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

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l
lt sin u dx cost dy sin t
t cos u
1. x = --du, y = . --du, so by FTCl, we have dt = -t- and dt = - t - . Vertical tangent lines occur when
1 tL 1 tL
dx = 0 {:} cost = 0. The parameter value corresponding to (x, y) = {0, 0) is t = 1, so tl1e nearest vertical tangent
dt .
occurs when t = ~.
Therefore, tl1e arc length between these points is
7r/2
L=
1
(- dx) 2 + (dy) - 2 dt= 1 1r/Z
1 dt dt 1
2
t . 2 t j '1f/Z
dt
cos ~dt = - ~ [ lnt]7f 12 = ln :!!.
t2 + t2 1 t 1 2
3. ln. terms ofx andy, we have x =?'cosO= (1 + csin B)cosB =cos O+ csinBcosB =cosO + ~cs in2B and
y = rsinB = (1 + csinO)sinB =sinO+ csin 2 B. Now - 1 $sinO $ 1 =>
-1 $sinO -t csin 2 0 $ 1 + c $2, so
-1 $ y $ 2. Furthermore, y = 2 when c = 1 and B = ~ , while y = -1 for c = 0 and 0 = 3 ; . Therefore, we need a viewing.
rectangle with - 1 $ y $ 2.
To find the x-values, iook at the equation x =cos B + ~ csin 20 and use the fact that sin 20 ~ 0 for 0 $ B $ ~ and
sin 28 $ 0 for - ~ $ B $ 0. [Because r =::, 1 + csin 0 is symmetric about the y-axis, we only need to consider
- ~ S 8. $ ~ .) So for - ~ $ B .$ 0, x has a maximum value when c = 0 and then x = cos 0 has a maximum value
of 1 at B = 0. Thus, the maximum value of x must occur on [ 0, ~ ] with c = 1. Then x = cos 0 + ~ sin 20 =>
~~ = -sinB +cos2B=- sinB + 1-2sin 2 9 =? j~ = -(2sin9-1)(sin9+1 ) = 0when sin9 = - 1or~
[but sin B # - 1 for 0 $ 9 $ H If sin 9 = ~, then B = * and
x = cos {f + ~sin 3 = ~V3. Thus, the maximum value of xis ~V3, and,
by symmetry, the minimum value is - ~ V3. Therefore, the smallest
viewing rectangle that contains every member of the family of polar curves
r = 1 + csinB, where 0 $ c $ 1, is [-~V3, ~V3] x [-1, 2].
2.1 c=t
5. Without Joss of generality, assume the hyperbola has equation :: - ~: = 1. Use implicit differentiation to get
2 : -
2 Yb2 y' = 0, soy' = b:x. The tangent line at the point (c, d) on the hyperbola has equation y- d = b:dc (x - c).
a a y . a
' 2
The tangent line intersects the asymptote y = !!. x when .!!.x - d = bzdc (x- c)
a a a
=> abdx- a 2 d 2 = b 2 cx - b 2 c 2 =>
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