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

CHAPTER 2 Review 167
;
3–20 Find the limit.
3. lim e x3 2x
x l1
x 2 2 9
5. lim
x l23 x 2 1 2x 2 3
sh 2 1d 3 1 1
7. lim
h l0 h
9. lim
r l9
sr
10. lim
4
sr 2 9d v l 4 1
u 4 2 1
11. lim
u l 1 u 3 1 5u 2 2 6u
13. lim
x l `
sx 2 2 9
2x 2 6
x 2 2 9
4. lim
x l3 x 2 1 2x 2 3
x 2 2 9
6. lim
x l1 1 x 2 1 2x 2 3
t 2 2 4
8. lim
t l2 t 3 2 8
4 2 v
| 4 2 v |
sx 1 6 2 x
12. lim
x l 3 x 3 2 3x 2
sx
14. lim
2 2 9
x l 2` 2x 2 6
1 2 2x 2 2 x 4
15. lim lnssin xd 16. lim
x l 2 x l 2` 5 1 x 2 3x 4
17. lim
x l `
(sx 2 1 4x 1 1 2 x)
19. lim
x l0 1 tan 21 s1yxd
18. lim e x2x2
xl`
20. lim
x l 1S 1
x 2 1 1 1
x 2 2 3x 1 2D
21–22 Use graphs to discover the asymptotes of the curve. Then
prove what you have discovered.
21. y − cos2 x
x 2 22. y − sx 2 1 x 1 1 2 sx 2 2 x
23. If 2x 2 1 < f sxd < x 2 for 0 , x , 3, find lim x l1 f sxd.
24. Prove that lim x l 0 x 2 coss1yx 2 d − 0.
25–28 Prove the statement using the precise definition of a limit.
25. lim
x l 2
s14 2 5xd − 4
27. lim
x l 2
sx 2 2 3xd − 22
29. Let
26. lim
x l 0
s 3 x − 0
if x , 0
f sxd −Hs2x
3 2 x if 0 < x , 3
sx 2 3d 2 if x . 3
(a) Evaluate each limit, if it exists.
(i) lim
x l0 1 f sxd
(ii) lim f sxd
x l0 2
(iv) lim f sxd (v) lim f sxd
x l3 2 x l3 1
(b) Where is f discontinuous?
(c) Sketch the graph of f.
30. Let
tsxd −
2x 2 x 2
2 2 x
x 2 4
2
28. lim − `
x l 4 1 sx 2 4
if 0 < x < 2
if 2 , x < 3
if 3 , x , 4
if x > 4
(iii) lim f sxd
x l0
(vi) lim f sxd
x l3
;
(a) For each of the numbers 2, 3, and 4, discover whether
t is continuous from the left, continuous from the
right, or continuous at the number.
(b) Sketch the graph of t.
31–32 Show that the function is continuous on its domain.
State the domain.
31. hsxd − xe sin x 32. tsxd − sx 2 2 9
x 2 2 2
33–34 Use the Intermediate Value Theorem to show that there
is a root of the equation in the given interval.
33. x 5 2 x 3 1 3x 2 5 − 0, s1, 2d
34. cossx − e x 2 2, s0, 1d
35. (a) Find the slope of the tangent line to the curve
y − 9 2 2x 2 at the point s2, 1d.
(b) Find an equation of this tangent line.
36. Find equations of the tangent lines to the curve
y − 2
1 2 3x
at the points with x-coordinates 0 and 21.
37. The displacement (in meters) of an object moving in a
straight line is given by s − 1 1 2t 1 1 4 t 2 , where t is measured
in seconds.
(a) Find the average velocity over each time period.
(i) f1, 3g (ii) f1, 2g (iii) f1, 1.5g (iv) f1, 1.1g
(b) Find the instantaneous velocity when t − 1.
38. According to Boyle’s Law, if the temperature of a confined
gas is held fixed, then the product of the pressure P
and the volume V is a constant. Suppose that, for a certain
gas, PV − 800, where P is measured in pounds per square
inch and V is measured in cubic inches.
(a) Find the average rate of change of P as V increases
from 200 in 3 to 250 in 3 .
(b) Express V as a function of P and show that the instantan
eous rate of change of V with respect to P is
inversely proportional to the square of P.
39. (a) Use the definition of a derivative to find f 9s2d, where
f sxd − x 3 2 2x.
(b) Find an equation of the tangent line to the curve
y − x 3 2 2x at the point (2, 4).
(c) Illustrate part (b) by graphing the curve and the tangent
line on the same screen.
40. Find a function f and a number a such that
s2 1 hd 6 2 64
lim
− f 9sad
h l0 h
41. The total cost of repaying a student loan at an interest rate
of r% per year is C − f srd.
(a) What is the meaning of the derivative f 9srd? What are
its units?
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