James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 4.6 Graphing with Calculus and Calculators 327
π
The family of functions
f sxd − sinsx 1 sin cxd
where c is a constant, occurs in applications
to frequency modulation (FM)
synthesis. A sine wave is modulated by
a wave with a different frequency
ssin cxd. The case where c − 2 is
studied in Example 4. Exercise 27
explores another special case.
1.2
0
_1.2
FIGURE 16
1.2
y=ƒ
y=fª(x)
0 π
_1.2
f·
FIGURE 17
1.2
f
π
this and locate them more accurately, we calculate that
f 9sxd − cossx 1 sin 2xd ? s1 1 2 cos 2xd
and graph both f and f 9 in Figure 16.
Using zoom-in and the First Derivative Test, we find the following approximate
values:
The second derivative is
Intervals of increase: s0, 0.6d, s1.0, 1.6d, s2.1, 2.5d
Intervals of decrease: s0.6, 1.0d, s1.6, 2.1d, s2.5, d
Local maximum values: f s0.6d < 1, f s1.6d < 1, f s2.5d < 1
Local minimum values: f s1.0d < 0.94, f s2.1d < 0.94
f 0sxd − 2s1 1 2 cos 2xd 2 sinsx 1 sin 2xd 2 4 sin 2x cossx 1 sin 2xd
Graphing both f and f 0 in Figure 17, we obtain the following approximate values:
Concave upward on: s0.8, 1.3d, s1.8, 2.3d
Concave downward on: s0, 0.8d, s1.3, 1.8d, s2.3, d
Inflection points: s0, 0d, s0.8, 0.97d, s1.3, 0.97d, s1.8, 0.97d, s2.3, 0.97d
Having checked that Figure 15 does indeed represent f accurately for 0 < x < ,
we can state that the extended graph in Figure 18 represents f accurately for
22 < x < 2.
Our final example is concerned with families of functions. This means that the functions
in the family are related to each other by a formula that contains one or more arbitrary
constants. Each value of the constant gives rise to a member of the family and the
idea is to see how the graph of the function changes as the constant changes.
n
_2π
2π
ExamplE 5 How does the graph of f sxd − 1ysx 2 1 2x 1 cd vary as c varies?
_1.2
SOLUTION The graphs in Figures 19 and 20 (the special cases c − 2 and c − 22)
show two very different-looking curves.
FIGURE 18
2
2
y= 1
≈+2x-2
_5 4
1
y=
≈+2x+2
_5 4
_2
FIGURE 19
FIGURE 20
c − 2 c − 22
_2
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