Deck 10: Further Applications of the Derivative

Question 1

Volume. A rectangular box with a square base is to be formed from a square piece of metal with 30-inch sides. If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box, the volume of the box is $V = ( 30 - 2 x ) ^ { 2 } x$ What value of x will maximize the volume of the box?

A)15
B)3
C)5
D)7
E)4

Answer 1

5

Question 2

Find the point on the graph of $f ( x ) = x ^ { 2 }$ that is closest to the point (6, 0.5). Round your answer to two decimal places.

A)(1.44, 2.07)
B)(1.82, 3.31)
C)(2.29, 5.24)
D)(1.26, 1.59)
E)(1.14, 1.30)

Answer 2

(1.44, 2.07)

Question 3

Find the dimensions of the rectangle of maximum area bounded by the x-axis and y-axis and the graph of $y = \frac { 4 - x } { 2 }$ .

A)length 1.5; width 1.25
B)length 2; width 1
C)length 0.5; width 1.75
D)length 1; width 1.5
E)none of the above

Answer 3

length 2; width 1

Question 4

If the total cost function for a product is $C ( x ) = 200 + 4 x + 0.09 x ^ { 2 }$ dollars. Find the minimum average cost.

A) $\$ 15.00$
B) $\$ 23.00$
C) $\$ 13.03$
D) $\$ 12.49$
E) $\$ 11.00$

Question 5

A firm can produce 100 units per week. If its total cost function is $C = 500 + 1400 x$ dollars, and its total revenue function is $R = 1500 x - x ^ { 2 }$ dollars, find the maximum profit.

A)$3886
B)$2000
C)$7007
D)$5503
E)$5836

Question 6

You are in a boat 2 miles from the nearest point on the coast. You are to go to point Q located 3 miles down the coast and 4 miles inland (see figure). You can row at a rate of 4 miles per hour and you can walk at a rate of 4 miles per hour. Toward what point on the coast should you row in order to reach point Q in the least time?

A)3 miles
B)8 miles
C)4 miles
D)1 mile
E)5 miles

Question 7

A firm has total revenue given by $R ( x ) = 300 x - 45.5 x ^ { 2 } - x ^ { 3 } \text { dollars }$ for x units of a product. Find the maximum revenue from sales of that product.

A)$600
B)$464
C)$303
D)$1100
E)$963

Question 8

If the total cost function for a product is $C ( x ) = 500 + 3 x + 0.08 x ^ { 2 }$ dollars, determine how many units x should be produced to minimize the average cost per unit?

A)18 units
B)500 units
C)55 units
D)79 units
E)88 units

Question 9

Find the length and width of a rectangle that has perimeter $32$ meters and a maximum area.

A)4, 12
B)1, 15
C)8, 8
D)9, 7
E)12, 4

Question 10

If the total revenue function for a blender is $R ( x ) = 35 x - 0.25 x ^ { 2 },$ determine how many units x must be sold to provide the maximum total revenue in dollars.

A)1225
B)3000
C)35
D)200
E)70

Question 11

A rectangular page is to contain $36$ square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

A) $8,8$
B)6, 6
C)4, 4
D)7, 7
E)5, 5

Question 12

Average costs. Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by $\bar { C } ( x ) = 5 x + \frac { 4500 } { x } , \quad x > 0$ , where x is the number of machines used. What is the minimum average cost?

A)$0
B)$30
C)$300
D)$150
E)$4505

Question 13

A rancher has 320 feet of fencing to enclose two adjacent rectangular corrals (see figure). What dimensions should be used so that the enclosed area will be a maximum?

A) $x = 40.00$ and $y = 53.33$
B) $x = 8.00$ and $y = 96.00$
C) $x = 16.00$ and $y = 106.67$
D) $x = 53.33$ and $y = 40.00$
E) $x = 24.00$ and $y = 64.00$

Question 14

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 24 feet. Round yours answers to two decimal places.

A)x = 6.72 feet and y = 3.36 feet
B)x = 3.36 feet and y = 7.68 feet
C)x = 2.24 feet and y = 9.12 feet
D)x = 5.72 feet and y = 4.65 feet
E)x = 7.72 feet and y = 2.08 feet

Question 15

A wooden beam has a rectangular cross section of height h and width w (see figure). The strength S of the beam is directly proportional to the width and the square of the height. What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 23 inches? Round your answers to two decimal places. [Hint: $S = k w h ^ { 2 },$ where $k > 0$ is the proportionality constant.]

