1. The profit, P, from producing a product is expressed as a function of the cost, C, of
producing the product and the revenue, R, from selling the product. Do you expect P to
be an increasing or decreasing function of R?
2. Lift ticket sales, S, at a ski resort are a function of the price, p, of a ticket and the current
number, n, of inches of snowpack on the mountain. Thus, S  f ( p, n) . Do you expect f
to be an increasing or a decreasing function of p?
3. The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a
function of the number of tickets sold, t, and the number of food items sold, f. Thus,
R  g (t , f ) . Find g (300, 600) .
4. The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a
function of the number of tickets sold, t, and the number of food items sold, f. Thus,
R  g (t , f ) . What is the revenue when ticket sales are 300 and food sales are 600?
Page 1
5. The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a
function of the number of tickets sold, t, and the number of food items sold, f. Thus,
R  g (t , f ) . What is the revenue when ticket sales are 400 and food sales are 1000?
6. The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a
function of the number of tickets sold, t, and the number of snacks sold, f. Thus,
R  g (t , f ) . Assuming there is only one price for tickets, what is that price?
7. The cost, C, of renting a car is $45 per day and $0.15 per mile. Write a cost function,
C  f (d , m) , for the total cost of renting a car where d is the number of days and m is
the number of miles.
8. A company sells two products. The fixed costs for the company are $3200, the variable
costs for the first product are $8 per unit, and the variable costs for the second product
are $11 per unit.. If the company produces q1 units of the first product and q2 units of
the second product, which of the following is a formula for the total cost, C, as a
function of q1 and q2 ?
A)
B)
C  3200  8q1  11q2
C)
D)
C  3200  8q1  11q2
C  3200(q1  q2 )  8q1  11q2
C  3200  8q1  11q2
9. A company sells two products. The fixed costs for the company are $2600, the variable
costs for the first product are $8 per unit, and the variable costs for the second product
are $12 per unit.. If the company produces q1 units of the first product and q2 units of
the second product, and the total cost is given by
Page 2
C  f (q1 , q2 )
, what is f (600,350) ?
10. A company sells two products. The fixed costs for the company are $3300, the variable
costs for the first product are $8 per unit, and the variable costs for the second product
are $10 per unit.. If the company produces q1 units of the first product and q2 units of
the second product, and the total cost is given by C  f (q1 , q2 ) , is C an increasing or
decreasing function of q2 ?
11. The fuel cost, C, for a 3000 mile trip depends on the price, p, per gallon of gasoline and
3000 p
C
m .
fuel economy in miles per gallon, m, of the car according to the formula
What is the cost when p = $2.00 and m = 20?
12. The fuel cost, C, for a 3000 mile trip depends on the price, p, per gallon of gasoline and
3000 p
C
m .
fuel economy in miles per gallon, m, of the car according to the formula
On the same set of axes, sketch a graph of C as a function of p with fixed mileage rates
of 10, 20, 30, and 40. What do you observe as mileage rates increase?
A) The graphs are similarly shaped curves that get progressively higher.
B) The graphs are all straight lines from the origin with slopes that get progressively
less steep.
C) The graphs are all straight lines from the origin with slopes that get progressively
steeper.
D) The graphs are similarly shaped curves that get progressively lower.
13. You are in a nicely heated cabin in the winter. Deciding that it's too warm, you open a
small window. Let T be the temperature in the room, t minutes after the window was
opened, x feet from the window. Is T an increasing or decreasing function of t?
A) Decreasing B) Increasing C) Neither
Page 3
14. The following table gives the number f(x, y) of grape vines, in thousands, of age x in
year y.
In one year a fungal disease killed most of the older grapevines, and in the following
year a long freeze killed most of the young vines. Which are these years?
15. A certain piece of electronic surveying equipment is designed to operate in temperatures
ranging from 0° C to 30° C. Its performance index, p(t, h), measured on a scale from 0
to 1, depends on both the temperature t and the humidity h of its surrounding
environment. Values of the function p = f(t, h) are given in the following table. (The
higher the value of p, the better the performance.)
What is the value of p(20, 75)?
Page 4
16. Yummy Potato Chip Company has manufacturing plants in N.Y. and N.J. The cost of
manufacturing depends on the quantities (in thousand of bags), q1 and q2, produced in
the N.Y. and N.J. factories respectively. Suppose the cost function is given by
C (q1 , q2 )  4q12  q1q2  q22  500
Find C (0,100) .
17. Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in
thousands of dollars), and the time, t, required to pay it back (in months). What is the
meaning of C(7, 48) = 230?
A) If you borrow $7,000 from the bank for 48 months (4 year loan), your monthly car
loan payment is $230.
B) If you borrow $4,000 from the bank for 48 months (7 year loan), your monthly car
loan payment is $230.
C) If you borrow $230 from the bank for 48 months (4 year loan), your monthly car
loan payment is $7.
D) If you borrow $7 from the bank for 48 months (4 year loan), your monthly car loan
payment is $230.
18. Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in
thousands of dollars), and the time, t, required to pay it back (in months). Is C an
increasing or decreasing function of t?
A) Decreasing
B) Increasing
19. The following figure shows contours for the function z  f ( x, y ) . Is z an increasing or
decreasing function of y?
Page 5
20. The following figure is a contour diagram for the demand for pork as a function of the
price of pork and the price of beef. Which axis corresponds to the price of pork?
A) The y-axis
B) The x-axis
21. Which of the following describes the contour diagram of f ( x, y )  5 x  y  1 ?
A) A set of lines extending from the point (0,1).
