Deck 12: Applications of the Derivative

Question 1

You have been hired as a marketing consultant to Big Book Publishing, Inc., and you have been approached to determine the best price for the hit calculus text by Whiner and Istanbul entitled Fun with Derivatives. You decide to make life easy and assume that the demand equation for Fun with Derivatives has the linear form $q = m p + b$
Where p is the price per book, q is the demand in annual sales, and m and b constants are certain constants you'll have to figure out. Your market studies reveal the following sales figures: when the price is set at $42 per book, the sales amount to 50,560 per year, when the price is set at $93 per book, the sales drop to 34,240 per year. Now estimate the unit price in order to maximize annual revenue and predict what Big Book Publishing's annual revenue will be at that price. Round your answers to the nearest dollar.

A) $p = \$ 77$ , $R = \$ 320,000$
B) $p = \$ 90$ , $R = \$ 3,200,000$
C) $p = \$ 100$ , $R = \$ 3,200,000$
D) $p = \$ 308$ , $R = \$ 64,000$
E) $p = \$ 136$ , $R = \$ 1,600,000$

Answer 1

$p = \$ 100$ , $R = \$ 3,200,000$

Question 2

You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form
$q = m p + b$
Where p is the price per hamburger, q is the demand in weekly sales, and m and b constants are certain constants you must determine. Your market studies reveal the following sales figures: when the price is set at $4 per hamburger, the sales amount to 5,920 per week, but when the price is set at $8 per hamburger, the sales drop to zero. Now estimate the unit price in order to maximize annual revenue and predict what the weekly revenue will be at that price. Round your answers to the nearest cent.

A) $p = \$ 5.00$ , $R = \$ 22,200.00$
B) $p = \$ 4.00$ , $R = \$ 5,920.00$
C) $p = \$ 4.00$ , $R = \$ 23,680.00$
D) $p = \$ 6.00$ , $R = \$ 17,760.00$
E) $p = \$ 7.00$ , $R = \$ 10,360.00$

Answer 2

$p = \$ 4.00$ , $R = \$ 23,680.00$

Question 3

The weekly sales in Honolulu Red Oranges is given by the following. $q = 950 - 19 p$
Calculate the price that gives a maximum weekly revenue. Round your answer to the nearest cent.

A)$50.00
B) $26.00
C) $25.00
D) $27.00
E) $12.50

Answer 3

$25.00

Question 4

The consumer demand curve for tissues is given by
$q = ( 72 - p ) ^ { 2 }$
Where p is the price per case of tissues and q is the demand in weekly sales. At what price should tissues be sold to maximize the revenue Round your answer to the nearest cent.

A)$48.00
B) $20.00
C) $72.00
D) $36.00
E) $24.00

Question 5

The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula $q = 92 e ^ { - 4 p ^ { 2 } + p }$
Where q is the demand in monthly sales and p is the retail price in yen. Determine the elasticity of demand E when the retail price is set at 2 yen.

A) 15
B) 2760
C) 60
D) 30
E) 16

Question 6

You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form
$q = m p + b$
Where p is the price per hamburger, q is the demand in weekly sales, and m and b constants are certain constants you must determine. Your market studies reveal the following sales figures: when the price is set at $4 per hamburger, the sales amount to 1,420 per week, but when the price is set at $5 per hamburger, the sales drop to zero. Use these data to calculate the demand equation.

A) $q = 1,420 p + 8,520$
B) $q = - 1,420 p + 5,680$
C) $q = - 1,420 p$
D) $q = - 1,420 p + 7,100$
E) $q = 1,420 p - 7,100$

Question 7

The consumer demand curve for tissues is given by
$q = ( 92 - p ) ^ { 2 }$
Where p is the price per case of tissues and q is the demand in weekly sales. Determine the elasticity of demand E when the price is set at $30.

Round the answer to the nearest hundredth.

A)0.01
B) 0.48
C) 1.94
D) 0.97
E) 0.65

Question 8

A general linear demand function has the form that follows. $q = n p + b$ (n and b constants, with $n \neq 0$ )

Obtain a formula for the price that maximizes revenue.

A) $p = \frac { b } { n }$
B) $p = - \frac { b } { 2 n }$
C) $p = b n$
D) $p = - \frac { n } { 2 b }$
E) $p = - \frac { 2 n } { b }$

Question 9

A general exponential demand function has the form to follow. $q = A e ^ { - m q }$ (A, m - nonzero constants)

Obtain a formula for the elasticity of demand at a unit price of p.

