Question 1

Find the interval on which $f ( x ) = x e ^ { - x }$ is increasing.

Accepted Answer

A)  $( - \infty , 1 ]$ 
B)  $( - \infty , 2 ]$ 
C)  $( - \infty , 3 ]$ 
D)  $[ 1,2 ]$ 
E)  $[ 1 , e ]$ 
F) $( - \infty , e ]$

G) $( - \infty , \infty )$

H) $\left( - \infty , \frac { 1 } { e } \right)$ 
A)  $( - \infty , 1 ]$ 
B)  $( - \infty , 2 ]$ 
C)  $( - \infty , 3 ]$ 
D)  $[ 1,2 ]$ 
E)  $[ 1 , e ]$ 
F) $( - \infty , e ]$

G) $( - \infty , \infty )$

H) $\left( - \infty , \frac { 1 } { e } \right)$

Question 2

Find the maximum and minimum values of $f ( x ) = \left| 9 - x ^ { 2 } \right| \text { where } - 2 \leq x \leq 4$

Accepted Answer

Maximum va

Question 3

Find the intervals on which $f ( x ) = 2 \sin x + \sin ^ { 2 } x$ is increasing.

Accepted Answer

The answer of Find the intervals on which \(f (...

Question 4

Use Newton's method to approximate the root of $f ( x ) = x ^ { 4 } - 4 x - 2$ that lies between $x = - 1$ and $x = 0$ .

Accepted Answer

The answer of Use Newton's method to approximate the root...

Question 5

Let $f ( x ) = 0.01 x ^ { 3 } - x ^ { 2 } + 5$ .(a) Use your graphing calculator to graph f using the default viewing rectangle.(b) Find the critical points of f and use them to determine an appropriate viewing rectangle (or rectangles) for f. The graph f again using the viewing rectangle.(c) Compare the graphs in parts (a) and (b)

Accepted Answer

(b) Critical points

Question 6

A rental estate management company manages an apartment complex with 20 units. The manager estimates that all 20 units can be rented if the rent is $150 per unit per month, and that for each increase in rent of $25, one apartment will be vacated. What rent should the manager charge to maximize revenue?

Accepted Answer

The answer of A rental estate management company manages an...

Question 7

A school band decides to raise with a car wash. Labor and material are donated, so pure profit is realized from the sale of the tickets. They know that if they charge $5 per ticket, they will sell 1000, while if they charge $4 per ticket, they will sell 1500. Assuming the demand function is linear, what should they charge to maximize their profit

Accepted Answer

The answer of A school band decides to raise with...

Question 8

Given $f ^ { \prime } ( x ) = \frac { 1 } { 2 \sqrt { x } }$ , and $f ( 1 ) = 2$ , find $f \left( \frac { 1 } { 4 } \right)$ .

Accepted Answer

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

Question 9

Two positive numbers have product 200. Find the minimum value of the sum of one number plus twice the other.

Accepted Answer

The answer of Two positive numbers have product 200. Find...

Question 10

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { x } { \sin x + \tan x }$ .

Accepted Answer

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

Question 11

Given $f ( x ) = x ^ { 3 } + 2 x - 4$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

Accepted Answer

A)  $x _ { n + 1 } = - \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$ 
B)  $x _ { n + 1 } = x _ { n } - \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$ 
C)  $x _ { n + 1 } = x _ { n } - \frac { x _ { n } ^ { 3 } + 2 x _ { n } - 4 } { 3 x _ { n } ^ { 2 } + 2 }$ 
D)  $x _ { n + 1 } = x _ { n } + \frac { 3 } { x _ { n } }$ 
E)  $x _ { n + 1 } = \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$ 
F) $x _ { n + 1 } = x _ { n } + \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$

G) $x _ { n + 1 } = x _ { n } + \frac { x _ { n } ^ { 3 } + 2 x _ { n } - 4 } { 3 x _ { n } ^ { 2 } + 2 }$

H)None of these 
A)  $x _ { n + 1 } = - \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$ 
B)  $x _ { n + 1 } = x _ { n } - \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$ 
C)  $x _ { n + 1 } = x _ { n } - \frac { x _ { n } ^ { 3 } + 2 x _ { n } - 4 } { 3 x _ { n } ^ { 2 } + 2 }$ 
D)  $x _ { n + 1 } = x _ { n } + \frac { 3 } { x _ { n } }$ 
E)  $x _ { n + 1 } = \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$ 
F) $x _ { n + 1 } = x _ { n } + \frac { x _ { n } ^ { 3 } + 2 x _ { n } } { 3 x _ { n } ^ { 2 } + 2 }$

G) $x _ { n + 1 } = x _ { n } + \frac { x _ { n } ^ { 3 } + 2 x _ { n } - 4 } { 3 x _ { n } ^ { 2 } + 2 }$

H)None of these

Question 12

Find each of the following limits or explain why it does not exist.(a) $\lim _ { x \rightarrow 0 } \frac { x - \sin x } { x ^ { 3 } }$ (b) $\lim _ { x \rightarrow \infty } e ^ { - x } x ^ { 2 }$ (c) $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \frac { 2 } { x } }$

Accepted Answer

The answer of Find each of the following limits or...