B) A set of lines extending from the origin.
C) A set of parallel lines, all with slope 5.
D)
1
A set of parallel lines, all with slope 5 .
22. The heat index tells you how hot it feels as a result of the combination of temperature
and humidity. The figure below gives a contour diagram for the heat index as a function
of temperature and humidity. If the humidity is 50%, what temperature feels like 120 ?
Page 6
23. The heat index tells you how hot it feels as a result of the combination of temperature
and humidity. The figure below gives a contour diagram for the heat index as a function
of temperature and humidity. Heat exhaustion is likely to occur when the heat index is
105 or higher. If the temperature is about 105 , at about what humidity level does heat
exhaustion become likely?
A) 10%
B) 14% C) 18% D) 22%
Page 7
24. The heat index tells you how hot it feels as a result of the combination of temperature
and humidity. The figure below gives a contour diagram for the heat index as a function
of temperature and humidity. Fix the humidity level at 50%. As the temperature rises,
does the heat index rise more rapidly at milder temperatures (70 to 90 ) or at very hot
temperatures (90 to 110 )?
A) milder temperatures
B) very hot temperatures
Page 8
25. The heat index tells you how hot it feels as a result of the combination of temperature
and humidity. The figure below gives a contour diagram for the heat index as a function
of temperature and humidity. Is the heat index an increasing or decreasing function of
temperature?
26. Which of the following describes the contour diagram for the function C  50d  0.2m
(with m on the vertical axis and d on the horizontal axis)?
A) A set of parallel lines, all with slopes –250
B) A set of parallel lines, all with slopes 10
C) A set of parallel lines, all with slopes –0.004
D) A set of parallel lines, all with slopes 50
27. Plants need varying amount of water and sunlight to grow. The following contour
diagrams each show the growth of a different plant as a function of the amount of water
and sunlight. Which diagram would represent a plant that does best in a tropical rain
forest floor (lots of water and little sunshine)?
Page 9
28. You build a campfire while up in the mountains. It is 45 F when you start the fire. Let
H ( x, t ) be the temperature x feet from the fire t minutes after you start it. The following
figure shows the contour diagram for H. How many F is it 8 feet from the fire after 5
minutes?
29. You build a campfire while up in the mountains. It is 45 F when you start the fire. Let
H ( x, t ) be the temperature x feet from the fire t minutes after you start it. The following
figure shows the contour diagram for H. Is H an increasing or decreasing function of x?
Page 10
30. Below is a contour diagram depicting D, the average fox population density as a
function of xE , kilometers east of the western end of England, and xN , kilometers north
of the same point.
Is D increasing or decreasing at the point (60, 25) in the southern direction?
31. The demand, D, for gasoline at Station A is a function of the price, a, of gas at Station A
and the price, b, of gas at Station B across the street. Thus, D  f (a, b) . Do you expect
D /  b to be positive or negative?
32. The amount, A  f ( w, h) , of your paycheck is a function of your hourly wage, w, and
the number of hours, h, you work. Would you expect fw(w,h) to be positive or negative?
33. The amount, A  f ( w, h) , of your paycheck is a function of your hourly wage, w, and
the number of hours, h, you work. What does the statement f h (9, 29)  9 mean in terms
of your paycheck?
A) An increase of 1 hour worked from 29 hours results in a wage increase of $9.
B) An increase of 1 hour worked from 9 hours results in a wage increase of $29.
C) An wage increase of $1 per hour from $29 per hour results in a wage increase of
$9.
D) An wage increase of $1 per hour from $9 per hour results in a wage increase of
$29.
Page 11
34. The number of tons, N, of a product produced at a factory in a day is a function of the
number of workers, w, and the number of hours worked, h. We have N  f ( w, h) .
Which of the following equations tells us that when the factory has 9 workers who work
for 8 hours, adding one additional worker will cause the amount produced to go up by
about 39 tons?
(9,8)  39
(9,8)  39
(9,8)  39
(9,8)  39
A) fh
B) fw
C) fh
D) fw
35.
( x, y )
The following is a contour diagram for z  f ( x, y ) . Is fy
positive or negative?
Page 12
36.
(1, 4)
The following is a contour diagram for z  f ( x, y ) . Estimate fy
.
A) –1.5
B) 0 C) 1.5
D) 3
37. A table of values for f ( x, y ) is given below. Is fy positive or negative?
Page 13
38. A table of values for f ( x, y ) is given below. Estimate fy (4,10). Use the next higher
point to make your estimate.
39. Estimate fx (3,3) from the contour diagram shown below. Use the next higher point to
make your estimate.
40.
For a function F (n, m) , we are given f (20,5)  3.1 , f n (20,5)  0.6 , and
f m (20,5)  –0.3
. Estimate f (23, 6) .
41. An airline's revenue, R  f ( x, y ) , is a function of the number of full price tickets, x, and
the number of discount tickets, y, sold. When 300 full price and 600 discount tickets are
f (300, 600)  200
f (300,600)  350
sold, R = $225,000. Use partial derivatives x
and y
to estimate revenue when x = 300 and y = 605.
Page 14
42. The amount in dollars, A, spent weekly by a household on food is a function of the
household's weekly income in dollars, I, and the number of people in the household, n.
Thus, A  f ( I , n) . If AI (1000, 4) = a > 0, then which of the following is true?
A) At a weekly income of $1000 and a household size of 4, an increase of 1 person in
the household produces a decrease of $a in the amount spent weekly on food.
B) At a weekly income of $1000 and a household size of 4, an increase of $1 in
weekly income produces a decrease of $a in the amount spent weekly on food.
