Question 1

Find the interval on which $f ( x ) = x + \cos x , 0 \leq x \leq 2 \pi$ 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 2

A water tank has the shape of an inverted circular cone with base radius 6 feet and height of 12 feet. The tank is full of water. If the water level is falling at the rate of $2 \text { inches } / \mathrm { m }$ , how fast is the tank losing water when the water is 8 feet deep?

Accepted Answer

The answer of A water tank has the shape of...

Question 3

Suppose that a baseball is tossed straight up and that its height as a function of time (in seconds) is given by the formula $h ( t ) = 128 t - 16 t ^ { 2 } \mathrm { ft }$ . What is the maximum height of the ball?

Accepted Answer

The answer of Suppose that a baseball is tossed straight...

Question 4

Find the most general antiderivative of the function $f ( x ) = \frac { 1 + 3 x } { x \sqrt { x } }$ .

Accepted Answer

A)  $x ^ { \frac { - 1 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } + C$ 
B)  $- 2 x ^ { \frac { - 1 } { 2 } } - 6 x ^ { \frac { 1 } { 2 } } + C$ 
C)  $- 2 x ^ { \frac { - 1 } { 2 } } + x ^ { \frac { 1 } { 2 } } + C$ 
D)  $- 2 x ^ { \frac { - 1 } { 2 } } + 3 x ^ { \frac { 1 } { 2 } } + C$ 
E)  $- 2 x ^ { \frac { - 1 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } + C$ 
F) $3 \sqrt { x } + C$

G) $2 x ^ { \frac { - 1 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } + C$

H) $x + 6 \sqrt { x } + C$ 
A)  $x ^ { \frac { - 1 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } + C$ 
B)  $- 2 x ^ { \frac { - 1 } { 2 } } - 6 x ^ { \frac { 1 } { 2 } } + C$ 
C)  $- 2 x ^ { \frac { - 1 } { 2 } } + x ^ { \frac { 1 } { 2 } } + C$ 
D)  $- 2 x ^ { \frac { - 1 } { 2 } } + 3 x ^ { \frac { 1 } { 2 } } + C$ 
E)  $- 2 x ^ { \frac { - 1 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } + C$ 
F) $3 \sqrt { x } + C$

G) $2 x ^ { \frac { - 1 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } + C$

H) $x + 6 \sqrt { x } + C$

Question 5

The graph of the first derivative $f ^ { \prime }$ of a function f is given below.   (a) On what intervals is f increasing? Explain.(b) At what values of x does f have a local maximum and local minimum? Explain.(c) On what intervals is f concave up? concave down? Explain
(d) Where are the coordinates of the inflection points of f? Explain

Accepted Answer

(a) @#IMG-DLM& and @#IMG-DLM& , where @#IMG-DLM& (b) f(x) has a local

Question 6

Find the largest interval on which the function $f ( \theta ) = \cos ^ { 2 } \theta$ is concave upward.

Accepted Answer

The answer of Find the largest interval on which the...

Question 7

The cost to produce x units of a certain product is given by $C ( x ) = 10,000 + 8 x + \frac { 1 } { 16 } x ^ { 2 }$ . Find the value of x that gives the minimum average cost.

Accepted Answer

A) 400 
B) 25 
C) 160,000 
D) 4000 
E) 500
F)64
G)40,000
H)Does not exist 
A) 400 
B) 25 
C) 160,000 
D) 4000 
E) 500
F)64
G)40,000
H)Does not exist

Question 8

Find all critical numbers for the function $f ( x ) = \frac { 1 } { x ^ { 2 } - 9 }$ .

Accepted Answer

A) 0 
B) -3 
C) 0, -3 
D) No Critical Number 
E) 3
F)3, -3
G)0, 3, -3
H)None of the above 
A) 0 
B) -3 
C) 0, -3 
D) No Critical Number 
E) 3
F)3, -3
G)0, 3, -3
H)None of the above

Question 9

A particle moves along the circle $x ^ { 2 } + y ^ { 2 } = 25$ . If x decreases at a rate 2 units per minute, at what rate is y changing when $x = 3$ ?

Accepted Answer

A)  $2 \text { unit } / \min$ 
B)  $\frac { 3 } { 2 } \text { unit } / \mathrm { min }$ 
C)  $3 \text { unit } / \mathrm { min }$ 
D)  $\frac { 1 } { 4 } \text { unit } / \mathrm { min }$ 
E)  $- 2 \text { unit } / \min$
F) $- \frac { 3 } { 2 } \text { unit } / \mathrm { min }$

G) $- 3 \text { unit } / \min$

H) $- \frac { 1 } { 4 } \text { unit } / \mathrm { min }$ 
A)  $2 \text { unit } / \min$ 
B)  $\frac { 3 } { 2 } \text { unit } / \mathrm { min }$ 
C)  $3 \text { unit } / \mathrm { min }$ 
D)  $\frac { 1 } { 4 } \text { unit } / \mathrm { min }$ 
E)  $- 2 \text { unit } / \min$
F) $- \frac { 3 } { 2 } \text { unit } / \mathrm { min }$

G) $- 3 \text { unit } / \min$

H) $- \frac { 1 } { 4 } \text { unit } / \mathrm { min }$

Question 10

Find the critical number(s) (if any) and test them to determine if $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 9$ has a local maximum or local minimum or neither at each critical number.

