Deck 4: Applications of Differentiation

Question 1

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.

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

Answer 1

B

Question 2

If $f ^ { \prime \prime } ( x ) = 12 x ^ { 2 } - 2 , f ^ { \prime } ( - 1 ) = - 4$ , and $f ( 0 ) = 2$ , find $f ( 1 )$ .

A)1
B)2
C)3
D)4
E)5
F)6
G)7
H)8

Answer 2

8

Question 3

A direction field for a function f is given below. Use this to sketch the antiderivative F that satisfies .

Answer 3

Question 4

Draw a direction field for the function . Use this direction field to sketch several members of the family of antiderivatives.

Question 5

(a) Draw a direction field for the function . Use this direction field to sketch several members of the family of antiderivatives.(b) Compute the general antiderivative for explicitly and sketch several specific antiderivatives. Compare these results with your sketches from part (a).

Question 6

Find the most general antiderivative of the function $f ( x ) = \frac { 2 + x ^ { 3 } } { x ^ { 3 } }$ .

A) $\frac { 2 + 3 x ^ { 2 } } { 3 x ^ { 2 } } + C$

B) $\frac { 2 x + \frac { 1 } { 4 } x ^ { 4 } } { x ^ { 4 } } + C$
C) $- \frac { 1 } { x ^ { 2 } } + 1 + C$
D) $- \frac { 1 } { x ^ { 2 } } + C$
E) $\frac { 1 } { x ^ { 2 } } + x + C$
F)
$\frac { 1 } { x ^ { 2 } } + 1 + C$

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

H) $\frac { 2 } { x ^ { 2 } } + C$

Question 7

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

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 8

Given $f ^ { \prime \prime } ( x ) = 2 , f ^ { \prime } ( 0 ) = 1$ , and $f ( 0 ) = \frac { 3 } { 2 }$ , find $f ( 1 )$ .

A) $\frac { 10 } { 3 }$

B) $\frac { 17 } { 3 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 3 } { 10 }$
E) $\frac { 17 } { 2 }$

F) $\frac { 13 } { 3 }$

G) $\frac { 23 } { 3 }$

H) $\frac { 21 } { 2 }$

Question 9

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

A) $2 \sqrt { x } + 2 \sqrt [ 3 ] { x ^ { 2 } } + C$

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

G) $2 \sqrt { x } + \sqrt [ 3 ] { x ^ { 2 } } + C$

H) $\sqrt { x } + 2 \sqrt [ 3 ] { x ^ { 2 } } + C$

Question 10

Find the most general antiderivative of $\sec ^ { 2 } x + \sin x$ .

A) $\frac { 1 } { 3 } \sec ^ { 3 } x + \sin x + C$

B) $\tan ^ { 2 } x - \sin x + C$
C) $\tan x + \cos x + C$
D) $\tan x - \cos x + C$
E) $\tan ^ { 2 } x + \sin x + C$
F) $\frac { 1 } { 3 } \sec ^ { 3 } x + \frac { 1 } { 2 } \sin ^ { 2 } x + C$

G) $\frac { 1 } { 3 } \sec ^ { 3 } x - \sin x + C$

H) $\frac { 1 } { 3 } \sec ^ { 3 } x - \frac { 1 } { 2 } \sin ^ { 2 } x + C$

Question 11

Find the most general antiderivative of the function $f ( x ) = \frac { 2 } { x ^ { 3 } }$ .

A) $\frac { 2 } { 3 x ^ { 2 } } + C$

B) $\frac { 1 } { x ^ { 4 } } + C$
C) $- \frac { 1 } { x ^ { 2 } } + C$
D) $\frac { 2 } { x ^ { 4 } } + C$
E) $\frac { 1 } { x ^ { 2 } } + C$
F) $- \frac { 2 } { x ^ { 4 } } + C$

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

H) $2 x ^ { 4 } + C$

Question 12

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

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 13

Find the most general antiderivative of the function $f ( x ) = \frac { 2 e ^ { x } e ^ { 2 x } } { e ^ { x } }$ .

A) $2 x + e ^ { x } + C$

B) $2 x - e ^ { x } + C$
C) $2 - e ^ { x } + C$
D) $\frac { 2 e ^ { x } - e ^ { 2 x } } { e ^ { x } } + C$
E) $2 x - 2 e ^ { x } + C$
F) $\frac { 2 e ^ { x } - \frac { 1 } { 2 } e ^ { 2 x } } { e ^ { x } } + C$

G) $2 + x e ^ { x - 1 } + C$

H) $2 x + \frac { 1 } { 1 + x } e ^ { x + 1 } + C$

Question 14

Find the most general antiderivative of the function.(a) (b) (c) (d)

Question 15

(a) Draw a direction field for the function . Use this direction field to sketch several members of the family of antiderivatives.(b) Compute the general antiderivative for explicitly and sketch several specific antiderivatives. Compare these results with your sketches from part (a).

