, find the slope of the secant line connecting the points whose x-coordinates are x = -4 and x = -3.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -4.

Accepted Answer

Slope of secant line

Question 5

Differentiate: $f ( x ) = \sqrt { x } + \frac { 5 } { \sqrt { x } }$

Accepted Answer

A)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 5 } { 2 \sqrt { x ^ { 3 } } }$ 
B)  0 
C)  5 
D)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 5 } { 2 \sqrt { x ^ { 3 } } }$ 
A)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 5 } { 2 \sqrt { x ^ { 3 } } }$ 
B)  0 
C)  5 
D)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 5 } { 2 \sqrt { x ^ { 3 } } }$

Question 6

If $f ( x ) = \frac { ( 3 - 5 x ) ^ { 3 } } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } }$ , then $f ^ { \prime } ( x ) = - 5 ( 2 x + 1 )$ .

Accepted Answer

A) True 
 B)False

Question 7

Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.

Accepted Answer

A)  (0, 0) and (-4, -8) 
B)  There are none. 
C)  (0, 0) 
D)  (2, 1) 
A)  (0, 0) and (-4, -8) 
B)  There are none. 
C)  (0, 0) 
D)  (2, 1)

Question 8

An equation for the tangent line to the curve $y = \left( 3 x ^ { 2 } + x - 1 \right) ^ { 5 }$ at the point where x = 0 is

Accepted Answer

A)  y = 5x - 1 
B)  y = 10x + 1 
C)  y = 5x + 1 
D)  y = 10x - 1 
A)  y = 5x - 1 
B)  y = 10x + 1 
C)  y = 5x + 1 
D)  y = 10x - 1

Question 9

An equation for the tangent line to the curve $y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 5 }$ at the point where x = 0 is

Accepted Answer

A)  y = 5x + 1 
B)  y = 5x - 1 
C)  y = 10x - 1 
D)  y = 10x + 1 
A)  y = 5x + 1 
B)  y = 5x - 1 
C)  y = 10x - 1 
D)  y = 10x + 1

Question 10

An efficiency study at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have produced $Q ( t ) = - t ^ { 3 } + 6 t ^ { 2 } + 18 t$ units t hours later. At what rate, in units/hour, is the worker's rate of production changing with respect to time at 9:00 A.M.?

Accepted Answer

The answer of An efficiency study at a certain factory...

Question 11

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 7 }$

Accepted Answer

The answer of Differentiate: \(f ( x ) = \frac...

Question 12

Find $f ^ { \prime\prime\prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 3 } { x ^ { 2 } } + \sqrt { 7 }$

Accepted Answer

A)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 16 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$ 
B)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$ 
C)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 5 } { 72 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$ 
D)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 3 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 3 } { x ^ { 3 } }$ 
A)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 16 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$ 
B)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 15 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$ 
C)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 5 } { 72 x ^ { 3 } \sqrt { 3 x } } + \frac { 72 } { x ^ { 5 } }$ 
D)  $f ^ { \prime \prime \prime } ( x ) = - \frac { 3 } { 8 x ^ { 3 } \sqrt { 3 x } } + \frac { 3 } { x ^ { 3 } }$

Question 13

If $f ( x ) = \frac { 4 - 5 x ^ { 2 } } { x ^ { 3 } + 6 x - 5 }$ , what is $f ^ { \prime } ( x )$ ?

Accepted Answer

The answer of If \(f ( x ) = \frac...

Question 14

In a certain factory, output Q is related to inputs x and y by the equation $Q = 3 x ^ { 3 } + 5 x ^ { 2 } y ^ { 2 } + 2 y ^ { 3 }$ . If the current levels of input are x = 255 and y = 155, use calculus to estimate the change in input y that should be made to offset a decrease of 0.6 unit in input x so that output will be maintained at its current level. Round your answer to two decimal places, if necessary.

Accepted Answer

A)  An increase of 0.37 
B)  A decrease of 0.37 
C)  It cannot be determined 
D)  No change 
A)  An increase of 0.37 
B)  A decrease of 0.37 
C)  It cannot be determined 
D)  No change

Question 15

Differentiate: $f ( x ) = \left( x ^ { 2 } + 1 \right) ( x + 6 )$

Accepted Answer

A)  $x ^ { 2 } + 1$ 
B)  $3 x ^ { 2 } + 12 x + 1$ 
C)  2x + 1 
D)  12x + 1 
A)  $x ^ { 2 } + 1$ 
B)  $3 x ^ { 2 } + 12 x + 1$ 
C)  2x + 1 
D)  12x + 1

Question 16

Find all points on the graph of the function $f ( x ) = x ^ { 3 } ( 3 x + 12 )$ where the tangent line is horizontal.

