Chapter 1
1.
()fx
is the age of Antarctic ice (in hundreds of years) at a depth of x meters below the
surface. Is f increasing or decreasing?
2.
A graph of
()y f x=
is given in the following figure.
A. What is
(0)f
(to the nearest whole number)?
B. What is the range of the function?
Chapter 1
3.
From the following table,
A. Find f(4)
B. Find the value(s) of x for which
( ) 0fx=
. If there is more than one, list them in
increasing order, separated by commas.
x
1
2
3
4
5
6
f(x)
0
3
7
6
4
0
Ans:
A. 6
B. 1, 6
4.
Let
2
( ) 2 1y f x x= = −
.
A. Find the value of y when x is zero.
B. Find f(3).
Ans:
A. –1
B. 17
Chapter 1
5.
The empirical function
()P g t=
graphed below represents the population P of a city (in
thousands of people) at time t. The _____ of the function is from 1900 to 1980, and the
_____ of the function is from approximately 35,000 to 70,000 people.
6.
Could the following table represent a linear function? Answer yes or no.
t
0
1
2
3
d
0
15
30
45
Chapter 1
Page 4
7.
A. Which two lines in the following figure have the same slope? Enter your answer as
“I and II,” etc.
B. Which two lines have the same y-intercept?
C. Which line has the largest slope?
D. Which line has the largest y-intercept?
8.
The average weight in pounds of American men in their sixties (in 1979) as a function
of their heights in inches is given in the following table. The formula that expresses the
weight w in terms of the height h is given by w = _____+_____h
height (h)
68
69
70
71
72
73
weight (w)
164
170
176
182
188
194
Chapter 1
9.
Suppose that
()y f t=
is the distance in miles traveled in t hours by a car moving at 70
miles per hour. Give a formula for the function
()ft
.
10.
Find the value for b in the following table of values for the linear function f.
x
0
5
10
15
20
()fx
10
15
a
b
c
11.
Find a formula for the linear function f.
.
x
0
100
200
300
400
()fx
10
20
?
?
?
A)
( ) 0.1 10f x x=+
B)
( ) 100 10f x x=+
C)
( ) 20 10f x x=+
D)
( ) 0.2 10f x x=+
12.
A car is worth $15,000 when it is 1 year old, and it is worth $8,000 when it is three
years old.
A. Write the value of the car, V (in dollars), as a function of the age of the car, a (in
years). Assume this is a linear function.
B. How much does the car depreciate in value each year?
C. How much was the car worth when it was first purchased?
Chapter 1
13.
The equation of the line through the points (1, 2) and (–1, –4) is:
Learning Objectives: Understand the properties and terminology of functions:
input/output, function notation, intercepts, increasing/decreasing.
difficulty: medium section: 1.2
14.
The bill for electricity is $200 when 55 kilowatt hours are used and $300 when 95
kilowatt hours are used.
A. The base cost (without using any electricity) is $______.
B. Each additional kilowatt hour used costs $_____.
Ans:
A. 62.50
B. 2.50
Learning Objectives: Interpret properties of linear functions: slope, intercepts.
difficulty: medium section: 1.2
15.
A school library opened in 1980. In January of 2000, they had 33,000 books. One
year later, they had 33,240 books. Assume they acquire the same number of books at
the start of each month.
A. How many books did they have in January of 2003?
B. How many books did they have in July of 1980?
Ans:
A. 33,720
B. 28,320
Learning Objectives: Build linear functions from data, words, or graphs.
difficulty: easy section: 1.2
16.
A school library opened in 1980. In January of 2000, they had 16,000 books. One
year later, they had 16,840 books. Assuming they acquire the same number of books
at the start of each month, give a linear formula for the number of books, N, in the
library as a function of the number of years, t, the library has been open.
Learning Objectives: Build linear functions from data, words, or graphs.
difficulty: easy section: 1.2
Chapter 1
17.
A furniture moving company charges a fixed amount plus a charge for each pound that
they move. A person who shipped 60 pounds of furniture was charged $280, while
someone else was charged $610 to ship 170 pounds.
A. Write a function that represents the moving cost, C, in terms of pounds, x, and fixed
cost.
B. Suppose the company changes their rates. They increase the per pound charge by
$1 but cut the fixed amount they charge by half. What is the new function that
represents the new moving cost, D?
C. Will someone who ships 170 pounds pay more or less with the new rates than they
would have with the original rates?
Chapter 1
18.
