Question 1

Given the graph of f, find any values of x at which f ' is not defined.
-

Accepted Answer

A)  x = 0, 1, 2 
B)  x = 0 
C)  x = 1 
D)  x = 2 
A)  x = 0, 1, 2 
B)  x = 0 
C)  x = 1 
D)  x = 2 B

Question 2

Find y'.
-$$y = ( 5 x - 5 ) ( 6 x + 1 )$$

Accepted Answer

A)  $30 x - 25$ 
B)  $60 x - 25$ 
C)  $60 x - 35$ 
D)  $60 x - 12.5$ 
A)  $30 x - 25$ 
B)  $60 x - 25$ 
C)  $60 x - 35$ 
D)  $60 x - 12.5$ B

Question 3

Find the second derivative.
-$$w = z ^ { - 5 } - \frac { 1 } { z }$$

Accepted Answer

A)  $30 z ^ { - 7 } + \frac { 2 } { z ^ { 3 } }$ 
B)  $30 z ^ { - 7 } - \frac { 2 } { z ^ { 3 } }$ 
C)  $- 5 z ^ { - 6 } + \frac { 1 } { z ^ { 2 } }$ 
D)  $- 5 z ^ { - 7 } - \frac { 2 } { z ^ { 3 } }$ 
A)  $30 z ^ { - 7 } + \frac { 2 } { z ^ { 3 } }$ 
B)  $30 z ^ { - 7 } - \frac { 2 } { z ^ { 3 } }$ 
C)  $- 5 z ^ { - 6 } + \frac { 1 } { z ^ { 2 } }$ 
D)  $- 5 z ^ { - 7 } - \frac { 2 } { z ^ { 3 } }$ B

Question 4

Find the slope of the curve at the indicated point.
-For a motorcycle traveling at speed $v$ (in $m p h$ ) when the brakes are applied, the distance $d$ (in feet) required to stop the motorcycle may be approximated by the formula $\mathrm { d } = 0.05 \mathrm { v } ^ { 2 } + \mathrm { v }$. Find the instantaneous rate of change of distance with respect to velocity when the speed is $46 \mathrm { mph }$.

Accepted Answer

A)  5.6 mph 
B)  47 mph 
C)  11.2 mph 
D)  4.6 mph 
A)  5.6 mph 
B)  47 mph 
C)  11.2 mph 
D)  4.6 mph

Question 5

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative.
-$$u ( 2 ) = 10 , u ^ { \prime } ( 2 ) = 4 , v ( 2 ) = - 3 , v ^ { \prime } ( 2 ) = - 5 \text {. }$$
$\frac { \mathrm { d } } { \mathrm { dx } } \left( \frac { \mathrm { v } } { \mathrm { u } } \right)$ at $\mathrm { x } = 2$

Accepted Answer

A)  $- \frac { 31 } { 50 }$ 
B)  $\frac { 19 } { 50 }$ 
C)  $- \frac { 19 } { 50 }$ 
D)  $- \frac { 19 } { 5 }$ 
A)  $- \frac { 31 } { 50 }$ 
B)  $\frac { 19 } { 50 }$ 
C)  $- \frac { 19 } { 50 }$ 
D)  $- \frac { 19 } { 5 }$

Question 6

Find the derivative.
-$$y = \frac { 1 } { x ^ { 2.4 } } - \frac { \pi } { \sqrt { x } }$$

Accepted Answer

A)  $- 3.4 x ^ { - 2.4 } + \frac { \pi } { 2 } x ^ { - 3 / 2 }$ 
B)  $- 2.4 x ^ { - 3.4 } + \pi x ^ { - 3 / 2 }$ 
C)  $- 2.4 x ^ { - 3.4 } + \frac { \pi } { 2 } x ^ { - 3 / 2 }$ 
D)  $- 2.4 x ^ { - 3.4 } - \frac { \pi } { 2 } x ^ { - 3 / 2 }$ 
A)  $- 3.4 x ^ { - 2.4 } + \frac { \pi } { 2 } x ^ { - 3 / 2 }$ 
B)  $- 2.4 x ^ { - 3.4 } + \pi x ^ { - 3 / 2 }$ 
C)  $- 2.4 x ^ { - 3.4 } + \frac { \pi } { 2 } x ^ { - 3 / 2 }$ 
D)  $- 2.4 x ^ { - 3.4 } - \frac { \pi } { 2 } x ^ { - 3 / 2 }$

