Question 1

Find the absolute maximum and minimum values of $f ( x ) = \frac { x } { x ^ { 2 } + 1 } + 1$ on the interval $[ 0,2 \pi ]$ .

Accepted Answer

Maximum va

Question 2

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 3

How many points of inflection does the function $f ( x ) = x ^ { 8 } - x ^ { 2 }$ have?

Accepted Answer

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

Question 4

Find the x-coordinate of the point on the parabola $y = \sqrt { x }$ nearest to the point (4, 0).

Accepted Answer

A)  $\frac { 7 } { 2 }$ 
B) 4 
C) 3 
D)  $\frac { 3 } { 4 }$ 
E) 
$\frac { 2 } { 7 }$
F)2
G)0

H)None of these 
A)  $\frac { 7 } { 2 }$ 
B) 4 
C) 3 
D)  $\frac { 3 } { 4 }$ 
E) 
$\frac { 2 } { 7 }$
F)2
G)0

H)None of these

Question 5

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

Accepted Answer

A)  $\sqrt { 3 } \mathrm { in } / \mathrm { min }$ 
B)  $6 \sqrt { 3 } \text { in } / \min$ 
C)  $2 \sqrt { 3 } \mathrm { in } / \mathrm { min }$ 
D)  $72 \text { in } / \mathrm { min }$ 
E)  $3 \sqrt { 3 } \text { in } / \min$ 
F) $9 \mathrm { in } / \mathrm { min }$

G) $36 \text { in } / \mathrm { min }$

H) $24 \text { in } / \mathrm { min }$ 
A)  $\sqrt { 3 } \mathrm { in } / \mathrm { min }$ 
B)  $6 \sqrt { 3 } \text { in } / \min$ 
C)  $2 \sqrt { 3 } \mathrm { in } / \mathrm { min }$ 
D)  $72 \text { in } / \mathrm { min }$ 
E)  $3 \sqrt { 3 } \text { in } / \min$ 
F) $9 \mathrm { in } / \mathrm { min }$

G) $36 \text { in } / \mathrm { min }$

H) $24 \text { in } / \mathrm { min }$

Question 6

Find the critical number(s) (if any) and test them to determine if $f ( x ) = \left| 4 - x ^ { 2 } \right|$ has a local maximum or local minimum or neither at each critical number.

Accepted Answer

f(x) has critical numbers @#IMG-DLM& . f

Question 7

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area can be enclosed?

Accepted Answer

The answer of A rancher with 300 ft of fence...

Question 8

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { \cos x - 1 } { x ^ { 2 } }$ .

Accepted Answer

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

Question 9

A plane flying horizontally at an altitude of 1 km and a speed of 500 km/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 km away from the station.

Accepted Answer

A)  $125 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
B)  $175 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
C)  $250 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
D)  $275 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
E)  $125 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$
F) $175 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$

G) $250 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$

H) $275 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$ 
A)  $125 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
B)  $175 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
C)  $250 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
D)  $275 \sqrt { 3 } \mathrm {~km} / \mathrm { h }$ 
E)  $125 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$
F) $175 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$

G) $250 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$

H) $275 \sqrt { 2 } \mathrm {~km} / \mathrm { h }$

Question 10

Is it true that $\int \ln x d x = x \ln x - x + C$ ? Explain why or why not.

Accepted Answer

The answer of Is it true that \(\int \ln x...

Question 11

Find all critical numbers for the function $f ( x ) = x - \cos x \text { for } 0 < x < 2 \pi$ .

Accepted Answer

A)  $\frac { \pi } { 2 }$ 
B)  $\frac { 3 \pi } { 2 }$ 
C) 0 
D) None of these 
E)  $\frac { \pi } { 3 }$ 
F) $\frac { \pi } { 4 }$

G)-1
H)No critical numbers 
A)  $\frac { \pi } { 2 }$ 
B)  $\frac { 3 \pi } { 2 }$ 
C) 0 
D) None of these 
E)  $\frac { \pi } { 3 }$ 
F) $\frac { \pi } { 4 }$

G)-1
H)No critical numbers

Question 12

If f is continuous over $[ 1,3 ]$ and differentiable over $( 1,3 )$ , suppose that $2 \leq f ^ { \prime } ( x ) \leq 5$ for all x in $[ 1,3 ]$ , and $f ( 1 ) = 4$ . What is the maximum possible value that f(3) can be? What is the minimum possible value that f(3) can be?

Accepted Answer

The maximum possible

Question 13

Find each of the following limits or explain why it does not exist.(a) $\lim _ { x \rightarrow 1 } \frac { x ^ { 3 } - 3 x + 2 } { 1 - x }$ (b) $\lim _ { x \rightarrow 0 ^ { + } } \frac { \cos x - 1 } { x ^ { 2 } }$ (c) $\lim _ { x \rightarrow 0 ^ { + } } x ^ { 2 } ( \ln x )$

Accepted Answer

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

Question 14

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 1$ . Use the graphs of f, $f ^ { \prime }$ , and $f ^ { \prime \prime }$ to estimate, to two decimal places,
(a) the critical numbers of f.(b) the interval(s) where f is decreasing.(c) the interval(s) where f is concave upward.

