Solução_Calculo_Stewart_6e

F.
TX.10SECTION 4.6 GRAPHING WITH CALCULUS AND CALCULATORS ¤ 185
From the graph of f 0 ,weestimatethatf is decreasing on (−∞, −15), increasing on (−15, 4.40), decreasing
on (4.40, 18.93), and increasing on (18.93, ∞), with local minimum values of f(−15) ≈−9,700,000 and
f(18.93) ≈−12,700,000 and local maximum value f(4.40) ≈ 53,800. Fromthegraphoff 00 ,weestimatethatf is CU on
(−∞, −11.34),CDon(−11.34, 0),CUon(0, 2.92),CDon(2.92, 15.08), andCUon(15.08, ∞). There is an inflection
point at (0, 0) and at about (−11.34, −6,250,000), (2.92, 31,800),and(15.08, −8,150,000).
5. f(x) =
x
x 3 − x 2 − 4x +1
⇒ f 0 (x) = −2x3 + x 2 +1
⇒ f 00 (x) = 2(3x5 − 3x 4 +5x 3 − 6x 2 +3x +4)
(x 3 − x 2 − 4x +1) 2 (x 3 − x 2 − 4x +1) 3
We estimate from the graph of f that y =0is a horizontal asymptote, and that there are vertical asymptotes at x = −1.7,
x =0.24,andx =2.46. Fromthegraphoff 0 , we estimate that f is increasing on (−∞, −1.7), (−1.7, 0.24),and(0.24, 1),
and that f is decreasing on (1, 2.46) and (2.46, ∞). There is a local maximum value at f(1) = − 1 3 . From the graph of f 00 ,we
estimate that f is CU on (−∞, −1.7), (−0.506, 0.24),and(2.46, ∞),andthatf is CD on (−1.7, −0.506) and (0.24, 2.46).
There is an inflection point at (−0.506, −0.192).
7. f(x) =x 2 − 4x +7cosx, −4 ≤ x ≤ 4. f 0 (x) =2x − 4 − 7sinx ⇒ f 00 (x) =2− 7cosx.
f(x) =0 ⇔ x ≈ 1.10; f 0 (x) =0 ⇔ x ≈−1.49, −1.07,or2.89; f 00 (x) =0 ⇔ x = ± cos −1 2
7
≈ ±1.28.
From the graphs of f 0 ,weestimatethatf is decreasing (f 0 < 0) on (−4, −1.49), increasing on (−1.49, −1.07), decreasing
on (−1.07, 2.89), and increasing on (2.89, 4), with local minimum values f(−1.49) ≈ 8.75 and f(2.89) ≈−9.99 and local
maximum value f(−1.07) ≈ 8.79 (notice the second graph of f). From the graph of f 00 ,weestimatethatf is CU (f 00 > 0)
on (−4, −1.28), CDon(−1.28, 1.28), and CU on (1.28, 4). Thereareinflection points at about (−1.28, 8.77)
and (1.28, −1.48).