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

320 Chapter 4 Applications of Differentiation
B. The y-intercept is f s0d − ln 4. To find the x-intercept we set
y − lns4 2 x 2 d − 0
We know that ln 1 − 0, so we have 4 2 x 2 − 1 ? x 2 − 3 and therefore the
x-intercepts are 6s3 .
C . Since f s2xd − f sxd, f is even and the curve is symmetric about the y-axis.
D. We look for vertical asymptotes at the endpoints of the domain. Since 4 2 x 2 l 0 1
as x l 2 2 and also as x l 22 1 , we have
lim lns4 2 x 2 d − 2`
x l2 2
lim lns4 2 x 2 d − 2`
x l22 1
Thus the lines x − 2 and x − 22 are vertical asymptotes.
E . f 9sxd − 22x
4 2 x 2
x=_2
y
(0, ln 4)
x=2
Since f 9sxd . 0 when 22 , x , 0 and f 9sxd , 0 when 0 , x , 2, f is increasing
on s22, 0d and decreasing on s0, 2d.
F . The only critical number is x − 0. Since f 9 changes from positive to negative at 0,
f s0d − ln 4 is a local maximum by the First Derivative Test.
0
{_œ„3, 0}
{œ„3, 0}
x
G. f 0sxd − s4 2 x 2 2
ds22d 1 2xs22xd 28 2 2x
−
s4 2 x 2 d 2 s4 2 x 2 d 2
FIGURE 11
y − lns4 2 x 2 d
Since f 0sxd , 0 for all x, the curve is concave downward on s22, 2d and has no
inflection point.
H. Using this information, we sketch the curve in Figure 11. n
y
Slant Asymptotes
Some curves have asymptotes that are oblique, that is, neither horizontal nor vertical. If
y=ƒ
lim f f sxd 2 smx 1 bdg − 0
x l `
ƒ-(mx+b)
0
FIGURE 12
y=mx+b
x
where m ± 0, then the line y − mx 1 b is called a slant asymptote because the vertical
distance between the curve y − f sxd and the line y − mx 1 b approaches 0, as
in Fig ure 12. (A similar situation exists if we let x l 2`.) For rational functions,
slant asymp totes occur when the degree of the numerator is one more than the degree of
the denominator. In such a case the equation of the slant asymptote can be found by long
division as in the following example.
ExamplE 6 Sketch the graph of f sxd − x 3
x 2 1 1 .
A. The domain is R − s2`, `d.
B. The x- and y-intercepts are both 0.
C. Since f s2xd − 2f sxd, f is odd and its graph is symmetric about the origin.
D. Since x 2 1 1 is never 0, there is no vertical asymptote. Since f sxd l ` as x l `
and f sxd l 2` as x l 2`, there is no horizontal asymptote. But long division
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