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

Section 4.6 Graphing with Calculus and Calculators 329
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1–8 Produce graphs of f that reveal all the important aspects
of the curve. In particular, you should use graphs of f 9 and
f 0 to estimate the intervals of increase and decrease, extreme
values, intervals of concavity, and inflection points.
1. f sxd − x 5 2 5x 4 2 x 3 2 28x 2 2 2x
2. f sxd − 22x 6 1 5x 5 1 140x 3 2 110x 2
3. f sxd − x 6 2 5x 5 1 25x 3 2 6x 2 2 48x
4. f sxd − x4 2 x 3 2 8
x 2 2 x 2 6
6. f sxd − 6 sin x 2 x 2 , 25 < x < 3
7. f sxd − 6 sin x 1 cot x, 2 < x <
8. f sxd − e x 2 0.186x 4
5.
f sxd −
x
x 3 1 x 2 1 1
9–10 Produce graphs of f that reveal all the important aspects
of the curve. Estimate the intervals of increase and decrease
and intervals of concavity, and use calculus to find these intervals
exactly.
9. f sxd − 1 1 1 x 1 8 x 2 1 1 x 3 10. f sxd − 1 x 8 2 2 3 108
x 4
11–12
(a) Graph the function.
(b) Use l’Hospital’s Rule to explain the behavior as x l 0.
(c) Estimate the minimum value and intervals of concavity.
Then use calculus to find the exact values.
11. f sxd − x 2 ln x 12. f sxd − xe 1yx
13–14 Sketch the graph by hand using asymptotes and intercepts,
but not derivatives. Then use your sketch as a guide to
producing graphs (with a graphing device) that display the
major features of the curve. Use these graphs to estimate the
maximum and minimum values.
sx 1 4dsx 2 3d2
13. f sxd −
x 4 sx 2 1d
14. f sxd − s2x 1 3d2 sx 2 2d 5
x 3 sx 2 5d 2
15. If f is the function considered in Example 3, use a computer
algebra system to calculate f 9 and then graph it to
confirm that all the maximum and minimum values are as
given in the example. Calculate f 0 and use it to estimate
the intervals of concavity and inflection points.
16. If f is the function of Exercise 14, find f 9 and f 0 and
use their graphs to estimate the intervals of increase and
decrease and concavity of f.
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17–22 Use a computer algebra system to graph f and to find
f 9 and f 0. Use graphs of these derivatives to estimate the
intervals of increase and decrease, extreme values, intervals
of concavity, and inflection points of f.
17. f sxd − x 3 1 5x 2 1 1
x 4 1 x 3 2 x 2 1 2
18. f sxd −
x 2y3
1 1 x 1 x 4
19. f sxd − sx 1 5 sin x , x < 20
20.
21.
22.
f sxd − x 2 tan 21 sx 2 d
f sxd − 1 2 e1yx
1 1 e 1yx
f sxd −
3
3 1 2 sin x
23–24 Graph the function using as many viewing rectangles
as you need to depict the true nature of the function.
23. f sxd − 1 2 cossx 4 d
x 8
24. f sxd − e x 1 ln| x 2 4 |
25–26
(a) Graph the function.
(b) Explain the shape of the graph by computing the limit as
x l 0 1 or as x l `.
(c) Estimate the maximum and minimum values and then use
calculus to find the exact values.
(d) Use a graph of f 0 to estimate the x-coordinates of the
inflection points.
25. f sxd − x 1yx
26. f sxd − ssin xd sin x
27. In Example 4 we considered a member of the family
of functions f sxd − sinsx 1 sin cxd that occur in FM
synthesis. Here we investigate the function with c − 3.
Start by graphing f in the viewing rectangle f0, g by
f21.2, 1.2g. How many local maximum points do you
see? The graph has more than are visible to the naked eye.
To discover the hidden maximum and minimum points
you will need to examine the graph of f 9 very carefully.
In fact, it helps to look at the graph of f 0 at the same
time. Find all the maximum and minimum values and
inflection points. Then graph f in the viewing rectangle
f22, 2g by f21.2, 1.2g and comment on symmetry.
28–35 Describe how the graph of f varies as c varies. Graph
several members of the family to illustrate the trends that you
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