Harley Davidson (ticker symbol HOG) stock prices dropped sharply in late 2008.
Series 1 in the graph below shows the actual prices at the end of each week. The trend
over time is approximately linear; and the graph of a possible linear model is given by
Series 2. Based on the data given, find the linear model and use it to approximate the
stock’s price on November 30, 2008, assuming the current trend continued.
0
5
10
15
20
25
30
35
40
45
1
2
3
4
5
6
7
8
9
10
11
D
o
l
l
a
r
s
Week Number
Harley Davidson (HOG)
Weekly Closing Prices
September 1 – November 15, 2008
Series1
Series2
19.
The height (in inches) and weight (in pounds) of 8 students is given in the following
table. Find a regression line for this data and use it to estimate the weight of a person
who is 5 feet 1 inches tall. Round to the nearest pound.
Height (inches)
64
68
62
70
69
65
73
71
Weight (pounds)
110
150
115
185
160
125
200
170
Chapter 1
20.
Do you expect the average rate of change in the number of smart phones in the U.S.
since 2000 to be positive or negative?
Ans:
positive
Learning Objectives: Understand interpretations of average rate of change on an
interval: increasing/decreasing, concavity, slope of secant line, average velocity.
difficulty: easy section: 1.3
21.
The population of Los Angeles, California was 2,811,801 in 1970 and was 3,448,613 in
1994. The average rate of change in the population of Los Angeles between 1970 and
1994 was _____ people per year.
Ans:
26,534
Learning Objectives: Find and give units for average rate of change of a function on an
interval. difficulty: easy section: 1.3
22.
The following table gives the number of students taking an applied calculus course at a
community college. Find the change in the number of students taking the course
between 2001 and 2004.
Year
1998
1999
2000
2001
2002
2003
2004
students
311
358
383
440
496
574
631
Ans:
191
23.
The following table gives the number of students taking an applied calculus course at a
community college.
A. Find the average rate of change in the number of students taking the course between
2000 and 2004 (in students per year).
B. If the average rate of change continues at the same rate as between 2000 and 2004,
in which year will the number of students taking the course first exceed 900?
Year
1998
1999
2000
2001
2002
2003
2004
students
312
360
384
439
487
566
636
Ans:
A. 63
B. 2009
Learning Objectives: Find and give units for average rate of change of a function on an
interval. difficulty: medium section: 1.3
Chapter 1
24.
The number of reported offenses of violent crime in the U.S. between 1983 and 1996 is
given in the following table.
A. Find the average rate of change between 1983 and 1992 (to the nearest integer).
B. Find the average rate of change between 1992 and 1996 (to the nearest integer).
25.
The total sales of household computers in the U.S., as measured by sales to retail
consumer dealers, in millions of dollars, was 2,385 in 1984 and 16,585 in 1997.
A. Find the average rate of change in sales between 1984 and 1997 (in millions of
dollars, to the nearest tenth).
B. Use your answer to estimate total sales in 2000 (in millions of dollars, to the nearest
tenth).
Chapter 1
26.
The distance
()d f t=
in feet that a golf ball will fall in t seconds if dropped from a very
high tower is given by the formula
2
( ) 16f t t=
. Make and label a table or a graph of
values of f(t) giving distances fallen for the time period
08t
. Using your table or
graph, the change in the height of the golf ball between times t = 5 and t = 7 is _____
feet, and the average rate of change in the height of the golf ball between times t = 5 and
t = 7 is____ feet per second.
Part A:
Part B:
interval. difficulty: medium section: 1.3
27.
Values for g(x) are given in the following table. Does it appear that g(x) is concave up
or concave down?
x
1
2
3
4
5
6
g(x)
–100
–90
–81
–73
–66
–60
Ans:
concave down
Learning Objectives: Understand interpretations of average rate of change on an
difficulty: easy section: 1.3
28.
Consider the following graph. Between point A and point B, the graph is: (mark all
that apply)
A)
decreasing
B)
increasing
C)
concave up
D)
concave down
Ans: A, C Learning Objectives: Understand interpretations of average rate of
velocity. difficulty: easy section: 1.3
Chapter 1
29.
Find the average rate of change of
2
( ) –9.8 80 14s t t t= + −
between t = 1 and t = 3.
Round to two decimal places.
A)
40.80
B)
81.60
C)
120.8
D)
43.92
Ans: A Learning Objectives: Find and give units for average rate of change of a
function on an interval. difficulty: medium section: 1.3
30.
