Question 1

Find the absolute maximum of the function $f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }$

Accepted Answer

A)  $\frac { 4 } { 3 }$ 
B) 1 
C)  $e ^ { - 1 }$ 
D)  $e ^ { - \frac { 1 } { 2 } }$ 
E) 2
F) $\frac { 1 } { 2 }$  
G) $\frac { 3 } { 2 }$ 
H)No absolute minimum 
A)  $\frac { 4 } { 3 }$ 
B) 1 
C)  $e ^ { - 1 }$ 
D)  $e ^ { - \frac { 1 } { 2 } }$ 
E) 2
F) $\frac { 1 } { 2 }$  
G) $\frac { 3 } { 2 }$ 
H)No absolute minimum

Question 2

How many points of inflection does the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 6 x ^ { 2 }$ have?

Accepted Answer

A) 7 
B) 3 
C) 2 
D) 6 
E) 0
F)1
G)5
H)4 
A) 7 
B) 3 
C) 2 
D) 6 
E) 0
F)1
G)5
H)4

Question 3

The function $f ( x ) = \frac { x + 1 } { x } , x \neq 0$ , is

Accepted Answer

A) Increasing on $( - 1,0 )$ 
B) Decreasing on $( 0,1 )$ 
C) Increasing on $( 0,1 )$ 
D) Decreasing on $( 1 , \infty )$ 
E) Increasing on $( 0 , \infty )$ 
F)Decreasing on $( 1 , \infty )$ 
 
G)Increasing on $( - \infty , 0 )$ 
 
H)Decreasing on $( - \infty , 0 )$ 
A) Increasing on $( - 1,0 )$ 
B) Decreasing on $( 0,1 )$ 
C) Increasing on $( 0,1 )$ 
D) Decreasing on $( 1 , \infty )$ 
E) Increasing on $( 0 , \infty )$ 
F)Decreasing on $( 1 , \infty )$ 
 
G)Increasing on $( - \infty , 0 )$ 
 
H)Decreasing on $( - \infty , 0 )$

Question 4

Use Newton's method to approximate the root of $f ( x ) = x ^ { 3 } - 4 x - 2$ that lies between $x = 1$ and $x = 2$ .

Accepted Answer

A) 1.673 
B) 1.7693 
C) 1.77 
D) 1 
E) 2.473
F)1.75
G)1.8693
H)None of these 
A) 1.673 
B) 1.7693 
C) 1.77 
D) 1 
E) 2.473
F)1.75
G)1.8693
H)None of these

Question 5

A cyclist traveling at 40 ft/s decelerates at a constant 4 ft/s $2$ . How many feet does she travel before coming to a complete stop?

Accepted Answer

The answer of A cyclist traveling at 40 ft/s decelerates...

Question 6

Find the absolute minimum and maximum values of the function $f ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } + 12 x + 7$ on the closed interval $[ 0,3 ]$ .

Accepted Answer

A) 0, 3 
B) 0, 5 
C) 3, 5 
D) 3, 9.75 
E) 3, 16
F)5, 7
G)7, 16
H)5, 10.25 
A) 0, 3 
B) 0, 5 
C) 3, 5 
D) 3, 9.75 
E) 3, 16
F)5, 7
G)7, 16
H)5, 10.25

Question 7

Find all critical numbers for the function $f ( x ) = \frac { x - 1 } { x + 1 }$ .

Accepted Answer

A) 1 
B)  $- 1,1$ 
C)  $1 , - 1 , - 2$ 
D) 0 
E)  $- 1$
F)2
G) $1 , - 1,2$ 
 ,

H)No critical numbers 
A) 1 
B)  $- 1,1$ 
C)  $1 , - 1 , - 2$ 
D) 0 
E)  $- 1$
F)2
G) $1 , - 1,2$ 
 ,

H)No critical numbers

Question 8

The graph of the derivative of $f ^ { \prime } ( x )$ of a function f(x) is given below.   Sketch a possible graph of $y = f ( x )$ on the interval $[ - 2,2 ]$ such that $f ( - 2 ) = 0$ . On your graph, label the critical point(s) and inflection point(s).

Accepted Answer

The answer of The graph of the derivative of \(f...

Question 9

At what rate is the volume of a cube increasing if its edges are 2 inches and are increasing at a rate 3 inches per minute?

