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

SECTION 13.3 ARC LENGTH AND CURVATURE D 161
47. :t [u (t) · v(t)] = u'(t) · v (t) + u (t) ·.v'(t) [by Formula 4 of Theorem 3]
= (cost, - sin t , 1) · (t, cost, sin t) + (sin t, cost, t) · (1,-sin t, cost)
= t cos t - cos t sin t + sin t + sin t - cos t sin t + t cos t
= 2tcost + 2sint - 2cost sint
49. By Formula 4 of Theorem 3, f' (t) = u ' (t) · v (t) + u (t) · v' (t), and v' (t) = (1, 2t, 3t 2 ), so
j'(2) = u'(2) · v{2) + u {2) · v'{2) = (3, 0, 4) · (2,4,8) + (1, 2, - 1) · (1, 4, 12) = 6 + 0 + 32 + 1 + 8- 12 = 35.
d .
51 . - [r(t) x r' (t)] = r' (t) ~ r' (t) + r (t) x r" (t) by Formula 5 of Theorem 3. But r' (t) x r' (t) = 0 (by Example 2 in
dt
Section 12.4). Thus,! [r (t) x r'(t)] = r(t) x r"(t).
53. ~ Jr (t)l = dd [r(t) · r(t)] 112 = ~ [r (t) · r (t)]- 1 1 2 [2r(t) · r '(t)] = -
1
( 1 )I r(t) · r'(t)
dt t · - r t
55. Since u (t) = r (t) · [r'(t) x r"(t)),
u' (t) = r 1 (t) · [r' (t) X r" (t)] + r (t) · ! [r' {t) X r 11 (t))
= 0 + r(t). · (r"(t) x r "(t) + r'(t) X r 111 (t)]
= r(t) · (r' (t) X r 111 (t)]
[since r' (t) ..l r' (t) x r" (t)]
[since r"(t) x r"(t) = 0]
13.3 Arc Length and Curvature
1. r (t) = (t,3cos t,3sint) =? r' (t) = (1,-3sint,3cost) =?
Jr'(t)l = ) 1 2 + (- 3 sint) 2 + (3cost) 2 = ) 1 + 9(sin 2 t + cos 2 t) = v'IO.
Then using Formula3, we have L = f~ 5 lr'(t)l dt = ts v'Wdt = .fi0t] ~ 5 ~ 10 .fiO.
3. r(t) = v'2t i + etj + e- tk =? r'(t) = v'2i + etj - e-tk =?
lr'{t)l = J ( v'2/ + (et) 2 + (-e- t)2 =:= J2 + e 2 t + e u = .J(et + e- t)2 = et + e- t [since et + e- t > OJ.
Then L = }~ 1 lr' (t)i dt = J~( et + e- t) dt = [et - e- t] ~ = e- e- 1 .
5. r(t) = i + t 2 j + t 3 k =? r'(t) = 2tj + 3t 2 k =? ir'(t)i = J4t 2 + 9t 4 = t J4 + 9t2 . [since t 2': OJ.
Then L = J 1 0
lr '(t)i dt = J; 2t tJ4 + 9t 2 dt = fa· t(4 + 9t 2 ) 3 1 = 2
17 (133 1 2 - 4 3 1 2 ) = 2
\ (13 3 / 2 - 8).
7. r(t) = (t 2 , t 3 , t 4 ) =? r 1 (t) = (2t, 3t 2 , 4t 3 ) => lr' (t)i = V(2t? + (3t 2 )2 + (4t 3 ) 2 = J4t2 + 9t4 + 1fit6, so
L = J~ lr'(t)l dt = J; J4t 2 + 9t 4 + 16t 6 dt ~ 18.6833.
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