Accepted Answer

(a) Critical numbers are -1, 0

Question 15

An open box is made from a 8 inch $\times$ 8 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. What size squares should be cut out to create a box with maximum volume?

Accepted Answer

A)  $2 \text { in. } \times 2 \text { in. }$ 
B)  $4 \text { in. } \times 4 \text { in. }$ 
C)  $1 \text { in. } \times 1 \text { in. }$ 
D)  $0 \text { in. } \times 0 \text { in. }$ 
E)  $\frac { 2 } { 3 } \text { in } \times \frac { 2 } { 3 } \text { in. }$
F) $\frac { 4 } { 3 } \text { in. } x \frac { 4 } { 3 } \text { in. }$

G) $1.5 \text { in. } \times 1.5 \text { in. }$

H)None of these 
A)  $2 \text { in. } \times 2 \text { in. }$ 
B)  $4 \text { in. } \times 4 \text { in. }$ 
C)  $1 \text { in. } \times 1 \text { in. }$ 
D)  $0 \text { in. } \times 0 \text { in. }$ 
E)  $\frac { 2 } { 3 } \text { in } \times \frac { 2 } { 3 } \text { in. }$
F) $\frac { 4 } { 3 } \text { in. } x \frac { 4 } { 3 } \text { in. }$

G) $1.5 \text { in. } \times 1.5 \text { in. }$

H)None of these

Question 16

A hog weighs 250 pounds. A high-yield diet allows the animal to gain 6 pounds a day at a cost of $0.56 a day. The market price for hogs is currently $0.75 per pound, but it is falling at a rate of $0.01 per day, and that price decline is expected to remain steady for the foreseeable future. When should the hog be sold in order to provide the farmer with the highest financial gain?

Accepted Answer

The answer of A hog weighs 250 pounds. A high-yield...

Question 17

Find all critical numbers for the function $f ( x ) = ( x + 3 ) ^ { 4 } ( x - 2 ) ^ { 3 }$ .

Accepted Answer

A)  $- \frac { 1 } { 7 }$ 
B)  $- 3 , - \frac { 1 } { 7 } , 2$ 
C) 3, 2 
D) 0 
E)  $- 3,2$ 
F)2
G) $- 3,2$ 
H)No critical numbers 
A)  $- \frac { 1 } { 7 }$ 
B)  $- 3 , - \frac { 1 } { 7 } , 2$ 
C) 3, 2 
D) 0 
E)  $- 3,2$ 
F)2
G) $- 3,2$ 
H)No critical numbers

Question 18

Using the language of calculus to explain the following economic principles for output price decisions.(a) Maximum profit requires the firm to choose the level of output at which marginal revenue is equal to marginal cost.(b) A change in fixed cost will not change the profit maximizing level of output.(c) It may pay a firm to expand its output if it is selling at a price greater than marginal cost, even if that price happens to be below average cost.

Accepted Answer

Let @#IMG-DLM& be the profit function, @#IMG-DLM& be the rev

Question 19

Find the absolute maximum of the function $f ( x ) = x ^ { 3 } - x ^ { 2 } - x$ on the interval $- 10 \leq x \leq 2$ .

Accepted Answer

A)  $\frac { 1 } { 9 }$ 
B)  $\frac { 1 } { 5 }$ 
C) 0 
D)  $\frac { 1 } { 7 }$ 
E) 1
F) $\frac { 1 } { 4 }$ 
G) $\frac { 1 } { 3 }$

H)2 
A)  $\frac { 1 } { 9 }$ 
B)  $\frac { 1 } { 5 }$ 
C) 0 
D)  $\frac { 1 } { 7 }$ 
E) 1
F) $\frac { 1 } { 4 }$ 
G) $\frac { 1 } { 3 }$

H)2

Question 20

Sketch the graph of a function f that is continuous on [0;4] and has the following properties: Absolute minimum at 0, absolute maximum at 4, local maximum at 2, and local minimum at 3.

Accepted Answer

The answer of Sketch the graph of a function f...

Find the absolute maximum and minimum values of $f ( x ) = \frac { x } { x ^ { 2 } + 1 } + 1$ on the interval $[ 0,2 \pi ]$ .

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?

How many points of inflection does the function $f ( x ) = x ^ { 8 } - x ^ { 2 }$ have?

Find the x-coordinate of the point on the parabola $y = \sqrt { x }$ nearest to the point (4, 0).

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

Find the critical number(s) (if any) and test them to determine if $f ( x ) = \left| 4 - x ^ { 2 } \right|$ has a local maximum or local minimum or neither at each critical number.

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area can be enclosed?

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { \cos x - 1 } { x ^ { 2 } }$ .

A plane flying horizontally at an altitude of 1 km and a speed of 500 km/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 km away from the station.

Is it true that $\int \ln x d x = x \ln x - x + C$ ? Explain why or why not.

Find all critical numbers for the function $f ( x ) = x - \cos x \text { for } 0 < x < 2 \pi$ .

