Section 1.5
1) A salesperson is paid $200 a week, plus $20 for every sale she makes. Let x = the number of
sales made in a week. Let y = her weekly income in dollars. Model this situation numerically,
graphically, and algebraically.
2) The graph shows the amount remaining to be paid
on a 30-year mortgage over the life of the mortgage.
a) What is the slope of the graph and explain the
meaning of the slope in this context?
b) What is the rate of change of the mortgage
balance during this time period?
c) Identify the horizontal and vertical axis intercepts
and explain their significance to the mortgage
holder.
d) What is the equation of the line?
3) According to a 1979 Build and Blood Pressure Study by the Society of
h
w
Actuaries, the average weight w (in pounds) of American men in their
68
167
sixties for various heights (in inches) is given in the following table.
69
172
70
176
a) Use your calculator to construct a scatter plot
71
181
of the data using the horizontal axis for heights.
72
186
73
191
b) Do the data fall on a line?
74
196
c) Find a linear model for these data.
75
200
d) Use the model to estimate the average weight of all men in their sixties
whose height is 71.25 inches.
e) Use the model to estimate the average weight of all men in their sixties who are 5’6” tall.
f) Use the model to estimate the average height of all men in their sixties who weigh 320 pounds.
4) Consider the following data from the World Almanac.
Year Total Personal Income
1990 $4,679,800,000,000
1991 $4,828,400,000,000
1992 $5,058,100,000,000
1993 $5,375,100,000,000
1994 $5,701,700,000,000
Align this data to make the numbers more manageable.
Section 2.1
1) Suppose $1000 is deposited into an account that earns 4% interest compounded annually and
no further deposits or withdrawals are made. Let A(t) denote the amount in the account t years
after the initial deposit. Find a formula for A(t) and graph it.
Section 2.1 (cont.)
2) The strength in any signal in a fiber optic cable, such as the type used for telephone and cable
TV lines, diminishes 15% for every 10 miles.
a) Find a model for the strength of a signal remaining after any number of miles.
b) How much of the signal is left after 100 miles?
c) How far does a signal go until its strength is down to 1% of the original level?
3) Total per capita health expenditures in the United
Year
Per Capita Health
States for the years 1970 to 1982 are given in the table.
Expenditures
a) Check to see if the increase is approximately
exponential.
1970
349
1972
428
b) Find an exponential model for the data.
1974
521
c) According to the model, what was the yearly
1976
665
1978
822
percentage change in expenditures?
1980
1054
1982
1348
Section 2.2
1) You have two investment choices, one with an APR of 4.9% compounded continuously and
the other with an APR of 5% compounded annually. Which is the better option?
2) An investment that has interest compounded continuously has an APY of 8.4%. What is the
APR (rounded to the nearest tenth of a percent)?
3) You need a new computer before you start graduate school in two years. How much should
you invest now in a C.D. paying 5.1% interest compounded monthly so that you will have the
$2500 cost of the computer in 2 years?
Section 2.3
1) A company that makes satellite dishes for TV’s has asked you to analyze the market. Their
biggest competitors are the cable companies. The table shows the percentage of U.S. households
with cable (Source: Neilson Media Research). Find a model for this data.
YEAR
‘77 ‘78 ‘79 ‘80 ‘81 ‘82 ‘83 ‘84 ‘85 ‘86 ‘87 ‘88 ‘89 ‘90 ‘91 ‘92 ‘93 ‘94
PERCENTAGE 16.6 17.9 19.4 22.6 28.3 35.0 40.5 43.7 46.2 48.1 50.5 53.8 57.1 59.0 60.6 61.5 62.5 63.4
Section 2.4
1) Data from 4 functions are shown in the table. One
function is linear, one is exponential, one is quadratic, and
one is cubic. Determine which function is which and use
regression to find the appropriate model.
x
3
3.5
4
4.5
5
f(x)
28.8
39.2
51.2
64.8
80.0
g(x)
4.39
5.01
5.71
6.51
7.43
h(x)
10.80
12.96
15.55
18.66
22.39
k(x)
28.8
39.2
49.6
60.0
70.4
Section 3.1
1) Find the average rate of change of y with respect to x on the graph of y = 2x +1 from the point
(2,5) to the point (10,21).
2) Find the average rate of change of y with respect to x on the graph of y = x2 from the point
(2,4) to the point (5, 25).
3) Suppose that the percentage of U.S. homes having record players is modeled by
Percentage =
90
1+.01e .28 x
where x is the number of years since 1975.
a) How did the percentage change from 1975 to 1995?
b) How rapidly did the percentage change from 1975 to 1995?
Section 3.2
1) In homework problem 17b from section 2.3, we saw that the price (in cents) for first class
postage can be modeled by the function y = .014x2 - 1.242x + 29.427 cents, x years after 1900.
a) Find the average rate of change of the price of first class postage from 1980 to 1990.
b) Estimate the rate of increase in first class postage in 1990.
2) Using the graph from problem 20 on page 189, answer the following:
a) In about 1987, the death rates from prostate cancer and colon/rectum cancer were the same.
Were the instantaneous rates of change of the cancers also equal in 1987?
b) In 1947, which type of cancer was growing most rapidly?
c) In 1990, which type of cancer was growing most rapidly?
3) Draw secant lines through neighboring points to find the tangent line at each of the indicated
points.
