Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

A)  $\left\{ \frac { \pi } { 2 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$ 
E)  $\{ 0 \}$ 
A)  $\left\{ \frac { \pi } { 2 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$ 
E)  $\{ 0 \}$

Question 3

A dehumidifier is used to reduce the moisture in a basement. The level of moisture reported on the display panel when the dehumidifier is turned on is 70%. The percent moisture, P(t), at time t is recorded for different times t in the table. $$\begin{array} { l l l l l l l } 
T & 0 & 2 & 6 & 15 & 20 & 30 \
P ( t ) & 70 & 67 & 60 & 53 & 51 & 50
\end{array}$$ Estimate P'(20).

Accepted Answer

A) -0.20% per minute 
B) -0.10% per minute 
C) 0.10% per minute 
D) 0.20% per minute 
E) 0.40% per minute 
A) -0.20% per minute 
B) -0.10% per minute 
C) 0.10% per minute 
D) 0.20% per minute 
E) 0.40% per minute

Question 4

Let $f ( x ) = \frac { 3 } { x ^ { 2 } + 1 }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $- \frac { 6 x } { x ^ { 2 } + 1 }$ 
B)  $\frac { 6 x } { x ^ { 2 } + 1 }$ 
C)  $- \frac { 6 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$ 
D)  $\frac { 6 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$ 
E)  $- \frac { 3 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$ 
A)  $- \frac { 6 x } { x ^ { 2 } + 1 }$ 
B)  $\frac { 6 x } { x ^ { 2 } + 1 }$ 
C)  $- \frac { 6 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$ 
D)  $\frac { 6 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$ 
E)  $- \frac { 3 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$

Question 5

The instantaneous rate of change of $f ( x ) = \frac { 8 } { x + 1 }$ at c = 3 is

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

A)  $2 x \cos x$ 
B)  $x ( \sin x + \cos x )$ 
C)  $x ( 2 \sin x + x \cos x )$ 
D)  $x ( x \sin x + 2 \cos x )$ 
E)  $x ( 2 \sin x + \cos x )$ 
A)  $2 x \cos x$ 
B)  $x ( \sin x + \cos x )$ 
C)  $x ( 2 \sin x + x \cos x )$ 
D)  $x ( x \sin x + 2 \cos x )$ 
E)  $x ( 2 \sin x + \cos x )$

Question 8

Let $f ( x ) = \frac { x ^ { 3 } - 2 x ^ { 2 } } { x }$ Then $f ^ { \prime } ( x ) = 0$ 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 9

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

Accepted Answer

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

Question 10

What is the equation of the normal line to the graph of $f ( x ) = x ^ { 2 } - 2 x$ at $x = 1$ ?

Accepted Answer

A) y = x 
B) y = -x 
C) y + 1 = x - 1 
D) x = 1 
E) y = -1 
A) y = x 
B) y = -x 
C) y + 1 = x - 1 
D) x = 1 
E) y = -1

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

A)  $\left\{ \frac { \pi } { 2 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$ 
E)  $\varnothing$ 
A)  $\left\{ \frac { \pi } { 2 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$ 
E)  $\varnothing$

Question 13

The sixth derivative $\frac { d ^ { 6 } } { d x ^ { 6 } } \left( x ^ { 6 } - 3 x ^ { 3 } + 5 \right)$ is

Accepted Answer

A) 720 
B)  $72 x$ 
C) 0 
D) -720 
E) -720 $x$ 
A) 720 
B)  $72 x$ 
C) 0 
D) -720 
E) -720 $x$

Question 14

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

Accepted Answer

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

Question 15

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ 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)  $( - \infty , - 3 )$ 
C)  $( - \infty , 3 )$ 
D) (-3,3) 
E)  $( 3 , \infty )$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , - 3 )$ 
C)  $( - \infty , 3 )$ 
D) (-3,3) 
E)  $( 3 , \infty )$

Question 16

Let $f ( x ) = x - 4 e ^ { x }$ 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)  $( - \infty , - \ln 4 )$ 
C)  $( - \ln 4 , \ln 4 )$ 
D)  $( - \ln 4 , \infty )$ 
E)  $\varnothing$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , - \ln 4 )$ 
C)  $( - \ln 4 , \ln 4 )$ 
D)  $( - \ln 4 , \infty )$ 
E)  $\varnothing$