A) $E = A m p$
B) $E = - A p$
C) $E = \frac { A p } { m }$
D) $E = - A e ^ { - m }$
E) $E = m p$

Question 10

As a new owner of the supermarket, you have inherited a large inventory of unsold imported Limburger cheese, and you would like to price it so that your revenue from selling it is as great as possible. Previous sale figures of the cheese are shown in the following table. $$\begin{array} { | l | l | l | l | } \hline \text { Price per pound, } \boldsymbol { p } & \$ 3.00 & \$ 4.00 & \$ 5.00 \\\hline \text { Monthly sales (pounds), } q & 407 & 301 & 223 \\\hline\end{array}$$ Use the sale figures for the $3 and $5 per pound prices to construct a demand function of the form $q = A e ^ { - b p }$ where A and b are constants you must determine. (The values of A, b were rounded.)

A) $q = 1,100 e ^ { - 1.3 p }$
B) $q = 1,000 e ^ { - 0.4 p }$
C) $q = 1,100 e ^ { - 0.2 p }$
D) $q = 900 e ^ { - 1.3 p }$
E) $q = 1,000 e ^ { - 0.3 p }$

Question 11

The population P is currently 20,000 and growing at a rate of 1,000 per year. What is the mathematical notation for the rate ?

A) $\frac { \mathrm { d } t } { \mathrm {~d} P }$
B) $\frac { \mathrm { d } P } { \mathrm {~d} t }$
C) $P$
D) $\mathrm { d } t$
E) $\mathrm { d } P$

Question 12

You have been hired as a marketing consultant to Big Book Publishing, Inc., and you have been approached to determine the best price for the hit calculus text by Whiner and Istanbul entitled Fun with Derivatives. You decide to make life easy and assume that the demand equation for Fun with Derivatives has the linear form $q = m p + b$
Where p is the price per book, q is the demand in annual sales, and m and b constants are certain constants you'll have to figure out. Your market studies reveal the following sales figures: when the price is set at $57 per book, the sales amount to 12,280 per year, when the price is set at $86 per book, the sales drop to 4,740 per year. Use these data to calculate the demand equation.

A) $q = 57 p - 271$
B) $q = 260 p$
C) $q = - 86 p + 260$
D) $q = 86 p - 2,540$
E) $q = - 260 p + 27,100$

Question 13

A general quadratic demand function has the form to follow. $q = c p ^ { 2 } + l p + r$ (c, l, r are constants, $l \neq 0$ )

Obtain a formula for the elasticity of demand at a unit price p.

A) $E = c p ^ { 2 } + l p + r$
B) $E = - \frac { 2 } { p + r }$
C) $E = - \frac { p ( 2 c p + l ) } { c \mathrm { p } ^ { 2 } + l p + r }$
D) $E = - l p$
E) $E = \frac { c p + l } { c p ^ { 2 } + l p + r }$

Question 14

The consumer demand curve for Professor Stefan Schwartzenegger dumbbells is given by $q = ( 552 - 8 p ) ^ { 2 }$
Where p is the price per dumbbell and q is the demand in weekly sales. Find the price Professor Schwartzenegger should charge for his dumbbells in order to maximize revenue.

A)$184
B) $25
C) $15
D) $23
E) $19

Question 15

A fried chicken franchise finds that the demand equation for its new roast chicken product, "Roasted Rooster", is given by
$p = \frac { 1080 } { q ^ { 1.5 } }$
Where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this price. Express q as a function of p, and find the elasticity of demand when the price is set at $5 per serving.

A) $\frac { 1 } { 36 }$
B) $\frac { 2 } { 3 }$
C) 36
D) $\frac { 11 } { 36 }$
E) $\frac { 3 } { 4 }$

Question 16

The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula $q = 120 e ^ { p - \frac { 5 p ^ { 2 } } { 2 } }$
Where q is the demand in monthly sales and p is the retail price in yen. At what price will revenue be a maximum Round the answer to the nearest hundredth.

A)66.99 yen
B) 0.56 yen
C) 0.78 yen
D) 0.99 yen
E) 1.22 yen

Question 17

A general linear demand function has the form to follow. $q = k p + b$ (k and b constants, with $k \neq 0$ )

Obtain a formula for the elasticity of the demand at a unit price p.

A) $E = - \frac { p } { k + b }$
B) $E = b + k p$
C) $E = - \frac { k p } { k p + b }$
D) $E = \frac { k p } { b }$
E) $E = - \frac { p } { p + b }$

Question 18

The weekly sales in Honolulu Red Oranges is given by the following equation. $q = 1,070 - 24 p$
Calculate the elasticity of demand for a price of $27 per orange. Round your answer to the two decimal places.

