Question 1

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 8 x ^ { 4 } + 7 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 4$ .

Accepted Answer

A)  $f ^ { ( 4 ) } ( x ) = 120 x - 192$ 
B)  $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 8 x$ 
C)  $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 8 x + 7$ 
D)  $f ^ { ( 4 ) } ( x ) = 60 x ^ { 2 } - 192 x + 42$ 
A)  $f ^ { ( 4 ) } ( x ) = 120 x - 192$ 
B)  $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 8 x$ 
C)  $f ^ { ( 4 ) } ( x ) = x ^ { 2 } - 8 x + 7$ 
D)  $f ^ { ( 4 ) } ( x ) = 60 x ^ { 2 } - 192 x + 42$ A

Question 2

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 6 }$

Accepted Answer

A) 2x 
B) -x 
C)  $\frac { 3 x ^ { 2 } + 12 x } { ( x - 6 ) ^ { 2 } }$ 
D)  $\frac { x ^ { 2 } - 12 x } { ( x - 6 ) ^ { 2 } }$ 
A) 2x 
B) -x 
C)  $\frac { 3 x ^ { 2 } + 12 x } { ( x - 6 ) ^ { 2 } }$ 
D)  $\frac { x ^ { 2 } - 12 x } { ( x - 6 ) ^ { 2 } }$ D

Question 3

Find the rate of change of the given function f (x)with respect for x for the prescribed value x = -2. f (x)= x3 + 3x + 9

Accepted Answer

A) 6 
B) -3 
C) 15 
D) 24 
A) 6 
B) -3 
C) 15 
D) 24 C

Question 4

Differentiate: $f ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }$

Accepted Answer

A) 7 
B)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$ 
C) 0 
D)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$ 
A) 7 
B)  $\frac { 1 } { 2 \sqrt { x } } + \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$ 
C) 0 
D)  $\frac { 1 } { 2 \sqrt { x } } - \frac { 7 } { 2 \sqrt { x ^ { 3 } } }$

Question 5

For $f ( x ) = - \frac { 3 } { \sqrt { x } }$ ,find the average rate of change of f (x)with respect to x as x changes from 144 to 145.Then use calculus to find the instantaneous rate of change at x = 144.Round your answer to six decimal places,if necessary.

Accepted Answer

A) Average rate of change: 0.000864; Instantaneous rate of change: 0.000868 
B) Average rate of change: -0.000864; Instantaneous rate of change: 0.125 
C) Average rate of change: 0.000864; Instantaneous rate of change: -0.125 
D) Average rate of change: -0.000864; Instantaneous rate of change: 0.000868 
A) Average rate of change: 0.000864; Instantaneous rate of change: 0.000868 
B) Average rate of change: -0.000864; Instantaneous rate of change: 0.125 
C) Average rate of change: 0.000864; Instantaneous rate of change: -0.125 
D) Average rate of change: -0.000864; Instantaneous rate of change: 0.000868

Question 6

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 7

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 8

You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is $14 ^ { 3 } = 2,744$ cubic centimeters.If your measurement of the side is accurate to within 2%,approximately how accurate is your calculation of this volume? Round to two decimal places,if necessary.

Accepted Answer

A) Maximum error in volume is about ±164.64 cm3 
B) Maximum error in volume is about ±0.84 cm3 
C) Maximum error in volume is about ±2,304.96 cm3 
D) Maximum error in volume is about ±11.76 cm3 
A) Maximum error in volume is about ±164.64 cm3 
B) Maximum error in volume is about ±0.84 cm3 
C) Maximum error in volume is about ±2,304.96 cm3 
D) Maximum error in volume is about ±11.76 cm3

Question 9

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x$ at x = 7 is

Accepted Answer

A) y = 16x - 7 
B) y = 16x - 49 
C) y = 16x - 343 
D) y = 16x - 686 
A) y = 16x - 7 
B) y = 16x - 49 
C) y = 16x - 343 
D) y = 16x - 686

Question 10

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 11

The equation of the line tangent to the graph of $f ( x ) = 3 \sqrt { x }$ at x = 1 is

Accepted Answer

A)  $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$ 
B)  $y = \frac { 3 } { 2 } x - 1$ 
C)  $y = \frac { 1 } { 2 } x - \frac { 1 } { 2 }$ 
D)  $y = \frac { 3 } { 2 } x + \frac { 3 } { 2 }$ 
A)  $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$ 
B)  $y = \frac { 3 } { 2 } x - 1$ 
C)  $y = \frac { 1 } { 2 } x - \frac { 1 } { 2 }$ 
D)  $y = \frac { 3 } { 2 } x + \frac { 3 } { 2 }$

