Question 1

The derivative of $f ( x ) = \frac { 4 } { x + 5 }$ at x = -4 is

Accepted Answer

A) -5 
B) -4 
C) 1 
D) 4 
E) 5 
A) -5 
B) -4 
C) 1 
D) 4 
E) 5 B

Question 2

The instantaneous rate of change of $f ( x ) = \sqrt { x } + 5$ at c = 9 is

Accepted Answer

A)  $- \frac { 1 } { 6 }$ 
B)  $- \frac { 1 } { 3 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  $\frac { 1 } { 3 }$ 
A)  $- \frac { 1 } { 6 }$ 
B)  $- \frac { 1 } { 3 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  $\frac { 1 } { 3 }$ D

Question 3

Let $e ^ { x } / x ^ { 4 }$ . Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\frac { e ^ { x } } { x ^ { 4 } }$ 
B)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 8 } }$ 
C)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 7 } }$ 
D)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 6 } }$ 
E)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 5 } }$ 
A)  $\frac { e ^ { x } } { x ^ { 4 } }$ 
B)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 8 } }$ 
C)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 7 } }$ 
D)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 6 } }$ 
E)  $\frac { e ^ { x } ( x - 4 ) } { x ^ { 5 } }$ E

Question 4

Let $f ( x ) = x ^ { 4 } - \frac { 4 } { x ^ { 4 } }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $4 x ^ { 3 } - \frac { 1 } { x ^ { 3 } }$ 
B)  $4 x ^ { 3 } - \frac { 4 } { x ^ { 5 } }$ 
C)  $4 x ^ { 3 } - \frac { 16 } { x ^ { 5 } }$ 
D)  $4 x ^ { 3 } + \frac { 4 } { x ^ { 5 } }$ 
E)  $4 x ^ { 3 } + \frac { 16 } { x ^ { 5 } }$ 
A)  $4 x ^ { 3 } - \frac { 1 } { x ^ { 3 } }$ 
B)  $4 x ^ { 3 } - \frac { 4 } { x ^ { 5 } }$ 
C)  $4 x ^ { 3 } - \frac { 16 } { x ^ { 5 } }$ 
D)  $4 x ^ { 3 } + \frac { 4 } { x ^ { 5 } }$ 
E)  $4 x ^ { 3 } + \frac { 16 } { x ^ { 5 } }$

Question 5

Let $f ( x ) = \frac { 1 } { 2 } - 4 x$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A) -4 
B)  $- \frac { 7 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $- \frac { 1 } { 2 }$ 
E) 4 
A) -4 
B)  $- \frac { 7 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $- \frac { 1 } { 2 }$ 
E) 4

Question 6

What is the equation of the normal line to the graph of $f ( x ) = 3 \sqrt { x } + 2$ at $x = 9$ ?

Accepted Answer

A)  $y = - 2 x$ 
B)  $y - 11 = - 2 ( x - 9 )$ 
C)  $y - 11 = 0.5 ( x - 9 )$ 
D)  $y = 0.5 x$ 
E)  $y = 0.5 x - 9$ 
A)  $y = - 2 x$ 
B)  $y - 11 = - 2 ( x - 9 )$ 
C)  $y - 11 = 0.5 ( x - 9 )$ 
D)  $y = 0.5 x$ 
E)  $y = 0.5 x - 9$

Question 7

The instantaneous rate of change of $f ( x ) = 3 \sqrt { x + 5 }$ at c = 4 is

Accepted Answer

A)  $- \frac { 3 } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E)  $\frac { 3 } { 2 }$ 
A)  $- \frac { 3 } { 2 }$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E)  $\frac { 3 } { 2 }$

Question 8

Let $f ( x ) = x + e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 3 )$ 
E)  $\varnothing$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 3 )$ 
E)  $\varnothing$

Question 9

Let $f ( x ) = ( - 3 x + 5 ) ^ { 2 }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $- 6 x$ 
B)  $- 6 x - 10$ 
C)  $18 x - 30$ 
D)  $18 x + 30$ 
E)  $18 x - 5$ 
A)  $- 6 x$ 
B)  $- 6 x - 10$ 
C)  $18 x - 30$ 
D)  $18 x + 30$ 
E)  $18 x - 5$