A)-1.54
B) -1.65
C) 1.65
D) 1.43
E) 1.54

Question 19

A study of about 1800 U.S. colleges and universities resulted in the demand equation $q = 11,684 - 2.3 p$
Where q is the enrollment at a college or university and p is the average annual tuition (plus fees) it charges. The study also found that the average tuition charged by universities and colleges was $2,771. What is the corresponding elasticity of demand Round your answer to the two decimal places.

A)0.72
B) 2.40
C) -1.20
D) 1.80
E) 1.20

Question 20

A general hyperbolic demand function has the form to follow. $q = \frac { m } { p ^ { z } }$ (s, m - nonzero constants)

Obtain a formula for the elasticity of demand at a unit price of p.

A) $E = s m p$
B) $E = m$
C) $E = s$
D) $E = \frac { s p } { m }$
E) $E = - m p$

Question 21

The automobile assembly plant you manage has a Cobb-Douglas production function given by
$P = 10 x ^ { 0.3 } y ^ { 0.7 }$
Where P is the number of automobiles the plant produces per year, x is the number of employees, and y is the daily operating budget (in dollars). You maintain a production level of 3,500 automobiles per year. If you currently employ 100 workers and are hiring new workers at a rate of 10 per year, how fast is your daily operating budget changing Round your answer to the nearest cent.

A)-$25.66 per year
B) $139.71 per year
C) -$2,566.04 per year
D) $25.66 per year
E) $17.96 per year

Question 22

A right conical circular vessel is being filled with green industrial waste at a rate of 200 cubic meters per second. How fast is the level rising after $100 \pi$ cubic meters have been poured in (The cone has height 75 m and radius 45 m at its brim. The volume of a cone of height h and cross-sectional radius r at its brim is given by $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ .) Round your answer to the one decimal place.

A) $1.2 \frac { \mathrm { m } } { \mathrm { s } }$
B) $6.0 \frac { \mathrm { m } } { \mathrm { s } }$
C) $0.4 \frac { \mathrm { m } } { \mathrm { s } }$
D) $6.3 \frac { \mathrm { m } } { \mathrm { s } }$
E) $2.0 \frac { \mathrm { m } } { \mathrm { s } }$

Question 23

A point on the graph of $y = \frac { 5 } { x }$
Is moving along the curve in such a way that its x-coordinate is increasing at a rate of 2 units per second. At what rate is the y-coordinate decreasing at the instant the y-coordinate is equal to 5

A)5 units per second
B) 2.5 units per second
C) 25 units per second
D) 10 units per second
E) 2 units per second

Question 24

A baseball diamond is a square with side 90 ft. A batter at the home base hits the ball and runs toward first base with a speed of $22 \frac { \mathrm { ft } } { \mathrm { sec } }$ . At what rate is his distance from third base increasing when he is halfway to first base ? Round your answer to the one decimal place.

A) $5.5 \frac { \mathrm { ft } } { \mathrm { sec } }$
B) $11.0 \frac { \mathrm { ft } } { \mathrm { sec } }$
C) $9.8 \frac { \mathrm { ft } } { \mathrm { sec } }$
D) $15.6 \frac { \mathrm { ft } } { \mathrm { sec } }$
E) $49.2 \frac { \mathrm { ft } } { \mathrm { sec } }$

Question 25

Demand for your tie-dyed T-shirts is given by the formula ? $p = 5 + \frac { 100 } { \sqrt { q } }$
Where p is the price in dollars you can charge to sell q T-shirts per month. If you currently sell T-shirts for $10 each and you raise price by $1 each month, how fast will the demand drop

A)80 T-shirts per month
B) 20 T-shirts per month
C) 160 T-shirts per month
D) 80,000 T-shirts per month
E) 800 T-shirts per month

Question 26

The area of a circular sun spot is growing at a rate of $600 \frac { \mathrm { km } ^ { 2 } } { \mathrm {~s} }$ . How fast is the radius growing at the instant when it equals 5,000 km Round your answer to three decimal places.