If f is continuous over $[ 1,3 ]$ and differentiable over $( 1,3 )$ , suppose that $2 \leq f ^ { \prime } ( x ) \leq 5$ for all x in $[ 1,3 ]$ , and $f ( 1 ) = 4$ . What is the maximum possible value that f(3) can be? What is the minimum possible value that f(3) can be?

Find each of the following limits or explain why it does not exist.(a) $\lim _ { x \rightarrow 1 } \frac { x ^ { 3 } - 3 x + 2 } { 1 - x }$ (b) $\lim _ { x \rightarrow 0 ^ { + } } \frac { \cos x - 1 } { x ^ { 2 } }$ (c) $\lim _ { x \rightarrow 0 ^ { + } } x ^ { 2 } ( \ln x )$

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 1$ . Use the graphs of f, $f ^ { \prime }$ , and $f ^ { \prime \prime }$ to estimate, to two decimal places, (a) the critical numbers of f.(b) the interval(s) where f is decreasing.(c) the interval(s) where f is concave upward.

An open box is made from a 8 inch $\times$ 8 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. What size squares should be cut out to create a box with maximum volume?

Find all critical numbers for the function $f ( x ) = ( x + 3 ) ^ { 4 } ( x - 2 ) ^ { 3 }$ .

Find the absolute maximum of the function $f ( x ) = x ^ { 3 } - x ^ { 2 } - x$ on the interval $- 10 \leq x \leq 2$ .

Sketch the graph of a function f that is continuous on [0;4] and has the following properties: Absolute minimum at 0, absolute maximum at 4, local maximum at 2, and local minimum at 3.

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 absolute maximum and minimum values of f(x)=xx2+1+1f ( x ) = \frac { x } { x ^ { 2 } + 1 } + 1f(x)=x2+1x​+1 on the interval [0,2π][ 0,2 \pi ][0,2π] .

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?

How many points of inflection does the function f(x)=x8−x2f ( x ) = x ^ { 8 } - x ^ { 2 }f(x)=x8−x2 have?

Find the x-coordinate of the point on the parabola y=xy = \sqrt { x }y=x​ nearest to the point (4, 0).

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

Find the critical number(s) (if any) and test them to determine if f(x)=∣4−x2∣f ( x ) = \left| 4 - x ^ { 2 } \right|f(x)=​4−x2​ has a local maximum or local minimum or neither at each critical number.

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area can be enclosed?

Find the value of the limit lim⁡x→0+cos⁡x−1x2\lim _ { x \rightarrow 0 ^ { + } } \frac { \cos x - 1 } { x ^ { 2 } }limx→0+​x2cosx−1​ .

A plane flying horizontally at an altitude of 1 km and a speed of 500 km/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 km away from the station.

Is it true that ∫ln⁡xdx=xln⁡x−x+C\int \ln x d x = x \ln x - x + C∫lnxdx=xlnx−x+C ? Explain why or why not.

Find all critical numbers for the function f(x)=x−cos⁡x for 0<x<2πf ( x ) = x - \cos x \text { for } 0 < x < 2 \pif(x)=x−cosx for 0<x<2π .

An open box is made from a 8 inch ×\times× 8 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. What size squares should be cut out to create a box with maximum volume?

Find all critical numbers for the function f(x)=(x+3)4(x−2)3f ( x ) = ( x + 3 ) ^ { 4 } ( x - 2 ) ^ { 3 }f(x)=(x+3)4(x−2)3 .

Find the absolute maximum of the function f(x)=x3−x2−xf ( x ) = x ^ { 3 } - x ^ { 2 } - xf(x)=x3−x2−x on the interval −10≤x≤2- 10 \leq x \leq 2−10≤x≤2 .

Sketch the graph of a function f that is continuous on [0;4] and has the following properties: Absolute minimum at 0, absolute maximum at 4, local maximum at 2, and local minimum at 3.

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 and minimum values of $f ( x ) = \frac { x } { x ^ { 2 } + 1 } + 1$ on the interval $[ 0,2 \pi ]$ .

How many points of inflection does the function $f ( x ) = x ^ { 8 } - x ^ { 2 }$ have?

Find the x-coordinate of the point on the parabola $y = \sqrt { x }$ nearest to the point (4, 0).

Find the critical number(s) (if any) and test them to determine if $f ( x ) = \left| 4 - x ^ { 2 } \right|$ has a local maximum or local minimum or neither at each critical number.

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { \cos x - 1 } { x ^ { 2 } }$ .

Is it true that $\int \ln x d x = x \ln x - x + C$ ? Explain why or why not.

Find all critical numbers for the function $f ( x ) = x - \cos x \text { for } 0 < x < 2 \pi$ .

An open box is made from a 8 inch $\times$ 8 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. What size squares should be cut out to create a box with maximum volume?

Find all critical numbers for the function $f ( x ) = ( x + 3 ) ^ { 4 } ( x - 2 ) ^ { 3 }$ .

Find the absolute maximum of the function $f ( x ) = x ^ { 3 } - x ^ { 2 } - x$ on the interval $- 10 \leq x \leq 2$ .