Question 1

Differentiate: $f ( x ) = x ^ { 4 } + 7$

Accepted Answer

A)  $4 x ^ { 3 }$ 
B)  $4 x ^ { 3 } + 7$ 
C)  $4 x ^ { 5 } + 7 x$ 
D)  $3 x ^ { 3 }$ 
A)  $4 x ^ { 3 }$ 
B)  $4 x ^ { 3 } + 7$ 
C)  $4 x ^ { 5 } + 7 x$ 
D)  $3 x ^ { 3 }$

Question 2

What is the rate of change of $f ( t ) = \frac { 7 t - 4 } { t + 3 }$ with respect to t when t = 3?

Accepted Answer

A)  $\frac { 25 } { 36 }$ 
B)  $\frac { 19 } { 6 }$ 
C)  6 
D)  $\frac { 5 } { 3 }$ 
A)  $\frac { 25 } { 36 }$ 
B)  $\frac { 19 } { 6 }$ 
C)  6 
D)  $\frac { 5 } { 3 }$

Question 3

Find an equation for the tangent line to the curve $x ^ { 2 } + y ^ { 3 } = x y + 1$ at the point (1, -1).

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } - 9 x + 16$ at the point (1, 8).

Accepted Answer

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

Question 6

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 7

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 8

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 5 x } { 6 } + \frac { 2 } { 3 x } - \sqrt [ 5 ] { x }$

Accepted Answer

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

Question 9

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

Question 10

An equation for the tangent line to the curve $y = \left( 6 x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 0 is

Accepted Answer

The answer of An equation for the tangent line to...

Question 11

Find an equation for the tangent line to the curve $y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 0.

Accepted Answer

A)  y = 14x + 1 
B)  y = 24x + 3 
C)  y = 3x + 1 
D)  y = 3x - 1 
A)  y = 14x + 1 
B)  y = 24x + 3 
C)  y = 3x + 1 
D)  y = 3x - 1

Question 12

Find all points on the graph of the function $f ( x ) = x ^ { 3 } ( 5 x + 20 )$ where the tangent line is horizontal.

Accepted Answer

(0, 0) and

Question 13

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 3 x$ at x = 2 is

Accepted Answer

A)  y = 7x - 4 
B)  y = 7x - 422 
C)  y = 7x - 2 
D)  y = 7x - 144 
A)  y = 7x - 4 
B)  y = 7x - 422 
C)  y = 7x - 2 
D)  y = 7x - 144

Question 14

If f (x) represents the price per barrel of oil in terms of time, what does $\frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ represent? What about $\lim _ { h \rightarrow 0 } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ ?

Accepted Answer

The average rate of change of

Question 15

If $x ^ { 2 } + 2 y ^ { 2 } = 5$ , then $\frac { d y } { d x } = 2 x$ .

Accepted Answer

A) True 
 B)False

Question 16

If f (x) represents the price per barrel of oil in terms of time, what does $\frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ represent? What about $\lim _ { h \rightarrow 0 } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ ?

Accepted Answer

The average rate of change of

Question 17

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

Accepted Answer

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

Question 18

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 7 } { 3 } x + \frac { 8 } { 7 x } - \sqrt [ 6 ] { x }$

Accepted Answer

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

Question 19

Find $\frac { d y } { d x }$ , where $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$ .

Accepted Answer

A)  $y ^ { 3 } - 6 x - 7$ 
B)  $\frac { 6 x - y ^ { 3 } } { 3 x y ^ { 2 } - 7 }$ 
C)  $y ^ { 3 } - 6 x$ 
D)  $\frac { 6 x ^ { 2 } } { y ^ { 3 } }$ 
A)  $y ^ { 3 } - 6 x - 7$ 
B)  $\frac { 6 x - y ^ { 3 } } { 3 x y ^ { 2 } - 7 }$ 
C)  $y ^ { 3 } - 6 x$ 
D)  $\frac { 6 x ^ { 2 } } { y ^ { 3 } }$

Question 20

Find $\frac { d y } { d x }$ , where $\sqrt { x } + \sqrt { y } = x y$ .

Accepted Answer

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

Differentiate: $f ( x ) = x ^ { 4 } + 7$

What is the rate of change of $f ( t ) = \frac { 7 t - 4 } { t + 3 }$ with respect to t when t = 3?

Find an equation for the tangent line to the curve $x ^ { 2 } + y ^ { 3 } = x y + 1$ at the point (1, -1).

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

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } - 9 x + 16$ at the point (1, 8).

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.

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 5 x } { 6 } + \frac { 2 } { 3 x } - \sqrt [ 5 ] { x }$

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

An equation for the tangent line to the curve $y = \left( 6 x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 0 is

Find an equation for the tangent line to the curve $y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 0.

Find all points on the graph of the function $f ( x ) = x ^ { 3 } ( 5 x + 20 )$ where the tangent line is horizontal.