Accepted Answer

A)  $12 \mathrm { in } ^ { 3 } / \mathrm { min }$ 
B)  $18 \mathrm { in } ^ { 3 } / \mathrm { min }$ 
C)  $2 \operatorname { in } ^ { 3 } / \min$ 
D)  $72 \mathrm { in } ^ { 3 } / \mathrm { min }$ 
E)  $4 \operatorname { in } ^ { 3 } / \min$ 
F) $9 \mathrm { in } ^ { 3 } / \mathrm { min }$

G) $36 \mathrm { in } ^ { 3 } / \mathrm { min }$

H) $24 \operatorname { in } ^ { 3 } / \min$ 
A)  $12 \mathrm { in } ^ { 3 } / \mathrm { min }$ 
B)  $18 \mathrm { in } ^ { 3 } / \mathrm { min }$ 
C)  $2 \operatorname { in } ^ { 3 } / \min$ 
D)  $72 \mathrm { in } ^ { 3 } / \mathrm { min }$ 
E)  $4 \operatorname { in } ^ { 3 } / \min$ 
F) $9 \mathrm { in } ^ { 3 } / \mathrm { min }$

G) $36 \mathrm { in } ^ { 3 } / \mathrm { min }$

H) $24 \operatorname { in } ^ { 3 } / \min$

Question 10

Choose the correct statement: If f(x) is continuous on (1, 3) then
(i) f(x) has no critical point over the interval.(ii) f(x) has no extreme value over the interval.(iii) f(x) might have extreme value over the interval.

Accepted Answer

The answer of Choose the correct statement: If f(x) is...

Question 11

The graph of f is given below.   (a) Where is f increasing?
(b) Where is f decreasing?
(c) Where is f concave upward?
(d) Where is f concave downward?

Accepted Answer

The answer of The graph of f is given below....

Question 12

Find the absolute maximum value for the function $f ( x ) = 2 \sin x \cos x \text { for } 0 \leq x \leq \pi$

Accepted Answer

A)  $1$ 
B)  - 2 
C)  $\frac { 1 } { 4 }$ 
D) 0 
E)  - 1
F)2
G) $\frac { 1 } { 2 }$

H)No absolute minimum 
A)  $1$ 
B)  - 2 
C)  $\frac { 1 } { 4 }$ 
D) 0 
E)  - 1
F)2
G) $\frac { 1 } { 2 }$

H)No absolute minimum

Question 13

Find the value of the limit $\lim _ { x \rightarrow \infty } x ^ { \frac { 2 } { x } }$ .

Accepted Answer

A) 2 
B)  $e$ 
C)  $\ln 2$ 
D) 0 
E) 1
F) $\infty$
G) $\frac { 1 } { 2 }$

H) $\sqrt { 2 }$ 
A) 2 
B)  $e$ 
C)  $\ln 2$ 
D) 0 
E) 1
F) $\infty$
G) $\frac { 1 } { 2 }$

H) $\sqrt { 2 }$

Question 14

Find the interval on which $f ( x ) = x - 2 \sin x , 0 \leq x \leq 2 \pi$ is increasing.

Accepted Answer

A)  $\left[ \frac { \pi } { 3 } , \frac { 5 \pi } { 3 } \right]$ 
B)  $\left[ 0 , \frac { \pi } { 3 } \right]$ 
C)  $\left[ \frac { 5 \pi } { 3 } , 2 \pi \right]$ 
D)  $\left[ 0 , \frac { \pi } { 2 } \right]$ 
E)  $\left[ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right]$ 
F) $\left[ \frac { 3 \pi } { 2 } , 2 \pi \right]$

G) $[ 0 , \pi ]$

H) $[ \pi , 2 \pi ]$ 
A)  $\left[ \frac { \pi } { 3 } , \frac { 5 \pi } { 3 } \right]$ 
B)  $\left[ 0 , \frac { \pi } { 3 } \right]$ 
C)  $\left[ \frac { 5 \pi } { 3 } , 2 \pi \right]$ 
D)  $\left[ 0 , \frac { \pi } { 2 } \right]$ 
E)  $\left[ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right]$ 
F) $\left[ \frac { 3 \pi } { 2 } , 2 \pi \right]$

G) $[ 0 , \pi ]$

H) $[ \pi , 2 \pi ]$

Question 15

Find the absolute maximum value for the function $f ( x ) = \frac { x - 1 } { x + 1 } \text { for } 0 \leq x \leq 1$

Accepted Answer

A)  $1$ 
B)  - 2 
C)  $\frac { 1 } { 4 }$ 
D) 0 
E)  - 1
F)2
G) $\frac { 1 } { 2 }$

H)No absolute minimum 
A)  $1$ 
B)  - 2 
C)  $\frac { 1 } { 4 }$ 
D) 0 
E)  - 1
F)2
G) $\frac { 1 } { 2 }$

H)No absolute minimum

Question 16

Given $f ( x ) = x ^ { 3 } + x - 4$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

Accepted Answer

The answer of Given \(f ( x ) = x...