A)w = 13.28 inches and h = 18.78 inches
B)w = 7.67 inches and h = 21.68 inches
C)w = 19.92 inches and h = 11.50 inches
D)w = 16.26 inches and h = 16.27 inches
E)w = 18.78 inches and h = 13.28 inches

Question 16

Average costs. Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by $\bar { C } ( x ) = 7 x + \frac { 1372 } { x } , \quad x > 0$ , where x is the number of machines used. How many machines give minimum average costs?

A)Using 14 machines gives the minimum average costs.
B)Using zero machines gives the minimum average costs.
C)Using 24 machines gives the minimum average costs.
D)Using 28 machines gives the minimum average costs.
E)Using 33 machines gives the minimum average costs.

Question 17

Minimum cost. From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $\$ 6$ per foot and the fence for the middle costs $\$ 4$ per foot. If each lot contains $2200$ square feet, find the dimensions of each lot that yield the minimum cost for the fence.

A)Dimensions are 61.86 ft for the side parallel to the divider and 35.56 ft for the other side.
B)Dimensions are 35.56 ft for the side parallel to the divider and 61.86 ft for the other side.
C)Dimensions are 46.90 ft for the side parallel to the divider and 46.90 ft for the other side.
D)Dimensions are 40.62 ft for the side parallel to the divider and 54.16 ft for the other side.
E)Dimensions are 54.16 ft for the side parallel to the divider and 40.62 ft for the other side.

Question 18

If the total revenue function for a blender is $R ( x ) = 40 x - 0.25 x ^ { 2 },$ find the maximum revenue.

A)$80
B)$1500
C)$40
D)$100
E)$1600

Question 19

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 400 square meters.

A)square base side $\frac { 20 \sqrt { 6 } } { 3 }$ ; height $\frac { 20 \sqrt { 6 } } { 3 }$
B)square base side $\frac { 20 \sqrt { 6 } } { 3 }$ ; height $\frac { 10 \sqrt { 6 } } { 3 }$
C)square base side $\frac { 10 \sqrt { 6 } } { 3 }$ ; height $\frac { 20 \sqrt { 6 } } { 3 }$
D)square base side $\frac { 10 \sqrt { 6 } } { 3 }$ ; height $\frac { 10 \sqrt { 6 } } { 3 }$
E)square base side $\frac { 200 \sqrt { 6 } } { 3 }$ ; height $\frac { 10 \sqrt { 6 } } { 3 }$

Question 20

A firm can produce 100 units per week. If its total cost function is $C = 600 + 1200 x$ dollars, and its total revenue function is $R = 1300 x - x ^ { 2 }$ dollars, how many units x should it produce to maximize its profit?

A)1250 units
B)650 units
C)93 units
D)50 units
E)100 units

Question 21

This problem contains a function and its graph, where $A = 20$ Use the graph to determine, as well as you can, the horizontal asymptote. Check your conclusion by using the function to determine the horizontal asymptote analytically. $$f ( x ) = \frac { 4 ( x - 3 ) } { x - 2 }$$

A) $y = - 3$
B) $y = 0$
C) $y = 4$
D) $y = - 1$
E) $y = - 20$

Question 22

Find any horizontal asymptotes for the given function. $y = \frac { 9 x - 9 } { x + 5 }$

A) $y = 1$
B) $y = 5$
C) $y = 0$
D) $y = 9$
E)no horizontal asymptotes

Question 23

Find any horizontal asymptotes for the given function. $y = \frac { 8 x ^ { 3 } } { 8 x ^ { 2 } + 6 }$

A) $y = 1$
B) $y = 8$
C) $y = 6$
D) $y = 0$
E)no horizontal asymptotes

Question 24

A power station is on one side of a river that is 0.5 mile wide, and a factory is 6.00 miles downstream on the other side of the river. It costs $\$$ 18 per foot to run overland power lines and $\$$ 25 per foot to run underwater power lines. Estimate the value of x that minimizes the cost.

A)4.00
B)0.52
C)5.00
D)0.00
E)1.52

Question 25

Analytically determine the location(s) of any vertical asymptote(s). $f ( x ) = \frac { 800 x + 6000 } { x ^ { 2 } - 38 x - 2867 }$

A) $x = 75.82$
B) $x = - 37.82$
C) $x = 75.82$ , $x = - 37.82$
D) $x = 0$
E)no vertical asymptotes

Question 26

Analytically determine the location(s) of any horizontal asymptote(s). $f ( x ) = \frac { 100 x + 6000 } { x ^ { 2 } - 17 x - 1373 }$

A) $y = 46.52$
B) $y = - 29.52$
C) $y = 46.52$ , $y = - 29.52$
D) $y = 0$
E)no horizontal asymptotes

Question 27

A travel agency will plan a tour for groups of size $13$ or larger. If the group contains exactly $13$ people, the price is $\$ 800$ per person. However, each person's price is reduced by $\$ 10$ for each additional person above the $13$ . If the travel agency incurs a price of $\$ 150$ per person for the tour, what size group will give the agency the maximum profit?

