Question 1

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

Accepted Answer

A)  $1 + \cot ^ { 2 } x$ 
B)  $-CSCS .X$ 
C)  $- \csc x \cot x$ 
D)  $\csc x \cot x$ 
E)  $\sec x \tan x$ 
A)  $1 + \cot ^ { 2 } x$ 
B)  $-CSCS .X$ 
C)  $- \csc x \cot x$ 
D)  $\csc x \cot x$ 
E)  $\sec x \tan x$

Question 2

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ Then $f ^ { \prime } ( x ) = 0$ for all x in

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

The slope of the tangent line to the graph $f ( x ) = 3 \sqrt { x }$ at the point (4,6) is

Accepted Answer

A)  $- \frac { 3 } { 4 }$ 
B)  $\frac { 3 } { 4 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 3 } { 2 }$ 
A)  $- \frac { 3 } { 4 }$ 
B)  $\frac { 3 } { 4 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 3 } { 2 }$

Question 6

Let $f ( x ) = 6 x - e ^ { x }$ . 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)  $( - \infty , \ln 6 )$ 
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 7

Let $f ( x ) = \pi ^ { 2 } e ^ { x }$ Then $f ^ { ( 10 ) } ( x )$ is

Accepted Answer

A)  $2 \pi e ^ { x }$ 
B) 0 
C)  $10 \pi e ^ { x }$ 
D)  $\pi e ^ { x }$ 
E)  $\pi ^ { 2 } e ^ { x }$ 
A)  $2 \pi e ^ { x }$ 
B) 0 
C)  $10 \pi e ^ { x }$ 
D)  $\pi e ^ { x }$ 
E)  $\pi ^ { 2 } e ^ { x }$

Question 8

An equation of the tangent line to the graph $f ( x ) = \frac { 3 } { 2 x - 1 }$ at the point (1, 3) is

Accepted Answer

A)  $y = - 6 x + 9$ 
B)  $y = - 6 x + 3$ 
C)  $y = - 6 x + 1$ 
D)  $y = - 6 x - 1$ 
E)  $y = - 6 x - 3$ 
A)  $y = - 6 x + 9$ 
B)  $y = - 6 x + 3$ 
C)  $y = - 6 x + 1$ 
D)  $y = - 6 x - 1$ 
E)  $y = - 6 x - 3$

Question 9

Let $f ( x ) = 6 x - e ^ { x }$ 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)  $( - \infty , \ln 6 )$ 
C)  $( \ln 6 , \infty )$ 
D)  $( - \infty , - \ln 6 )$ 
E)  $( - \infty , - \ln 6 )$. 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , \ln 6 )$ 
C)  $( \ln 6 , \infty )$ 
D)  $( - \infty , - \ln 6 )$ 
E)  $( - \infty , - \ln 6 )$.

Question 10

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

Accepted Answer

A) -tan x 
B) cot x 
C) -cot x - csc x 
D) -csc x(cot x + csc x) 
E) -csc x(cot x - csc x) 
A) -tan x 
B) cot x 
C) -cot x - csc x 
D) -csc x(cot x + csc x) 
E) -csc x(cot x - csc x)

Question 11

Let $f ( x ) = \sin x - e ^ { x }$ If $x \in [ - \pi , 3 \pi ]$ then ƒ has a horizontal tangent line for each x in

Accepted Answer

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

Question 12

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ 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)  $( 4 , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0,4 )$ 
D)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
E)  $( - \infty , \infty )$ 
A)  $( 4 , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( 0,4 )$ 
D)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
E)  $( - \infty , \infty )$

Question 13

Let $f ( x ) = \left\{ \begin{array} { c c } 
x ^ { 2 } & x \leq 0 \
- x ^ { 2 } & x > 0
\end{array} \right.$ 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 14

