Chapter 8
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
(100000,7, 20)
.
f
N
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:
Estimate the value of
(100000,7, 20)
f
A
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.
Ans: A, C Learning Objectives: Estimate and interpret partial derivatives.
difficulty: medium section: 8.3
Chapter 8
47.
The table below gives values of a function f(x, y) near x = 1, y = 2.
Estimate
.
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 statement:
I(80,3) 0.022f=
, and include units in your answer.
49.
Find
x
f
if
32
( , )f x y x y=
.
A)
22
3xy
B)
3
2xy
C)
2
6xy
D)
2
5xy
50.
Find
y
f
if
22
( , ) 7f x y x xy y= + +
.
A)
B)
C)
272x x y++
D)
2 7 2xy++
Chapter 8
51.
If
2
1π
3
V r h=
,
2π
3
Vrh
r
=
.
A)
True
B)
False
52.
If
2
4π
3
V r h=
,
8π
3
Vr
h
=
.
A)
True
B)
False
order partial derivatives. difficulty: easy section: 8.4
53.
If
2
( , ) xy
f x y x e=
, find fx(1, 3).
54.
The cost of building a new fence is
20 120C f g=+
, where f is the number of linear
feet of fence and g is the number of gates. Find
/C
f.
Ans:
20
derivatives. difficulty: easy section: 8.4
55.
Use the values of
( , )f x y
in the following table to estimate fyx (3,2). Use the next
higher point to make your estimate.
Ans:
0
a contour diagram. difficulty: medium section: 8.2
Chapter 8
56.
Given the following contour diagram of
( , )f x y
, estimate fx (1,2). 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 %
interest, compounded continuously, is given by the formula
rt
P Be−
=
. Which of the
following gives the amount principal can be changed by and still maintain the same
balance if there is an increase of 1% in the interest rate?
A)
–Br
rt
e−
B)
rt
e−
C)
–Bt
rt
e−
D)
rt
Brte−
−
58.
Suppose the Cobb-Douglas production function for a company is given by
0.70 0.40
50P L K=
, 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 20 thousand dollars and the company employs 16
workers? Round to the nearest ton.
Chapter 8
59.
Suppose the Cobb-Douglas production function for a company is given by
0.70 0.40
50P L K=
, 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.
The volume of a cylinder is given by
2
( , )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
, calculate
2/V
s2 .
62.
For
, calculate
2/V
h
s
63.
For
2
( , ) x
f x y y
=
, calculate fxx (x,y).
Chapter 8
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
100
( , , ) .
rt
B f P r t Pe==
Find
/.Bt
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.
Find
.
P
V
66.
Find the following partial derivative:
p
H
(2, 2) if
3
( , ) .
4
P
H P T PT
=+
Give your
answer to 4 decimal places.
67.
Find the following partial derivative: fxy if
96
( , )f x y x y=
.
A)
95
6
xy
f x y=
B)
86
9
xy
f x y=
C)
85
6
xy
f x y=
D)
85
9
xy
f x y=
E)
85
54
xy
f x y=
Chapter 8
68.
A contour diagram of
( , )f x y
is given in the following figure. Is there a global
minimum near the point (5,3)?
Chapter 8
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 7 at (1, 3.8), a global maximum of
about 15 at (5.2, 4.9), and a global minimum of about -7 at (3, 1.6).
B)
The function has a global maximum of about 7 at (1, 3.8), a local maximum of
about 15 at (5.2, 4.9), and a global minimum of about -7 at (3, 1.6).
C)
The function has a local minimum of about 7 at (1, 3.8), a global minimum of
about 15 at (5.2, 4.9), and a global maximum of about -7 at (3, 1.6).
D)
The function has a global maximum of about 7 at (1, 3.8), a global maximum of
about 15 at (5.2, 4.9), and a global minimum of about -7 at (3,1.6).
Ans: A Learning Objectives: Identify critical points and classify as minimum
values, maximum values, or neither. difficulty: easy section: 8.5
Chapter 8
Page 29
70.
The following table gives values for a function
( , )f x y
. For
0 40x
and
0 20y
, there is a global minimum of at most _____ near the point (_____,_____).
71.
The function
22
( , ) 2 2 10f x y x xy y y= + + +
has a local __________ (maximum /
minimum / neither) at the critical point, where x = _____ and y = _____.
Part A:
minimum
Part B:
5
Part C:
72.
The function
2
( , ) 3 15f x y x xy y= + −
has a local __________ (maximum / minimum /
neither) at the critical point, where x = _____ and y = _____.
Part A:
neither
Part B:
5
Part C:
values, or neither. difficulty: easy section: 8.5
73.
The function
3 2 2
( , ) 2 3 8f x y x y x y= + − + −
has a critical point which is a local
minimum when x = _____ and y = _____.
Part A:
4/3
Part B:
-3/2
values, or neither. difficulty: medium section: 8.5
Part B:
40
Part C:
15
values, or neither. difficulty: medium section: 8.5
Chapter 8
Page 30
74.