Question 13

At what value of x does the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ change from increasing to decreasing?

Accepted Answer

A) 3 
B) 1 
C) 4 
D)  - 3 
E) 2
F)0
G) - 2
H) - 1 
A) 3 
B) 1 
C) 4 
D)  - 3 
E) 2
F)0
G) - 2
H) - 1

Question 14

Find the value of the limit $\lim _ { x \rightarrow 2 } \left( \frac { 1 } { x - 2 } - \frac { 4 } { x ^ { 2 } - 4 } \right)$ . Use L'Hospital's Rule, if applicable.

Accepted Answer

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

Question 15

Find the y-coordinate of the point of inflection of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ .

Accepted Answer

A) 2. 
B)  - 2 
C) 0 
D) 2 
E)  - 1
F)1
G) $- \frac { 2 } { 3 }$ 
H) $\frac { 2 } { 3 }$ 
A) 2. 
B)  - 2 
C) 0 
D) 2 
E)  - 1
F)1
G) $- \frac { 2 } { 3 }$ 
H) $\frac { 2 } { 3 }$

Question 16

Find the absolute maximum and minimum values of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 4$ on the interval $[ - 2,1 ]$ .

Accepted Answer

Maximum va

Question 17

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow \infty } \frac { \sqrt { 1 + x ^ { 2 } } } { x }$ , then find the limit.

Accepted Answer

Because when L'Hospi

Question 18

Find all critical numbers for the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 }$ .

Accepted Answer

A)  $- 3$ 
B)  $3$ 
C) 0, 3 
D) 0 
E) 0, 4
F)2

G) $0 , - 3$ 
H)No critical numbers 
A)  $- 3$ 
B)  $3$ 
C) 0, 3 
D) 0 
E) 0, 4
F)2

G) $0 , - 3$ 
H)No critical numbers

Question 19

A rancher intends to fence o a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed?

Accepted Answer

The answer of A rancher intends to fence o a...

Question 20

A chocolate cream pie is thrown vertically up from the ground with velocity 72 ft/s. Find the amount of time in seconds until it hits the ground.

Accepted Answer

A) 6.5 
B) 4.5 
C) 6 
D) 7.5 
E) 5.5
F)7
G)5
H)4 
A) 6.5 
B) 4.5 
C) 6 
D) 7.5 
E) 5.5
F)7
G)5
H)4

Find the interval on which $f ( x ) = x e ^ { - x }$ is increasing.

Find the maximum and minimum values of $f ( x ) = \left| 9 - x ^ { 2 } \right| \text { where } - 2 \leq x \leq 4$

Find the intervals on which $f ( x ) = 2 \sin x + \sin ^ { 2 } x$ is increasing.

Use Newton's method to approximate the root of $f ( x ) = x ^ { 4 } - 4 x - 2$ that lies between $x = - 1$ and $x = 0$ .

Given $f ^ { \prime } ( x ) = \frac { 1 } { 2 \sqrt { x } }$ , and $f ( 1 ) = 2$ , find $f \left( \frac { 1 } { 4 } \right)$ .

Two positive numbers have product 200. Find the minimum value of the sum of one number plus twice the other.

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { x } { \sin x + \tan x }$ .

Given $f ( x ) = x ^ { 3 } + 2 x - 4$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

Find each of the following limits or explain why it does not exist.(a) $\lim _ { x \rightarrow 0 } \frac { x - \sin x } { x ^ { 3 } }$ (b) $\lim _ { x \rightarrow \infty } e ^ { - x } x ^ { 2 }$ (c) $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \frac { 2 } { x } }$

At what value of x does the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ change from increasing to decreasing?

Find the value of the limit $\lim _ { x \rightarrow 2 } \left( \frac { 1 } { x - 2 } - \frac { 4 } { x ^ { 2 } - 4 } \right)$ . Use L'Hospital's Rule, if applicable.

Find the y-coordinate of the point of inflection of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ .

Find the absolute maximum and minimum values of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 4$ on the interval $[ - 2,1 ]$ .

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow \infty } \frac { \sqrt { 1 + x ^ { 2 } } } { x }$ , then find the limit.

Find all critical numbers for the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 }$ .