Accepted Answer

(0, 0) and

Question 17

Find $f ^ { \prime\prime\prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 2 } { x ^ { 4 } } + \sqrt { 2 }$

Accepted Answer

The answer of Find \(f ^ { \prime\prime\prime } (...

Question 18

Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.

Accepted Answer

A)  There are none. 
B)  (2, 1) 
C)  (0, 0) and (-4, -8) 
D)  (0, 0) 
A)  There are none. 
B)  (2, 1) 
C)  (0, 0) and (-4, -8) 
D)  (0, 0)

Question 19

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 39 t ^ { 2 } + 459 t + 2$ . Find the velocity and acceleration of the object at time t. When is the object stationary?

Accepted Answer

A)  $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 9 and 17 
B)  $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 13 
C)  $v ( t ) = 3 t ^ { 2 } - 26 t + 459$ ; a(t) = 6t - 26; t = 9 
D)  $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 9 
A)  $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 9 and 17 
B)  $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 13 
C)  $v ( t ) = 3 t ^ { 2 } - 26 t + 459$ ; a(t) = 6t - 26; t = 9 
D)  $v ( t ) = 3 t ^ { 2 } - 78 t + 459$ ; a(t) = 6t - 78; t = 9

Question 20

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = x ^ { 3 } + 4$ .

Accepted Answer

The answer of Find \(f ^ { \prime \prime }...

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 5 t - 2$ . What is the object's acceleration?

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 3$ . Find the velocity and acceleration of the object at time t. When is the object stationary?

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 x ^ { 2 } - 5 x + 8 } { 6 - 5 x ^ { 3 } }$ at the point (1, 7).

For f (x) = 1 - x², find the slope of the secant line connecting the points whose x-coordinates are x = -4 and x = -3.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -4.

Differentiate: $f ( x ) = \sqrt { x } + \frac { 5 } { \sqrt { x } }$

If $f ( x ) = \frac { ( 3 - 5 x ) ^ { 3 } } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } }$ , then $f ^ { \prime } ( x ) = - 5 ( 2 x + 1 )$ .

Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.

An equation for the tangent line to the curve $y = \left( 3 x ^ { 2 } + x - 1 \right) ^ { 5 }$ at the point where x = 0 is

An equation for the tangent line to the curve $y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 5 }$ at the point where x = 0 is

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 7 }$

Find $f ^ { \prime\prime\prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 3 } { x ^ { 2 } } + \sqrt { 7 }$

If $f ( x ) = \frac { 4 - 5 x ^ { 2 } } { x ^ { 3 } + 6 x - 5 }$ , what is $f ^ { \prime } ( x )$ ?

Differentiate: $f ( x ) = \left( x ^ { 2 } + 1 \right) ( x + 6 )$

Find all points on the graph of the function $f ( x ) = x ^ { 3 } ( 3 x + 12 )$ where the tangent line is horizontal.

Find $f ^ { \prime\prime\prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 2 } { x ^ { 4 } } + \sqrt { 2 }$

Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 39 t ^ { 2 } + 459 t + 2$ . Find the velocity and acceleration of the object at time t. When is the object stationary?

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = x ^ { 3 } + 4$ .

Functions, Graphs, and Limits

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

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Exam 2: Differentiation: Basic Concepts

The displacement function of a moving object is described by s(t)=t2+5t−2s ( t ) = t ^ { 2 } + 5 t - 2s(t)=t2+5t−2 . What is the object's acceleration?

An object moves along a line in such a way that its position at time t is s(t)=t3−6t2+9t+3s ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 3s(t)=t3−6t2+9t+3 . Find the velocity and acceleration of the object at time t. When is the object stationary?

Find the equation of the tangent line to the curve f(x)=4x2−5x+86−5x3f ( x ) = \frac { 4 x ^ { 2 } - 5 x + 8 } { 6 - 5 x ^ { 3 } }f(x)=6−5x34x2−5x+8​ at the point (1, 7).

For f (x) = 1 - x2, find the slope of the secant line connecting the points whose x-coordinates are x = -4 and x = -3.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -4.

Differentiate: f(x)=x+5xf ( x ) = \sqrt { x } + \frac { 5 } { \sqrt { x } }f(x)=x​+x​5​

If f(x)=(3−5x)3(x2+x−1)2f ( x ) = \frac { ( 3 - 5 x ) ^ { 3 } } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } }f(x)=(x2+x−1)2(3−5x)3​ , then f′(x)=−5(2x+1)f ^ { \prime } ( x ) = - 5 ( 2 x + 1 )f′(x)=−5(2x+1) .