Section 4.5
1) Suppose that gas costs $1.20 per gallon and that we are using gas at a rate of 3 gallons per
minute. What is the rate of change of cost with respect to time, in dollars per minute?
Section 6.1
1) An office worker assembles advertising portfolios. As fatigue sets in the number of portfolios
he can assemble per hour decreases. Using regression, it is determined that he can assemble
f ( t ) = 20 − t 2 portfolios/hour t hours after he begins work. How many portfolios can he
assemble in 3 hours?
2) Find the area under the graph of f ( x ) = 20 − x 2 and above the x-axis between
x = 0 and x = 3.
Section 6.3
1) Given a function f that is continuous (or piecewise continuous) and nonnegative from x = a to
x = b, find the area of the region below the graph of f and above the x-axis from a to b.
Section 7.3
1) To prepare for your future retirement (in 45 years), you begin investing $100 per month in an
annuity with a fixed rate of return of 7.5%.
a) Assuming that interest is also compounded monthly, what will the annuity be worth at the end
of 45 years?
b) Assuming a continuous stream, what will the annuity be worth at the end of 45 years?
c) How much would you have to invest now, in one lump sum instead of a continuous stream, in
order to build to the same future value as in part b? Assume that interest is compounded
continuously.
d) How much would you have to invest now, in one lump sum instead of a monthly stream, in
order to build to the same future value as in part a? Assume monthly compounding of interest.
Section 7.4
1) Assume the quantity demanded (in thousands) of TI-92 calculators for various per-calculator
prices is given in the table. Find the amount consumers are willing and able to spend for 30,000
calculators.
Price of TI-92 (in $) 180
Quantity demanded
30
(in thousands)
190
25
200
20
210
15
225
10
275
5
Section 9.1
1. Suppose you are a beef producer and you want to know how much beef people will buy. This
depends on two things: i) how much money people have,
and ii) the price of beef.
Let I = household income (in thousands of dollars per year)
p = price of beef (in dollars per pound)
C = consumption of beef (in pounds per week per household)
C is a function of both I and p as shown in the table:
Quantity of beef bought (in pounds/week/household)
I, income
p, price (in $/lb.)
(thousand $/yr.) 3.00
3.50
4.00
4.50
20
2.65 2.59
2.51
2.43
40
4.14 4.05
3.94
3.88
60
5.11 5.00
4.97
4.84
80
5.35 5.29
5.19
5.07
100
5.79 5.77
5.60
5.53
Source: An Introduction
rd
to Positive Economics, 3 edition, by Richard G. Lipsey
a) Write a sentence interpreting each of the following mathematical notations:
i) C(20, 3.50) = 2.59
ii) C(80, p)
iii) C(I, 4.50)
b) Find an appropriate cross-sectional model for the consumption of beef as a function of the
price of beef when the household income is $20,000/yr.
c) Use your answer to (b) to estimate the consumption of beef when household income is
$20,000/yr. and the price of beef is $3.90/lb.
2. The amount of money A (in dollars) in an account is given by A( P , r , n , t ) = P (1 +
r nt
) ,
n
where P is the original principal (in dollars), r is the annual percentage rate (expressed as a
decimal), n is the number of times per year interest is compounded, and t is the number of years
interest is added.
a) Find A(100, .03, 12, 5)
b) Write the cross-sectional model for r = .02 and n = 12.
c) Find and interpret A(1000, r, n, 7).
Section 9.3
1) It is estimated that the weekly output of a certain plant is given by the function
Q ( x , y ) = 1200 x + 500 y + x 2 y − x 3 − y 2 units, where x is the number of skilled workers
and y the number of unskilled workers employed at the plant.
∂Q
∂x
∂Q
b) Find
∂y
a) Find
c) Find and interpret
∂Q(100,50)
∂x
d) When there are 30 skilled workers and 60 unskilled workers employed at the plant, how
rapidly is weekly output increasing with respect to the number of skilled workers.
2) Referring to the beef consumption example of section 9.1, when the household income is
$20,000 and the price of beef is $4.50, how rapidly is beef consumption changing with respect to
household income?
3) Referring to the function Q ( x , y ) = 1200 x + 500 y + x 2 y − x 3 − y 2 from example 1, find
the second partial derivatives Qxx ( x , y ), Qxy ( x , y ), Q yx ( x , y ), and Q yy ( x , y ) .
Section 10.2
1) Evaluate the determinant of each matrix:
a)
3 10
2 1
b)
6 −1
1 0
2) The annual cost for labor and specialized robotics equipment for an automobile manufacturer,
in millions of dollars, is given by C ( x , y ) = 5 x + 5 xy + 7.5 y − 40 x − 45 y + 135 , where x
is the amount (in millions of dollars) spent each year on labor and y is the amount (in millions of
dollars) spent each year on robotics equipment. Determine how much should be spent on each,
per year, to minimize cost. Determine the minimum cost.
2
2
3) Determine the relative extrema of f ( x , y ) = x 3 + 3 x 2 − y 2 + 2 y + 4
4) The only grocery store in a small rural community carries two brands of frozen apple juice, a
local brand that it obtains at the cost of 30 cents per can and a well-known national brand that it
obtains at the cost of 40 cents per can. The grocer estimates that if the local brand is sold for x
cents per can and the national brand for y cents per can, approximately 70 - 5x + 4y cans of the
local brand and 80 + 6x - 7y cans of the national brand will be sold each day. How should the
grocer price each brand to maximize the profit from the sale of the juice?