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Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS D 95 ·<br />

1 3 1 s Bth I · s ·<br />

27. sinx = x- 3!x +Six - · · ·. y e A ternatmg enes<br />

Estimation Theorem, the error in the approximation<br />

sin x = x - ~x 3 is less than I ;,x 5 1 < 0.01 ¢:}<br />

lx 5 l < 120(0.01) ¢:} lxl < (1.2) 115 ~ 1.037. The curves<br />

y = x - ix 3 andy= sinx- 0.01 intersect at x ~ 1.043, so<br />

the graph confirms our estimate. Since both the sine function<br />

0.9<br />

r-------------~~~~--~<br />

y=sinx+ o.oV /<br />

/ / y=x-kx 3<br />

/ ~ 'fooox-O.QJ<br />

0.9 "-L,__L,.:...__ _.__________./ 1.2<br />

0.8<br />

and the given approximation are odd functions, we ne<strong>ed</strong> to check the esti~ate only for x > 0. Thus, the desir<strong>ed</strong> range of<br />

. values for xis - 1.037 < x < 1.037.<br />

x 3 x 5 x 7<br />

29. arctan x = x - 3<br />

+<br />

5 - 7<br />

+ · · · . By the Alternating Series<br />

Estimation Theorem, the error is less than l -~ x 7 1 < 0.05 ¢:}<br />

y =arctan x + 0.05<br />

jx 7 j < 0.35 ¢:} lxl < (0.35? 17 ~ 0.8607. The curves<br />

y = x- tx 3 + kx 5 andy= arctanx + 0.05 intersect at<br />

x ~ 0.9245, so the graph confirms our estimate. Since both the<br />

arctangent function and the given approximation are odd functions,<br />

we ne<strong>ed</strong> to check the estimate only for x > 0. Thus, the desir<strong>ed</strong><br />

range of values for xis - 0.86 < x < 0.86.<br />

- 1<br />

y = x - .!. x 3 + .!. xs<br />

3 5<br />

31. Let s(t) be the position function of the car, and for convenience set s(O) = 0. The velocity of the car is v(t) = s' (t) and th~<br />

acceleration is a(t) = s" (t), so the second degree Taylor Polynomial is T2(t) = s(O) + v(O)t + a~O) t 2 = 20t + t 2 . We<br />

estimate the distance travel<strong>ed</strong> during the next second to be s(1) ~ T2(1) = 20 + 1 =21m. The function T 2 (t) would not be<br />

accurate over a full minute, since the car could not possibly maintain an acceleration of 2 m/s 2 for that long (if it did, its final<br />

spe<strong>ed</strong> would be 140 m/ s ~ 313 mi/ hl).<br />

33. E = ~2 - (D!d)2 = ~2- D2(1:d/ D)2 = ; 2 [1 - (1+ ~r2l<br />

We use the Binomial Series to expand (1 + d/ D ) - 2 :.<br />

when D is much larger than d; that is, when P is far away from the dipole.<br />

35. (a) If the water is deep, then 27rd/ L is large, and we know that tanh x -+ 1 as x -+ oo. So we can approximate<br />

tanh(27rd/ L) ~ 1, andsov 2 ~ f!L/ (27r) ¢:} v ~ ..jgLj (21r).<br />

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