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

chapter 4 Review 359
18. The most general antiderivative of f sxd − x 22 is
Fsxd − 2 1 x 1 C
19. If f 9sxd exists and is nonzero for all x, then f s1d ± f s0d.
20. If lim
xl`
f sxd − 1 and lim
xl`
tsxd − `, then
21. lim
x l 0
x
e x − 1
lim f f sxdg tsxd − 1
xl`
EXERCISES
1–6 Find the local and absolute extreme values of the function
on the given interval.
1. f sxd − x 3 2 9x 2 1 24x 2 2, f0, 5g
16. f s0d − 0, f is continuous and even,
f 9sxd − 2x if 0 , x , 1, f 9sxd − 21 if 1 , x , 3,
f 9sxd − 1 if x . 3
2. f sxd − xs1 2 x , f21, 1g
3. f sxd − 3x 2 4 , f22, 2g
x 2 1 1
4. f sxd − sx 2 1 x 1 1 , f22, 1g
5. f sxd − x 1 2 cos x, f2, g
6. f sxd − x 2 e 2x , f21, 3g
7–14 Evaluate the limit.
7. lim
x l 0
e x 2 1
tan x
8. lim
xl 0
tan 4x
x 1 sin 2x
17. f is odd, f 9sxd , 0 for 0 , x , 2,
f 9sxd . 0 for x . 2, f 0sxd . 0 for 0 , x , 3,
f 0sxd , 0 for x . 3, lim f sxd − 22
x l `
18. The figure shows the graph of the derivative f 9of a function f.
(a) On what intervals is f increasing or decreasing?
(b) For what values of x does f have a local maximum or
minimum?
(c) Sketch the graph of f 0.
(d) Sketch a possible graph of f.
y
y=f ª(x)
9. lim
xl 0
e 2x 2 e 22x
lnsx 1 1d
10. lim
xl `
e 2x 2 e 22x
lnsx 1 1d
_2
_1
0 1 2 3 4 5 6 7 x
11. lim
x l 2` sx 2 2 x 3 de 2x
1S 13. lim
x
x l 1
x 2 1 2 1
ln xD
12. lim sx 2 d csc x
xl2 x
14. lim xdcos
x lsy2d 2stan
19–34 Use the guidelines of Section 4.5 to sketch the curve.
19. y − 2 2 2x 2 x 3
15–17 Sketch the graph of a function that satisfies the given
conditions.
15. f s0d − 0, f 9s22d − f 9s1d − f 9s9d − 0,
lim f sxd − 0, lim
x l `
f sxd − 2`,
x l6
f 9sxd , 0 on s2`, 22d, s1, 6d, and s9, `d,
f 9sxd . 0 on s22, 1d and s6, 9d,
f 0sxd . 0 on s2`, 0d and s12, `d,
f 0sxd , 0 on s0, 6d and s6, 12d
20. y − 22x 3 2 3x 2 1 12x 1 5
21. y − 3x 4 2 4x 3 1 2 22. y − x
1 2 x 2
23. y −
25. y −
1
xsx 2 3d 24. y − 1 2 x 2 1
2 sx 2 2d 2
sx 2 1d3
x 2 26. y − s1 2 x 1 s1 1 x
27. y − xs2 1 x 28. y − x 2y3 sx 2 3d 2
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