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

Section 4.6 Graphing with Calculus and Calculators 325
10
y=ƒ
The y-axis appears to be a vertical asymptote and indeed it is because
lim
x l 0
x 2 1 7x 1 3
x 2
− `
y=1
_20 20
_5
FIGURE 8
Figure 7 also allows us to estimate the x-intercepts: about 20.5 and 26.5. The
exact values are obtained by using the quadratic formula to solve the equation
x 2 1 7x 1 3 − 0; we get x − (27 6 s37 )y2.
To get a better look at horizontal asymptotes, we change to the viewing rectangle
f220, 20g by f25, 10g in Figure 8. It appears that y − 1 is the horizontal asymptote
and this is easily confirmed:
lim
x l 6`
x 2 1 7x 1 3
x 2
− lim
x l 6`S1 1 7 x 1 3 x 2D − 1
_3 0
2
To estimate the minimum value we zoom in to the viewing rectangle f23, 0g by
f24, 2g in Figure 9. The cursor indicates that the absolute minimum value is about
23.1 when x < 20.9, and we see that the function decreases on s2`, 20.9d and s0, `d
and increases on s20.9, 0d. The exact values are obtained by differentiating:
y=ƒ
FIGURE 9
_4
f 9sxd − 2 7 x 2 2 6 x 3 − 2 7x 1 6
x 3
This shows that f 9sxd . 0 when 2 6 7 , x , 0 and f 9sxd , 0 when x , 2 6 7 and when
x . 0. The exact minimum value is f (2 6 7 ) − 2 37
12 < 23.08.
Figure 9 also shows that an inflection point occurs somewhere between x − 21 and
x − 22. We could estimate it much more accurately using the graph of the second
deriv ative, but in this case it’s just as easy to find exact values. Since
f 0sxd − 14
x 1 18
3 x 4
− 2s7x 1 9d
x 4
we see that f 0sxd . 0 when x . 2 9 7 sx ± 0d. So f is concave upward on s2 9 7, 0d and
s0, `d and concave downward on s2`, 2 9 7 d. The inflection point is s2 9 7, 2 71
27 d.
The analysis using the first two derivatives shows that Figure 8 displays all the
major aspects of the curve.
n
ExamplE 3 Graph the function f sxd −
x 2 sx 1 1d 3
sx 2 2d 2 sx 2 4d 4 .
10
y=ƒ
_10 10
FIGURE 10
_10
SOLUTION Drawing on our experience with a rational function in Example 2, let’s
start by graphing f in the viewing rectangle f210, 10g by f210, 10g. From Figure 10
we have the feeling that we are going to have to zoom in to see some finer detail and
also zoom out to see the larger picture. But, as a guide to intelligent zooming, let’s
first take a close look at the expression for f sxd. Because of the factors sx 2 2d 2 and
sx 2 4d 4 in the denominator, we expect x − 2 and x − 4 to be the vertical asymptotes.
Indeed
lim
x l2
x 2 sx 1 1d 3
sx 2 2d 2 4
− `
sx 2 4d
and lim
x l 4
x 2 sx 1 1d 3
sx 2 2d 2 sx 2 4d 4
− `
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