Accepted Answer

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

Question 11

What is the length of the largest interval on which the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ is decreasing?

Accepted Answer

A)  $3 \sqrt { 2 }$ 
B) 1 
C) 4 
D)  $2 \sqrt { 2 }$ 
E) 3
F) 
$2 \sqrt { 3 }$ 
G) $\sqrt { 3 }$  
H)2 
A)  $3 \sqrt { 2 }$ 
B) 1 
C) 4 
D)  $2 \sqrt { 2 }$ 
E) 3
F) 
$2 \sqrt { 3 }$ 
G) $\sqrt { 3 }$  
H)2

Question 12

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln \cos ( 3 x ) } { \ln \cos ( 5 x ) }$ .

Accepted Answer

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

Question 13

(a) Draw a direction field for the function $f ( x ) = 9 x ^ { 2 }$ . Use this direction field to sketch several members of the family of antiderivatives.(b) Compute the general antiderivative for $f ( x ) = 9 x ^ { 2 }$ explicitly and sketch several specific antiderivatives. Compare these results with your sketches from part (a).

Accepted Answer

The answer of (a) Draw a direction field for the...

Question 14

How many points of inflection does the function $f ( x ) = 3 x ^ { 5 } - 10 x ^ { 3 } + 5$ have?

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Find the position function s (t) given acceleration $a ( t ) = 3 t$ if $v ( 2 ) = 0$ and $s ( 2 ) = 1$ .

Accepted Answer

The answer of Find the position function s (t) given...

Question 17

A rectangular storage container with an open top is to have a volume of 10 m $3$ : The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the dimensions of the container needed to minimize the cost.

Accepted Answer

The answer of A rectangular storage container with an open...

Question 18

It is desired to make a rectangular pen with diameter equal to 40 feet. It is not necessary, however, to achieve maximum area. In fact, any area that is greater than or equal to 75% of the maximum area will be satisfactory. Under these conditions, find the maximum acceptable length of the longer side.

Accepted Answer

The answer of It is desired to make a rectangular...

Question 19

Suppose that f is a continuous function and $f ( 0 ) = 0 , f ( 2 ) = f ( - 2 ) = 1 , f ^ { \prime } ( 0 ) = 0 , f ^ { \prime } ( x ) > 0$ for $x > 0 , f ^ { \prime } ( x )  0$ for $| x |  2$ , and $\lim _ { x \rightarrow \infty } f ( x ) = \lim _ { x \rightarrow - \infty } f ( x ) = 2$ . Sketch the graph of $y = f ( x )$ .

Accepted Answer

The answer of Suppose that f is a continuous function...

Question 20

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

Accepted Answer

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

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

A water tank has the shape of an inverted circular cone with base radius 6 feet and height of 12 feet. The tank is full of water. If the water level is falling at the rate of $2 \text { inches } / \mathrm { m }$ , how fast is the tank losing water when the water is 8 feet deep?

Suppose that a baseball is tossed straight up and that its height as a function of time (in seconds) is given by the formula $h ( t ) = 128 t - 16 t ^ { 2 } \mathrm { ft }$ . What is the maximum height of the ball?

Find the most general antiderivative of the function $f ( x ) = \frac { 1 + 3 x } { x \sqrt { x } }$ .

Find the largest interval on which the function $f ( \theta ) = \cos ^ { 2 } \theta$ is concave upward.

The cost to produce x units of a certain product is given by $C ( x ) = 10,000 + 8 x + \frac { 1 } { 16 } x ^ { 2 }$ . Find the value of x that gives the minimum average cost.

Find all critical numbers for the function $f ( x ) = \frac { 1 } { x ^ { 2 } - 9 }$ .

A particle moves along the circle $x ^ { 2 } + y ^ { 2 } = 25$ . If x decreases at a rate 2 units per minute, at what rate is y changing when $x = 3$ ?

Find the critical number(s) (if any) and test them to determine if $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 9$ has a local maximum or local minimum or neither at each critical number.

What is the length of the largest interval on which the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ is decreasing?

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln \cos ( 3 x ) } { \ln \cos ( 5 x ) }$ .

How many points of inflection does the function $f ( x ) = 3 x ^ { 5 } - 10 x ^ { 3 } + 5$ have?

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

Find the position function s (t) given acceleration $a ( t ) = 3 t$ if $v ( 2 ) = 0$ and $s ( 2 ) = 1$ .

A rectangular storage container with an open top is to have a volume of 10 m $3$ : The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the dimensions of the container needed to minimize the cost.

