Question 1

Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ does not exist if x is

Accepted Answer

A) -9 
B) -5 
C) 5 
D) 6 
E) 9 
A) -9 
B) -5 
C) 5 
D) 6 
E) 9

Question 2

The derivative of $f ( x ) = 3 x ^ { 2 } - 1$ at x = -1 is

Accepted Answer

A) -6 
B) -3 
C) -1 
D) 3 
E) 6 
A) -6 
B) -3 
C) -1 
D) 3 
E) 6

Question 3

Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\frac { \sqrt { 9 - x } } { 2 }$ 
B)  $\frac { 2 } { \sqrt { 9 - x } }$ 
C)  $\frac { 1 } { 2 \sqrt { 9 - x } }$ 
D)  $- \frac { 1 } { 2 \sqrt { 9 - x } }$ 
E)  $- \frac { 2 } { \sqrt { 9 - x } }$ 
A)  $\frac { \sqrt { 9 - x } } { 2 }$ 
B)  $\frac { 2 } { \sqrt { 9 - x } }$ 
C)  $\frac { 1 } { 2 \sqrt { 9 - x } }$ 
D)  $- \frac { 1 } { 2 \sqrt { 9 - x } }$ 
E)  $- \frac { 2 } { \sqrt { 9 - x } }$

Question 4

Let $y = e ^ { x } \sin x$ Then $y\prime\prime$ is

Accepted Answer

A)  $2 e ^ { x } \cos x$ 
B)  $2 e ^ { x } \sin x$ 
C)  $- 2 e ^ { x } \cos x$ 
D)  $- 2 e ^ { x } \sin x$ 
E)  $e ^ { x } ( \sin x + \cos x )$ 
A)  $2 e ^ { x } \cos x$ 
B)  $2 e ^ { x } \sin x$ 
C)  $- 2 e ^ { x } \cos x$ 
D)  $- 2 e ^ { x } \sin x$ 
E)  $e ^ { x } ( \sin x + \cos x )$

Question 5

The second derivative $\frac { d ^ { 2 } } { d x ^ { 2 } } \left( \left( x ^ { 2 } - 1 \right) ( x + 2 ) \right)$ is

Accepted Answer

A)  $3 x ^ { 2 } + 4 x - 1$ 
B)  $2 x$ 
C) 0 
D)  $6 x + 4$ 
E)  $6 x - 4$ 
A)  $3 x ^ { 2 } + 4 x - 1$ 
B)  $2 x$ 
C) 0 
D)  $6 x + 4$ 
E)  $6 x - 4$

Question 6

Let $f ( x ) = 4 x ^ { 3 } e ^ { x }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $12 x ^ { 2 } e ^ { x }$ 
B)  $12 x ^ { 2 } + e ^ { x }$ 
C)  $4 x ^ { 2 } ( x + 3 )$ 
D)  $4 x ^ { 2 } e ^ { x } ( x + 3 )$ 
E)  $4 x ^ { 3 } e ^ { x - 1 }$ 
A)  $12 x ^ { 2 } e ^ { x }$ 
B)  $12 x ^ { 2 } + e ^ { x }$ 
C)  $4 x ^ { 2 } ( x + 3 )$ 
D)  $4 x ^ { 2 } e ^ { x } ( x + 3 )$ 
E)  $4 x ^ { 3 } e ^ { x - 1 }$

Question 7

Let $f ( x ) = \frac { 3 } { \sqrt { x } } + 4$ . Then does not exist if x is

Accepted Answer

A) 0 
B) 1 
C) 4 
D) 9 
E) 16 
A) 0 
B) 1 
C) 4 
D) 9 
E) 16

Question 8

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

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , e )$ 
E)  $\left( - \infty , e ^ { 2 } \right)$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , e )$ 
E)  $\left( - \infty , e ^ { 2 } \right)$

Question 9

Let $f ( x ) = \frac { \sin x } { e ^ { x } }$ . Then $f ^ { \prime } ( x )$ is

Accepted Answer

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

Question 10

Let $f ( x ) = \lfloor x \rfloor$ 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 11

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

Accepted Answer

A) -2x 
B)  $- x ^ { 2 }$ 
C) 2x 
D)  $x ^ { 2 }$ 
E)  $4 - 2 x$ 
A) -2x 
B)  $- x ^ { 2 }$ 
C) 2x 
D)  $x ^ { 2 }$ 
E)  $4 - 2 x$

Question 12

The slope of the tangent line to the graph $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point (-1,1) is