A) $\frac { \mathrm { km } } { \mathrm { s } }$
B) $0.019 \frac { \mathrm { km } } { \mathrm { s } }$
C) $0.038 \frac { \mathrm { km } } { \mathrm { s } }$
D) $8.333 \frac { \mathrm { km } } { \mathrm { s } }$
E) $0.060 \frac { \mathrm { km } } { \mathrm { s } }$

Question 27

My aunt and I were approaching the same intersection, she from the south and I from the west. She was travelling at a steady speed of 20 mph, while I was approaching the intersection at 80 mph. At a certain instant in time, I was $\frac { 1 } { 10 }$ of a mile from the intersection, while she was $\frac { 1 } { 20 }$ of a mile from it. How fast were we approaching each other at that instant Round your answer to the nearest whole number.

A)88 miles per hour
B) 82 miles per hour
C) 80 miles per hour
D) 78 miles per hour
E) 85 miles per hour

Question 28

A spherical party balloon is being inflated by helium pumped in at a rate 3 cubic feet per minute. How fast is the radius growing at the instant when the radius has reached 1 foot Round your answer to the two decimal places.
The volume of a sphere of radius r is

$V = \frac { 4 } { 3 } \pi r ^ { 3 }$ .

A)0.72 feet per minute
B) 0.76 feet per minute
C) 0.14 feet per minute
D) 0.64 feet per minute
E) 0.24 feet per minute

Question 29

The base of a 35-foot ladder is being pulled away from a wall at a rate of 12 feet per second. How fast is the top of the ladder sliding down the wall at the instance when the base of the ladder is 21 feet from the wall ?

A) $9 \frac { \mathrm { ft } } { \mathrm { s } }$
B) $48 \frac { \mathrm { ft } } { \mathrm { s } }$
C) $36 \frac { \mathrm { ft } } { \mathrm { s } }$
D) $16 \frac { \mathrm { ft } } { \mathrm { s } }$
E) $3 \frac { \mathrm { ft } } { \mathrm { s } }$

Question 30

A point is moving along the circle
$x ^ { 2 } + ( y - 3 ) ^ { 2 } = 5$
In such a way that its x-coordinate is decreasing at a rate of 2 units per second. At what rate is the y-coordinate decreasing at the instant when the point has reached $( - 2,4 )$

A)8 units per second
B) 6 units per second
C) 3.5 units per second
D) 5 units per second
E) 4 units per second

Question 31

The HMS Dreadnaught is 20 miles north of Montauk and steaming due north at 10 mph, while the USS Mona Lisa is 25 miles east of Montauk and steaming due east at an even 15 mph. How fast is their distance apart increasing Round your answer to the nearest whole number.

A)15 miles per hour
B) 23 miles per hour
C) 36 miles per hour
D) 52 miles per hour
E) 18 miles per hour

Question 32

Assume that the demand function for tuna in a small coastal town is given by $p = \frac { 50,000 } { q ^ { 1.5 } }$
Where q is the number of pounds of tuna that can be sold in 1 month at the price of p dollars per pound. The town's fishery finds that the demand for tuna is currently 900 pounds per month and is increasing at a rate of 200 pounds per month. How fast is the price changing Round your answer to the three decimal places.

A)-$0.206 per pound per month
B) -$5.556 per pound per month
C) -$0.617 per pound per month
D) -$0.412 per pound per month
E) -$0.274 per pound per month

Question 33

The radius of a circular puddle is growing at a rate of $8 \frac { \mathrm { cm } } { \mathrm { s } }$ . How fast is the area growing at the instant when it equals 20 cm² Round your answer to the nearest whole number.

A) $127 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
B) $1005 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
C) $3 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
D) $72 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$
E) $63 \frac { \mathrm { cm } ^ { 2 } } { \mathrm {~s} }$

Question 34

There are presently $N = 300$ cases of Bangkok flu, and the number is growing by 40 new cases every month. Rewrite the rate in mathematical notation.

A) $\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { 2 } { 15 }$
B) $\frac { \mathrm { d } N } { \mathrm {~d} t } = 40$
C) $\mathrm { d } N = 40$
D) $\frac { \mathrm { d } N } { \mathrm {~d} t } = 7.50$
E) $N = 40$

Question 35

A cylindrical bucket is being filled with paint at a rate of $4 \frac { \mathrm { cm } ^ { 3 } } { \min }$ . How fast is the level rising when the bucket starts to overflow The bucket has a height of 40 cm and a radius of 10 cm. Round your answer to the three decimal places.