C) At a weekly income of $1000 and a household size of 4, an increase of 1 person in
the household produces an increase of $a in the amount spent weekly on food.
D) At a weekly income of $1000 and a household size of 4, an increase of $1 in
weekly income produces an increase of $a in the amount spent weekly on food.
43. Suppose that the price P (in dollars) to purchase a used car is a function of C, its original
P
?
cost (in dollars), and its age A (in years). So P = f(C, A). What is the sign of C
A) Positive B) Negative
44. The consumption of beef, C (in pounds per week per household) is given by the function
C = f(I, p), where I is the household income in thousands of dollars per year, and p is the
price of beef in dollars per pound.
Do you expect f I to be positive or negative?
A) Positive B) Negative
45. The monthly mortgage payment in dollars, P, for a house is a function of three variables
P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is
the number of years before the mortgage is paid off. It is given that:
Estimate the value of
f
.
A (100000,7, 20)
Page 15
46. The monthly mortgage payment in dollars, P, for a house is a function of three variables
P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is
the number of years before the mortgage is paid off. It is given that:
f
A (100000,7, 20)
Estimate the value of
and interpret your answer in terms of a mortgage
payment. Select all answers that apply.
A) We are currently borrowing $100,000 at 7% interest rate on a 20-year mortgage.
B) The monthly payment will go up by approximately $0.007753 for each extra
percentage point charged.
C) The monthly payment will go up by approximately $0.007753 for each extra dollar
we borrow.
D) The monthly payment will go up by approximately $0.007753 for each extra year
of the mortgage.
E) The monthly payment will go down by approximately $0.007753 for each extra
dollar we borrow.
47. The table below gives values of a function f(x, y) near x = 1, y = 2.
Estimate
f y (0,1.5)
.
48. The consumption of beef, C (in pounds per week per household) is given by the function
C = f(I, p), where I is the household income in thousands of dollars per year, and p is the
price of beef in dollars per pound. Explain the meaning of the
f (80,3)  –1.6
statement: p
, and include units in your answer.
49.
50.
51.
4 3
f
Find x if f ( x, y)  x y .
3 3
3 2
4 2
A) 4x y B) 3x y
C) 12x y
3 2
D) 7x y
2
2
f
Find y if f ( x, y)  x  5xy  y .
2
A) 5 y  2 x B) 5 x  2 y C) x  5x  2 y
V 2
1
 πrh
V  πr 2 h
3
If
, r 3
.
Page 16
D) 2 x  5  2 y
52.
V
If
53.
4 2 V 8
 πr
πr h
3
, h 3 .
2 xy
If f ( x, y)  x e , find fx(1, 3).
54. The cost of building a new fence is C  15 f  80 g , where f is the number of linear feet
of fence and g is the number of gates. Find C /  g.
55. Use the values of f ( x, y ) in the following table to estimate fyx (2,3). Use the next
higher point to make your estimate.
56. Given the following contour diagram of f ( x, y ) , estimate fx (1,1). Use the next higher
point to make your estimate.
57. The amount of principal, $P, needed to obtain a balance of $B after t years at r %
 rt
interest, compounded continuously, is given by the formula P  Be . Which of the
following gives the amount principal can be changed by and still maintain the same
balance if the account is held an additional year?
 rt
 rt
 rt
 rt
A) e
B) -Bt e
C) -Br e
D)  Brte
Page 17
58. Suppose the Cobb-Douglas production function for a company is given by
P  50 L0.70 K 0.40 , where P is production in tons, L is the number of workers, and K is the
capital investment, in thousands of dollars. How many tons are produced if the
company has an initial investment of 30 thousand dollars and the company employs 17
workers? Round to the nearest ton.
59. Suppose the Cobb-Douglas production function for a company is given by
P  50 L0.70 K 0.40 , where P is production in tons, L is the number of workers, and K is the
capital investment, in thousands of dollars. Find P /  L if the company has a capital
investment of 25 thousand dollars and the company employs 15 workers. Round to the
nearest tenth.
60.
2
The volume of a cylinder is given by V  f (r , h)   r h , where r is the radius of the
base and h is the height, both in centimeters. At a radius of 5 centimeters and a height
of 8 centimeters, how much does the volume increase for each 1 cm increase in the
height?
A) 40  cm3 B) 80  cm3 C) 50  cm3 D) 25  cm3
61. For V  s 2 h , calculate  2V /  h2 .
62.
2
2
For V  s h , calculate  V /  h  s
63.
f ( x, y) 
For
x2
y , calculate fyy (x,y).
64. If $P is invested in a bank account earning r% interest a year, compounded
continuously, the balance, $B, at the end of t years is given by
B  f ( P, r , t )  Pe
rt
100
.
Find B / t.
65. The ideal gas law states that PV  RT for a fixed amount of gas, called a mole of gas,
where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the
temperature (in degrees Kelvin) and R is a positive constant.
P
.
Find T
66.
Find the following partial derivative:
to 4 decimal places.
Hp
H ( P, T ) 
(2, 2) if
Page 18
3P
.
3P  T Give your answer
67.
3 4
Find the following partial derivative: fxy if f ( x, y)  x y .
A) f xy  4 x3 y3
D) f xy  3x2 y3
B)
f xy  3x2 y 4
C)
f xy  4 x2 y3
E)
f xy  12 x2 y3
68. A contour diagram of f ( x, y ) is given in the following figure. Is there a global
minimum near the point (1,4)?