Question 10

The fifth derivative $\frac { d ^ { 5 } } { d x ^ { 5 } } \left( 5 x ^ { 4 } - e ^ { x } \right)$ is

Accepted Answer

A)  $60 x ^ { 3 } - e ^ { x }$ 
B)  $180 x ^ { 2 } - e ^ { x }$ 
C)  $360 x - e ^ { x }$ 
D)  $360 - e ^ { x }$ 
E)  $- e ^ { x }$ 
A)  $60 x ^ { 3 } - e ^ { x }$ 
B)  $180 x ^ { 2 } - e ^ { x }$ 
C)  $360 x - e ^ { x }$ 
D)  $360 - e ^ { x }$ 
E)  $- e ^ { x }$

Question 11

Let $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + x - 5$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $6 x - 8$ 
B)  $6 x$ 
C)  $3 x ^ { 2 } - 8 x$ 
D)  $- 6 x$ 
E)  $- 6$ 
A)  $6 x - 8$ 
B)  $6 x$ 
C)  $3 x ^ { 2 } - 8 x$ 
D)  $- 6 x$ 
E)  $- 6$

Question 12

Let $f ( x ) = - x ^ { 2 }$ Which of the following is true?

Accepted Answer

A)  $f$ is discontinuous at 0. 
B)  $f$ does not have a derivative at 0. 
C)  $f$ has a corner at (0,0). 
D)  $f$ has a has a horizontal tangent line at (0,0). 
E)  $f$ has a vertical tangent line at (0,0). 
A)  $f$ is discontinuous at 0. 
B)  $f$ does not have a derivative at 0. 
C)  $f$ has a corner at (0,0). 
D)  $f$ has a has a horizontal tangent line at (0,0). 
E)  $f$ has a vertical tangent line at (0,0).

Question 13

Let $f ( x ) = \frac { 1 - x } { 1 + x }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $- \frac { 2 } { ( 1 + x ) ^ { 2 } }$ 
B)  $\frac { 2 } { ( 1 + x ) ^ { 2 } }$ 
C)  $- \frac { 2 x } { ( 1 + x ) ^ { 2 } }$ 
D)  $\frac { 2 x } { ( 1 + x ) ^ { 2 } }$ 
E)  $- \frac { 2 } { 1 + x }$ 
A)  $- \frac { 2 } { ( 1 + x ) ^ { 2 } }$ 
B)  $\frac { 2 } { ( 1 + x ) ^ { 2 } }$ 
C)  $- \frac { 2 x } { ( 1 + x ) ^ { 2 } }$ 
D)  $\frac { 2 x } { ( 1 + x ) ^ { 2 } }$ 
E)  $- \frac { 2 } { 1 + x }$

Question 14

Let $f ( x ) = 2 x ^ { 3 }$ Which of the following is true?

Accepted Answer

A)  $f$ is discontinuous at 0. 
B)  $f$ does not have a derivative at 0. 
C)  $f$ has a corner at (0,0). 
D)  $f$ has a has a horizontal tangent line at (0,0). 
E)  $f$ has a vertical tangent line at (0,0). 
A)  $f$ is discontinuous at 0. 
B)  $f$ does not have a derivative at 0. 
C)  $f$ has a corner at (0,0). 
D)  $f$ has a has a horizontal tangent line at (0,0). 
E)  $f$ has a vertical tangent line at (0,0).

Question 15

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ for all x in

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , \ln 6 )$ 
C)  $( \ln 6 , \infty )$ 
D)  $( - \infty , 6 )$ 
E)  $( - \infty , - \ln 6 )$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , \ln 6 )$ 
C)  $( \ln 6 , \infty )$ 
D)  $( - \infty , 6 )$ 
E)  $( - \infty , - \ln 6 )$

Question 16

Let $f ( x ) = x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

Accepted Answer

A) -e 
B) -1 
C) 0 
D) 1 
E) e 
A) -e 
B) -1 
C) 0 
D) 1 
E) e

Question 17

Let $f ( x ) = 3 x e ^ { x }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $3 e ^ { x }$ 
B)  $3 e ^ { x } + 3 x ^ { 2 } e ^ { x - 1 }$ 
C)  $3 x e ^ { x - 1 }$ 
D)  $3 e ^ { x } ( x + 2 )$ 
E)  $3 e ^ { x } ( x + 1 )$ 
A)  $3 e ^ { x }$ 
B)  $3 e ^ { x } + 3 x ^ { 2 } e ^ { x - 1 }$ 
C)  $3 x e ^ { x - 1 }$ 
D)  $3 e ^ { x } ( x + 2 )$ 
E)  $3 e ^ { x } ( x + 1 )$