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ 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)  $( 4 , \infty )$ 
B)  $( - \infty , 0 )$ 
C) (0,4) 
D)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
E)  $( - \infty , \infty )$ 
A)  $( 4 , \infty )$ 
B)  $( - \infty , 0 )$ 
C) (0,4) 
D)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
E)  $( - \infty , \infty )$

Question 15

Let $y = x \sin x$ Then $y\prime\prime$ is

Accepted Answer

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

Question 16

Let $y = 3 \sin x - 2 \cos x$ Then $y\prime\prime$ is

Accepted Answer

A)  $- 3 \sin x - 2 \cos x$ 
B)  $- 3 \sin x + 2 \cos x$ 
C)  $3 \sin x - 2 \cos x$ 
D)  $3 \sin x + 2 \cos x$ 
E)  $2 \sin x - 3 \cos x$ 
A)  $- 3 \sin x - 2 \cos x$ 
B)  $- 3 \sin x + 2 \cos x$ 
C)  $3 \sin x - 2 \cos x$ 
D)  $3 \sin x + 2 \cos x$ 
E)  $2 \sin x - 3 \cos x$

Question 17

Let $f ( x ) = 3 x - 2$ 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 18

The slope of the tangent line to the graph $f ( x ) = \frac { 2 } { x + 4 }$ at the point (-3, 2) is

Accepted Answer

A) -3 
B) -2 
C) 2 
D) 3 
E) 4 
A) -3 
B) -2 
C) 2 
D) 3 
E) 4

Question 19

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

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( - 1 , \infty )$ 
D)  $( - \infty , 0 ) \cup ( 1 , \infty )$ 
E)  $( 1 , \infty )$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , 0 )$ 
C)  $( - 1 , \infty )$ 
D)  $( - \infty , 0 ) \cup ( 1 , \infty )$ 
E)  $( 1 , \infty )$

Question 20

The instantaneous rate of change of $f ( x ) = 4 x - 3$ at c = 5 is

Accepted Answer

A) -3 
B) 4 
C) 5 
D) 17 
E) 20 
A) -3 
B) 4 
C) 5 
D) 17 
E) 20

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

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ Then $f ^ { \prime } ( x ) = 0$ for all x in

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

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

The slope of the tangent line to the graph $f ( x ) = 3 \sqrt { x }$ at the point (4,6) is

Let $f ( x ) = 6 x - e ^ { x }$ . 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 $f ( x ) = \pi ^ { 2 } e ^ { x }$ Then $f ^ { ( 10 ) } ( x )$ is

An equation of the tangent line to the graph $f ( x ) = \frac { 3 } { 2 x - 1 }$ at the point (1, 3) is

Let $f ( x ) = 6 x - e ^ { x }$ 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 $f ( x ) = \frac { 1 + \cos x } { \sin x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \sin x - e ^ { x }$ If $x \in [ - \pi , 3 \pi ]$ then ƒ has a horizontal tangent line for each x in

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ 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 $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & x \leq 0 \\- x ^ { 2 } & x > 0\end{array} \right.$ Which of the following is true?

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ 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 $y = x \sin x$ Then $y\prime\prime$ is

Let $y = 3 \sin x - 2 \cos x$ Then $y\prime\prime$ is

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

The slope of the tangent line to the graph $f ( x ) = \frac { 2 } { x + 4 }$ at the point (-3, 2) is

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

The instantaneous rate of change of $f ( x ) = 4 x - 3$ at c = 5 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

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Exam 3: The Derivative

Let f(x)=sin⁡x1−cos⁡2xf ( x ) = \frac { \sin x } { 1 - \cos ^ { 2 } x }f(x)=1−cos2xsinx​ Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=12x2−3x+4f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4f(x)=21​x2−3x+4 Then f′(x)=0f ^ { \prime } ( x ) = 0f′(x)=0 for all x in

Let f(x)=sin⁡x−xf ( x ) = \sin x - xf(x)=sinx−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)=x2cos⁡xf ( x ) = x ^ { 2 } \cos xf(x)=x2cosx Then f′(x)f ^ { \prime } ( x )f′(x) is