Page 19
69. A contour diagram of f ( x, y ) is given in the following figure. Which one of the
following statements is true?
A) The function has a local maximum of about 5 at (1, 3.8), a global maximum of
about 14 at (5.2, 4.9), and a global minimum of about -5 at (3, 1.6).
B) The function has a global maximum of about 5 at (1, 3.8), a local maximum of
about 14 at (5.2, 4.9), and a global minimum of about -5 at (3, 1.6).
C) The function has a local minimum of about 5 at (1, 3.8), a global minimum of
about 14 at (5.2, 4.9), and a global maximum of about -5 at (3, 1.6).
D) The function has a global maximum of about 5 at (1, 3.8), a global maximum of
about 14 at (5.2, 4.9), and a global minimum of about -5 at (3,1.6).
70. The following table gives values for a function f ( x, y ) . For 0  x  40 and 0  y  20 ,
there is a global minimum of at most _____ near the point (_____,_____).
71.
2
2
The function f ( x, y)  x  2 xy  2 y  8 y has a local __________ (maximum /
minimum / neither) at the critical point, where x = _____ and y = _____.
Page 20
72.
2
The function f ( x, y)  x  3xy  15 y has a local __________ (maximum / minimum /
neither) at the critical point, where x = _____ and y = _____.
73.
3
2
2
The function f ( x, y)  x  y  2 x  3 y  3 has a critical point which is a local
minimum when x = _____ and y = _____.
74.
2
2
The function f ( x, y)  450  2 x  6 x  2 xy  3 y  12 y has a local maximum value of
_____ when x = _____ and y = _____.
75.
Two products are manufactured in quantities q1 and q2 and sold at prices $8 and $12
2
2
respectively. The cost of producing them is given by C  q1  2q2  8 . Find the
maximum profit that can be made.
76.
p
A company sells two styles of jeans in quantities q1 and q2 at prices 1 and p2
respectively, with p1 and p2 in dollars. The quantities demanded depend on both p1
and p2 according to the formulas
q1  100  6 p1  4 p2
and
q2  140  5 p1  7 p2 .
Write a formula for total sales revenue as a function of p1 and p2 , and use it to
determine what prices the company should charge to maximize sales revenue. They
should charge $_____ for style q1 and $_____ for style q2 .
77.
2
2
Suppose f ( x, y)  x  Ax  y  By  C . Then f ( x, y ) has a local minimum value of
13 at the point (2, 3) when A = _____, B = _____, and C = _____.
78.
f ( x, y )  xy 
8
8
 2.
2
x
y
Find all the critical points of the function
Classify these critical points as local maxima, local minima, or neither.
79. Is (0, 0) a critical point of the following function?
A)
B)
C)
D)
80.
Yes: neither a maximum nor a minimum.
Yes: (global) minimum.
Yes: (global) maximum.
No.
3
3
Let h( x, y)  x  y  3xy  1.
Determine all local critical points. Are the local extrema also global extrema?
Page 21
81.
3
3
Let h( x, y)  x  y  3xy  5.
Which figure best represents the contours of this function?
A)
C)
B)
D)
82.
2
2
Suppose that f ( x, y)  x  xy  y .
Find and classify the critical point(s) as local maxima, local minima, or neither.
83.
3
2
Find all the critical points of f ( x, y)  x  3x  y  8 y and classify each as maximum,
minimum, or neither.
Select all possible choices.
A) The point (1, 4) is a local minimum.
B) The point (-1, –4) is neither a maximum nor a minimum.
C) The point (-1, 4) is neither a maximum nor a minimum.
D) The point (1, 4) is a local maximum.
E) The point (1, -4) is a local maximum.
Page 22
84.
3
2
The function f ( x, y)  x  3x  y  5 y has a critical point that is neither a maximum
nor a minimum at (–1, 2.5). Which of the following is a sketch of the level curves of f
near this point?
A)
85.
B)
3
2
Find the critical points of f ( x, y)  x  15xy  y and classify each as maximum,
minimum or neither.
86. The contour diagram of f is shown below. Which of the points A, B, C, D, and E appear
to be critical points? Select all that apply.
A) A
B) B C) C
D) D
E) E
87.
4
3
2 2
3
4
Consider the function f ( x, y)  x  2 x y  4 x y  7 xy  y  2.
Check that (0,0) is a critical point of f and classify it as a local minimum, local
maximum or neither.
88.
3
2
The point (2, 1) is a critical point of g ( x, y)  3x  72 xy  36 xy .
Classify it either as a local minimum, local maximum, or neither.
89.
 ( x a ) ( y b )
The function f ( x, y)  e
where a and b are constants is sometimes referred to
as a "bump function" and is used to construct functions which take on maximum values
at certain points. Show that f(x, y) has a maximum at (a, b).
90.
2
2
Find a and b so that f ( x, y)  ax  bxy  y has a critical point at (1, 3).
2
2
Page 23
91. The Perfect House company produces two types of bathtub, the Hydro Deluxe model
and the Singing Bird model. The company noticed that demand and prices are related. In
particular,
for Hydro Deluxe: demand = 1800 - price of Hydro Deluxe + price of Singing Bird
for Singing Bird: demand = 1400 + price of Hydro Deluxe -2(price of Singing Bird).
The costs of manufacturing the Hydro Deluxe and Singing Bird are $500 and $300 per
unit respectively. Determine the price of each model that gives the maximum profit.
