Projects - Cengage Learning

Appendix C C-53
1
y
Figure A
A graph of f(x) 1
for 0 x 1.
-1
1
-1
Figure B
A graph of
y
1
f(x) b 1, 0 x 1 .
1, 1 x 0
1
x
PROJECT
x
Making Waves
The cosine function is the prototypical example of a function that is both even
and periodic with domain all real numbers. Similarly, the sine function is the
prototypical example of a function that is both odd and periodic with domain all
real numbers. In many applications, especially in physics and engineering, even
and odd periodic phenomena are often represented by more basic waves. In this
project we examine square waves, sawtooth waves, and triangular waves.
We begin with a square wave. Consider the function f defined by f(x) 1,
for 0 x 1, graphed in Figure A. We want to extend this function to be an
odd and periodic function with domain all real numbers. First we extend this
function to be an odd function. We get
f(x) b 1, 0 x 1
1, 1 x 0
This extended version of f is an odd function, since its domain, (1, 0) (0, 1),
is symmetric about zero and for each x in its domain f(x) f(x). For example,
if x 1 , then f 1 1 and f 1 1 22 1 f 1 1 2
22 1
22. Notice the graph of
this extended version of f, shown in Figure B, is symmetric about the origin as
is true for any odd function.
Next, we extend again to obtain a periodic version of this odd function.
We have
f(x) b 1, 0 x 1 and f(x 2) f(x)
1, 1 x 0
for all noninteger real numbers x. Note that f is still an odd function and is also
a periodic function with period 2. Its graph is shown in Figure C.
y
-4
-3
-2
-1
1
1 2 3 4
x
-1
Figure C
A graph of
f(x) b 1, 0 x 1
1, 1 x 0
and f(x 2) f(x) for noninteger real numbers x.
To complete our task, we define f(x) 0 for all integers x. So we obtain
f(x) b 1, 0 x 1
1, 1 x 0 , f(x 2) f(x), for noninteger real number x,
and f(x) 0 for integer x.
This final version of f is an odd periodic function of period 2 with domain
all real numbers. Its graph, shown in Figure D, is called a square wave.