Question 16

Find the most general antiderivative of the function.(a) (b) (c) (d)

Question 17

Find the most general antiderivative of the function $f ( x ) = 9 x ^ { 2 }$ .

A) $3 x ^ { 3 } + C$

B) $2 x + C$
C) $\frac { x } { 2 } + C$
D) $2 x ^ { 2 } + C$
E) $\frac { x ^ { 3 } } { 3 } + C$
F) $3 x + C$

G) $\frac { x } { 3 } + C$

H) $3 x ^ { 2 } + C$

Question 18

Find the most general antiderivative of the function.(a) (b) (c) (d)

Question 19

Find the most general antiderivative of the function $f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + 5$ .

A) $x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } + 5 x + C$

B) $\frac { 1 } { 2 } x ^ { 4 } - x ^ { 3 } + 5 x + C$
C) $2 x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } + 5 x + C$
D) $\frac { 1 } { 2 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } + C$
E) $\frac { 1 } { 2 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } + 5 x + C$
F) $\frac { 1 } { 4 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } + 5 x + C$

G) $\frac { 1 } { 2 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 2 } + 5 x + C$

H) $6 x ^ { 2 } - 2 x + C$

Question 20

A direction field for a function f is given below. Use this to sketch the antiderivative F that satisfies .

Question 21

A car is traveling at 50 miles per hour when the brakes are fully applied, producing a constant deceleration of 22 feet per second . What is the distance covered before the car comes to a stop?

Question 22

A ball is thrown directly upward at a speed of 64 feet per second from a cliff 80 feet above the ground.(a) Find expressions for the velocity and height of the ball t seconds after it was released.(b) At what time does the ball reach its highest point? How high above the ground at the base of the cliff does it reach?
(c) When does the ball strike the ground at the base of the cliff? What is its velocity at that instant?

Question 23

Is it true that ? Explain why or why not.

Question 24

Is it true that ? Explain why or why not.

Question 25

An astronaut stands on a platform 3 meters above the moon's surface and throws a rock directly upward with an initial velocity of 32 m/s. Given that the acceleration due to gravity on the moon's surface is 1.6 m/s , how high above the surface of the moon will the rock travel?

Question 26

A custard pie is thrown vertically up from the ground with velocity 48 ft/s. Find the greatest height that it attains.

Question 27

Find f(x) if .

Question 28

If , find the value of .

Question 29

A stone falls from a cliff and reaches the ground at a speed 180 miles per hour. What is the height of the cliff above the ground? (Assume there is no air resistance.)

Question 30

Is it true that ? Explain why or why not.

Question 31

If Newton's method is used to solve $2 x ^ { 3 } + 2 x + 1 = 0$ with first approximation $x _ { 1 } = - 1$ , what is the second approximation, $x _ { 2 }$ ?

A)-0.500
B)-0.525
C)-0.550
D)-0.575
E)-0.600
F)-0.625
G)-0.650
H)-0.675

Question 32

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

Question 33

Find f(x) if .

Question 34

Is it possible for the velocity of an object to be zero at the same time that its acceleration is not zero? Explain.

Question 35

Use Newton's method to find the root of $6 x ^ { 3 } - x ^ { 2 } - 19 x + 6 = 0$ that lies between 0 and 1.

A)0.316
B)0.333
C)0.158
D)0.167
E)0.474
F)0.500
G)0.079
H)0.084

Question 36

Use Newton's method with the initial approximation $x _ { 1 } = 2$ to find $x _ { 2 }$ , the second approximation to a root of the equation $x ^ { 5 } - 34 = 0$ .

A) $\frac { 79 } { 40 }$

B) $\frac { 81 } { 40 }$
C) $\frac { 77 } { 40 }$
D) $\frac { 161 } { 80 }$
E) $\frac { 83 } { 40 }$
F) $\frac { 157 } { 80 }$

G)
$\frac { 163 } { 80 }$

H) $\frac { 159 } { 80 }$

Question 37

Find the position function s (t) given acceleration if and .

Question 38

Is it true that ? Explain why or why not.

