Question 1

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  False

Question 2

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

Accepted Answer

A)  y = 12x - 11 
B)  y = 12x 
C)  y = 3x + 1 
D)  y = 12x - 1 
A)  y = 12x - 11 
B)  y = 12x 
C)  y = 3x + 1 
D)  y = 12x - 1 A

Question 3

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)  $- \frac { 80 } { 11 } x ^ { 3 }$ D

Question 4

If the displacement of a moving object is $s ( t ) = 3 t ^ { 3 }$ , the acceleration is 18t.

Accepted Answer

A) True 
 B)False

Question 5

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

Accepted Answer

The answer of If the total cost of manufacturing q...

Question 6

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = \frac { 3 } { 1 + x ^ { 3 } }$ .

Accepted Answer

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

Question 7

If $f ( x ) = 3 x ^ { 5 } - 7 x ^ { 3 } + 2 x ^ { 2 } + 5$ , then $f ^ { \prime \prime \prime } ( x ) = 180 x ^ { 2 } - 42$ .

Accepted Answer

A) True 
 B)False

Question 8

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 2 u ^ { 2 } - 3$ and $u = x ^ { 2 } + x - 1$ .

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 9

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have assembled $f ( x ) = - x ^ { 3 } + 10 x ^ { 2 } - 3 x$ transistor radios x hours later. Approximately how many radios will the worker assemble between 9:00 and 9:15 A.M.?

Accepted Answer

Approximat

Question 10

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

Question 11

Differentiating $f ( x ) = x ^ { 3 } - 3 x + 1$ gives $3 x ^ { 2 }$ .

Accepted Answer

A) True 
 B)False

Question 12

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at (1, 2).

Accepted Answer

A)  Not defined 
B)  y = 2 
C)  x = 1 
D)  y = 2x 
A)  Not defined 
B)  y = 2 
C)  x = 1 
D)  y = 2x

Question 13

The temperature in degrees Fahrenheit inside an oven t minutes after turning it on can be modeled with the function $F ( t ) = \frac { 400 t + 70 } { t + 1 }$ . Find $F ^ { \prime } ( 5 )$ and interpret what it tells us about the temperature. Round your answer to 2 decimal places.

Accepted Answer

@#LAT-DLM& ; After 5 minutes,

Question 14

Differentiating $f ( x ) = \frac { 1 } { 3 } x ^ { 7 } - 2 x ^ { 5 } + 9 x - 8$ gives $\frac { 7 x ^ { 6 } } { 3 } - 10 x ^ { 4 } + 9$ .

Accepted Answer

A) True 
 B)False

Question 15

What is the rate of change of $f ( t ) = \frac { 8 t - 3 } { t + 6 }$ with respect to t when t = 45?

Accepted Answer

A)  $\frac { 1 } { 51 }$ 
B)  $- \frac { 1 } { 51 }$ 
C)  51 
D)  -51 
A)  $\frac { 1 } { 51 }$ 
B)  $- \frac { 1 } { 51 }$ 
C)  51 
D)  -51

Question 16

Differentiate: $f ( x ) = x ^ { 8 } + 2$

Accepted Answer

A)  $8 x ^ { 7 } + 2$ 
B)  $8 x ^ { 9 } + 2 x$ 
C)  $8 x ^ { 7 }$ 
D)  $7 x ^ { 7 }$ 
A)  $8 x ^ { 7 } + 2$ 
B)  $8 x ^ { 9 } + 2 x$ 
C)  $8 x ^ { 7 }$ 
D)  $7 x ^ { 7 }$

Question 17

If $f ( x ) = \frac { 5 x - 1 } { 7 x + 5 }$ , what is $f ^ { \prime } ( x )$ ?

Accepted Answer

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

Question 18

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: -0.125 
B)  Average rate of change: -0.000864; Instantaneous rate of change: 0.000868 
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: -0.125 
B)  Average rate of change: -0.000864; Instantaneous rate of change: 0.000868 
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 19

For f (x) = 6 - x2, find the slope of the secant line connecting the points whose x-coordinates are x = -1 and x = -0.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -1.

Accepted Answer

Slope of secant line

Question 20

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

Accepted Answer

A) True 
 B)False

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

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

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

If the displacement of a moving object is $s ( t ) = 3 t ^ { 3 }$ , the acceleration is 18t.

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = \frac { 3 } { 1 + x ^ { 3 } }$ .

If $f ( x ) = 3 x ^ { 5 } - 7 x ^ { 3 } + 2 x ^ { 2 } + 5$ , then $f ^ { \prime \prime \prime } ( x ) = 180 x ^ { 2 } - 42$ .

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 2 u ^ { 2 } - 3$ and $u = x ^ { 2 } + x - 1$ .

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?

Differentiating $f ( x ) = x ^ { 3 } - 3 x + 1$ gives $3 x ^ { 2 }$ .

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at (1, 2).

The temperature in degrees Fahrenheit inside an oven t minutes after turning it on can be modeled with the function $F ( t ) = \frac { 400 t + 70 } { t + 1 }$ . Find $F ^ { \prime } ( 5 )$ and interpret what it tells us about the temperature. Round your answer to 2 decimal places.