Question 18

Let $f ( x ) = e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 3 )$ 
E)  $\varnothing$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 3 )$ 
E)  $\varnothing$

Question 19

Let $f ( x ) = \left\{ \begin{array} { l l } 
2 x + 1 & x  0
\end{array} \right.$ Which of the following is true?

Accepted Answer

A)  $f$ is discontinuous at 0. 
B)  $f$ has a derivative at 0. 
C)  $f$ has a corner at (0,0). 
D)  $f$ has a has a horizontal tangent line at (0,0). 
E)  $f$ has a vertical tangent line at (0,0). 
A)  $f$ is discontinuous at 0. 
B)  $f$ has a derivative at 0. 
C)  $f$ has a corner at (0,0). 
D)  $f$ has a has a horizontal tangent line at (0,0). 
E)  $f$ has a vertical tangent line at (0,0).

Question 20

The instantaneous rate of change of $f ( x ) = \frac { 3 } { x + 1 }$ at c = 2 is

Accepted Answer

A)  -3 
B) 
 $- \frac { 1 } { 3 }$ 
C) 
 $\frac { 1 } { 3 }$ 
D)  1 
E)  3 
A)  -3 
B) 
 $- \frac { 1 } { 3 }$ 
C) 
 $\frac { 1 } { 3 }$ 
D)  1 
E)  3

The derivative of $f ( x ) = \frac { 4 } { x + 5 }$ at x = -4 is

The instantaneous rate of change of $f ( x ) = \sqrt { x } + 5$ at c = 9 is

Let $e ^ { x } / x ^ { 4 }$ . Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = x ^ { 4 } - \frac { 4 } { x ^ { 4 } }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \frac { 1 } { 2 } - 4 x$ Then $f ^ { \prime } ( x )$ is

What is the equation of the normal line to the graph of $f ( x ) = 3 \sqrt { x } + 2$ at $x = 9$ ?

The instantaneous rate of change of $f ( x ) = 3 \sqrt { x + 5 }$ at c = 4 is

Let $f ( x ) = x + e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

Let $f ( x ) = ( - 3 x + 5 ) ^ { 2 }$ Then $f ^ { \prime } ( x )$ is

The fifth derivative $\frac { d ^ { 5 } } { d x ^ { 5 } } \left( 5 x ^ { 4 } - e ^ { x } \right)$ is

Let $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + x - 5$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = - x ^ { 2 }$ Which of the following is true?

Let $f ( x ) = \frac { 1 - x } { 1 + x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = 2 x ^ { 3 }$ Which of the following is true?

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ for all x in

Let $f ( x ) = x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

Let $f ( x ) = 3 x e ^ { x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

Let $f ( x ) = \left\{ \begin{array} { l l } 2 x + 1 & x < 0 \\- x + 1 & x > 0\end{array} \right.$ Which of the following is true?

The instantaneous rate of change of $f ( x ) = \frac { 3 } { x + 1 }$ at c = 2 is

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Exam 3: The Derivative

The derivative of f(x)=4x+5f ( x ) = \frac { 4 } { x + 5 }f(x)=x+54​ at x = -4 is

The instantaneous rate of change of f(x)=x+5f ( x ) = \sqrt { x } + 5f(x)=x​+5 at c = 9 is

Let ex/x4e ^ { x } / x ^ { 4 }ex/x4 . Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=x4−4x4f ( x ) = x ^ { 4 } - \frac { 4 } { x ^ { 4 } }f(x)=x4−x44​ Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=12−4xf ( x ) = \frac { 1 } { 2 } - 4 xf(x)=21​−4x Then f′(x)f ^ { \prime } ( x )f′(x) is

What is the equation of the normal line to the graph of f(x)=3x+2f ( x ) = 3 \sqrt { x } + 2f(x)=3x​+2 at x=9x = 9x=9 ?