Accepted Answer

A) -3 
B) -2 
C) -1 
D) 1 
E) 2 
A) -3 
B) -2 
C) -1 
D) 1 
E) 2

Question 13

Let $f ( x ) = \frac { 8 } { \sqrt { 5 - x } }$ Then $f ^ { \prime } ( x )$ does not exist if x is

Accepted Answer

A) -5 
B) 1 
C) 2 
D) 3 
E) 5 
A) -5 
B) 1 
C) 2 
D) 3 
E) 5

Question 14

The derivative of $f ( x ) = \sqrt { x - 4 }$ at x = 8 is

Accepted Answer

A)  $- \frac { 1 } { 8 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 1 } { 8 }$ 
A)  $- \frac { 1 } { 8 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 4 }$ 
E)  $\frac { 1 } { 8 }$

Question 15

Let $f ( x ) = \cos x - 2 x$ If $x \in [ 0,2 \pi ]$ then ƒ has a horizontal tangent line for each x in

Accepted Answer

A)  $\left\{ \frac { \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { 5 \pi } { 6 } \right\}$ 
C)  $\left\{ \frac { 7 \pi } { 6 } \right\}$ 
D)  $\left\{ \frac { 11 \pi } { 6 } \right\}$ 
E)  $\varnothing$ 
A)  $\left\{ \frac { \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { 5 \pi } { 6 } \right\}$ 
C)  $\left\{ \frac { 7 \pi } { 6 } \right\}$ 
D)  $\left\{ \frac { 11 \pi } { 6 } \right\}$ 
E)  $\varnothing$

Question 16

Let $f ( x ) = x - \frac { 2 } { x }$ . Then $f ^ { \prime } ( x )$ does not exist if x is

Accepted Answer

A) -2 
B) -1 
C) 0 
D) 1 
E) 2 
A) -2 
B) -1 
C) 0 
D) 1 
E) 2

Question 17

Let $f ( x ) = \sin x \cos x$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $\sin ^ { 2 } x - \cos ^ { 2 } x$ 
B)  $\cos ^ { 2 } x - \sin ^ { 2 } x$ 
C) 1 
D)  $2 \sin x \cos x$ 
E)  $- 2 \sin x \cos x$ 
A)  $\sin ^ { 2 } x - \cos ^ { 2 } x$ 
B)  $\cos ^ { 2 } x - \sin ^ { 2 } x$ 
C) 1 
D)  $2 \sin x \cos x$ 
E)  $- 2 \sin x \cos x$

Question 18

Let $f ( x ) = \frac { e ^ { x } } { x }$ Assuming $x \neq 0 , f ^ { \prime } ( x )$ is

Accepted Answer

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

Question 19

Let $f ( x ) = - 4 x + 2$ . 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 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 horizontal tangent line at (0,0). 
E)  $f$ has a vertical tangent line at (0,0).

Question 20

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

Accepted Answer

A)  $2 x - 4$ 
B)  $2 x + 4$ 
C)  $- 2 x - 4$ 
D)  $- 2 x + 4$ 
E)  $2 x + 8$ 
A)  $2 x - 4$ 
B)  $2 x + 4$ 
C)  $- 2 x - 4$ 
D)  $- 2 x + 4$ 
E)  $2 x + 8$

Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ does not exist if x is

The derivative of $f ( x ) = 3 x ^ { 2 } - 1$ at x = -1 is

Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ is

Let $y = e ^ { x } \sin x$ Then $y\prime\prime$ is

The second derivative $\frac { d ^ { 2 } } { d x ^ { 2 } } \left( \left( x ^ { 2 } - 1 \right) ( x + 2 ) \right)$ is

Let $f ( x ) = 4 x ^ { 3 } e ^ { x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \frac { 3 } { \sqrt { x } } + 4$ . Then does not exist if x is

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

Let $f ( x ) = \frac { \sin x } { e ^ { x } }$ . Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

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

The slope of the tangent line to the graph $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point (-1,1) is

Let $f ( x ) = \frac { 8 } { \sqrt { 5 - x } }$ Then $f ^ { \prime } ( x )$ does not exist if x is

The derivative of $f ( x ) = \sqrt { x - 4 }$ at x = 8 is

Let $f ( x ) = \cos x - 2 x$ If $x \in [ 0,2 \pi ]$ then ƒ has a horizontal tangent line for each x in

Let $f ( x ) = x - \frac { 2 } { x }$ . Then $f ^ { \prime } ( x )$ does not exist if x is

Let $f ( x ) = \sin x \cos x$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \frac { e ^ { x } } { x }$ Assuming $x \neq 0 , f ^ { \prime } ( x )$ is

Let $f ( x ) = - 4 x + 2$ . Which of the following is true?