Differentiating $f ( x ) = \frac { 1 } { 3 } x ^ { 7 } - 2 x ^ { 5 } + 9 x - 8$ gives $\frac { 7 x ^ { 6 } } { 3 } - 10 x ^ { 4 } + 9$ .

What is the rate of change of $f ( t ) = \frac { 8 t - 3 } { t + 6 }$ with respect to t when t = 45?

Differentiate: $f ( x ) = x ^ { 8 } + 2$

If $f ( x ) = \frac { 5 x - 1 } { 7 x + 5 }$ , what is $f ^ { \prime } ( x )$ ?

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.

For f (x) = 6 - x², find the slope of the secant line connecting the points whose x-coordinates are x = -1 and x = -0.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -1.

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

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

Exam 2: Differentiation: Basic Concepts

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 equation for the tangent line to the curve y=(x3+x−1)3y = \left( x ^ { 3 } + x - 1 \right) ^ { 3 }y=(x3+x−1)3 at the point where x = 1 is

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 the displacement of a moving object is s(t)=3t3s ( t ) = 3 t ^ { 3 }s(t)=3t3 , the acceleration is 18t.

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

Find f′′(x)f ^ { \prime \prime } ( x )f′′(x) , where f(x)=31+x3f ( x ) = \frac { 3 } { 1 + x ^ { 3 } }f(x)=1+x33​ .

If f(x)=3x5−7x3+2x2+5f ( x ) = 3 x ^ { 5 } - 7 x ^ { 3 } + 2 x ^ { 2 } + 5f(x)=3x5−7x3+2x2+5 , then f′′′(x)=180x2−42f ^ { \prime \prime \prime } ( x ) = 180 x ^ { 2 } - 42f′′′(x)=180x2−42 .

Find dydx\frac { d y } { d x }dxdy​ if y=u3+2u2−3y = u ^ { 3 } + 2 u ^ { 2 } - 3y=u3+2u2−3 and u=x2+x−1u = x ^ { 2 } + x - 1u=x2+x−1 .

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?

Differentiating f(x)=x3−3x+1f ( x ) = x ^ { 3 } - 3 x + 1f(x)=x3−3x+1 gives 3x23 x ^ { 2 }3x2 .

Find the equation of the tangent line to the graph of f(x)=x2+1f ( x ) = x ^ { 2 } + 1f(x)=x2+1 at (1, 2).

Differentiating f(x)=13x7−2x5+9x−8f ( x ) = \frac { 1 } { 3 } x ^ { 7 } - 2 x ^ { 5 } + 9 x - 8f(x)=31​x7−2x5+9x−8 gives 7x63−10x4+9\frac { 7 x ^ { 6 } } { 3 } - 10 x ^ { 4 } + 937x6​−10x4+9 .

What is the rate of change of f(t)=8t−3t+6f ( t ) = \frac { 8 t - 3 } { t + 6 }f(t)=t+68t−3​ with respect to t when t = 45?

Differentiate: f(x)=x8+2f ( x ) = x ^ { 8 } + 2f(x)=x8+2

If f(x)=5x−17x+5f ( x ) = \frac { 5 x - 1 } { 7 x + 5 }f(x)=7x+55x−1​ , what is f′(x)f ^ { \prime } ( x )f′(x) ?

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.

For f (x) = 6 - x2, find the slope of the secant line connecting the points whose x-coordinates are x = -1 and x = -0.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -1.

If f(x)=x2−3x+51−3xf ( x ) = \frac { x ^ { 2 } - 3 x + 5 } { \sqrt { 1 - 3 x } }f(x)=1−3x​x2−3x+5​ , then f′(x)=2x−31−3xf ^ { \prime } ( x ) = \frac { 2 x - 3 } { \sqrt { 1 - 3 x } }f′(x)=1−3x​2x−3​ .

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

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

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

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

If the displacement of a moving object is $s ( t ) = 3 t ^ { 3 }$ , the acceleration is 18t.

Find $f ^ { \prime \prime } ( x )$ , where $f ( x ) = \frac { 3 } { 1 + x ^ { 3 } }$ .

If $f ( x ) = 3 x ^ { 5 } - 7 x ^ { 3 } + 2 x ^ { 2 } + 5$ , then $f ^ { \prime \prime \prime } ( x ) = 180 x ^ { 2 } - 42$ .

Find $\frac { d y } { d x }$ if $y = u ^ { 3 } + 2 u ^ { 2 } - 3$ and $u = x ^ { 2 } + x - 1$ .

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?

Differentiating $f ( x ) = x ^ { 3 } - 3 x + 1$ gives $3 x ^ { 2 }$ .

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 1$ at (1, 2).

Differentiating $f ( x ) = \frac { 1 } { 3 } x ^ { 7 } - 2 x ^ { 5 } + 9 x - 8$ gives $\frac { 7 x ^ { 6 } } { 3 } - 10 x ^ { 4 } + 9$ .

What is the rate of change of $f ( t ) = \frac { 8 t - 3 } { t + 6 }$ with respect to t when t = 45?

Differentiate: $f ( x ) = x ^ { 8 } + 2$

If $f ( x ) = \frac { 5 x - 1 } { 7 x + 5 }$ , what is $f ^ { \prime } ( x )$ ?

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.

For f (x) = 6 - x², find the slope of the secant line connecting the points whose x-coordinates are x = -1 and x = -0.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -1.

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