Question 39

On the surface of the moon, a ball is thrown directly upward at a speed of 64 feet per second from a cliff 80 feet above the ground. The acceleration due to lunar gravity is 5:2 ft/s .(a) Find expressions for the velocity and height of the ball t seconds after it was released.(b) At what time does the ball reach its highest point? How high above the ground at the base of the cliff does it reach?
(c) When does the ball strike the ground at the base of the cliff? What is its velocity at that instant?

Question 40

A ball is dropped from a tower. When it strikes the ground it is traveling downward at a rate of 60 feet per second. How tall is the tower?

Question 41

Use Newton's method to approximate the root of that lies between and .

Question 42

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

Question 43

Sketch the graph of on the interval . Suppose that Newton's method is used to approximate the positive root of f with initial approximation .(a) On your sketch, draw the tangent lines that you would use to find and , and estimate the numerical values of and .(b) To approximate the negative root of f, use as the starting approximation. As before, draw the tangent lines that you would use to find and , and estimate the numerical values of and .(c) Suppose that you had used as the starting point for approximating the negative root. Discuss what would happen.

Question 44

(a) Explain why Newton's Method is unable to find a root for if or .(b) Using ,use Newton's Method to approximate a root of this equation to four decimal places.

Question 45

Find, correct to six decimal places, the root of .

Question 46

Sketch the graph of . Clearly the only x-intercept is zero. However, Newton's method fails to converge here. Explain this failure.

Question 47

Given , use Newton's method to find the iterative formula for .

Question 48

Given , use Newton's method to find the iterative formula for .

Question 49

Use Newton's Method to approximate all real roots of .

Question 50

If Newton's method is used to find the cube root of a number a with first approximation , find an expression for .

Question 51

Given , use Newton's method to find the iterative formula for .

Question 52

Sketch the graph of on . For each initial approximation given below, determine graphically what happens if Newton's Method is used to approximate the roots of .(a) (b) (c) (d) (e)

Question 53

Use Newton's method to approximate the root of $f ( x ) = x ^ { 3 } + 4 x + 2$ that lies between $x = - 1$ and $x = 0$ .

A)0.473
B)0.372
C)0.563
D)1
E)-0.473
F)-0.372
G)-0.563
H)None of these

Question 54

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

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 55

There are 50 apple trees in an orchard. Each tree produces 100 lbs of apples. For each additional tree planted in the orchard, the output per tree drops by 1 lb. How many trees should be added to the existing orchard in order to maximize the total output of trees?

A)15
E)5
B)10
F)20
C)25
G)40
D)0
H)None of these

Question 56

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

A) $x _ { n + 1 } = x _ { n } + \frac { 1 } { 2 x _ { n } }$

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


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

H)None of these

Question 57

Use Newton's method to find correct to four decimal places.

Question 58

Find two positive numbers whose sum is 12 and the product of one number and the square of the other number is a maximum.

A)12, 0
E)4, 8
B)6, 6
F)2, 10
C)1, 11
G)3, 9
D)5, 7
H)None of these

Question 59

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

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 60

Use Newton's method to approximate the root of that lies between and .

Question 61

A rancher wishes to fence in a rectangular corral enclosing 1300 square yards and must divide it in half with a fence down the middle. If the perimeter fence costs $5 per yard and the fence down the middle costs $3 per yard, determine the dimensions of the corral so that the fencing cost will be as small as possible.

A) $10 \sqrt { 10 } \times 13 \sqrt { 10 }$

B) $13 \sqrt { 10 } \times 13 \sqrt { 10 }$
C) $10 \sqrt { 13 } \times 13 \sqrt { 13 }$
D) $10 \times 10$
E) $10 \sqrt { 10 } \times 10 \sqrt { 10 }$

F) $10 \sqrt { 13 } \times 10 \sqrt { 13 }$ .

G) $13 \times 13$

Question 62

A revenue function is given by $R ( x ) = 6000 x - 0.2 x ^ { 3 }$ . where x is the number of units sold. Find the number of units x that produces maximum revenue.

A)200
B)300
C)1000
D)100
E)500
F)3000
G)400
H)Does not exist

Question 63

A farmer intends to fence o a rectangular pen for his pig Wilbur, using the barn as one of the sides. If the enclosed area is to be 50 square feet, what is smallest amount of fence needed, in feet?

A)30
B)40
C)20
D)10
E)50
F)8
G) $20 \sqrt { 2 }$
H)45

Question 64

A plastic right cylinder with closed ends is to hold V cubic feet. If there is no waste in construction, find the ratio between the height and diameter that results in the minimum use of materials.