Question 17

Find the difference between the local maximum and the local minimum values of the function $f ( x ) = x ^ { 3 } - 3 x + 27$ .

Accepted Answer

A) 4 
B) 1 
C) 9 
D) 2 
E) 6
F)5
G)8
H)3 
A) 4 
B) 1 
C) 9 
D) 2 
E) 6
F)5
G)8
H)3

Question 18

Find the absolute maximum value of the function $f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }$ .

Accepted Answer

The answer of Find the absolute maximum value of the...

Question 19

Find the value of the limit $\lim _ { x \rightarrow 1 ^ { - } } \frac { \frac { \pi } { 2 } - \arcsin x } { \sqrt { 1 - x ^ { 2 } } }$ .

Accepted Answer

The answer of Find the value of the limit \(\lim...

Question 20

Find the value of the limit $\lim _ { x \rightarrow 1 ^ { + } } \left( \frac { 1 } { x - 1 } - \frac { 1 } { \ln x } \right)$ .

Accepted Answer

The answer of Find the value of the limit \(\lim...

Find the absolute maximum of the function $f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }$

How many points of inflection does the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 6 x ^ { 2 }$ have?

The function $f ( x ) = \frac { x + 1 } { x } , x \neq 0$ , is

Use Newton's method to approximate the root of $f ( x ) = x ^ { 3 } - 4 x - 2$ that lies between $x = 1$ and $x = 2$ .

A cyclist traveling at 40 ft/s decelerates at a constant 4 ft/s $2$ . How many feet does she travel before coming to a complete stop?

Find the absolute minimum and maximum values of the function $f ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } + 12 x + 7$ on the closed interval $[ 0,3 ]$ .

Find all critical numbers for the function $f ( x ) = \frac { x - 1 } { x + 1 }$ .

At what rate is the volume of a cube increasing if its edges are 2 inches and are increasing at a rate 3 inches per minute?

Choose the correct statement: If f(x) is continuous on (1, 3) then (i) f(x) has no critical point over the interval.(ii) f(x) has no extreme value over the interval.(iii) f(x) might have extreme value over the interval.

The graph of f is given below. (a) Where is f increasing? (b) Where is f decreasing? (c) Where is f concave upward? (d) Where is f concave downward?

Find the absolute maximum value for the function $f ( x ) = 2 \sin x \cos x \text { for } 0 \leq x \leq \pi$

Find the value of the limit $\lim _ { x \rightarrow \infty } x ^ { \frac { 2 } { x } }$ .

Find the interval on which $f ( x ) = x - 2 \sin x , 0 \leq x \leq 2 \pi$ is increasing.

Find the absolute maximum value for the function $f ( x ) = \frac { x - 1 } { x + 1 } \text { for } 0 \leq x \leq 1$

Given $f ( x ) = x ^ { 3 } + x - 4$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

Find the difference between the local maximum and the local minimum values of the function $f ( x ) = x ^ { 3 } - 3 x + 27$ .

Find the absolute maximum value of the function $f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }$ .

Find the value of the limit $\lim _ { x \rightarrow 1 ^ { - } } \frac { \frac { \pi } { 2 } - \arcsin x } { \sqrt { 1 - x ^ { 2 } } }$ .

Find the value of the limit $\lim _ { x \rightarrow 1 ^ { + } } \left( \frac { 1 } { x - 1 } - \frac { 1 } { \ln x } \right)$ .

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Exam 4: Applications of Differentiation

Find the absolute maximum of the function f(x)=e−x1+x2f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }f(x)=1+x2e−x​

How many points of inflection does the function f(x)=x4−4x3+6x2f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 6 x ^ { 2 }f(x)=x4−4x3+6x2 have?

The function f(x)=x+1x,x≠0f ( x ) = \frac { x + 1 } { x } , x \neq 0f(x)=xx+1​,x=0 , is

Use Newton's method to approximate the root of f(x)=x3−4x−2f ( x ) = x ^ { 3 } - 4 x - 2f(x)=x3−4x−2 that lies between x=1x = 1x=1 and x=2x = 2x=2 .

A cyclist traveling at 40 ft/s decelerates at a constant 4 ft/s 222 . How many feet does she travel before coming to a complete stop?

Find the absolute minimum and maximum values of the function f(x)=4x3−15x2+12x+7f ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } + 12 x + 7f(x)=4x3−15x2+12x+7 on the closed interval [0,3][ 0,3 ][0,3] .