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

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Exam 3: The Derivative

Let f(x)=9−xf ( x ) = \sqrt { 9 - x }f(x)=9−x​ Then f′(x)f ^ { \prime } ( x )f′(x) does not exist if x is

The derivative of f(x)=3x2−1f ( x ) = 3 x ^ { 2 } - 1f(x)=3x2−1 at x = -1 is

Let f(x)=9−xf ( x ) = \sqrt { 9 - x }f(x)=9−x​ Then f′(x)f ^ { \prime } ( x )f′(x) is

Let y=exsin⁡xy = e ^ { x } \sin xy=exsinx Then y′′y\prime\primey′′ is

The second derivative d2dx2((x2−1)(x+2))\frac { d ^ { 2 } } { d x ^ { 2 } } \left( \left( x ^ { 2 } - 1 \right) ( x + 2 ) \right)dx2d2​((x2−1)(x+2)) is

Let f(x)=4x3exf ( x ) = 4 x ^ { 3 } e ^ { x }f(x)=4x3ex Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=3x+4f ( x ) = \frac { 3 } { \sqrt { x } } + 4f(x)=x​3​+4 . Then does not exist if x is

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

Let f(x)=sin⁡xexf ( x ) = \frac { \sin x } { e ^ { x } }f(x)=exsinx​ . Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=⌊x⌋f ( x ) = \lfloor x \rfloorf(x)=⌊x⌋ Which of the following is true?

Let f(x)=4−x2f ( x ) = 4 - x ^ { 2 }f(x)=4−x2 . Then f′(x)f ^ { \prime } ( x )f′(x) is

The slope of the tangent line to the graph f(x)=1x2f ( x ) = \frac { 1 } { x ^ { 2 } }f(x)=x21​ at the point (-1,1) is

Let f(x)=85−xf ( x ) = \frac { 8 } { \sqrt { 5 - x } }f(x)=5−x​8​ Then f′(x)f ^ { \prime } ( x )f′(x) does not exist if x is

The derivative of f(x)=x−4f ( x ) = \sqrt { x - 4 }f(x)=x−4​ at x = 8 is

Let f(x)=cos⁡x−2xf ( x ) = \cos x - 2 xf(x)=cosx−2x If x∈[0,2π]x \in [ 0,2 \pi ]x∈[0,2π] then ƒ has a horizontal tangent line for each x in

Let f(x)=x−2xf ( x ) = x - \frac { 2 } { x }f(x)=x−x2​ . Then f′(x)f ^ { \prime } ( x )f′(x) does not exist if x is

Let f(x)=sin⁡xcos⁡xf ( x ) = \sin x \cos xf(x)=sinxcosx Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=exxf ( x ) = \frac { e ^ { x } } { x }f(x)=xex​ Assuming x≠0,f′(x)x \neq 0 , f ^ { \prime } ( x )x=0,f′(x) is

Let f(x)=−4x+2f ( x ) = - 4 x + 2f(x)=−4x+2 . Which of the following is true?

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

Preparing for Calculus

Limits and Continuity

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

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Vector Functions

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Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ does not exist if x is

The derivative of $f ( x ) = 3 x ^ { 2 } - 1$ at x = -1 is

Let $f ( x ) = \sqrt { 9 - x }$ Then $f ^ { \prime } ( x )$ is

Let $y = e ^ { x } \sin x$ Then $y\prime\prime$ is

The second derivative $\frac { d ^ { 2 } } { d x ^ { 2 } } \left( \left( x ^ { 2 } - 1 \right) ( x + 2 ) \right)$ is

Let $f ( x ) = 4 x ^ { 3 } e ^ { x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \frac { 3 } { \sqrt { x } } + 4$ . Then does not exist if x is

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

Let $f ( x ) = \frac { \sin x } { e ^ { x } }$ . Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

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

The slope of the tangent line to the graph $f ( x ) = \frac { 1 } { x ^ { 2 } }$ at the point (-1,1) is

Let $f ( x ) = \frac { 8 } { \sqrt { 5 - x } }$ Then $f ^ { \prime } ( x )$ does not exist if x is

The derivative of $f ( x ) = \sqrt { x - 4 }$ at x = 8 is

Let $f ( x ) = \cos x - 2 x$ If $x \in [ 0,2 \pi ]$ then ƒ has a horizontal tangent line for each x in

Let $f ( x ) = x - \frac { 2 } { x }$ . Then $f ^ { \prime } ( x )$ does not exist if x is

Let $f ( x ) = \sin x \cos x$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \frac { e ^ { x } } { x }$ Assuming $x \neq 0 , f ^ { \prime } ( x )$ is

Let $f ( x ) = - 4 x + 2$ . Which of the following is true?

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