Question 17

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

Accepted Answer

A)  $\frac { 4 x ^ { 3 } } { e ^ { x } }$ 
B)  $\frac { x ^ { 4 } } { e ^ { x } }$ 
C)  $\frac { x ^ { 3 } ( 4 - x ) } { e ^ { x } }$ 
D)  $\frac { x ^ { 3 } ( 4 - x ) } { e ^ { 2 x } }$ 
E)  $\frac { x ^ { 4 } - 3 } { e ^ { x } }$ 
A)  $\frac { 4 x ^ { 3 } } { e ^ { x } }$ 
B)  $\frac { x ^ { 4 } } { e ^ { x } }$ 
C)  $\frac { x ^ { 3 } ( 4 - x ) } { e ^ { x } }$ 
D)  $\frac { x ^ { 3 } ( 4 - x ) } { e ^ { 2 x } }$ 
E)  $\frac { x ^ { 4 } - 3 } { e ^ { x } }$

Question 18

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

Accepted Answer

A) -6 
B) -ln 6 
C) 0 
D) ln 6 
E) 6 
A) -6 
B) -ln 6 
C) 0 
D) ln 6 
E) 6

Question 19

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

Accepted Answer

A)  $\left\{ \frac { \pi } { 2 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$ 
E)  $\varnothing$ 
A)  $\left\{ \frac { \pi } { 2 } \right\}$ 
B)  $\{ \pi \}$ 
C)  $\left\{ \frac { 3 \pi } { 2 } \right\}$ 
D)  $\{ 2 \pi \}$ 
E)  $\varnothing$

Question 20

Let $f ( x ) = \frac { 8 } { \sqrt { 4 - x } }$ Then $f ^ { \prime } ( x )$ is

Accepted Answer

A)  $- \frac { 4 } { ( 4 - x ) ^ { \frac { 3 } { 2 } } }$ 
B)  $- \frac { 4 } { ( 4 - x ) ^ { \frac { 1 } { 2 } } }$ 
C)  $\frac { 4 } { ( 4 - x ) ^ { \frac { 1 } { 2 } } }$ 
D)  $\frac { 4 } { ( 4 - x ) ^ { \frac { 3 } { 2 } } }$ 
E)  $\frac { 8 } { ( 4 - x ) ^ { \frac { 3 } { 2 } } }$ 
A)  $- \frac { 4 } { ( 4 - x ) ^ { \frac { 3 } { 2 } } }$ 
B)  $- \frac { 4 } { ( 4 - x ) ^ { \frac { 1 } { 2 } } }$ 
C)  $\frac { 4 } { ( 4 - x ) ^ { \frac { 1 } { 2 } } }$ 
D)  $\frac { 4 } { ( 4 - x ) ^ { \frac { 3 } { 2 } } }$ 
E)  $\frac { 8 } { ( 4 - x ) ^ { \frac { 3 } { 2 } } }$

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

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

A dehumidifier is used to reduce the moisture in a basement. The level of moisture reported on the display panel when the dehumidifier is turned on is 70%. The percent moisture, P(t), at time t is recorded for different times t in the table. T 0 2 6 15 20 30 P(t) 70 67 60 53 51 50 Estimate P'(20).

Let $f ( x ) = \frac { 3 } { x ^ { 2 } + 1 }$ Then $f ^ { \prime } ( x )$ is

The instantaneous rate of change of $f ( x ) = \frac { 8 } { x + 1 }$ at c = 3 is

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

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

Let $f ( x ) = \frac { x ^ { 3 } - 2 x ^ { 2 } } { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

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

What is the equation of the normal line to the graph of $f ( x ) = x ^ { 2 } - 2 x$ at $x = 1$ ?