Find all critical numbers for the function f(x)=x−1x+1f ( x ) = \frac { x - 1 } { x + 1 }f(x)=x+1x−1​ .

At what rate is the volume of a cube increasing if its edges are 2 inches and are increasing at a rate 3 inches per minute?

Choose the correct statement: If f(x) is continuous on (1, 3) then (i) f(x) has no critical point over the interval.(ii) f(x) has no extreme value over the interval.(iii) f(x) might have extreme value over the interval.

The graph of f is given below. (a) Where is f increasing? (b) Where is f decreasing? (c) Where is f concave upward? (d) Where is f concave downward?

Find the absolute maximum value for the function f(x)=2sin⁡xcos⁡x for 0≤x≤πf ( x ) = 2 \sin x \cos x \text { for } 0 \leq x \leq \pif(x)=2sinxcosx for 0≤x≤π

Find the value of the limit lim⁡x→∞x2x\lim _ { x \rightarrow \infty } x ^ { \frac { 2 } { x } }limx→∞​xx2​ .

Find the interval on which f(x)=x−2sin⁡x,0≤x≤2πf ( x ) = x - 2 \sin x , 0 \leq x \leq 2 \pif(x)=x−2sinx,0≤x≤2π is increasing.

Find the absolute maximum value for the function f(x)=x−1x+1 for 0≤x≤1f ( x ) = \frac { x - 1 } { x + 1 } \text { for } 0 \leq x \leq 1f(x)=x+1x−1​ for 0≤x≤1

Given f(x)=x3+x−4f ( x ) = x ^ { 3 } + x - 4f(x)=x3+x−4 , use Newton's method to find the iterative formula for xn+1x _ { n + 1 }xn+1​ .

Find the difference between the local maximum and the local minimum values of the function f(x)=x3−3x+27f ( x ) = x ^ { 3 } - 3 x + 27f(x)=x3−3x+27 .

Find the absolute maximum value of the function f(x)=e−x1+x2f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }f(x)=1+x2e−x​ .

Find the value of the limit lim⁡x→1−π2−arcsin⁡x1−x2\lim _ { x \rightarrow 1 ^ { - } } \frac { \frac { \pi } { 2 } - \arcsin x } { \sqrt { 1 - x ^ { 2 } } }limx→1−​1−x2​2π​−arcsinx​ .

Find the value of the limit lim⁡x→1+(1x−1−1ln⁡x)\lim _ { x \rightarrow 1 ^ { + } } \left( \frac { 1 } { x - 1 } - \frac { 1 } { \ln x } \right)limx→1+​(x−11​−lnx1​) .

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the absolute maximum of the function $f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }$

How many points of inflection does the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 6 x ^ { 2 }$ have?

The function $f ( x ) = \frac { x + 1 } { x } , x \neq 0$ , is

Use Newton's method to approximate the root of $f ( x ) = x ^ { 3 } - 4 x - 2$ that lies between $x = 1$ and $x = 2$ .

A cyclist traveling at 40 ft/s decelerates at a constant 4 ft/s $2$ . How many feet does she travel before coming to a complete stop?

Find the absolute minimum and maximum values of the function $f ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } + 12 x + 7$ on the closed interval $[ 0,3 ]$ .

Find all critical numbers for the function $f ( x ) = \frac { x - 1 } { x + 1 }$ .

Find the absolute maximum value for the function $f ( x ) = 2 \sin x \cos x \text { for } 0 \leq x \leq \pi$

Find the value of the limit $\lim _ { x \rightarrow \infty } x ^ { \frac { 2 } { x } }$ .

Find the interval on which $f ( x ) = x - 2 \sin x , 0 \leq x \leq 2 \pi$ is increasing.

Find the absolute maximum value for the function $f ( x ) = \frac { x - 1 } { x + 1 } \text { for } 0 \leq x \leq 1$

Given $f ( x ) = x ^ { 3 } + x - 4$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

Find the difference between the local maximum and the local minimum values of the function $f ( x ) = x ^ { 3 } - 3 x + 27$ .

Find the absolute maximum value of the function $f ( x ) = \frac { e ^ { - x } } { 1 + x ^ { 2 } }$ .

Find the value of the limit $\lim _ { x \rightarrow 1 ^ { - } } \frac { \frac { \pi } { 2 } - \arcsin x } { \sqrt { 1 - x ^ { 2 } } }$ .

Find the value of the limit $\lim _ { x \rightarrow 1 ^ { + } } \left( \frac { 1 } { x - 1 } - \frac { 1 } { \ln x } \right)$ .