A)26
B)14
C)39
D)22
E)32

Question 28

Find the speed v, in miles per hour, that will minimize costs on a 125-mile delivery trip. The cost per hour for fuel is $C = \frac { v ^ { 2 } } { 800 }$ dollars, and the driver is paid $W = \$ 8$ dollars per hour. (Assume there are no costs other than wages and fuel.)

A)675 mph
B)125 mph
C)800 mph
D)80 mph
E)40 mph

Question 29

A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist, where $A = 51$ Confirm your results analytically. $$f ( x ) = \frac { 17 x ^ { 2 } } { ( x - 2 ) ^ { 2 } }$$

A) $x = 1$
B) $x = 2$
C) $x = 0$
D) $x = 5$
E)no vertical asymptotes

Question 30

This problem contains a function and its graph, where $\mathrm { A } = 10$ Use the graph to determine, as well as you can, the vertical asymptote. Check your conclusion by using the function to determine the vertical asymptote analytically. $$f ( x ) = \frac { 2 ( x - 3 ) } { x - 2 }$$

A) $x = - 5$
B) $x = 0$
C) $x = - 1$
D) $x = 2$
E)no vertical asymptote

Question 31

Find any horizontal asymptotes for the given function. $y = \frac { 18 x } { 9 - x ^ { 2 } }$

A) $y = 2$
B) $y = 18$
C) $y = 9$
D) $y = 0$
E)no horizontal asymptotes

Question 32

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist, where $$A = 15$$ Confirm your results analytically. $$f ( x ) = \frac { 5 x ^ { 2 } } { ( x - 2 ) ^ { 2 } }$$

A) $y = 2$
B) $y = 5$
C) $y = 1$
D) $y = 0$
E)no horizontal asymptotes

Question 33

Analytically determine the location of any vertical asymptotes. $f ( x ) = \frac { x - 80 } { x ^ { 2 } + 1800 }$

A) $x = 8.944272$
B) $x = 0.044444$
C) $x = 42.426407$
D) $x = - 42.426407$
E)no vertical asymptotes

Question 34

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically. $f ( x ) = \frac { 24 } { x + 2 }$

A) $y = 4$
B) $y = - 2$
C) $y = 0$
D) $y = 2$
E)no horizontal asymptotes

Question 35

Analytically determine the location(s) of any horizontal asymptote(s). $f ( x ) = \frac { x - 80 } { x ^ { 2 } + 1100 }$

A) $y = 0$
B) $y = 0.072727$
C) $y = 33.166248$
D) $y = - 33.166248$
E)no horizontal asymptotes

Question 36

A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist. Confirm your results analytically. $f ( x ) = \frac { 8 } { x + 2 }$

A) $x = 2$
B) $x = 8$
C) $x = 6$
D) $x = - 2$
E)no vertical asymptotes

Question 37

p is in dollars and q is the number of units. Find the elasticity of the demand function $5 p + 4 q = 152$ at the price $p = \$ 30$ .

A)-75.00
B)1.00
C)75.00
D)-1.25
E)1.25

Question 38

Find any vertical asymptotes for the given function. $y = \frac { 5 x - 5 } { x + 2 }$

A) $x = 1$
B) $x = 2$
C) $x = - 2$
D) $x = 0$
E)no vertical asymptotes

Question 39

Suppose the sales S (in billions of dollars per year) for Proctor & Gamble for the years 1999 through 2004 can be modeled by $S = 2.5931 t ^ { 2 } - 1.5682 t + 39.831,1999 \leq t \leq 2004$ where t represents the year. During which year were the sales increasing at the lowest rate?

A)Sales are increasing at the lowest rate in the year 2004.
B)Sales are increasing at the lowest rate in the year 1999.
C)Sales are increasing at the lowest rate in the year 2000.
D)Sales are increasing at the lowest rate in the year 2002.
E)Sales are increasing at the lowest rate in the year 2001.