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 3 x$ at x = 2 is

If f (x) represents the price per barrel of oil in terms of time, what does $\frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ represent? What about $\lim _ { h \rightarrow 0 } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ ?

If $x ^ { 2 } + 2 y ^ { 2 } = 5$ , then $\frac { d y } { d x } = 2 x$ .

If f (x) represents the price per barrel of oil in terms of time, what does $\frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ represent? What about $\lim _ { h \rightarrow 0 } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ ?

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

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 7 } { 3 } x + \frac { 8 } { 7 x } - \sqrt [ 6 ] { x }$

Find $\frac { d y } { d x }$ , where $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$ .

Find $\frac { d y } { d x }$ , where $\sqrt { x } + \sqrt { y } = x y$ .

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

Differentiate: f(x)=x4+7f ( x ) = x ^ { 4 } + 7f(x)=x4+7

What is the rate of change of f(t)=7t−4t+3f ( t ) = \frac { 7 t - 4 } { t + 3 }f(t)=t+37t−4​ with respect to t when t = 3?

Find an equation for the tangent line to the curve x2+y3=xy+1x ^ { 2 } + y ^ { 3 } = x y + 1x2+y3=xy+1 at the point (1, -1).

Find the equation of the line that is tangent to the curve f(x)=5+2x2−x3f ( x ) = 5 + 2 x ^ { 2 } - x ^ { 3 }f(x)=5+2x2−x3 at the point (1, 6).

Find the equation of the tangent to the graph of f(x)=x2−9x+16f ( x ) = x ^ { 2 } - 9 x + 16f(x)=x2−9x+16 at the point (1, 8).

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.

Differentiate: f(x)=23x6−5x6+23x−x5f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 5 x } { 6 } + \frac { 2 } { 3 x } - \sqrt [ 5 ] { x }f(x)=32​x6−65x​+3x2​−5x​

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

An equation for the tangent line to the curve y=(6x2+x−1)3y = \left( 6 x ^ { 2 } + x - 1 \right) ^ { 3 }y=(6x2+x−1)3 at the point where x = 0 is

Find an equation for the tangent line to the curve y=(7x2+x−1)3y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 3 }y=(7x2+x−1)3 at the point where x = 0.

Find all points on the graph of the function f(x)=x3(5x+20)f ( x ) = x ^ { 3 } ( 5 x + 20 )f(x)=x3(5x+20) where the tangent line is horizontal.

The equation of the line tangent to the graph of f(x)=x2+3xf ( x ) = x ^ { 2 } + 3 xf(x)=x2+3x at x = 2 is

If x2+2y2=5x ^ { 2 } + 2 y ^ { 2 } = 5x2+2y2=5 , then dydx=2x\frac { d y } { d x } = 2 xdxdy​=2x .

Find the equation of the tangent line to the curve f(x)=x3−x2+7f ( x ) = x ^ { 3 } - x ^ { 2 } + 7f(x)=x3−x2+7 at the point (1, 7).

Differentiate: f(x)=23x6−73x+87x−x6f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 7 } { 3 } x + \frac { 8 } { 7 x } - \sqrt [ 6 ] { x }f(x)=32​x6−37​x+7x8​−6x​

Find dydx\frac { d y } { d x }dxdy​ , where xy3−3x2=7yx y ^ { 3 } - 3 x ^ { 2 } = 7 yxy3−3x2=7y .

Find dydx\frac { d y } { d x }dxdy​ , where x+y=xy\sqrt { x } + \sqrt { y } = x yx​+y​=xy .

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

Differentiate: $f ( x ) = x ^ { 4 } + 7$

What is the rate of change of $f ( t ) = \frac { 7 t - 4 } { t + 3 }$ with respect to t when t = 3?

Find an equation for the tangent line to the curve $x ^ { 2 } + y ^ { 3 } = x y + 1$ at the point (1, -1).

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

Find the equation of the tangent to the graph of $f ( x ) = x ^ { 2 } - 9 x + 16$ at the point (1, 8).

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.

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 5 x } { 6 } + \frac { 2 } { 3 x } - \sqrt [ 5 ] { x }$

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

An equation for the tangent line to the curve $y = \left( 6 x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 0 is

Find an equation for the tangent line to the curve $y = \left( 7 x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 0.

Find all points on the graph of the function $f ( x ) = x ^ { 3 } ( 5 x + 20 )$ where the tangent line is horizontal.

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 3 x$ at x = 2 is

If $x ^ { 2 } + 2 y ^ { 2 } = 5$ , then $\frac { d y } { d x } = 2 x$ .

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

Differentiate: $f ( x ) = \frac { 2 } { 3 } x ^ { 6 } - \frac { 7 } { 3 } x + \frac { 8 } { 7 x } - \sqrt [ 6 ] { x }$

Find $\frac { d y } { d x }$ , where $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$ .

Find $\frac { d y } { d x }$ , where $\sqrt { x } + \sqrt { y } = x y$ .