Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 2 REVIEW ¤ 79
8. See Theorem 2.5.10.
9. See Definition 2.7.1.
10. See the paragraph containing Formula 3 in Section 2.7.
11. (a) The average rate of change of y with respect to x over the interval [x 1 ,x 2 ] is
f(x2) − f(x1)
x 2 − x 1
.
f(x 2) − f(x 1)
(b) The instantaneous rate of change of y with respect to x at x = x 1 is lim
.
x 2 →x 1 x 2 − x 1
12. See Definition 2.7.2. The pages following the definition discuss interpretations of f 0 (a) as the slope of a tangent line to the
graph of f at x = a and as an instantaneous rate of change of f(x) with respect to x when x = a.
13. See the paragraphs before and after Example 6 in Section 2.8.
14. (a) A function f is differentiable at a number a if its derivative f 0 exists
(c)
at x = a;thatis,iff 0 (a) exists.
(b) See Theorem 2.8.4. This theorem also tells us that if f is not
continuous at a,thenf is not differentiable at a.
15. See the discussion and Figure 7 on page 159.
1. False. Limit Law 2 applies only if the individual limits exist (these don’t).
3. True. Limit Law 5 applies.
x(x − 5) sin(x − 5)
5. False. Consider lim or lim .Thefirst limit exists and is equal to 5. By Example 3 in Section 2.2,
x→5 x − 5 x→5 x − 5
we know that the latter limit exists (and it is equal to 1).
7. True. A polynomial is continuous everywhere, so lim
x→b
p(x) exists and is equal to p(b).
9. True. See Figure 8 in Section 2.6.
11. False. Consider f(x) =
1/(x − 1) if x 6= 1
2 if x =1
13. True. Use Theorem 2.5.8 with a =2, b =5,andg(x) =4x 2 − 11. Notethatf(4) = 3 is not needed.
15. True, by the definition of a limit with ε =1.
17. False. See the note after Theorem 4 in Section 2.8.
19. False.
d 2 2
y
dy
dx is the second derivative while is the first derivative squared. For example, if y = x,
2 dx
then d 2 2
y dy
dx =0,but =1.
2 dx