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

Section 1.2 Mathematical Models: A Catalog of Essential Functions 31
When we sketch algebraic functions in Chapter 4, we will see that their graphs can
assume a variety of shapes. Figure 17 illustrates some of the possibilities.
y
2
y
y
_3
1
x
1
1
0
5
x
0
1
x
FIGURE 17
(a) ƒ=xœ„„„„ x+3
(b) ©=$œ„„„„„„ ≈-25
(c) h(x)=x@?#(x-2)@
An example of an algebraic function occurs in the theory of relativity. The mass of a
particle with velocity v is
m − f svd −
m 0
s1 2 v 2 yc 2
where m 0 is the rest mass of the particle and c − 3.0 3 10 5 kmys is the speed of light in a
vacuum.
The Reference Pages are located at
the back of the book.
Trigonometric Functions
Trigonometry and the trigonometric functions are reviewed on Reference Page 2 and also
in Appendix D. In calculus the convention is that radian measure is always used (except
when otherwise indicated). For example, when we use the function f sxd − sin x, it is
understood that sin x means the sine of the angle whose radian measure is x. Thus the
graphs of the sine and cosine functions are as shown in Figure 18.
y
y
π
_
2
1
3π
2
_π
π
_
2
1
π
3π
_π
_1
0
π
2
π
2π
5π
2
3π
x
_1
0
π
2
3π
2
2π
5π
2
x
(a) ƒ=sin x
(b) ©=cos x
FIGURE 18
Notice that for both the sine and cosine functions the domain is s2`, `d and the range
is the closed interval f21, 1g. Thus, for all values of x, we have
21 < sin x < 1 21 < cos x < 1
or, in terms of absolute values,
| sin x | < 1 | cos x | < 1
Also, the zeros of the sine function occur at the integer multiples of ; that is,
sin x − 0 when x − n n an integer
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