The function
22
( , ) 600 2 6 2 3 2f x y x x xy y y= − + + − +
has a local maximum value of
_____ when x = _____ and y = _____.
75.
Two products are manufactured in quantities
1
q
and
2
q
and sold at prices $8 and $12
respectively. The cost of producing them is given by
22
12
26C q q= + +
. Find the
maximum profit that can be made.
Ans:
$28
values, or neither. difficulty: medium section: 8.5
76.
A company sells two styles of jeans in quantities
1
q
and
2
q
at prices
1
p
and
2
p
respectively, with
1
p
and
2
p
in dollars. The quantities demanded depend on both
1
p
and
2
p
according to the formulas
1 1 2
80 6 4q p p
= − +
and
2 1 2
140 5 7q p p= + −
.
Write a formula for total sales revenue as a function of
1
p
and
2
p
, and use it to
determine what prices the company should charge to maximize sales revenue. They
should charge $_____ for style
1
q
and $_____ for style
2
q
.
Part A:
27.36
Part B:
27.59
values, or neither. difficulty: medium section: 8.5
77.
Suppose
22
( , )f x y x Ax y By C= + + + +
. Then
( , )f x y
has a local minimum value of
9 at the point (2, 3) when A = _____, B = _____, and C = _____.
Part A:
-4
Part B:
-6
Part C:
22
values, or neither. difficulty: medium section: 8.5
Part A:
607
Part B:
2
Part C:
1
values, or neither. difficulty: medium section: 8.5
Chapter 8
78.
Find all the critical points of the function
22
88
( , ) .f x y xy xy
= + +
Classify these critical points as local maxima, local minima, or neither.
Ans:
(2, 2); (-2, -2) are local minima.
Learning Objectives: Identify critical points and classify as minimum values, maximum
values, or neither. difficulty: hard section: 8.5
79.
Is (0, 0) a critical point of the following function?
A)
Yes: (global) maximum.
B)
Yes: neither a maximum nor a minimum.
C)
Yes: (global) minimum.
D)
No.
Ans: B Learning Objectives: Identify critical points and classify as minimum
values, maximum values, or neither. difficulty: medium section: 8.5
80.
Let
33
( , ) 6 10.h x y x y xy= + + +
Determine all local critical points. Are the local extrema also global extrema?
Ans:
(0, 0) is a critical point that is neither a maximum nor a minimum; (–2, –2) is a
local maximum. There is no global extremum.
Learning Objectives: Identify critical points and classify as minimum values, maximum
values, or neither. difficulty: medium section: 8.5
Chapter 8
Page 32
81.
Let
33
( , ) 3 2.h x y x y xy= + + +
Which figure best represents the contours of this function?
A)
B)
C)
D)
Chapter 8
values, maximum values, or neither. difficulty: easy section: 8.5
82.
Suppose that
22
( , ) 6 .f x y x xy y= + +
Find and classify the critical point(s) as local maxima, local minima, or neither.
Ans:
(0,0) is the only critical point. It is neither a maximum nor a minimum..
values, or neither. difficulty: easy section: 8.5
83.
Find all the critical points of
32
( , ) 3 6f x y x x y y= − + −
and classify each as maximum,
minimum, or neither.
Select all possible choices.
A)
The point (1, 3) is a local minimum.
B)
The point (-1, –3) is neither a maximum nor a minimum.
C)
The point (-1, 3) is neither a maximum nor a minimum.
D)
The point (1, 3) is a local maximum.
E)
The point (1, -3) is a local maximum.
values, maximum values, or neither. difficulty: easy section: 8.5
84.
The function
32
( , ) 3 5f x y x x y y= − + −
has a local minimum at (1, 2.5). Which of the
following is a sketch of the level curves of f near this point?
A)
B)
values, maximum values, or neither. difficulty: easy section: 8.5
Chapter 8
85.
Find the critical points of
32
( , ) 3f x y x xy y= − +
and classify each as maximum,
minimum or neither.
Learning Objectives: Identify critical points and classify as minimum values, maximum
values, or neither. difficulty: medium section: 8.5
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
Ans: A, B, C Learning Objectives: Identify critical points and classify as minimum
values, maximum values, or neither. difficulty: easy section: 8.5
87.
Consider the function
4 3 2 2 3 4
( , ) 2 8 7 4.f x y x x y x y xy y= + − + − +
Check that (0,0) is a critical point of f and classify it as a local minimum, local
maximum or neither.
Ans:
The point (0,0) is neither a maximum nor a minimum.
Learning Objectives: Identify critical points and classify as minimum values, maximum
values, or neither. difficulty: easy section: 8.5
88.
The point (0, 2) is a critical point of
32
( , ) 4 96 48 .g x y x xy xy= − +
Classify it either as a local minimum, local maximum, or neither.
Ans:
Neither a maximum nor a minimum.
Learning Objectives: Identify critical points and classify as minimum values, maximum
values, or neither. difficulty: medium section: 8.5