92. A company has two manufacturing plants which manufacture the same item. Suppose
2
2
the cost function is given by C (q1 , q2 )  6q1  q1q2  q2 , where q and q are the
1
2
quantities (measured in thousands) produced in each plant. The total demand q1 + q2 is
related to the price, p, by p  110  0.5(q1  q2 ).
How much should each plant produce in order to maximize the company's profit?
93.
2
2
Suppose that f ( x, y)  2 x  2.0000 y  x.
Find and classify (as local maxima, minima, or neither) all critical points of f.
94. Use Lagrange multipliers to find the maximum or minimum values of f ( x, y )  5 xy ,
subject to the constraint 4 x  y  38 .  = _____ and the maximum/minimum value of
f ( x, y ) is _____.
95.
2
2
Use Lagrange multipliers to find the maximum or minimum values of f ( x, y)  x  y ,
subject to the constraint 2 x  y  20 .  = _____ and the maximum/minimum value of
f ( x, y ) is _____.
96. Use Lagrange multipliers to find the maximum and minimum values of
f ( x, y )  2 x  2 y , subject to the constraint x 2  y 2  8 . For the minimum value,  =
_____ (use the decimal form), and the minimum value of f ( x, y ) is _____.
Page 24
97. The following figure shows contours of f ( x, y ) and the constraint g ( x, y )  c . What is
the maximum value of f subject to this constraint ?
98. A company operates two plants that make the same product. If Plant 1 produces
quantity x of the product and Plant 2 produces quantity y of the product, the cost
2
2
functions are C1  8.4  0.03 x and C2  5.2  0.04 y . The total market demand
q  x  y for the product is related to the selling price p by p  60  0.4q . What
quantity should Plant 1 produce to maximize the company's profit?
99.
The quantity, Q, of a good produced depends on the quantities x1 and x2 of the two
main materials used:
Q  x10.5 x20.5
.
Material x1 costs $50 per unit, and material
the cost of producing 100 units of the good.
A) C  50 x1  75 x2
B) x10.5 x20.5  100
Page 25
x2 costs $75 per unit. We want to minimize
What is the objective function?
C) 50 x1  75 x2  100
D) C  50 x10.5 75 x20.5
100.
The quantity, Q, of a good produced depends on the quantities x1 and x2 of the two
main materials used:
Q  x10.5 x20.5 .
Material x1 costs $100 per unit, and material x2 costs $75 per unit. We want to
minimize the cost of producing 100 units of the good. What is the constraint function?
A) C  100 x1  75x2
C) 100 x1  75 x2  100
0.5
0.5
B) x1 x2  100
D) C  100 x10.5 75 x20.5
101.
The quantity, Q, of a good produced depends on the quantities x1 and x2 of the two
main materials used:
Q  x10.5 x20.5
.
Material x1 costs $65 per unit, and material x2 costs $60 per unit. We want to minimize
the cost of producing 100 units of the good. Using Lagrange multipliers, we see that the
minimum cost of ________ occurs when x1 = _____ and x2 = _____.
102.
0.75 0.25
The production function for a company is P  300 x y , where P is the amount
produced given x units of labor and y units of equipment. Each unit of labor costs $800
and each unit of equipment costs $350. Assuming the goal of the company is to
maximize production given a fixed budget of $45,000, what are the objective and
constraint functions?
0.75
0.25
A)
objective: f ( x, y )  x  y ; constraint: 300(800 x 350 y )  45, 000
0.75
0.25
objective: f ( x, y)  800 x 350 y ; constraint: x  y  45, 000
0.75 0.25
C)
objective: f ( x, y)  300 x y ; constraint: 800 x  350 y  45, 000
B)
D)
103.
0.75 0.25
objective: f ( x, y )  800 x  350 y ; constraint: 300 x y  45, 000
0.75 0.25
The production function for a company is P  300 x y , where P is the amount
produced given x units of labor and y units of equipment. Each unit of labor costs $800
and each unit of equipment costs $400. Assuming the goal of the company is to
maximize production given a fixed budget of $55,000, what is the meaning of the
Lagrange multiplier  ?
A) The additional units that can be produced if the budget is increased by $1
B) The amount it would cost the company to produce one more unit
C) The additional units of labor allowed if the budget is increased by $1
D) The amount it would cost the company to obtain one additional unit of equipment
Page 26
104.
0.75 0.25
The production function for a company is P  300 x y , where P is the amount
produced given x units of labor and y units of equipment. Each unit of labor costs $900
and each unit of equipment costs $300. Assuming the goal of the company is to
minimize cost given a fixed production goal of 7000 units produced, what are the
objective and constraint functions?
0.75
0.25
A)
objective: f ( x, y )  x  y ; constraint: 300(900 x 300 y )  7000
0.75
0.25
objective: f ( x, y)  900 x 300 y ; constraint: x  y  7000
0.75 0.25
C)
objective: f ( x, y)  300 x y ; constraint: 900 x  300 y  7000
B)
D)
105.
0.75 0.25
objective: f ( x, y )  900 x  300 y ; constraint: 300 x y  7000
0.75 0.25
The production function for a company is P  300 x y , where P is the amount
produced given x units of labor and y units of equipment. Each unit of labor costs $900
and each unit of equipment costs $300. Assuming the goal of the company is to
minimize cost given a fixed production goal of 7000 units produced, what is the
meaning of the Lagrange multiplier  ?
A) The additional units that can be produced if the budget is increased by $1
B) The amount it would cost the company to produce one more unit
C) The additional units of labor allowed if the budget is increased by $1
D) The amount it would cost the company to obtain one additional unit of equipment
106. Find the maximum value of f ( x, y )  8 xy subject to the constraint equation 2 x  y  16
using Lagrange multipliers.