The instantaneous rate of change of f(x)=3x+5f ( x ) = 3 \sqrt { x + 5 }f(x)=3x+5​ at c = 4 is

Let f(x)=x+exf ( x ) = x + e ^ { x }f(x)=x+ex Then which of the following is the largest interval, of those listed, for which f′(x)<0f ^ { \prime } ( x ) < 0f′(x)<0 for all x in that interval?

Let f(x)=(−3x+5)2f ( x ) = ( - 3 x + 5 ) ^ { 2 }f(x)=(−3x+5)2 Then f′(x)f ^ { \prime } ( x )f′(x) is

The fifth derivative d5dx5(5x4−ex)\frac { d ^ { 5 } } { d x ^ { 5 } } \left( 5 x ^ { 4 } - e ^ { x } \right)dx5d5​(5x4−ex) is

Let f(x)=x3−4x2+x−5f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + x - 5f(x)=x3−4x2+x−5 Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=−x2f ( x ) = - x ^ { 2 }f(x)=−x2 Which of the following is true?

Let f(x)=1−x1+xf ( x ) = \frac { 1 - x } { 1 + x }f(x)=1+x1−x​ Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=2x3f ( x ) = 2 x ^ { 3 }f(x)=2x3 Which of the following is true?

Let f(x)=6x−exf ( x ) = 6 x - e ^ { x }f(x)=6x−ex Then f′(x)=0f ^ { \prime } ( x ) = 0f′(x)=0 for all x in

Let f(x)=x−exf ( x ) = x - e ^ { x }f(x)=x−ex Then f′(x)=0f ^ { \prime } ( x ) = 0f′(x)=0 if x is

Let f(x)=3xexf ( x ) = 3 x e ^ { x }f(x)=3xex Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=exf ( x ) = e ^ { x }f(x)=ex Then which of the following is the largest interval, of those listed, for which f′(x)<0f ^ { \prime } ( x ) < 0f′(x)<0 for all x in that interval?

Let f(x)={2x+1x<0−x+1x>0f ( x ) = \left\{ \begin{array} { l l } 2 x + 1 & x < 0 \\- x + 1 & x > 0\end{array} \right.f(x)={2x+1−x+1​x<0x>0​ Which of the following is true?

The instantaneous rate of change of f(x)=3x+1f ( x ) = \frac { 3 } { x + 1 }f(x)=x+13​ at c = 2 is

Preparing for Calculus

Limits and Continuity

More About Derivatives

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Applications of the Integral

Techniques of Integration

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The derivative of $f ( x ) = \frac { 4 } { x + 5 }$ at x = -4 is

The instantaneous rate of change of $f ( x ) = \sqrt { x } + 5$ at c = 9 is

Let $e ^ { x } / x ^ { 4 }$ . Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = x ^ { 4 } - \frac { 4 } { x ^ { 4 } }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \frac { 1 } { 2 } - 4 x$ Then $f ^ { \prime } ( x )$ is

What is the equation of the normal line to the graph of $f ( x ) = 3 \sqrt { x } + 2$ at $x = 9$ ?

The instantaneous rate of change of $f ( x ) = 3 \sqrt { x + 5 }$ at c = 4 is

Let $f ( x ) = x + e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

Let $f ( x ) = ( - 3 x + 5 ) ^ { 2 }$ Then $f ^ { \prime } ( x )$ is

The fifth derivative $\frac { d ^ { 5 } } { d x ^ { 5 } } \left( 5 x ^ { 4 } - e ^ { x } \right)$ is

Let $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + x - 5$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = - x ^ { 2 }$ Which of the following is true?

Let $f ( x ) = \frac { 1 - x } { 1 + x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = 2 x ^ { 3 }$ Which of the following is true?

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ for all x in

Let $f ( x ) = x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

Let $f ( x ) = 3 x e ^ { x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = e ^ { x }$ Then which of the following is the largest interval, of those listed, for which $f ^ { \prime } ( x ) < 0$ for all x in that interval?

Let $f ( x ) = \left\{ \begin{array} { l l } 2 x + 1 & x < 0 \\- x + 1 & x > 0\end{array} \right.$ Which of the following is true?

The instantaneous rate of change of $f ( x ) = \frac { 3 } { x + 1 }$ at c = 2 is