Question 12

When a certain commodity is sold for p dollars per unit,consumers will buy $D ( p ) = \frac { 31,500 } { p }$ units per month.It is estimated that t months from now,the price of the commodity will be $p ( t ) = t ^ { 2 / 3 } + 5.15$ dollars per unit.The approximate rate at which the monthly demand will be changing with respect to time in 27 months is

Accepted Answer

A) -32 units per month 
B) -35 units per month 
C) -132 units per month 
D) 35 units per month 
A) -32 units per month 
B) -35 units per month 
C) -132 units per month 
D) 35 units per month

Question 13

Suppose the output at a certain factory is $$Q = 4 x ^ { 3 } + 5 x ^ { 3 } y ^ { 4 } + 3 y ^ { 3 }$$ units,where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor.The current labor force consists of 20 hours of skilled labor and 10 hours of unskilled labor.Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level.Round you answer to two decimal places,if necessary.

Accepted Answer

A) -2.67 hours 
B) 2.67 hours 
C) -1 hours 
D) -0.38 hours 
A) -2.67 hours 
B) 2.67 hours 
C) -1 hours 
D) -0.38 hours

Question 14

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 7:00 A.M.will have assembled $f ( x ) = - x ^ { 3 } + 7 x ^ { 2 } - 3 x$ transistor radios x hours later.Approximately how many radios will the worker assemble between 9:00 and 9:30 A.M.?

Accepted Answer

A) Approximately 7 radios 
B) Approximately 13 radios 
C) Approximately 14 radios 
D) Approximately 390 radios 
A) Approximately 7 radios 
B) Approximately 13 radios 
C) Approximately 14 radios 
D) Approximately 390 radios

Question 15

The function $f ( x ) = \frac { x } { 2 x + 1 } - 5$ will decrease by approximately 0.6 as x decreases from 3 to 2.7.

Accepted Answer

A) True 
 B)False

Question 16

The largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula $S = 4 \pi r ^ { 2 }$ to be no greater than 6 percent is about:

Accepted Answer

A) 2% 
B) 6% 
C) 3% 
D) 1% 
A) 2% 
B) 6% 
C) 3% 
D) 1%

Question 17

What is the rate of change of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$ with respect to t when t = 2?

Accepted Answer

A)  $\frac { 9 } { 4 }$ 
B) 4 
C)  $\frac { 19 } { 16 }$ 
D)  $\frac { 17 } { 4 }$ 
A)  $\frac { 9 } { 4 }$ 
B) 4 
C)  $\frac { 19 } { 16 }$ 
D)  $\frac { 17 } { 4 }$

Question 18

Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .

Accepted Answer

A)  $60 x ^ { 2 } + 11$ 
B)  $80 x ^ { 3 }$ 
C)  $60 x ^ { 2 } - 100$ 
D)  $- \frac { 80 } { 11 } x ^ { 3 }$ 
A)  $60 x ^ { 2 } + 11$ 
B)  $80 x ^ { 3 }$ 
C)  $60 x ^ { 2 } - 100$ 
D)  $- \frac { 80 } { 11 } x ^ { 3 }$

Question 19

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

Accepted Answer

A) True 
 B)False

Question 20

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 9 t ^ { 2 } + 15 t + 2$ .Find the velocity and acceleration of the object at time t.When is the object stationary?

Accepted Answer

A)  $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 3 
B)  $v ( t ) = 3 t ^ { 2 } - 6 t + 15$ ; a(t)= 6t - 6; t = 1 
C)  $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 1 
D)  $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 1 and 5 
A)  $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 3 
B)  $v ( t ) = 3 t ^ { 2 } - 6 t + 15$ ; a(t)= 6t - 6; t = 1 
C)  $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 1 
D)  $v ( t ) = 3 t ^ { 2 } - 18 t + 15$ ; a(t)= 6t - 18; t = 1 and 5

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 8 x ^ { 4 } + 7 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 4$ .

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 6 }$

Find the rate of change of the given function f (x)with respect for x for the prescribed value x = -2. f (x)= x³ + 3x + 9

Differentiate: $f ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }$

For $f ( x ) = - \frac { 3 } { \sqrt { x } }$ ,find the average rate of change of f (x)with respect to x as x changes from 144 to 145.Then use calculus to find the instantaneous rate of change at x = 144.Round your answer to six decimal places,if necessary.

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

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

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x$ at x = 7 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

The equation of the line tangent to the graph of $f ( x ) = 3 \sqrt { x }$ at x = 1 is

The function $f ( x ) = \frac { x } { 2 x + 1 } - 5$ will decrease by approximately 0.6 as x decreases from 3 to 2.7.

The largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula $S = 4 \pi r ^ { 2 }$ to be no greater than 6 percent is about:

What is the rate of change of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$ with respect to t when t = 2?

Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .

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

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 9 t ^ { 2 } + 15 t + 2$ .Find the velocity and acceleration of the object at time t.When is the object stationary?

Functions,graphs,and Limits

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

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Exam 2: Differentiation: Basic Concepts

Find f(4)(x)f ^ { ( 4 ) } ( x )f(4)(x) if f(x)=x5−8x4+7x3−7x2+7x−4f ( x ) = x ^ { 5 } - 8 x ^ { 4 } + 7 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 4f(x)=x5−8x4+7x3−7x2+7x−4 .

Differentiate: f(x)=x2x−6f ( x ) = \frac { x ^ { 2 } } { x - 6 }f(x)=x−6x2​

Find the rate of change of the given function f (x)with respect for x for the prescribed value x = -2. f (x)= x3 + 3x + 9

Differentiate: f(x)=x+7xf ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }f(x)=x​+x​7​

For f(x)=−3xf ( x ) = - \frac { 3 } { \sqrt { x } }f(x)=−x​3​ ,find the average rate of change of f (x)with respect to x as x changes from 144 to 145.Then use calculus to find the instantaneous rate of change at x = 144.Round your answer to six decimal places,if necessary.

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​

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

The equation of the line tangent to the graph of f(x)=x2+2xf ( x ) = x ^ { 2 } + 2 xf(x)=x2+2x at x = 7 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

The equation of the line tangent to the graph of f(x)=3xf ( x ) = 3 \sqrt { x }f(x)=3x​ at x = 1 is

The function f(x)=x2x+1−5f ( x ) = \frac { x } { 2 x + 1 } - 5f(x)=2x+1x​−5 will decrease by approximately 0.6 as x decreases from 3 to 2.7.

The largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula S=4πr2S = 4 \pi r ^ { 2 }S=4πr2 to be no greater than 6 percent is about:

What is the rate of change of f(t)=7t−5t+2f ( t ) = \frac { 7 t - 5 } { t + 2 }f(t)=t+27t−5​ with respect to t when t = 2?

Use implicit differentiation to find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ for 4x5+11y=1004 x ^ { 5 } + 11 y = 1004x5+11y=100 .

If f(x)=1−3x2f ( x ) = \sqrt { 1 - 3 x ^ { 2 } }f(x)=1−3x2​ ,then f′′(x)=−3(1−3x2)3/2f ^ { \prime \prime } ( x ) = \frac { - 3 } { \left( 1 - 3 x ^ { 2 } \right) ^ { 3 / 2 } }f′′(x)=(1−3x2)3/2−3​ .

An object moves along a line in such a way that its position at time t is s(t)=t3−9t2+15t+2s ( t ) = t ^ { 3 } - 9 t ^ { 2 } + 15 t + 2s(t)=t3−9t2+15t+2 .Find the velocity and acceleration of the object at time t.When is the object stationary?

Functions,graphs,and Limits

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Find $f ^ { ( 4 ) } ( x )$ if $f ( x ) = x ^ { 5 } - 8 x ^ { 4 } + 7 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 4$ .

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 6 }$

Find the rate of change of the given function f (x)with respect for x for the prescribed value x = -2. f (x)= x³ + 3x + 9

Differentiate: $f ( x ) = \sqrt { x } + \frac { 7 } { \sqrt { x } }$

For $f ( x ) = - \frac { 3 } { \sqrt { x } }$ ,find the average rate of change of f (x)with respect to x as x changes from 144 to 145.Then use calculus to find the instantaneous rate of change at x = 144.Round your answer to six decimal places,if necessary.

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

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

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x$ at x = 7 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

The equation of the line tangent to the graph of $f ( x ) = 3 \sqrt { x }$ at x = 1 is

The function $f ( x ) = \frac { x } { 2 x + 1 } - 5$ will decrease by approximately 0.6 as x decreases from 3 to 2.7.

The largest percentage error you can allow in the measurement of the radius of a sphere if you want the error in the calculation of its surface area using the formula $S = 4 \pi r ^ { 2 }$ to be no greater than 6 percent is about:

What is the rate of change of $f ( t ) = \frac { 7 t - 5 } { t + 2 }$ with respect to t when t = 2?

Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .

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

An object moves along a line in such a way that its position at time t is $s ( t ) = t ^ { 3 } - 9 t ^ { 2 } + 15 t + 2$ .Find the velocity and acceleration of the object at time t.When is the object stationary?