107. For a production function f ( x, y ) , the maximum production cost of $400 is given by
f (15,32)  90 units, with   0.4 . Estimate the production if the budget cost is
lowered to $390.
108. A mop company can produce P ( x, y ) mops using x units of capital and y units of labor,
with production costs C ( x, y ) dollars. With a budget of $350,000, the maximum
production is 65,000 mops, using $300,000 in capital and $50,000 in labor. The
Lagrange multiplier is   0.4 . Which of the following is true?
A) The objective function is P ( x, y ) and the constraint is C ( x, y )  350, 000 .
The objective function is P ( x, y ) and the constraint is C ( x, y )  65, 000 .
C) The objective function is C ( x, y ) and the constraint is P( x, y )  350, 000 .
D) The objective function is C ( x, y ) and the constraint is P( x, y )  65, 000 .
B)
Page 27
109. A mop company can produce P ( x, y ) mops using x units of capital and y units of labor,
with production costs C ( x, y ) dollars. With a budget of $400,000, the maximum
production is 65,000, using $250,000 in capital and $150,000 in labor. The Lagrange
multiplier is   0.2 . What are the units for  ?
A) Number of budget dollars per mop
B) Number of mops per budget dollar
C) Number of units of capital per mop
D) Number of units of labor per budget dollar
110. A mop company can produce P ( x, y ) mops using x units of capital and y units of labor,
with production costs C ( x, y ) dollars. With a budget of $450,000, the maximum
production is 65,000, using $300,000 in capital and $150,000 in labor. The Lagrange
multiplier is   0.2 . What is the practical meaning of the statement   0.2 ?
A) If the number of units of labor is increased by 1, we expect the number of mops to
increase by about 0.2.
B) If the number of units of capital is increased by 1, we expect the budget to increase
by about $0.2.
C) If the budget is increased by $1, we expect the number of mops to increase by
about 0.2.
D) If the number of mops is increased by 1, we expect the budget to increase by about
$0.2.
111. The quantity, Q, of a good produced depends on the number of workers, W, and the
amount of capital invested, K, according to the Cobb-Douglas function
Q  9W 2 / 3 K 1/ 3 .
In addition, we know that labor costs are $16 per employee, capital costs are $8 per unit,
and the budget is $2500. The maximum production level is about _____ units, where W
= _____ and K = _____. Round to the nearest whole number.
112. The quantity, Q, of a good produced depends on the number of workers, W, and the
amount of capital invested, K, according to the Cobb-Douglas function
Q  9W 2 / 3 K 1/ 3 .
If the maximum production level at a budget of $2500 is 971 and   0.35 , estimate the
production if the budget is increased by $100.
113.
2
2
Suppose that you want to find the maximum and minimum values of f ( x, y)  x  y
subject to the constraint x + 4y = 3.
Use the method of Lagrange multipliers to find the exact location(s) of any extrema.
Page 28
114. Suppose the quantity, q, of a good produced depends on the number of workers, w, and
the amount of capital, k, invested and is represented by the Cobb-Douglas function
3
1
q  6 w 4 k 4 . In addition, labor costs are $20 per worker and capital costs are $20 per
unit, and the budget is $2720. Using Lagrange multipliers, find the optimum number of
workers.
115. The owner of a jewelry store has to decide how to allocate a budget of $540,000. He
notices that the earnings of the company depend on investment in inventory x1 (in
thousands of dollars) and expenditure x2 on advertising (in thousands of dollars)
2
1
3
3
according to the function f ( x, y)  6 x1 x2 .
How should the owner allocate the $540,000 between inventory and advertising to
maximize his earnings?
116. The Green Leaf Bakery makes two types of chocolate cakes, Delicious and Extra
Delicious. Each Delicious requires 0.1 lb of European chocolate, while each Extra
Delicious requires 0.2 lb. Currently there are only 235 lb of chocolate available each
2
2
month. Suppose the profit function is given by: p( x, y)  154 x  0.2 x  200 y  0.1y ,
where x is the number of Delicious cakes and y is the number of Extra Delicious cakes
that the bakery produces each month.
(a) How many of each cake should the bakery produce each month to maximize profit?
(b) What is the value of ? What does it mean?
(c) It will cost $17.00 to get an extra pound of European chocolate. Should the bakery
buy it?
117.
3
2
Let f ( x, y)  kx  75kx  y , where k  0. Find the critical points of f.
118. A person's weight, w, is a function of the number of calories consumed, c, and the
number of calories burned, b. Thus, w  f (c, b) . Would you expect fc (c,b) to be
positive or negative?
119. A television salesman earns a fixed salary of $9.00 per hour plus a $35 commission for
each television he sells. If h is the number of hours worked and s is the number of
televisions sold, find a formula for E (h, s ) , the salesman's total earnings, and use it to
calculate E (30, 26) .
Page 29
120. A television salesman earns a fixed salary of $8.00 per hour plus a $30 commission for
each television he sells. If h is the number of hours worked and s is the number of
televisions sold, find a formula for E (h, s ) , the salesman's total earnings. Is E an
increasing or decreasing function of h?
121. Given the following table of values for f ( x, y ) , estimate fx (10,0.2). Use the next
higher point to make your estimate.
122. Which of the following contour diagrams is more likely to show the population density
of a region of a small town where the center of the diagram is the town center?
I.
II.
123. The monthly car payment for a new car is a function of three variables, P  f (r , A, t ) ,
where r is the annual interest rate, A is the amount borrowed in dollars, and t is time in
( r , A, t )
months before the car is paid off. Would you expect ft
to be increasing or
decreasing?