A) $0.801 \frac { \mathrm { cm } } { \mathrm { min } }$
B) $0.407 \frac { \mathrm { cm } } { \mathrm { min } }$
C) $0.210 \frac { \mathrm { cm } } { \mathrm { min } }$
D) $0.013 \frac { \mathrm { cm } } { \mathrm { min } }$
E) $0.998 \frac { \mathrm { cm } } { \mathrm { min } }$

Question 36

A rather flimsy spherical balloon is designed to pop at the instant its radius has reached 5 cm. Assuming the balloon is filled with helium at a rate of $10 \frac { \mathrm { cm } ^ { 3 } } { \mathrm {~s} }$ , calculate how fast the diameter is growing at the instant it pops. Round your answer to the three decimal places.
The volume of a sphere of radius r is $V = \frac { 4 } { 3 } \pi r ^ { 3 }$ .

A) $0.255 \frac { \mathrm { cm } } { \mathrm { s } }$
B) $0.064 \frac { \mathrm { cm } } { \mathrm { s } }$
C) $0.318 \frac { \mathrm { cm } } { \mathrm { s } }$
D) $0.032 \frac { \mathrm { cm } } { \mathrm { s } }$
E) $0.424 \frac { \mathrm { cm } } { \mathrm { s } }$

Question 37

A study found that the divorce rate d (given as a percentage) appears to depend on the ratio r of available men to available women. This function can be approximated by $$d ( r ) = \left\{ \begin{array} { l l } - 40 r + 100 & \text { if } r \leq 1.3 \\\frac { 101 r } { 4 } - \frac { 118 } { 4 } & \text { if } r > 1.3\end{array} \right.$$
There are currently 0.8 available men per available woman in Littleville, and this ratio is increasing by 0.1 per year. At what percent is the divorce rate decreasing

A)0.4% per year
B) 0.08% per year
C) 10% per year
D) 4% per year
E) 10.1% per year

Question 38

The volume of paint in a right cylindrical can is given by $V = 2 t ^ { 2 } - t$ , where t is time in seconds and V is the volume in cm³. How fast is the level rising when the height is 3 cm The can has a height of 5 cm and a radius of 3 cm. [Hint: To get h as a function of t, first solve the volume $V = \pi r ^ { 2 } h$ for h.] Round your answer to the nearest whole number. ?

A) $6 \frac { \mathrm { cm } } { \mathrm { s } }$
B) $2 \frac { \mathrm { cm } } { \mathrm { s } }$
C) $4 \frac { \mathrm { cm } } { \mathrm { s } }$
D) $1 \frac { \mathrm { cm } } { \mathrm { s } }$
E) $3 \frac { \mathrm { cm } } { \mathrm { s } }$

Question 39

You can now sell 120 cups of lemonade at 40 cents per cup, but demand is dropping at a rate of 6 cups per week each week. Assuming that raising the price doesn't affect the demand, how fast do you have to raise your price if you want to keep the revenue constant ?

A)6 cents per week
B) 2 cents per week
C) 1 cent per week
D) 4 cents per week
E) 5 cents per week

Question 40

Find the coordinates of all relative and absolute extrema. $$g ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - 48 }$$

A)(-16, 16), (0, 0)
B) (-12, -18), (12, 18), (0, 0)
C) (0, 0), (12, 18)
D) (-12, -18), (12, 18)
E) (0, 0)

Question 41

Calculate $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ . $y = \frac { 10 } { x }$

A) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 10 } { x ^ { 2 } }$
B) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { x ^ { 2 } } { 10 }$
C) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 10 } { x ^ { 3 } }$
D) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 20 } { x ^ { 3 } }$
E) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 3 } } { 10 }$

Question 42

Find the coordinates of all relative and absolute extrema.
​ ​
​

Question 43

Combined SAT scores in the United States could be approximated by
​ ​
in the years 1964 - 1986. Here t is the number of years since 1964, and T is the combined SAT score average for the United States. Find all points of inflection of the graph of T, and interpret the result.
​
Enter your answers rounded to the nearest hundredth.

Question 44

Minimize $S = x + y$ with $x y = 25$ and both $\chi$ and $y > 0$ .

A) $S = - 15$
B) $S = 10$
C) $S = 25$
D) $S = 8$
E) $S = 5$

Question 45

The graph of a function is given.
​ ​
Find the coordinates of all points of inflection of this function (if any).
​
Please enter your answer as ordered pairs in the form (x, y) separated by commas.