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

Functions and Models

Limits and Derivatives

Differentiation Rules

Integrals

Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

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Partial Derivatives

Multiple Integrals

Vector Calculus

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Exam 4: Applications of Differentiation

Find the interval on which f(x)=x+cos⁡x,0≤x≤2πf ( x ) = x + \cos x , 0 \leq x \leq 2 \pif(x)=x+cosx,0≤x≤2π is increasing.

A water tank has the shape of an inverted circular cone with base radius 6 feet and height of 12 feet. The tank is full of water. If the water level is falling at the rate of 2 inches /m2 \text { inches } / \mathrm { m }2 inches /m , how fast is the tank losing water when the water is 8 feet deep?

Suppose that a baseball is tossed straight up and that its height as a function of time (in seconds) is given by the formula h(t)=128t−16t2fth ( t ) = 128 t - 16 t ^ { 2 } \mathrm { ft }h(t)=128t−16t2ft . What is the maximum height of the ball?

Find the most general antiderivative of the function f(x)=1+3xxxf ( x ) = \frac { 1 + 3 x } { x \sqrt { x } }f(x)=xx​1+3x​ .

Find the largest interval on which the function f(θ)=cos⁡2θf ( \theta ) = \cos ^ { 2 } \thetaf(θ)=cos2θ is concave upward.

The cost to produce x units of a certain product is given by C(x)=10,000+8x+116x2C ( x ) = 10,000 + 8 x + \frac { 1 } { 16 } x ^ { 2 }C(x)=10,000+8x+161​x2 . Find the value of x that gives the minimum average cost.

Find all critical numbers for the function f(x)=1x2−9f ( x ) = \frac { 1 } { x ^ { 2 } - 9 }f(x)=x2−91​ .

A particle moves along the circle x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25 . If x decreases at a rate 2 units per minute, at what rate is y changing when x=3x = 3x=3 ?

Find the critical number(s) (if any) and test them to determine if f(x)=3x4−8x3+9f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 9f(x)=3x4−8x3+9 has a local maximum or local minimum or neither at each critical number.

What is the length of the largest interval on which the function f(x)=x3−3x2−9xf ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 xf(x)=x3−3x2−9x is decreasing?

Find the value of the limit lim⁡x→∞ln⁡cos⁡(3x)ln⁡cos⁡(5x)\lim _ { x \rightarrow \infty } \frac { \ln \cos ( 3 x ) } { \ln \cos ( 5 x ) }limx→∞​lncos(5x)lncos(3x)​ .

How many points of inflection does the function f(x)=3x5−10x3+5f ( x ) = 3 x ^ { 5 } - 10 x ^ { 3 } + 5f(x)=3x5−10x3+5 have?

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​ .

Find the position function s (t) given acceleration a(t)=3ta ( t ) = 3 ta(t)=3t if v(2)=0v ( 2 ) = 0v(2)=0 and s(2)=1s ( 2 ) = 1s(2)=1 .

A rectangular storage container with an open top is to have a volume of 10 m 333 : The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the dimensions of the container needed to minimize the cost.

Find the value of the limit lim⁡x→∞(ln⁡x)2x\lim _ { x \rightarrow \infty } \frac { ( \ln x ) ^ { 2 } } { x }limx→∞​x(lnx)2​ .

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 $f ( x ) = x + \cos x , 0 \leq x \leq 2 \pi$ is increasing.

A water tank has the shape of an inverted circular cone with base radius 6 feet and height of 12 feet. The tank is full of water. If the water level is falling at the rate of $2 \text { inches } / \mathrm { m }$ , how fast is the tank losing water when the water is 8 feet deep?

Suppose that a baseball is tossed straight up and that its height as a function of time (in seconds) is given by the formula $h ( t ) = 128 t - 16 t ^ { 2 } \mathrm { ft }$ . What is the maximum height of the ball?

Find the most general antiderivative of the function $f ( x ) = \frac { 1 + 3 x } { x \sqrt { x } }$ .

Find the largest interval on which the function $f ( \theta ) = \cos ^ { 2 } \theta$ is concave upward.

The cost to produce x units of a certain product is given by $C ( x ) = 10,000 + 8 x + \frac { 1 } { 16 } x ^ { 2 }$ . Find the value of x that gives the minimum average cost.

Find all critical numbers for the function $f ( x ) = \frac { 1 } { x ^ { 2 } - 9 }$ .

A particle moves along the circle $x ^ { 2 } + y ^ { 2 } = 25$ . If x decreases at a rate 2 units per minute, at what rate is y changing when $x = 3$ ?

Find the critical number(s) (if any) and test them to determine if $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 9$ has a local maximum or local minimum or neither at each critical number.

What is the length of the largest interval on which the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x$ is decreasing?

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln \cos ( 3 x ) } { \ln \cos ( 5 x ) }$ .

How many points of inflection does the function $f ( x ) = 3 x ^ { 5 } - 10 x ^ { 3 } + 5$ have?

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

Find the position function s (t) given acceleration $a ( t ) = 3 t$ if $v ( 2 ) = 0$ and $s ( 2 ) = 1$ .

A rectangular storage container with an open top is to have a volume of 10 m $3$ : The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the dimensions of the container needed to minimize the cost.

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