A) $1 : \pi$

B). $1 : \frac { \pi } { 2 }$
C) $1 : 2 \pi$
D) $1 : 2$
E) $2 : \pi$
F) $2 : \frac { \pi } { 3 }$

G) $1 : \frac { \pi } { 3 }$

H) $1 : 3$

Question 65

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

Question 66

There are 50 apple trees in an orchard. Each tree produces 100 lbs of apples. For each additional tree planted in the orchard, the output per tree drops by 2 lb. How many trees should be added to the existing orchard in order to maximize the total output of trees?

A)15
E)5
B)10
F)20
C)25
G)40
D)0
H)None of these

Question 67

Farmer Brown wants to fence in a rectangular plot in a large field, using a rock wall which is already there as the north boundary. The fencing for the east and west sides of the plot will cost $3/yard, but she needs to use special fencing which will cost $5 / yard on the south side of the plot. If the area of the plot is to be 600 square yards, find the dimensions for the plot which will minimize the cost of the fencing. Dimensions below are listed east by south.

A) $10 \sqrt { 5 }$ by $12 \sqrt { 5 }$ yards
B) $12 \sqrt { 5 }$ by $10 \sqrt { 5 }$ yards
C)10 by 60 yards
D)60 by 10 yards
E) $8 \sqrt { 5 }$ by $15 \sqrt { 5 }$ yards
F) $15 \sqrt { 5 }$ by $8 \sqrt { 5 }$ yards
G)15 by 40 yards
H)40 by 15 yards

Question 68

A revenue function is given by $R ( x ) = 6000 x - 0.4 x ^ { 3 }$ . where x is the number of units sold. Find the marginal revenue when $x = 50$ .

A)4000
B)300
C)250,000
D)5000
E)500
F)3000
G)40,000
H)Does not exist

Question 69

Suppose that a baseball is tossed straight up and that its height as a function of time (in seconds) is given by the formula . What is the maximum height of the ball?

Question 70

Suppose that C(x) is the number of dollars in the total cost of producing x tables (x > 6) and $C ( x ) = 25 + 4 x + 18 x ^ { - 1 }$ . Find the marginal cost when $x = 15$ .

A)$3.72
B)$3.75
C)$3.79
D)$3.82
E)$3.89
F)$3.92
G)$3.97
H)$4.00

Question 71

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?

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 72

A company has cost function $C ( x ) = 1000 - 10 x + x ^ { 2 }$ . Find the average cost of producing 100 units.

A)150
B)200
C)210
D)120
E)250
F)90
G)180
H)100

Question 73

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.

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

Question 74

A company has a cost function $C ( x ) = 1000 - 10 x + x ^ { 2 }$ . Find the marginal cost of producing 100 units.

A)210
B)120
C)190
D)150
E)250
F)200
G)100
H)180

Question 75

A farmer has 20 feet of fence, and he wishes to make from it a rectangular pen for his pig Wilbur, using a barn as one of the sides. In square feet, what is the maximum area possible for this pen?

A)64
E)56
B)75
F)25
C)60
G)40
D)32
H)50

Question 76

A company has a cost function $C ( x ) = 200 - 50 x + x ^ { 2 }$ and demand function $p ( x ) = 50 - x$ . How many units should it make to maximize its profit?

A)15
B)25
C)8
D)5
E)30
F)10
G)20
H)35

Question 77

Find the shortest distance from the point $( 4,0 )$ to a point on the parabola $y ^ { 2 } = 2 x$ .

A)1
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D)2
E) $\sqrt { 5 }$
F) $\sqrt { 6 }$
G) $\sqrt { 7 }$
H) $2 \sqrt { 2 }$

Question 78

Find the x-coordinate of the point on the graph of $y = \frac { 1 } { x ^ { 2 } + 3 }$ that has the maximum slope.

A)-2
B)-1
C)3
D)0
E)2
F)1
G)-3
H)None of these

Question 79

Find the point on the line $y = 2 x - 3$ that is nearest to the origin.

A) $( 0.5 , - 2 )$

B) $( 7.5 , - 1.5 )$

C) $( 0.875 , - 1.25 )$
D) $( 1 , - 0.5 )$
E) $( 1.1 , - 0.8 )$
F) $( 1.2 , - 0.6 )$
G) $( 1.25 , - 0.5 )$

H) $( 1.5,0 )$

Question 80

A square is to be cut from each corner of a piece of paper which is 8 cm by 10 cm, and the sides are to be folded up to create an open box. What should the side of the square be for maximum volume? (State your answer correct to two decimal places.)

A)1.35 cm
E)1.52 cm
B)1.39 cm
F)1.55 cm
C)1.41 cm
G)1.60 cm
D)1.47 cm
H)1.62 cm