Question 46

Sketch the graph of the function, labeling all relative and absolute extrema and points of inflection, and vertical and horizontal asymptotes. Check your graph using technology.
$$k ( x ) = \frac { 4 } { 5 } x - ( x - 1 ) ^ { \frac { 4 } { 5 } }$$

A)
B)
C)
D)
E)

Question 47

Calculate $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ . $y = 4 x ^ { - 5 } + 5 \ln x$

A) $\frac { 4 } { x ^ { 10 } } - \frac { 5 } { x }$
B) $120 x ^ { 7 }$
C) $\frac { 120 } { x ^ { 7 } } - \frac { 5 } { x ^ { 2 } }$
D) $\frac { 120 } { x ^ { 7 } } + \frac { 5 } { x ^ { 2 } }$
E) $\frac { 100 } { x ^ { 5 } } - \frac { 1 } { x ^ { 2 } }$

Question 48

Calculate $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ . $y = 15 e ^ { - ( x - 5 ) } - x$

A) $15 x ^ { 2 } e ^ { - x }$
B) $- x$
C) $15 e ^ { - ( x - 5 ) }$
D) $15 ( - x + 5 ) ^ { 2 } e ^ { - ( x - 5 ) }$
E) $15 ( x - 5 ) e ^ { - ( x - 5 ) }$

Question 49

Maximize $P = x y$ with $x + y = 14$ .

A) $P = - 49$
B) $P = 49$
C) $P = 14$
D) $P = 7$
E) $P = 2,401$

Question 50

Combined SAT scores in the United States could be approximated by
$T ( t ) = - 0.0101 t ^ { 3 } + 0.5856 t ^ { 2 } - 9.6 t + 1038$ $( 0 \leq t \leq 23 )$
In the years 1967 - 1990. Here t is the number of years since 1967, and T is the combined SAT score average for the United States. Find all points of inflection of the graph of T, and interpret the result. Round coordinates to two decimal places if necessary.

A)(11.75, 989.90), (26.90, 1007.45)
B) (19.33, 998.67)
C) (23, 1004.56)
D) (0, 0)
E) (0, 0), (23, 1004.56)

Question 51

Maximize $P = x y$ with $3 x + 4 y = 72$ .

A) $P = 72$
B) $P = 4$
C) $P = 216$
D) $P = 108$
E) $P = 1,200$

Question 52

You manage a small antique store that owns a collection of Louis XVI jewelry boxes. Their value v is increasing according to the formula $v = \frac { 12,000 } { 1 + 500 e ^ { - 0.5 t } }$
Where t is the number of years from now. You anticipate an inflation rate of 1% per year, so that the present value of an item that will be worth $v in t years' time is given by
$p = v ( 1.01 ) ^ { - t }$
In how many years from now will the greatest rate of increase of the present value of your antiques be attained Round your answer to two decimal places.

A)t = 24.55 years
B) t = 3.07 years
C) t = 11.27 years
D) t = 12.27 years
E) t = 6.14 years

Question 53

Calculate $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ . $y = - \frac { 14 } { x ^ { 2 } }$

A) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 14 } { x ^ { 4 } }$
B) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } } { 14 }$
C) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 3 } } { 28 }$
D) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 28 } { x ^ { 3 } }$
E) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 84 } { x ^ { 4 } }$

Question 54

The graph of the second derivative, $$f ^ { \prime \prime } ( x )$$ , is given.
Determine the x-coordinates of all points of inflection of f(x), if any. (Assume that f(x) is defined and continuous everywhere in $[ - 3,3 ]$ .)

A) $x = - 2 , x = 0 , x = 2$
B) $x = - 2.2 , x = 2.2$
C) $x = - 1.2 , x = 0 , x = 1.2$
D) $x = - 1.2 , x = 1.2$
E) $x = 0$

Question 55

Minimize $F = x ^ { 2 } + y ^ { 2 }$ with $x + 2 y = 20$ .

A) $F = 2$
B) $F = 20$
C) $F = 80$
D) $F = 72$
E) $F = 60$

Question 56

Calculate .
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Question 57

A company finds that the number of new products it develops per year depends on the size if its annual R&D budget, x (in thousands of dollars), according to the following formula. Round your answer to one decimal places. $n = - 1 + 9 x + 4 x ^ { 2 } - 0.3 x ^ { 3 }$
Find $n ^ { \prime \prime } ( 2 )$ .

A)5.4
B) 9.4
C) 7.4
D) -3.6
E) 4.4

Question 58

Calculate $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ . $y = - 14 x ^ { 2 } + 15 x$

A) $- 7 x ^ { 3 } + 7.5 x ^ { 2 }$
B) $- 14 x ^ { 2 } + 15 x$
C) $- 7 x ^ { 3 } - 210 x ^ { 2 }$
D) $- 28$
E) $28 x$

Question 59

The graph of a function $f ( x ) = x ^ { 2 } \left( x ^ { 2 } - 24 \right)$ is given.