Question 40

Find all vertical asymptotes for the given function. $y = \frac { 5 x } { 4 - x ^ { 2 } }$

A) $x = 0$
B) $x = \pm 2$
C) $x = 2$
D) $x = - 2$
E)no vertical asymptotes

Question 41

Find the limit: $\lim _ { x \rightarrow 2 ^ { + } } \frac { x - 9 } { x - 2 }$

A) $\infty$
B) $- \infty$
C)0
D)-1
E)1

Question 42

For the function $f ( x ) = \frac { - x } { \sqrt { 4 x ^ { 2 } - 8 } }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. $\begin{array}{lllllll}x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5}\\f(x)\end{array}$

A)-1.15
B)0.75
C)-0.5
D)1.93
E)does not exist

Question 43

Use analytic methods to find the limit as $x \rightarrow - \infty$ for the given function. $f ( x ) = \frac { 7000 x } { 3300 - 4 x }$

A) $- 1750$
B) $825$
C) $- 825$
D) $1750$
E)does not exist

Question 44

Sketch the graph of the relation $x ^ { 2 } y = 1$ using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)

Question 45

Match the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 2 }$ with one of the following graphs.

A)
B)
C)
D)
E)

Question 46

The cost C (in millions of dollars) for the federal government to seize p% of a type of illegal drug as it enters the country is modeled by $C = 584 / ( 100 - p )$ , for $0 \leq p \leq 100$ . Find the limit of C as $p \rightarrow 100 ^ { - }$ .

A)584
B) $100$
C) $- 100$
D) $\infty$
E) $0$

Question 47

Sketch the graph of the relation $x y ^ { 2 } = 4$ using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)

Question 48

For the function $f ( x ) = \frac { 4 x + 7 } { 3 x - 7 }$ , use a graphing utility to complete the table and estimate the limit as x approaches infinity. $\begin{array}{lllllll}x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5}\\f(x)\end{array}$

A)0.75
B)1.333333
C)2.333333
D)1.75
E)-0.25

Question 49

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 3 x - 3 } { - 7 x - 4 }$

A) $\frac { 3 } { 4 }$
B) $- \frac { 3 } { 7 }$
C)1
D)0
E)does not exist

Question 50

A business has a cost (in dollars) of $C = 0.8 x + 100$ for producing x units. What is the limit of $\bar { C }$ as x approaches infinity?

A) $\infty$
B)$0.80
C)$100.80
D)$100.00
E)$99.20

Question 51

Match the function $f ( x ) = \frac { 2 x ^ { 2 } } { x ^ { 2 } + 2 }$ with one of the following graphs.

A)
B)
C)
D)
E)

Question 52

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 5 x ^ { 2 } - 4 x - 14 } { 1 - 3 x - 6 x ^ { 2 } }$

A) $- \frac { 5 } { 6 }$
B)14
C)-14
D) $\frac { 5 } { 6 }$
E) $\frac { 4 } { 3 }$

Question 53

Sketch the graph of the function $f ( x ) = \frac { 2 + x } { 2 - x }$ using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)

Question 54

Sketch the graph of the function $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 4 }$ using any extrema, intercepts, symmetry, and asymptotes.

A)
B)
C)
D)
E)

Question 55

Find the limit: $\lim _ { x \rightarrow 0 ^ { - } } \left( x ^ { 1 } + \frac { 1 } { x } \right)$

A)0
B) $\infty$
C)1
D)-1
E) $- \infty$

Question 56

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 7 x + 4 } { 8 x ^ { 2 } + 8 }$

A) $\infty$
B)1
C)0
D) $\frac { 7 } { 8 }$
E) $\frac { 1 } { 2 }$

Question 57

Sketch the graph of the equation given below. Use intercepts, extrema, and asymptotes as sketching aids. $g ( x ) = \frac { 2 x ^ { 2 } - 6 } { ( x - 1 ) ^ { 2 } }$

A)
B)
C)
D)
E)

Question 58

Find the limit. $\lim _ { x \rightarrow -\infty } \frac { 4 x ^ { 2 } } { x + 3 }$

A) $0$
B) $- \frac { 4 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $\infty$
E) $- \infty$

Question 59

Use a table utility with x-values larger than 10,000 to investigate $\lim _ { x \rightarrow + \infty } f ( x )$ .What does the table indicate about $\lim _ { x \rightarrow + \infty } f ( x ) ?$ $f ( x ) = \frac { 16 x ^ { 3 } - 5 x } { 5 - 2 x ^ { 3 } }$

A) $- 8$
B) $8$
C) $16$
D)0
E)does not exist

Question 60

Use analytic methods to find the limit as $x \rightarrow + \infty$ for the given function. $f ( x ) = \frac { 3000 x } { 3900 - 15 x }$

A) $200$
B) $260$
C) $- 260$
D) $- 200$
E)does not exist

Question 61

Analyze and sketch a graph of the function $f ( x ) = \frac { x ^ { 4 } } { x ^ { 4 } + 1 }$ .