Find all points on the graph of the function f(x)=x2x+2f ( x ) = \frac { x ^ { 2 } } { x + 2 }f(x)=x+2x2​ where the tangent line is horizontal.

An equation for the tangent line to the curve y=(3x2+x−1)5y = \left( 3 x ^ { 2 } + x - 1 \right) ^ { 5 }y=(3x2+x−1)5 at the point where x = 0 is

An equation for the tangent line to the curve y=(7x2+x−1)5y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 5 }y=(7x2+x−1)5 at the point where x = 0 is

Differentiate: f(x)=x2x−7f ( x ) = \frac { x ^ { 2 } } { x - 7 }f(x)=x−7x2​

Find f′′′(x)f ^ { \prime\prime\prime } ( x )f′′′(x) if f(x)=13x−3x2+7f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 3 } { x ^ { 2 } } + \sqrt { 7 }f(x)=3x​1​−x23​+7​

If f(x)=4−5x2x3+6x−5f ( x ) = \frac { 4 - 5 x ^ { 2 } } { x ^ { 3 } + 6 x - 5 }f(x)=x3+6x−54−5x2​ , what is f′(x)f ^ { \prime } ( x )f′(x) ?

Differentiate: f(x)=(x2+1)(x+6)f ( x ) = \left( x ^ { 2 } + 1 \right) ( x + 6 )f(x)=(x2+1)(x+6)

Find all points on the graph of the function f(x)=x3(3x+12)f ( x ) = x ^ { 3 } ( 3 x + 12 )f(x)=x3(3x+12) where the tangent line is horizontal.

Find f′′′(x)f ^ { \prime\prime\prime } ( x )f′′′(x) if f(x)=13x−2x4+2f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 2 } { x ^ { 4 } } + \sqrt { 2 }f(x)=3x​1​−x42​+2​

Find all points on the graph of the function f(x)=x2x+2f ( x ) = \frac { x ^ { 2 } } { x + 2 }f(x)=x+2x2​ where the tangent line is horizontal.

An object moves along a line in such a way that its position at time t is s(t)=t3−39t2+459t+2s ( t ) = t ^ { 3 } - 39 t ^ { 2 } + 459 t + 2s(t)=t3−39t2+459t+2 . Find the velocity and acceleration of the object at time t. When is the object stationary?

Find f′′(x)f ^ { \prime \prime } ( x )f′′(x) , where f(x)=x3+4f ( x ) = x ^ { 3 } + 4f(x)=x3+4 .

Functions, Graphs, and Limits

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

The displacement function of a moving object is described by $s ( t ) = t ^ { 2 } + 5 t - 2$ . What is the object's acceleration?

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 3$ . Find the velocity and acceleration of the object at time t. When is the object stationary?

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 x ^ { 2 } - 5 x + 8 } { 6 - 5 x ^ { 3 } }$ at the point (1, 7).

For f (x) = 1 - x², find the slope of the secant line connecting the points whose x-coordinates are x = -4 and x = -3.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -4.

Differentiate: $f ( x ) = \sqrt { x } + \frac { 5 } { \sqrt { x } }$

If $f ( x ) = \frac { ( 3 - 5 x ) ^ { 3 } } { \left( x ^ { 2 } + x - 1 \right) ^ { 2 } }$ , then $f ^ { \prime } ( x ) = - 5 ( 2 x + 1 )$ .

Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.

An equation for the tangent line to the curve $y = \left( 3 x ^ { 2 } + x - 1 \right) ^ { 5 }$ at the point where x = 0 is

An equation for the tangent line to the curve $y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 5 }$ at the point where x = 0 is

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 7 }$

Find $f ^ { \prime\prime\prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 3 } { x ^ { 2 } } + \sqrt { 7 }$

If $f ( x ) = \frac { 4 - 5 x ^ { 2 } } { x ^ { 3 } + 6 x - 5 }$ , what is $f ^ { \prime } ( x )$ ?

Differentiate: $f ( x ) = \left( x ^ { 2 } + 1 \right) ( x + 6 )$

Find all points on the graph of the function $f ( x ) = x ^ { 3 } ( 3 x + 12 )$ where the tangent line is horizontal.

Find $f ^ { \prime\prime\prime } ( x )$ if $f ( x ) = \frac { 1 } { \sqrt { 3 x } } - \frac { 2 } { x ^ { 4 } } + \sqrt { 2 }$

Find all points on the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x + 2 }$ where the tangent line is horizontal.

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 39 t ^ { 2 } + 459 t + 2$ . Find the velocity and acceleration of the object at time t. When is the object stationary?

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = x ^ { 3 } + 4$ .