The slope of the tangent line to the graph f(x)=3xf ( x ) = 3 \sqrt { x }f(x)=3x​ at the point (4,6) is

Let f(x)=6x−exf ( x ) = 6 x - e ^ { x }f(x)=6x−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)=π2exf ( x ) = \pi ^ { 2 } e ^ { x }f(x)=π2ex Then f(10)(x)f ^ { ( 10 ) } ( x )f(10)(x) is

An equation of the tangent line to the graph f(x)=32x−1f ( x ) = \frac { 3 } { 2 x - 1 }f(x)=2x−13​ at the point (1, 3) is

Let f(x)=6x−exf ( x ) = 6 x - e ^ { x }f(x)=6x−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)=1+cos⁡xsin⁡xf ( x ) = \frac { 1 + \cos x } { \sin x }f(x)=sinx1+cosx​ Then f′(x)f ^ { \prime } ( x )f′(x) is

Let f(x)=sin⁡x−exf ( x ) = \sin x - e ^ { x }f(x)=sinx−ex If x∈[−π,3π]x \in [ - \pi , 3 \pi ]x∈[−π,3π] then ƒ has a horizontal tangent line for each x in

Let f(x)=x3−6x2f ( x ) = x ^ { 3 } - 6 x ^ { 2 }f(x)=x3−6x2 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)={x2x≤0−x2x>0f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & x \leq 0 \\- x ^ { 2 } & x > 0\end{array} \right.f(x)={x2−x2​x≤0x>0​ Which of the following is true?

Let f(x)=x3−6x2f ( x ) = x ^ { 3 } - 6 x ^ { 2 }f(x)=x3−6x2 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 y=xsin⁡xy = x \sin xy=xsinx Then y′′y\prime\primey′′ is

Let y=3sin⁡x−2cos⁡xy = 3 \sin x - 2 \cos xy=3sinx−2cosx Then y′′y\prime\primey′′ is

Let f(x)=3x−2f ( x ) = 3 x - 2f(x)=3x−2 Which of the following is true?

The slope of the tangent line to the graph f(x)=2x+4f ( x ) = \frac { 2 } { x + 4 }f(x)=x+42​ at the point (-3, 2) 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 for all x in

The instantaneous rate of change of f(x)=4x−3f ( x ) = 4 x - 3f(x)=4x−3 at c = 5 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

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

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

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 3 x + 4$ Then $f ^ { \prime } ( x ) = 0$ for all x in

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

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

The slope of the tangent line to the graph $f ( x ) = 3 \sqrt { x }$ at the point (4,6) is

Let $f ( x ) = 6 x - e ^ { x }$ . 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 $f ( x ) = \pi ^ { 2 } e ^ { x }$ Then $f ^ { ( 10 ) } ( x )$ is

An equation of the tangent line to the graph $f ( x ) = \frac { 3 } { 2 x - 1 }$ at the point (1, 3) is

Let $f ( x ) = 6 x - e ^ { x }$ 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 $f ( x ) = \frac { 1 + \cos x } { \sin x }$ Then $f ^ { \prime } ( x )$ is

Let $f ( x ) = \sin x - e ^ { x }$ If $x \in [ - \pi , 3 \pi ]$ then ƒ has a horizontal tangent line for each x in

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ 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 $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & x \leq 0 \\- x ^ { 2 } & x > 0\end{array} \right.$ Which of the following is true?

Let $f ( x ) = x ^ { 3 } - 6 x ^ { 2 }$ 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 $y = x \sin x$ Then $y\prime\prime$ is

Let $y = 3 \sin x - 2 \cos x$ Then $y\prime\prime$ is

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

The slope of the tangent line to the graph $f ( x ) = \frac { 2 } { x + 4 }$ at the point (-3, 2) is

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

The instantaneous rate of change of $f ( x ) = 4 x - 3$ at c = 5 is