A)
B)
C)
D)
E)

Question 62

Sketch the graph of the function given below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. $y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } - 2 }$

A)
B)
C)
D)
E)

Question 63

Analyze and sketch a graph of the function $f ( x ) = \frac { x } { x ^ { 4 } + 1 }$ .

A)
B)
C)
D)
E)

Question 64

Compare dy and $\Delta y$ for $y = 3 x ^ { 2 } - 2$ at x = 0 with dx = -0.06. Give your answers to four decimal places.

A) $d y = - 0.0100 ; \quad \Delta y = 0.0106$
B) $d y = 0.0000 ; \quad \Delta y = 0.0108$
C) $d y = - 0.0300 , \quad \Delta y = 0.0107$
D) $d y = 0.0200 ; \quad \Delta y = 0.0105$
E) $d y = 0.0200 ; \quad \Delta y = 0.0107$

Question 65

Analyze and sketch a graph of the function $y = x \sqrt { 16 - x }$ .

A)
B)
C)
D)
E)

Question 66

Analyze and sketch a graph of the function $f ( x ) = \frac { x ^ { 2 } - 4 x + 12 } { x + 2 }$ .

A)
B)
C)
D)
E)none of the above

Question 67

Complete the table for the function $y = \frac { 1 } { x }$ . Let x = 4. $\begin{array} { l } d x=\Delta x & d y & \Delta y& \Delta y-d y & d y / \Delta y\\5.00000 \\2.50000 \\0.50000\end{array}$

A) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\5.00000 & - 0.3125 & - 0.13889 & 0.17361 & 2.24998 \\2.50000 & - 0.15625 & - 0.09615 & 0.0601 & 1.62507 \\0.50000 & - 0.06235 & - 0.02778 & 0.00347 & 1.12712\end{array}$
B) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\5.00000 & - 0.3125 & - 0.13889 & 0.17361 & 2.24998 \\2.50000 & - 0.15625 & 0.21385 & 0.0601 & 1.62928 \\0.50000 & - 0.03125 & - 0.02778 & 0.00347 & 1.12491\end{array}$
C) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\5.00000 & - 0.3125 & - 0.13889 & 0.17361 & 2.24998 \\2.50000 & - 0.15625 & - 0.09615 & 0.0601 & 1.62507 \\0.50000 & - 0.03125 & - 0.02778 & 0.00347 & 1.12491\end{array}$
D) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\5.00000 & - 0.2614 & - 0.13889 & 0.17361 & 2.24998 \\2.50000 & - 0.15625 & - 0.09615 & 0.0601 & 1.62507 \\0.50000 & - 0.06235 & - 0.02778 & 0.00347 & 1.12712\end{array}$
E) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\5.00000 & - 0.2614 & - 0.13889 & 0.17361 & 2.24998 \\2.50000 & - 0.15625 & 0.21385 & 0.0601 & 1.62928 \\0.50000 & - 0.06235 & - 0.02778 & 0.00347 & 1.12712\end{array}$

Question 68

The revenue R for a company selling x units is $R = 900 x - 0.3 x ^ { 2 }$ . Use differentials to approximate the change in revenue if sales increase from $x = 1000$ to $x = 1100$ units.

A)24,000 dollars
B)30,000 dollars
C)25,000 dollars
D)29,000 dollars
E)40,000 dollars

Question 69

Compare dy and $\Delta y$ for $y = 3 x ^ { 2 } - 3$ at x = 2 with $\Delta x =$ dx = -0.07. Give your answers to four decimal places.

A) $d y = - 0.8500 , \quad \Delta y = - 0.8255$
B) $d y = - 0.8400 ; \quad \Delta y = - 0.8253$
C) $d y = - 0.8700 , \quad \Delta y = - 0.8254$
D) $d y = - 0.8200 , \quad \Delta y = - 0.8256$
E) $d y = - 0.8200 , \quad \Delta y = - 0.8254$

Question 70

Analyze and sketch a graph of the function $y = 2 - x - x ^ { 3 }$ .

A)
B)
C)
D)
E)

Question 71

Analyze and sketch a graph of the function $y = x \sqrt { 1 - x ^ { 2 } }$ .