Page 30
124. A manufacturer sells two products. The first sells for $7 and the second sells for $3.
The manufacturer's costs are given by the equation
C (q1 , q2 )  3q1  2q2  90 ,
where q1 and q2 are the respective quantities produced of each product. Let  (q1 , q2 )
represent the profit at a production level of q1 and q2 units. Find the value of  / q2 .
125.
2
2
( x, y )
If f ( x, y)  x y  y x  3x , find fx
.
A) 2xy + x2 B) 2xy + y2 + 3 C) 4 xy  3
126. If P  5e  rt , find  2 P /  r2.
 rt
 rt
 rt
A) 5r2 e
B) 5t2 e
C) -5r2 e
127.
D) 4xy
D) -5t2 e
 rt
2
If P  4e , find  P /  t  r.
 rt
 rt
 rt
A) 4rte
B) 4rte
C) 4(rt  1)e
 rt
 rt
D) 4(rt  1)e
128.
2
( x, y ) 2 x sin y
If f ( x, y)  x cos y , does fyx
=
?
129.
2
2
The critical point of f ( x, y)  x  3xy  y  9 occurs at x = _____, y = _____, and is a
local ________ (maximum / minimum / neither).
130. A company's cost function to produce x units of one product and y units of a second
product is given by
C ( x, y)  900  4 x 2  4 xy  3 y 2  24 y
.
The minimum cost occurs when x = _____ and y = _____.
131. A company manufactures x units of one item and y units of another. The total cost, C,
in dollars of producing these two items is given by the function
C  6 x 2  xy  4 y 2  1500 .
Use Lagrange multipliers to find the minimum cost subject to the constraint that 100
items (total) must be produced. The minimum cost occurs when x = _____ and y =
_____. Round your answers to the nearest whole number.
132. A company manufactures x units of one item and y units of another. The total cost, C,
in dollars of producing these two items is given by the function
C  6 x2  xy  4 y 2  2500 .
Use Lagrange multipliers to find the minimum cost subject to the constraint that 100
items (total) must be produced. What is the value of  to one decimal place?
Page 31
133. The number, N, of children that can be enrolled in a private school is a function of the
number of certified teachers, T, and the number of aides, A, available, according to the
formula
N (T , A)  35T 0.7 A0.3 .
Teachers average an annual salary of $34,000 and aides average an annual salary of
$21,000. The annual budget for salaries is B = $760,000. If we want to maximize
enrollment, which of the following is true?
0.7 0.3
A)
The objective function is N (T , A)  35T A , and the constraint is
34, 000T  21, 000 A  760, 000 .
B)
0.7 0.3
The objective function is N (T , A)  35T A , and the constraint is
34, 000T  21, 000 A  55, 000 .
C) The objective function is B(T , A)  34, 000T  21, 000 A , and the constraint is
35T 0.7 A0.3  760,000 .
D) The objective function is B(T , A)  34, 000T  21, 000 A , and the constraint is
35T 0.7 A0.3  55,000 .
134. The number, N, of children that can be enrolled in a private school is a function of the
number of certified teachers, T, and the number of aides, A, available, according to the
formula
N (T , A)  35T 0.7 A0.3 .
Teachers average an annual salary of $33,000 and aides average an annual salary of
$20,000. The annual budget for salaries is B = $720,000. Using Lagrange multipliers to
maximize enrollment, we find that T = _____, A = _____, and  = _____. Round T and
A to the nearest whole number; round  to 5 decimal places.
135. Find the least squares line for the five points (1,3), (4,5), (2,4), (2,3), and (0,2) by
completing the following steps:
5
5
First, compute
SX   xi
i 1
b
Next, compute
,
SY   yi
i 1
5
5
,
SXX   xi2
( SXX )( SY )  ( SX )( SXY )
n( SXX )  ( SX )2
i 1
,
m
and
and
SXY   xi yi
i 1
.
n( SXY )  ( SX )( SY )
n( SXX )  ( SX ) 2 .
Rounding b and m to two decimal places, we get y = _____ + _____ x.
136. Consider the four points A = (1, 0), B = (2, 4), C = (3, 5) and D = (4, 2) in the xy-plane.
Find m and b in the line of best fit y = mx + b for these points.
Page 32
Answer Key
1. increasing
section: 9.1
2. decreasing
section: 9.1
3. $3300
section: 9.1
4. $3300
section: 9.1
5. $4700
section: 9.1
6. $8.00
section: 9.1
7. C  45d  0.15m
section: 9.1
8. A
section: 9.1
9. $11,600
section: 9.1
10. increasing
section: 9.1
11. $300.00
section: 9.1
12. B
section: 9.1
13. A
section: 9.1
14. 1982 and 1983
section: 9.1
15. 0.59
section: 9.1
16. 10,500
section: 9.1
17. A
section: 9.1
18. A
section: 9.1
19. increasing
section: 9.2
20. B
section: 9.2
21. C
section: 9.2
22. 100
section: 9.2
Page 33
23. C
section: 9.2
24. B
section: 9.2
25. increasing
section: 9.2
26. A
section: 9.2
27. III
section: 9.2
28. 45
section: 9.2
29. decreasing
section: 9.2
30. The function is decreasing in the southern direction since as we go south the number of
foxes decreases.