Find the coordinates of all points of inflection of this function (if any).

A) $( 0,0 ) , ( - \sqrt { 24 } , 0 ) , ( \sqrt { 24 } , 0 )$
B) $( 0,0 )$
C) $( - \sqrt { 12 } , - 144 ) , ( \sqrt { 12 } , - 144 )$
D) $( - 2 , - 80 ) , ( 2 , - 80 )$
E) $( - 2,80 ) , ( 2 , - 80 )$

Question 60

In 1965 the economist F.M. Scherer modeled the number, n, of patens produced by a firm as a function of the size, s, of the firm (measured in annual sales in millions of dollars). He came up with the following equation based on a study of 448 large firms. Round your answer to two decimal places.
$n = - 3.93 + 132.61 s - 22.88 s ^ { 2 } + 1.311 s ^ { 3 }$
Find $\left. \frac { \mathrm { d } ^ { 2 } n } { \mathrm { ds } ^ { 2 } } \right| _ {= 4 }$ .

A)-12.3
B) -14.3
C) 31.46
D) -17.3
E) -9.3

Question 61

The fruit yield per tree in an orchard that contains 60 trees is 240 pounds per tree each year. Due to crowding, the yield decreases by 2 pounds per season per every additional tree planted. How many additional trees should be planted for a maximum total annual yield ?

A)25 additional trees
B) 30 additional trees
C) 90 additional trees
D) 20 additional trees
E) 60 additional trees

Question 62

For a rectangle with perimeter 32 to have the largest area, what dimensions should it have ?

A) $8 \times 7$
B) $16 \times 16$
C) $7 \times 9$
D) $8 \times 8$
E) $8 \times 9$

Question 63

For a rectangle with area 25 to have the smallest perimeter, what dimensions should it have ?

A) $5 \times 4$
B) $4 \times 6$
C) $5 \times 6$
D) $25 \times 1$
E) $5 \times 5$

Question 64

Find the exact location of all the relative and absolute extrema of the function. $f ( x ) = \frac { x ^ { 2 } + 81 } { x ^ { 2 } - 81 } ; - 86 \leq x \leq 86 , x \neq \pm 9$

A) $( 0 , - 81 )$ - absolute minimum
B) $( 0 , - 1 )$ - relative minimum
C) $( 0 , - 1 )$ - absolute maximum
D) $( 0,81 )$ - absolute maximum
E) $( 0 , - 1 )$ - relative maximum

Question 65

The Chocolate Box Co. is going to make open-topped boxes out of $6 ^ { \prime \prime } \times 14 ^ { \prime \prime }$ rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way Round your answer to the nearest whole number.

A)63 cubic inches
B) 38 cubic inches
C) 50 cubic inches
D) 25 cubic inches
E) 76 cubic inches

Question 66

Fair Weather Airlines will accept only bags for which the sum of the length and width is 51 inches, while the sum of length, height, and twice the width is 102 inches. What is the largest volume of the bag that it will accept Round your answer to the nearest whole number.

A)14,739 cubic inches
B) 19,652 cubic inches
C) 24,565 cubic inches
D) 9,826 cubic inches
E) 4,913 cubic inches

Question 67

Find the exact location of all the relative and absolute extrema of the function with domain $( - \infty , \infty )$ .
$f ( x ) = e ^ { - 5 x ^ { 2 } }$

A) $( 0,1 )$ - absolute maximum
B) $( 1,5 )$ - absolute maximum
C) $( 0,2 )$ - relative minimum
D) $( 1,0 )$ - absolute maximum
E) $( 0,1 )$ - absolute minimum

Question 68

Maximize with and , and , , and .

Question 69

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model:
$C ( q ) = 3,500 + 150 q ^ { 2 }$
Where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost to the firm (in dollars) of this reduction. Government clean air subsidies amount to $750 per pound of pollutant removed. How many pounds of pollutant should the firm remove each day to minimize net cost (cost minus subsidy)

A)2 pounds
B) 10 pounds
C) 2.5 pounds
D) 5 pounds
E) 75 pounds

Question 70

Find the exact location of all the relative and absolute extrema of the function with domain $( 0 , \infty )$ . $f ( x ) = x \ln \left( x ^ { 3 } \right)$

A) $\left( \frac { 1 } { e } , \frac { 3 } { e } \right)$ - absolute minimum
B) $\left( \frac { 1 } { e } , - \frac { 3 } { e } \right)$ - relative maximum
C) $\left( \frac { 3 } { e } , - \frac { 1 } { 3 e } \right)$ - absolute minimum
D) $\left( \frac { 1 } { e } , - \frac { 1 } { 3 e } \right)$ - relative minimum
E) $\left( \frac { 1 } { e } , - \frac { 3 } { e } \right)$ - absolute minimum

Question 71

I want to fence in a rectangular vegetable patch. The fencing for the east and west sides costs $4 per foot, while the fencing for the north and south sides costs only $3 per foot. I have a budget of $120 for the project. What is the largest area I can enclose ?