A)
B)
C)
D)
E)

Question 72

Sketch the graph of the function below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. $y = x ^ { 4 } - 4 x ^ { 3 } + 16 x$

A)
B)
C)
D)
E)

Question 73

Use the graph $f ^ { \prime \prime }$ to sketch the graph of $f$ .

A)
B)
C)
D)
E)

Question 74

Compare dy and $\Delta y$ for $y = 3 x ^ { 4 } + 1$ at x = 1 with dx = 0.03. Give your answers to four decimal places.

A)dy = 0.3800 ; $\Delta y$ = 0.3762
B)dy = 0.3500 ; $\Delta y$ = 0.3763
C)dy = 0.3300 ; $\Delta y$ = 0.3764
D)dy = 0.3600 ; $\Delta y$ = 0.3765
E)dy = 0.3800 ; $\Delta y$ = 0.3764

Question 75

Complete the table for the function $y = \sqrt { x }$ . Let x = 4. $\begin{array} { l } d x=\Delta x \quad d y \quad \Delta y \quad \Delta y-d y \quad d y / \Delta y\\1.00000 \\0.50000 \\0.10000\end{array}$

A) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\1.00000 & 0.25 & 0.23607 & - 0.01393 & 1.05901 \\0.50000 & 0.125 & 0.12132 & - 0.00368 & 1.03033 \\0.10000 & - 0.0061 & 0.02485 & - 0.00015 & 1.00825\end{array}$
B) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\1.00000 & 0.25 & 0.23607 & - 0.01393 & 1.05901 \\0.50000 & 0.125 & 0.43132 & - 0.00368 & 1.03454 \\0.10000 & 0.025 & 0.02485 & - 0.00015 & 1.00604\end{array}$
C) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\1.00000 & 0.3011 & 0.23607 & - 0.01393 & 1.05901 \\0.50000 & 0.125 & 0.43132 & - 0.00368 & 1.03454 \\0.10000 & - 0.0061 & 0.02485 & - 0.00015 & 1.00825\end{array}$
D) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\1.00000 & 0.3011 & 0.23607 & - 0.01393 & 1.05901 \\0.50000 & 0.125 & 0.12132 & - 0.00368 & 1.03033 \\0.10000 & - 0.0061 & 0.02485 & - 0.00015 & 1.00825\end{array}$
E) $\begin{array} { l l l l l } d x = \Delta x & d y & \Delta y & \Delta y - d y & d y / \Delta y \\1.00000 & 0.25 & 0.23607 & - 0.01393 & 1.05901 \\0.50000 & 0.125 & 0.12132 & - 0.00368 & 1.03033 \\0.10000 & 0.025 & 0.02485 & - 0.00015 & 1.00604\end{array}$

Question 76

Find the differential dy of the function $y = 2 x ^ { 2 } - 3 x + 2$ .

A) $\left( 4 x ^ { 2 } - 3 x + 2 \right) d x$
B) $( 2 x - 3 ) d x$
C) $\left( \frac { 2 } { 3 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + 2 x \right) d x$
D) $\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 2 x \right) d x$
E) $( 4 x - 3 ) d x$

Question 77

Use the graph $f ^ { \prime }$ to sketch the graph of $f$ .

A)
B)
C)
D)
E)

Question 78

Analyze and sketch a graph of the function $y = x ^ { 3 } - 12 x ^ { 2 } + 3$ .

A)
B)
C)
D)
E)

Question 79

Find the differential dy of the function $y = - 2 x ^ { 5 / 6 }$ .

A) $x ^ { 5 / 6 } \ln ( - 2 )$
B) $\frac { 5 } { 6 } x ^ { - 1 / 6 } d x$
C) $- 2 x ^ { - 1 / 6 } d x$
D) $- \frac { 5 } { 3 } x ^ { - 1 / 6 } d x$
E) $- \frac { 5 } { 6 } x ^ { - 1 / 6 } d x$

Question 80

An employee of a delivery company earns $\$$ 25.00 per hour driving a delivery van in an area where gasoline costs $\$$ 2.90 per gallon. When the van is driven at a constant speed s (in miles per hour, with $45 \leq s \leq 60$ ), the van gets $\frac { 290 } { s }$ miles per gallon. Determine the most economical speed s for a 100-mile trip on an interstate highway.

A)The most economical speed is 52.0 mph.
B)The most economical speed is 48.0 mph.
C)The most economical speed is 25.0 mph.
D)The most economical speed is 50.0 mph.
E)The most economical speed is 53.0 mph.