section: 9.2
31. positive
section: 9.3
32. positive
section: 9.3
33. A
section: 9.3
34. B
section: 9.3
35. positive
section: 9.3
36. C
section: 9.3
37. negative
section: 9.3
38. –3
section: 9.3
39. 5
section: 9.3
40. 4.6
section: 9.3
41. $226,000
section: 9.3
42. D
section: 9.3
43. A
section: 9.3
44. A
section: 9.3
Page 34
45. f
A (100000,7, 20)
is approximately 0.007753.
section: 9.3
46. A, C
section: 9.3
47. f y (0,1.5)  –2
.
section: 9.3
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
f (80,3)  –1.6
The statement p
means that if the household income is $80,000 and the
price of beef is $3 per pound, then the amount of beef consumed per week will decrease
by approximately 1.6 pounds if the cost of beef increases by $1 per pound.
section: 9.3
A
section: 9.4
B
section: 9.4
True
section: 9.4
False
section: 9.4
5e3
section: 9.4
80
section: 9.4
0
section: 9.4
2
section: 9.4
C
section: 9.4
1416 tons
section: 9.4
56.3
section: 9.4
D
section: 9.4
61. 0
section: 9.4
62. 2s
section: 9.4
63. 2x2/y3
section: 9.4
64. B
 P(0.01r )e0.01rt
t
section: 9.4
Page 35
65. P R

T V .
section: 9.4
66. H P (2, 2) 
0.0938.
67.
68.
69.
70A.
70B.
70C.
71A.
71B.
71C.
72A.
72B.
72C.
73A.
73B.
74A.
74B.
74C.
75.
76A.
76B.
77A.
77B.
77C.
78.
79.
80.
section: 9.4
E
section: 9.4
no
section: 9.5
A
section: 9.5
0.2
40
15
section: 9.5
minimum
4
–4
section: 9.5
neither
5
–10/3
section: 9.5
4/3
-3/2
section: 9.5
477
3
3
section: 9.5
$26
section: 9.5
30.57
29.66
section: 9.5
-4
-6
26
section: 9.5
(2, 2); (-2, -2) are local minima.
section: 9.5
B
section: 9.5
(0, 0) is a critical point that is neither a maximum nor a minimum; (–1, –1) is a local
maximum. There is no global extremum.
section: 9.5
Page 36
81. D
section: 9.5
82. (0,0) is the only critical point. It is a local minimum..
section: 9.5
83. A, C
section: 9.5
84. A
section: 9.5
85. The point (0, 0) is neither a maximum nor a minimum.
 75 1125 
 ,

4  is a local minimum.
The point  2
section: 9.5
86. A, B, C
section: 9.5
87. The point (0,0) is neither a maximum nor a minimum.
section: 9.5
88. Local minimum.
section: 9.5
89. Since fx(a, b) = 0 and fy(a, b) = 0, (a, b) is a critical point of f(x, y). To determine the
nature of the critical point, we need to calculate the second derivatives. We get fxx(a, b)
= -2 and fyy(a, b) = -2. Calculating the mixed partial derivatives, we get fxy(a, b) = 0.
2
Therefore, D  (2)(2)  0  4 . Since fxx(a, b) < 0, it follows that (a, b) is a local
90.
91.
92.
93.
94A.
94B.
95A.
95B.
96A.
96B.
97.
maximum of f(x, y).
section: 9.5
a = 9, b = –6
section: 9.5
Hydro Deluxe: $2750.00 Singing Bird: $1750.00
section: 9.5
The first plant should produce 3143 items and the second plant should produce 34,571
items in order to maximize profit.
section: 9.5
1 
 ,0
 4  is a global minimum.
section: 9.5
23.75
451.25
section: 9.6
8
80
section: 9.6
–0.5
–8
section: 9.6
400
section: 9.6
Page 37
98. 652
section: 9.6
99. A
section: 9.6
100. B
section: 9.6
101A. $12,490.00
101B. 96.08
101C. 104.08
section: 9.6
102. C
section: 9.6
103. A
section: 9.6
104. D
section: 9.6
105. B
section: 9.6
106. 256
section: 9.6
107. 86
section: 9.6
108. A
section: 9.6
109. B
section: 9.6
110. C
section: 9.6
111A. 938 units of the good
111B. 104 workers
111C. 104 units of capital
section: 9.6
112. 1006
section: 9.6
113.  3 12 
 , 
 17 17 
section: 9.6
114. 102
section: 9.6
115. The owner should spend $360,000.00 on the inventory and $180,000.00 on the
advertising.
section: 9.6
116. (a) x  381.11, y  984.44.
(b)   15.56. This means that the profit will increase by $15.56 for every extra pound of
European chocolate.
Page 38
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
129A.
129B.
129C.
130A.
130B.
131A.
131B.
132.
133.
134A.
134B.
134C.
135A.
(c) If the bakery has to spend $17.00 to get that extra pound of chocolate, but will get
back $15.56 (the value of ) in return, it is a bad deal.
section: 9.6
 5,0 and 5,0 .
The critical points of f are
section: 9 review
positive
section: 9 review
$1180.00
section: 9 review
increasing
section: 9 review
–0.6
section: 9 review
II
section: 9 review
decreasing
section: 9 review
1
section: 9 review
B
section: 9 review
B
section: 9 review
C
section: 9 review
yes
section: 9 review
0
0
neither
section: 9 review
3
6
section: 9 review
39
61
section: 9 review
527.8
section: 9 review
A
section: 9 review
15
11
0.00067
section: 9 review
2.09
Page 39
135B. 0.73
section: Deriving the formula for a regression line
136. m  0.7, b  1
section: Deriving the formula for a regression line
Page 40