Question 7

Find the derivative.
-$$y = 2 - 8 x ^ { 3 }$$

Accepted Answer

A)  $- 24 x$ 
B)  $2 - 24 x ^ { 2 }$ 
C)  $- 16 x ^ { 2 }$ 
D)  $- 24 x ^ { 2 }$ 
A)  $- 24 x$ 
B)  $2 - 24 x ^ { 2 }$ 
C)  $- 16 x ^ { 2 }$ 
D)  $- 24 x ^ { 2 }$

Question 8

Provide an appropriate response.
-Does the curve $y = x ^ { 3 } + 4 x - 10$ have a tangent whose slope is $- 2$ ? If so, find an equation for the line and the point of tangency. If not, why not?

Accepted Answer

The curve has no tangent whose

Question 9

$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$
-$$g ( x ) = \frac { x } { x + 3 }$$

Accepted Answer

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

Question 10

Find an equation for the tangent to the curve at the given point.
-$$h ( x ) = t ^ { 3 } - 16 t + 4 , ( 4,4 )$$

Accepted Answer

A)  $y = 4$ 
B)  $y = 32 t + 4$ 
C)  $\mathrm { y } = 36 \mathrm { t } - 124$ 
D)  $y = 32 t - 124$ 
A)  $y = 4$ 
B)  $y = 32 t + 4$ 
C)  $\mathrm { y } = 36 \mathrm { t } - 124$ 
D)  $y = 32 t - 124$

Question 11

Find the derivative.
-$$y = \sqrt [ 9 ] { x ^ { 2 } } + x ^ { e }$$

Accepted Answer

A)  $\frac { 2 } { 9 x ^ { 7 / 9 } } + e x ^ { e }$ 
B)  $\frac { 2 x ^ { 7 / 9 } } { 9 } + e x ^ { e - 1 }$ 
C)  $\frac { 2 } { 9 x ^ { 7 / 9 } } + e x ^ { e - 1 }$ 
D)  $\frac { 2 x ^ { 7 / 9 } } { 9 } + e x ^ { e }$ 
A)  $\frac { 2 } { 9 x ^ { 7 / 9 } } + e x ^ { e }$ 
B)  $\frac { 2 x ^ { 7 / 9 } } { 9 } + e x ^ { e - 1 }$ 
C)  $\frac { 2 } { 9 x ^ { 7 / 9 } } + e x ^ { e - 1 }$ 
D)  $\frac { 2 x ^ { 7 / 9 } } { 9 } + e x ^ { e }$

Question 12

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

Accepted Answer

A)  $y = 5 x - 4$ 
B)  $y = - 3 x + 4$ 
C)  $y = 5 x + 4$ 
D)  $y = 3 x + 4$ 
A)  $y = 5 x - 4$ 
B)  $y = - 3 x + 4$ 
C)  $y = 5 x + 4$ 
D)  $y = 3 x + 4$

Question 13

Given the graph of f, find any values of x at which f ' is not defined.
-Use the following information to graph the function f over the closed interval [-5, 6]. 
i) The graph of f is made of closed line segments joined end to end.
Ii) The graph starts at the point (-5, 1).
Iii) The derivative of f is the step function in the figure shown here.

