Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 1 REVIEW ¤ 37
17. (a) The terms of f are a mixture of odd and even powers of x,sof is neither even nor odd.
(b) The terms of f are all odd powers of x,sof is odd.
(c) f(−x) =e −(−x)2 = e −x2 = f(x),sof is even.
(d) f(−x) =1+sin(−x) =1− sin x. Nowf(−x) 6=f(x) and f(−x) 6=−f(x),sof is neither even nor odd.
19. f(x) =lnx, D =(0, ∞); g(x) =x 2 − 9, D = R.
(a) (f ◦ g)(x) =f(g(x)) = f(x 2 − 9) = ln(x 2 − 9).
Domain: x 2 − 9 > 0 ⇒ x 2 > 9 ⇒ |x| > 3 ⇒ x ∈ (−∞, −3) ∪ (3, ∞)
(b) (g ◦ f)(x) =g(f(x)) = g(ln x) =(lnx) 2 − 9. Domain: x>0,or(0, ∞)
(c) (f ◦ f)(x) =f(f(x)) = f(ln x) =ln(lnx). Domain: ln x>0 ⇒ x>e 0 =1,or(1, ∞)
(d) (g ◦ g)(x) =g(g(x)) = g(x 2 − 9) = (x 2 − 9) 2 − 9. Domain: x ∈ R,or(−∞, ∞)
21. Many models appear to be plausible. Your choice depends on whether you
think medical advances will keep increasing life expectancy, or if there is
bound to be a natural leveling-off of life expectancy. A linear model,
y =0.2493x − 423.4818, gives us an estimate of 77.6 years for the
year 2010.
23. We need to know the value of x such that f(x) =2x +lnx =2.Sincex =1gives us y =2, f −1 (2) = 1.
25. (a) e 2ln3 = e ln 3 2
=3 2 =9
(b) log 10 25 + log 10 4=log 10 (25 · 4) = log 10 100 = log 10 10 2 =2
(c) tan
arcsin 1 2 =tan
π
= √ 1 6 3
(d) Let θ =cos −1 4 5 ,socos θ = 4 5 .Thensin cos −1 4 5
=sinθ =
√
1 − cos2 θ =
1 −
4 2 9
5
= = 3 . 25 5
27. (a) The population would reach 900 in about 4.4 years.
100,000
(b) P =
100 + 900e ⇒ 100P −t +900Pe−t = 100,000 ⇒ 900Pe −t = 100,000 − 100P ⇒
e −t 100,000 − 100P
1000 − P
1000 − P
= ⇒ −t =ln
⇒ t = − ln
,orln
900P
9P
9P
required for the population to reach a given number P .
9 · 900
(c) P =900 ⇒ t =ln
=ln81≈ 4.4 years, as in part (a).
1000 − 900
9P
1000 − P
;thisisthetime