20
CHAPTER 0
.
.
Preliminaries
020
76.
The spin rate of a golf ball hit with a 9 iron has been measured
at 9100 rpm for a 120compression ball and at 10,000 rpm
for a 60compression ball. Most golfers use 90compression
balls. If the spin rate is a linear function of compression, find
the spin rate for a 90compression ball. Professional golfers
often use 100compression balls. Estimate the spin rate of a
100compression ball.
77.
The chirping rate of a cricket depends on the temperature. A
species of tree cricket chirps 160 times per minute at 79
◦
F and
100 times per minute at 64
◦
F. Find a linear function relating
temperature to chirping rate.
78.
When describing how to measure temperature by counting
cricket chirps, most guides suggest that you count the number
of chirps in a 15second time period. Use exercise 77 to explain
why this is a convenient period of time.
79.
A person has played a computer game many times. The statis
tics show that she has won 415 times and lost 120 times, and
the winning percentage is listed as 78%. How many times in a
row must she win to raise the reported winning percentage to
80%?
EXPLORATORY EXERCISES
1.
Suppose you have a machine that will proportionally enlarge a
photograph. For example, it could enlarge a 4
×
6 photograph
to 8
×
12 by doubling the width and height. You could make
an 8
×
10 picture by cropping 1 inch off each side. Explain
how you would enlarge a 3
1
2
×
5 picture to an 8
×
10
.
A friend
returns from Scotland with a 3
1
2
×
5 picture showing the Loch
Ness monster in the outer
1
4
on the right. If you use your proce
duretomakean8
×
10 enlargement,doesNessiemakethecut?
2.
Solve
the
equation

x
−
2
 + 
x
−
3
 =
1
.
(Hint:
It’s
an
unusual
solution,
in
that
it’s
more
than
just
a
couple
of
numbers.)
Then,
solve
the
equation
x
+
3
−
4
√
x
−
1
+
x
+
8
−
6
√
x
−
1
=
1. (Hint: If you
make the correct substitution, you can use your solution to the
previous equation.)
0.2
GRAPHING CALCULATORS AND COMPUTER
ALGEBRA SYSTEMS
The relationships between functions and their graphs are central topics in calculus. Graphing
calculators and userfriendly computer software allow you to explore these relationships for
a much wider variety of functions than you could with pencil and paper alone. This section
presents a general framework for using technology to explore the graphs of functions.
Recall that the graphs of linear functions are straight lines and the graphs of quadratic
polynomials are parabolas. One of the goals of this section is for you to become more
familiar with the graphs of other functions. The best way to become familiar is through
experience, by working example after example.
EXAMPLE 2.1
Generating a Calculator Graph
Use your calculator or computer to sketch a graph of
f
(
x
)
=
3
x
2
−
1
.
y
x
4
2
4
2
20
40
60
FIGURE 0.26a
y
=
3
x
2
−
1
y
x
2
1
1
2
4
8
FIGURE 0.26b
y
=
3
x
2
−
1
Solution
You should get an initial graph that looks something like that in Figure
0.26a. This is simply a parabola opening upward. A graph is often used to search for
important points, such as
x
intercepts,
y
intercepts or peaks and troughs. In this case,
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