The table gives information about the number of cases of pancreatic cancer diagnosed in
the United States.
Year
1997
2002
2004
Number of cases
27,000
30,300
31,860
a) Find the average rate of change in number of cases from 1997 to 2002.
b) Find the average rate of change in number of cases from 2002 to 2004.
c) Is the average rate of change increasing or decreasing?
Learning Objectives: Find and give units for average rate of change of a function on an
interval. difficulty: medium section: 1.3
31.
Values of a linear cost function are given in the following table. Find a formula for the
cost function.
q
0
100
200
300
()Cq
140
225
310
395
Chapter 1
32.
A textbook company had fixed costs of $15,000 and variable costs of $20 for a certain
book. The company sells the books for $30 each. Find a formula for the profit
function,
()q
.
Learning Objectives: Understand cost, revenue, and profit functions and break even
33.
A textbook company had fixed costs of $15,000 and variable costs of $20 for a certain
book. The company sells the books for $40 each. What is the break-even point for the
company (to the nearest book)?
Ans:
750
Learning Objectives: Understand cost, revenue, and profit functions and break even
points. difficulty: hard section: 1.4
34.
A $3000 pump depreciates linearly . It is worth $2200 in 4 years. Find a formula for
the value of the pump, V, as a function of time, t (in years) since it was purchased.
Learning Objectives: Understand cost, revenue, and profit functions and break even
points. difficulty: medium section: 1.4
35.
A $2000 pump depreciates linearly. It is worth $1200 in 4 years. How many years
will it be before the pump is worth nothing?
Ans:
10
Learning Objectives: Understand cost, revenue, and profit functions and break even
points. difficulty: medium section: 1.4
36.
A premium ice cream company finds that at a price of $5.50, demand for their ice cream
cones is 2500. For each $0.25 increase in price, the demand decreases by 50. Graph
the revenue function and find the price that will maximize revenue.
Ans:
$9.00
taxation, budget constraints. difficulty: medium section: 1.4
Chapter 1
37.
The following graph shows the quantity of goods purchased by consumers at various
prices. If the price is $15 per item, how many items do consumers purchase?
A)
5
B)
8
C)
12
D)
15
demand, taxation, budget constraints. difficulty: easy section: 1.4
Chapter 1
38.
The following figure gives both supply and demand curves for a certain product. If the
price is $50 per item, how many items will the consumers buy?
A)
1000
B)
2400
C)
1700
D)
4000
demand, taxation, budget constraints. difficulty: easy section: 1.4
Chapter 1
39.
The following figure gives both supply and demand curves for a certain product. If the
price is $75 per item, would you expect the market pressures to push the price higher or
lower?
A)
higher
B)
lower
demand, taxation, budget constraints. difficulty: easy section: 1.4
40.
Suppose that
()Sq
is the price per unit (in dollars) of widgets which will induce
producers to supply q thousand widgets to the market, and suppose that
()Dq
is the
price per unit at which consumers will buy q thousand units. Which is larger, S(150) or
S(100)?
A)
S(100)
B)
S(150)
break even points. difficulty: easy section: 1.4
41.
Suppose that
()Sq
is the price per unit (in dollars) of widgets which will induce
producers to supply q thousand widgets to the market, and suppose that
()Dq
is the
price per unit at which consumers will buy q thousand units. If
(150) 10D=
and
(100) 10S=
, what do you predict about the future selling price of widgets (currently at
$10)?
A)
It will rise.
B)
It will fall.
break even points. difficulty: medium section: 1.4
Chapter 1
42.
A teenager has $36 to spend at a carnival on both food and rides. Food costs (on
average) $4 per item, and rides cost (on average) $2 each. Let f be the number of food
items purchased and r be the number of rides purchased. What is the equation of the
teenager’s budget constraint?
A)
36 4 2fr=−
B)
36 2 4fr=+
C)
36 4 2fr=+
D)
36( ) 4 2fr+ = +
Ans: C Learning Objectives: Understand economic applications to supply and
demand, taxation, budget constraints. difficulty: medium section: 1.4
43.
Production costs for manufacturing t-shirts consist of a fixed cost of $12,000 plus
variable costs of $2 per shirt. Each t-shirt sells for $7 dollars. Find the total profit,
π( )q
, as a function of the number of shirts produced, q.