Accepted Answer

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

Question 14

Find the derivative of the function.
-$$y = \frac { x ^ { 2 } + 8 x + 3 } { \sqrt { x } }$$

Accepted Answer

A)  $y ^ { \prime } = \frac { 2 x + 8 } { 2 x ^ { 3 / 2 } }$ 
B)  $y ^ { \prime } = \frac { 3 x ^ { 2 } + 8 x - 3 } { x }$ 
C)  $y ^ { \prime } = \frac { 2 x + 8 } { x }$ 
D)  $y ^ { \prime } = \frac { 3 x ^ { 2 } + 8 x - 3 } { 2 x ^ { 3 / 2 } }$ 
A)  $y ^ { \prime } = \frac { 2 x + 8 } { 2 x ^ { 3 / 2 } }$ 
B)  $y ^ { \prime } = \frac { 3 x ^ { 2 } + 8 x - 3 } { x }$ 
C)  $y ^ { \prime } = \frac { 2 x + 8 } { x }$ 
D)  $y ^ { \prime } = \frac { 3 x ^ { 2 } + 8 x - 3 } { 2 x ^ { 3 / 2 } }$

Question 15

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative.
-$$\mathrm { u } ( 2 ) = 7 , \mathrm { u } ^ { \prime } ( 2 ) = 2 , \mathrm { v } ( 2 ) = - 1 , \mathrm { v } ^ { \prime } ( 2 ) = - 4 \text {. }$$
$\frac { \mathrm { d } } { \mathrm { dx } } ( \mathrm { uv } )$ at $\mathrm { x } = 2$

Accepted Answer

A)  $- 30$ 
B)  $- 26$ 
C)  18 
D)  30 
A)  $- 30$ 
B)  $- 26$ 
C)  18 
D)  30

Question 16

Calculate the derivative of the function. Then find the value of the derivative as specified.
-$$g ( x ) = 3 x ^ { 2 } - 4 x ; g ^ { \prime } ( 3 )$$

Accepted Answer

A)  $g ^ { \prime } ( x ) = 2 x - 4 ; g ^ { \prime } ( 3 ) = 2$ 
B)  $g ^ { \prime } ( x ) = 6 x ; g ^ { \prime } ( 3 ) = 18$ 
C)  $g ^ { \prime } ( x ) = 3 x - 4 ; g ^ { \prime } ( 3 ) = 5$ 
D)  $g ^ { \prime } ( x ) = 6 x - 4 ; g ^ { \prime } ( 3 ) = 14$ 
A)  $g ^ { \prime } ( x ) = 2 x - 4 ; g ^ { \prime } ( 3 ) = 2$ 
B)  $g ^ { \prime } ( x ) = 6 x ; g ^ { \prime } ( 3 ) = 18$ 
C)  $g ^ { \prime } ( x ) = 3 x - 4 ; g ^ { \prime } ( 3 ) = 5$ 
D)  $g ^ { \prime } ( x ) = 6 x - 4 ; g ^ { \prime } ( 3 ) = 14$

Question 17

The graph of a function is given. Choose the answer that represents the graph of its derivative
-The graph of a function is given. Choose the answer that represents the graph of its derivative.

Accepted Answer

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

Question 18

Estimate the slope of the curve at the indicated point.
-

Accepted Answer

A)  - 12 
B)  2 
C)  -2 
D)  12 
A)  - 12 
B)  2 
C)  -2 
D)  12

Question 19

Find the slope of the curve at the indicated point.
-Find equations of all tangents to the curve $f ( x ) = \frac { 1 } { x + 1 }$ that have slope $- 1$.

Accepted Answer

A)  $y = - x + 3$ 
B)  $y = - x + 1 , y = - x - 3$ 
C)  $y = - x + 3 , y = - x + 1$ 
D)  $y = - x + 1$ 
A)  $y = - x + 3$ 
B)  $y = - x + 1 , y = - x - 3$ 
C)  $y = - x + 3 , y = - x + 1$ 
D)  $y = - x + 1$

Question 20

Differentiate the function and find the slope of the tangent line at the given value of the independent variable.
-$$f ( x ) = 7 x + \frac { 3 } { x } , x = 9$$