A rancher intends to fence o a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed?

A chocolate cream pie is thrown vertically up from the ground with velocity 72 ft/s. Find the amount of time in seconds until it hits the ground.

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

Exam 4: Applications of Differentiation

Find the interval on which f(x)=xe−xf ( x ) = x e ^ { - x }f(x)=xe−x is increasing.

Find the maximum and minimum values of f(x)=∣9−x2∣ where −2≤x≤4f ( x ) = \left| 9 - x ^ { 2 } \right| \text { where } - 2 \leq x \leq 4f(x)=​9−x2​ where −2≤x≤4

Find the intervals on which f(x)=2sin⁡x+sin⁡2xf ( x ) = 2 \sin x + \sin ^ { 2 } xf(x)=2sinx+sin2x is increasing.

Use Newton's method to approximate the root of f(x)=x4−4x−2f ( x ) = x ^ { 4 } - 4 x - 2f(x)=x4−4x−2 that lies between x=−1x = - 1x=−1 and x=0x = 0x=0 .

Given f′(x)=12xf ^ { \prime } ( x ) = \frac { 1 } { 2 \sqrt { x } }f′(x)=2x​1​ , and f(1)=2f ( 1 ) = 2f(1)=2 , find f(14)f \left( \frac { 1 } { 4 } \right)f(41​) .

Two positive numbers have product 200. Find the minimum value of the sum of one number plus twice the other.

Find the value of the limit lim⁡x→0+xsin⁡x+tan⁡x\lim _ { x \rightarrow 0 ^ { + } } \frac { x } { \sin x + \tan x }limx→0+​sinx+tanxx​ .

Given f(x)=x3+2x−4f ( x ) = x ^ { 3 } + 2 x - 4f(x)=x3+2x−4 , use Newton's method to find the iterative formula for xn+1x _ { n + 1 }xn+1​ .

At what value of x does the function f(x)=x3−3x2−9xf ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 xf(x)=x3−3x2−9x change from increasing to decreasing?

Find the value of the limit lim⁡x→2(1x−2−4x2−4)\lim _ { x \rightarrow 2 } \left( \frac { 1 } { x - 2 } - \frac { 4 } { x ^ { 2 } - 4 } \right)limx→2​(x−21​−x2−44​) . Use L'Hospital's Rule, if applicable.

Find the y-coordinate of the point of inflection of the function f(x)=x3−3x2f ( x ) = x ^ { 3 } - 3 x ^ { 2 }f(x)=x3−3x2 .

Find the absolute maximum and minimum values of f(x)=2x3+3x2+4f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 4f(x)=2x3+3x2+4 on the interval [−2,1][ - 2,1 ][−2,1] .

Explain why L'Hospital's Rule is not applicable to evaluate lim⁡x→∞1+x2x\lim _ { x \rightarrow \infty } \frac { \sqrt { 1 + x ^ { 2 } } } { x }limx→∞​x1+x2​​ , then find the limit.

Find all critical numbers for the function f(x)=x4−4x3f ( x ) = x ^ { 4 } - 4 x ^ { 3 }f(x)=x4−4x3 .

A rancher intends to fence o a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed?

A chocolate cream pie is thrown vertically up from the ground with velocity 72 ft/s. Find the amount of time in seconds until it hits the ground.

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 interval on which $f ( x ) = x e ^ { - x }$ is increasing.

Find the maximum and minimum values of $f ( x ) = \left| 9 - x ^ { 2 } \right| \text { where } - 2 \leq x \leq 4$

Find the intervals on which $f ( x ) = 2 \sin x + \sin ^ { 2 } x$ is increasing.

Use Newton's method to approximate the root of $f ( x ) = x ^ { 4 } - 4 x - 2$ that lies between $x = - 1$ and $x = 0$ .

Given $f ^ { \prime } ( x ) = \frac { 1 } { 2 \sqrt { x } }$ , and $f ( 1 ) = 2$ , find $f \left( \frac { 1 } { 4 } \right)$ .

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { x } { \sin x + \tan x }$ .

Given $f ( x ) = x ^ { 3 } + 2 x - 4$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

At what value of x does the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ change from increasing to decreasing?

Find the value of the limit $\lim _ { x \rightarrow 2 } \left( \frac { 1 } { x - 2 } - \frac { 4 } { x ^ { 2 } - 4 } \right)$ . Use L'Hospital's Rule, if applicable.

Find the y-coordinate of the point of inflection of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ .

Find the absolute maximum and minimum values of $f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + 4$ on the interval $[ - 2,1 ]$ .

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow \infty } \frac { \sqrt { 1 + x ^ { 2 } } } { x }$ , then find the limit.

Find all critical numbers for the function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 }$ .