Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 2 REVIEW ¤ 83
√
45. (a) f 0 f(x + h) − f(x) 3 − 5(x + h) − 3 − 5x
(x) =lim
=lim
h→0 h
h→0 h
[3 − 5(x + h)] − (3 − 5x)
=lim
h→0
3 √ = lim
h − 5(x + h)+ 3 − 5x h→0
3 − 5(x + h)+
√ 3 − 5x
3 − 5(x + h)+
√ 3 − 5x
−5
3 − 5(x + h)+
√ 3 − 5x
=
−5
2 √ 3 − 5x
(b) Domain of f: (the radicand must be nonnegative) 3 − 5x ≥ 0
5x ≤ 3 ⇒ x ∈
−∞, 3 5
⇒
Domain of f 0 :exclude 3 because it makes the denominator zero;
5
x ∈
−∞, 3 5
(c) Our answer to part (a) is reasonable because f 0 (x) is always negative and f
is always decreasing.
47. f is not differentiable: at x = −4 because f is not continuous, at x = −1 because f has a corner, at x =2because f is not
continuous, and at x =5because f has a vertical tangent.
49. C 0 (1990) is the rate at which the total value of US currency in circulation is changing in billions of dollars per year. To
estimate the value of C 0 (1990), we will average the difference quotients obtained using the times t =1985and t = 1995.
Let A =
B =
C(1985) − C(1990)
1985 − 1990
C(1995) − C(1990)
1995 − 1990
C 0 (1990) =
=
=
187.3 − 271.9
−5
409.3 − 271.9
5
= −84.6
−5
= 137.4
5
=16.92 and
=27.48. Then
C(t) − C(1990)
lim
≈ A + B 16.92 + 27.48
= = 44.4 =22.2 billion dollars/year.
t→1990 t − 1990 2
2 2
51. |f(x)| ≤ g(x) ⇔ −g(x) ≤ f(x) ≤ g(x) and lim
x→a
g(x) = 0 = lim
x→a
−g(x).
Thus, by the Squeeze Theorem, lim
x→a
f(x) =0.