A)
( ) 2 12,000qq
=−
B)
( ) 7 12,000qq
=−
C)
( ) 9 12,000qq
=−
D)
( ) 5 12,000qq
=−
Ans: D Learning Objectives: Understand cost, revenue, and profit functions and
break even points. difficulty: medium section: 1.4
44.
Production costs for manufacturing T-shirts consist of a fixed cost of $18,000 plus
variable costs of $4 per shirt. Each T-shirt sells for $12 dollars. What is the marginal
cost?
A)
$4
B)
$12
C)
$8
D)
$18,000
Ans: A Learning Objectives: Understand cost, revenue, and profit functions and
break even points. difficulty: medium section: 1.4
45.
Production costs for manufacturing T-shirts consist of a fixed cost of $18,000 plus
variable costs of $2 per shirt. Each T-shirt sells for $8 dollars. How many T-shirts
must be sold for the company to break even?
A)
2,250
B)
3,000
C)
9,000
D)
5,625
Ans: B Learning Objectives: Understand cost, revenue, and profit functions and
break even points. difficulty: medium section: 1.4
Chapter 1
46.
The demand and supply curves for a certain product are given in terms of price, p, by
( ) 310 11D p p=−
and
( ) 4 110S p p=−
.
What is the equilibrium price?
A)
$7
B)
$2
C)
$28
D)
$200
break even points. difficulty: hard section: 1.4
47.
The demand and supply curves for a certain product are given in terms of price, p, by
( ) 380 8D p p=−
and
( ) 6 110S p p=−
.
What is the equilibrium quantity?
A)
35
B)
100
C)
2
D)
270
demand, taxation, budget constraints. difficulty: hard section: 1.4
48.
The cost of producing q items in a tortilla factory is given by
( ) 3000 0.71C q q=+
dollars. The revenue from sales of q items is
( ) 0.89R q q=
dollars.
A. For what values of q does the tortilla factory make a profit?
B. Write a formula for profit as a function of q.
points. difficulty: medium section: 1.4
49.
A population is growing according to the formula
400(1.065)t
P=
, where P is the
population at year t. What is the initial population?
A)
507
B)
426
C)
107
D)
400
Chapter 1
50.
A population is growing according to the formula
325(1.05)t
P=
, where P is the
population at year t. What is the annual growth rate?
A)
3.25%
B)
3.41%
C)
5%
D)
10.25%
51.
A population is growing according to the formula
250(1.07)t
P=
. What is the
population in year 9?
A)
460
B)
2,408
C)
7,013
D)
252
exponential functions: percent growth/decay rate, base, initial quantity.
difficulty: easy section: 1.5
52.
A population is growing according to the formula
250(1.05)t
P=
, where P is the
population in year t. How many years will it take for the population to exceed 1000?
A)
151
B)
28
C)
29
D)
150
exponential functions: percent growth/decay rate, base, initial quantity.
difficulty: hard section: 1.5
53.
A town has 2400 people initially. Find the formula for the population of the town, P, in
terms of the number of years, t, if the town grows by 70 people a year.
A)
2400 70Pt=+
B)
2400(70)t
P=
C)
2400(0.7)t
P=
D)
2400 (0.7)t
P=+
Ans: A Learning Objectives: Determine a formula for an exponential function
from data, graphs, or words. difficulty: medium section: 1.5
Chapter 1
54.
A town has 1200 people initially. Find the formula for the population of the town, P, in
terms of the number of years, t, if the town grows at an annual rate of 8% a year.
A)
1200 8Pt=+
B)
1200(0.08)t
P=
C)
1200(1.08)t
P=
D)
1000 (1.08)t
P=+
Ans: C Learning Objectives: Determine a formula for an exponential function
55.
A town has 2400 people initially. Find the formula for the population of the town, P, in
terms of the number of years, t, if the town shrinks by 70 people a year.
A)
2400 70Pt=−
B)
2400(0.3)t
P=
C)
2400(0.7)t
P=
D)
2400 (0.7)t
P=−
Ans: A Learning Objectives: Determine a formula for an exponential function
from data, graphs, or words. difficulty: medium section: 1.5
56.
A town has 800 people initially. Find the formula for the population of the town, P, in
terms of the number of years, t, if the town shrinks at an annual rate of 13% a year.
A)
800 13Pt=−
B)
800( 0.13)t
P=−
C)
800(0.87)t
P=
D)
800 (0.87)t
P=−
Ans: C Learning Objectives: Determine a formula for an exponential function
from data, graphs, or words. difficulty: medium section: 1.5