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

Section 4.6 Graphing with Calculus and Calculators 323
61–64 Find an equation of the slant asymptote. Do not sketch
the curve.
61. y − x2 1 1
x 1 1
63. y − 2x 3 2 5x 2 1 3x
x 2 2 x 2 2
62. y − 4x 3 2 10x 2 2 11x 1 1
x 2 2 3x
64. y − 26x 4 1 2x 3 1 3
2x 3 2 x
65–70 Use the guidelines of this section to sketch the curve. In
guideline D find an equation of the slant asymptote.
65. y − x 2
x 2 1
67. y − x 3 1 4
x 2 68. y −
69. y − 1 1 1 2 x 1 e2x
2
1 1 5x 2 2x
66. y −
x 2 2
x 3
sx 1 1d 2
70. y − 1 2 x 1 e 11xy3
71. Show that the curve y − x 2 tan 21 x has two slant asymptotes:
y − x 1 y2 and y − x 2 y2. Use this fact to help
sketch the curve.
72. Show that the curve y − sx 2 1 4x has two slant asymptotes:
y − x 1 2 and y − 2x 2 2. Use this fact to help
sketch the curve.
73. Show that the lines y − sbyadx and y − 2sbyadx are slant
asymptotes of the hyperbola sx 2 ya 2 d 2 sy 2 yb 2 d − 1.
74. Let f sxd − sx 3 1 1dyx. Show that
lim f f sxd 2 x 2 g − 0
x l 6`
This shows that the graph of f approaches the graph of
y − x 2 , and we say that the curve y − f sxd is asymptotic
to the parabola y − x 2 . Use this fact to help sketch the
graph of f.
75. Discuss the asymptotic behavior of f sxd − sx 4 1 1dyx in
the same manner as in Exercise 74. Then use your results
to help sketch the graph of f.
76. Use the asymptotic behavior of f sxd − sin x 1 e 2x
to sketch its graph without going through the curvesketching
procedure of this section.
You may want to read “Graphing
Calculators and Computers” at
www.stewartcalculus.com if you
haven’t already. In particular, it
explains how to avoid some of the
pitfalls of graphing devices by choosing
appropriate viewing rectangles.
The method we used to sketch curves in the preceding section was a culmination of much
of our study of differential calculus. The graph was the final object that we produced.
In this section our point of view is completely different. Here we start with a graph
produced by a graphing calculator or computer and then we refine it. We use calculus
to make sure that we reveal all the important aspects of the curve. And with the use of
graphing devices we can tackle curves that would be far too complicated to consider
without technology. The theme is the interaction between calculus and calculators.
ExamplE 1 Graph the polynomial f sxd − 2x 6 1 3x 5 1 3x 3 2 2x 2 . Use the graphs
of f 9 and f 0 to estimate all maximum and minimum points and intervals of concavity.
SOLUtion If we specify a domain but not a range, many graphing devices will deduce
a suitable range from the values computed. Figure 1 shows the plot from one such
device if we specify that 25 < x < 5. Although this viewing rectangle is useful for
showing that the asymptotic behavior (or end behavior) is the same as for y − 2x 6 , it is
obviously hiding some finer detail. So we change to the viewing rectangle f23, 2g by
f250, 100g shown in Figure 2.
41,000
100
y=ƒ
_5 5
_1000
FIGURE 1
y=ƒ
_3 2
FIGURE 2
_50
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