A)12 square feet
B) 150 square feet
C) 120 square feet
D) 300 square feet
E) 75 square feet

Question 72

Hercules films is deciding on the price of the video release of its film "Son of Frankenstein". Its marketing people estimate that at a price of p dollars, it can sell a total of $q = 10,000 - 200 p$ copies. What price will bring in the greatest revenue Round your answer to the nearest cent

A)$25.75
B) $25.00
C) $37.50
D) $24.50
E) $77.25

Question 73

Locate all maxima in the graph.

Select all correct answers.

A) $( - 5 , - 5 )$
B) $( 6,7 )$
C) $( 3,0 )$
D) $( 0,0 )$
E) $( - 1,8 )$

Question 74

Maximize $P = x y z$ with $x + y = 9$ and $z + y = 9$ , and $\chi$ , $y$ , and $z > 0$ .

A) $P = 105$
B) $P = 102$
C) $P = 108$
D) $P = 324$
E) $P = 9$

Question 75

A packaging company is going to make closed boxes, with square bases, that hold 2,197 cubic centimeters. What are the dimensions of the box that can be built with the least material

A) $6.5 \times 6.5 \times 52$ cm
B) $3.25 \times 3.25 \times 208$ cm
C) $13 \times 13 \times 13$ cm
D) $1 \times 1 \times 2,197$ cm
E) $26 \times 26 \times 3.25$ cm

Question 76

The FeatureRich Software Company sells its graphing program, Dogwood, with a volume discount. If a customer buys x copies, then that customer pays $600 \sqrt { x }$ . It costs the company $30,000 to develop the program and $5 to manufacture each copy. If just one customer were to buy all the copies of Dogwood, how many copies would the customer have to buy for FeatureRich Software's average profit per copy to be maximized

A)2,000 copies
B) 2,500 copies
C) 5,000 copies
D) 10,000 copies
E) 11,000 copies

Question 77

Your automobile assembly plant has a Cobb-Douglas production function given by
$q = x ^ { 0.5 } y ^ { 0.5 }$
Where q is the number of automobiles it produced per year, x is the number of employees, and y is the daily operating budget (in dollars). Annual operating costs amount to an average of $30,000 per employee plus the operating budget of $\$ 300 y$ . Assume you wish to produce 1,500 automobiles per year at a minimum cost. How many employees should you hire

A)180 employees
B) 75 employees
C) 30 employees
D) 150 employees
E) 225 employees

Question 78

The demand for rubies at Royal Ruby Retailers is given by
$q = - \frac { 4 } { 3 } p + 80$
where p is the price RRR charges (in dollars) and q is the number of rubies RRR sells per week. At what price should RRR sell its rubies to maximize its weekly revenue

Please enter your answer in dollars without the units.

Question 79

Find the exact location of all the relative and absolute extrema of the function with domain $( - \infty , \infty )$ . $g ( x ) = \frac { 1 } { 4 } x ^ { 4 } - \frac { 2 } { 3 } x ^ { 3 } + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 }$

A) $\left( 0 , \frac { 1 } { 5 } \right)$ - absolute maximum
B) $\left( 0 , \frac { 1 } { 5 } \right)$ - absolute minimum
C) $\left( 0 , \frac { 1 } { 3 } \right)$ - relative maximum
D) $\left( 0 , \frac { 1 } { 2 } \right)$ - absolute minimum
E) $( 0,1 )$ - relative maximum

Question 80

Find the exact location of all the absolute extrema of the function with domain $( - 7 , \infty )$ . ? $f ( x ) = x ^ { 4 } - 8 x ^ { 3 }$

A) $( 6,432 )$ - absolute minimum
B) $( - 6 , - 432 )$ - relative maximum
C) $( 6,0 )$ - absolute maximum
D) $( 6 , - 432 )$ - absolute minimum
E) $( 6 , - 432 )$ - relative minimum