Accepted Answer

A)  $\frac { 188 } { 27 }$ 
B)  $\frac { 190 } { 27 }$ 
C)  $\frac { 190 } { 3 }$ 
D)  $\frac { 188 } { 3 }$ 
A)  $\frac { 188 } { 27 }$ 
B)  $\frac { 190 } { 27 }$ 
C)  $\frac { 190 } { 3 }$ 
D)  $\frac { 188 } { 3 }$

Given the graph of f, find any values of x at which f ' is not defined. -

Find y'. - $y = ( 5 x - 5 ) ( 6 x + 1 )$

Find the second derivative. - $w = z ^ { - 5 } - \frac { 1 } { z }$

Find the derivative. - $y = \frac { 1 } { x ^ { 2.4 } } - \frac { \pi } { \sqrt { x } }$

Find the derivative. - $y = 2 - 8 x ^ { 3 }$

Provide an appropriate response. -Does the curve $y = x ^ { 3 } + 4 x - 10$ have a tangent whose slope is $- 2$ ? If so, find an equation for the line and the point of tangency. If not, why not?

$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$ - $g ( x ) = \frac { x } { x + 3 }$

Find an equation for the tangent to the curve at the given point. - $h ( x ) = t ^ { 3 } - 16 t + 4 , ( 4,4 )$

Find the derivative. - $y = \sqrt [ 9 ] { x ^ { 2 } } + x ^ { e }$

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

Find the derivative of the function. - $y = \frac { x ^ { 2 } + 8 x + 3 } { \sqrt { x } }$

Calculate the derivative of the function. Then find the value of the derivative as specified. - $g ( x ) = 3 x ^ { 2 } - 4 x ; g ^ { \prime } ( 3 )$

The graph of a function is given. Choose the answer that represents the graph of its derivative -The graph of a function is given. Choose the answer that represents the graph of its derivative.

Estimate the slope of the curve at the indicated point. -

Find the slope of the curve at the indicated point. -Find equations of all tangents to the curve $f ( x ) = \frac { 1 } { x + 1 }$ that have slope $- 1$ .

Differentiate the function and find the slope of the tangent line at the given value of the independent variable. - $f ( x ) = 7 x + \frac { 3 } { x } , x = 9$

Functions

Limits and Derivatives

Applications of Derivatives

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Filters

Exam 3: Differentiation

Given the graph of f, find any values of x at which f ' is not defined. -

Find y'. - y=(5x−5)(6x+1)y = ( 5 x - 5 ) ( 6 x + 1 )y=(5x−5)(6x+1)

Find the second derivative. - w=z−5−1zw = z ^ { - 5 } - \frac { 1 } { z }w=z−5−z1​

Find the derivative. - y=1x2.4−πxy = \frac { 1 } { x ^ { 2.4 } } - \frac { \pi } { \sqrt { x } }y=x2.41​−x​π​

Find the derivative. - y=2−8x3y = 2 - 8 x ^ { 3 }y=2−8x3

Provide an appropriate response. -Does the curve y=x3+4x−10y = x ^ { 3 } + 4 x - 10y=x3+4x−10 have a tangent whose slope is −2- 2−2 ? If so, find an equation for the line and the point of tangency. If not, why not?

Find an equation for the tangent to the curve at the given point. - h(x)=t3−16t+4,(4,4)h ( x ) = t ^ { 3 } - 16 t + 4 , ( 4,4 )h(x)=t3−16t+4,(4,4)

Find the derivative. - y=x29+xey = \sqrt [ 9 ] { x ^ { 2 } } + x ^ { e }y=9x2​+xe

Find an equation for the tangent to the curve at the given point. - y=x−x2,(2,−2)y = x - x ^ { 2 } , ( 2 , - 2 )y=x−x2,(2,−2)

Find the derivative of the function. - y=x2+8x+3xy = \frac { x ^ { 2 } + 8 x + 3 } { \sqrt { x } }y=x​x2+8x+3​

Calculate the derivative of the function. Then find the value of the derivative as specified. - g(x)=3x2−4x;g′(3)g ( x ) = 3 x ^ { 2 } - 4 x ; g ^ { \prime } ( 3 )g(x)=3x2−4x;g′(3)