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

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

The sixth derivative $\frac { d ^ { 6 } } { d x ^ { 6 } } \left( x ^ { 6 } - 3 x ^ { 3 } + 5 \right)$ is

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

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ 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 $f ( x ) = x - 4 e ^ { x }$ 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 $f ( x ) = \frac { x ^ { 4 } } { e ^ { x } }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

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

Let $f ( x ) = \frac { 8 } { \sqrt { 4 - x } }$ Then $f ^ { \prime } ( x )$ is

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Exam 3: The Derivative

Let y=excos⁡xy = e ^ { x } \cos xy=excosx Then y′′y \prime\primey′′ is

Let f(x)=sin⁡x−exf ( x ) = \sin x - e ^ { x }f(x)=sinx−ex If x∈[0,2π]x \in [ 0,2 \pi ]x∈[0,2π] , then ƒ has a horizontal tangent line for each x in

A dehumidifier is used to reduce the moisture in a basement. The level of moisture reported on the display panel when the dehumidifier is turned on is 70%. The percent moisture, P(t), at time t is recorded for different times t in the table. T 0 2 6 15 20 30 P(t) 70 67 60 53 51 50 Estimate P'(20).

Let f(x)=3x2+1f ( x ) = \frac { 3 } { x ^ { 2 } + 1 }f(x)=x2+13​ Then f′(x)f ^ { \prime } ( x )f′(x) is

The instantaneous rate of change of f(x)=8x+1f ( x ) = \frac { 8 } { x + 1 }f(x)=x+18​ at c = 3 is

The slope of the tangent line f(x)=−12x2f ( x ) = - \frac { 1 } { 2 } x ^ { 2 }f(x)=−21​x2 to the graph at the point (2,-2) is

Let f(x)=x2sin⁡xf ( x ) = x ^ { 2 } \sin xf(x)=x2sinx Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=x3−2x2xf ( x ) = \frac { x ^ { 3 } - 2 x ^ { 2 } } { x }f(x)=xx3−2x2​ Then f′(x)=0f ^ { \prime } ( x ) = 0f′(x)=0 if x is

Let f(x)=x−2xf ( x ) = x - \frac { 2 } { x }f(x)=x−x2​ . Then f′(x)f ^ { \prime } ( x )f′(x) is

What is the equation of the normal line to the graph of f(x)=x2−2xf ( x ) = x ^ { 2 } - 2 xf(x)=x2−2x at x=1x = 1x=1 ?

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

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

The sixth derivative d6dx6(x6−3x3+5)\frac { d ^ { 6 } } { d x ^ { 6 } } \left( x ^ { 6 } - 3 x ^ { 3 } + 5 \right)dx6d6​(x6−3x3+5) is

Let f(x)=2cos⁡x+xf ( x ) = 2 \cos x + xf(x)=2cosx+x 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)=12x2−3x+4f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4f(x)=21​x2−3x+4 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)=x−4exf ( x ) = x - 4 e ^ { x }f(x)=x−4ex 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)=x4exf ( x ) = \frac { x ^ { 4 } } { e ^ { x } }f(x)=exx4​ Then f′(x)f ^ { \prime } ( x )f′(x) is

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

Let f(x)=cos⁡x+exf ( x ) = \cos x + e ^ { x }f(x)=cosx+ex 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)=84−xf ( x ) = \frac { 8 } { \sqrt { 4 - x } }f(x)=4−x​8​ Then f′(x)f ^ { \prime } ( x )f′(x) is

Preparing for Calculus

Limits and Continuity

More About Derivatives

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

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

Let $f ( x ) = \frac { 3 } { x ^ { 2 } + 1 }$ Then $f ^ { \prime } ( x )$ is

The instantaneous rate of change of $f ( x ) = \frac { 8 } { x + 1 }$ at c = 3 is

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

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

Let $f ( x ) = \frac { x ^ { 3 } - 2 x ^ { 2 } } { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

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

What is the equation of the normal line to the graph of $f ( x ) = x ^ { 2 } - 2 x$ at $x = 1$ ?

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

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

The sixth derivative $\frac { d ^ { 6 } } { d x ^ { 6 } } \left( x ^ { 6 } - 3 x ^ { 3 } + 5 \right)$ is

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

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ 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 $f ( x ) = x - 4 e ^ { x }$ 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 $f ( x ) = \frac { x ^ { 4 } } { e ^ { x } }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = 6 x - e ^ { x }$ Then $f ^ { \prime } ( x ) = 0$ if x is

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

Let $f ( x ) = \frac { 8 } { \sqrt { 4 - x } }$ Then $f ^ { \prime } ( x )$ is