Solução_Calculo_Stewart_6e

F.
422 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
67. If f 0 is continuous and f 0 (t) 6= 0for a ≤ t ≤ b, then either f 0 (t) > 0 for all t in [a, b] or f 0 (t) < 0 for all t in [a, b]. Thus,f
is monotonic (in fact, strictly increasing or strictly decreasing) on [a, b]. It follows that f has an inverse. Set F = g ◦ f −1 ,
that is, define F by F (x) =g(f −1 (x)). Thenx = f(t) ⇒ f −1 (x) =t,soy = g(t) =g(f −1 (x)) = F (x).
dy
69. (a) φ =tan −1 ⇒
dx
d dy
= d ẏ
=
dt dx dt ẋ
t
fact that s =
dx
dt
0
dφ
dt = d
dy
1 d dy
dt tan−1 =
dx 1+(dy/dx) 2 dt dx
ÿẋ − ẍẏ
⇒ dφ
ÿẋ
ẋ 2 dt = 1 − ẍẏ
=
1+(ẏ/ẋ) 2 ẋ 2
. But dy
dx = dy/dt
dx/dt = ẏ
ẋ
⇒
ẋÿ − ẍẏ
. Using the Chain Rule, and the
ẋ 2 + ẏ2 2
+
dy
2
dt dt ⇒
ds = dx
2
dt dt + dy
2
dt = ẋ2 + ẏ 21/2 ,wehavethat
dφ
ds = dφ/dt
ẋÿ − ẍẏ
ds/dt = 1 ẋÿ − ẍẏ
=
ẋ 2 + ẏ 2 (ẋ 2 + ẏ 2 )
1/2
(ẋ 2 + ẏ 2 ) .Soκ = dφ
3/2 ds = ẋÿ − ẍẏ |ẋÿ − ẍẏ|
=
(ẋ 2 + ẏ 2 ) 3/2 (ẋ 2 + ẏ 2 ) . 3/2
(b) x = x and y = f(x) ⇒ ẋ =1, ẍ =0and ẏ = dy
dx , ÿ = d2 y
dx . 2
1 · (d 2 y/dx 2 ) − 0 · (dy/dx)
d 2 y/dx 2
So κ =
=
[1 + (dy/dx) 2 ] 3/2 [1 + (dy/dx) 2 ] . 3/2
71. x = θ − sin θ ⇒ ẋ =1− cos θ ⇒ ẍ =sinθ,andy =1− cos θ ⇒ ẏ =sinθ ⇒ ÿ =cosθ. Therefore,
cos θ − cos 2 θ − sin 2 θ κ =
[(1 − cos θ) 2 +sin 2 θ] = cos θ − (cos 2 θ +sin 2 θ) |cos θ − 1|
3/2 (1 − 2cosθ +cos 2 θ +sin 2 = . The top of the arch is
θ)
3/2 (2 − 2cosθ)
3/2
characterized by a horizontal tangent, and from Example 2(b) in Section 10.2, the tangent is horizontal when θ =(2n − 1)π,
so take n =1and substitute θ = π into the expression for κ: κ =
73. The coordinates of T are (r cos θ, r sin θ). SinceTP was unwound from
arc TA, TP has length rθ. Also∠PTQ = ∠PTR− ∠QT R = 1 2 π − θ,
so P has coordinates x = r cos θ + rθ cos 1
2 π − θ = r(cos θ + θ sin θ),
y = r sin θ − rθ sin 1
2 π − θ = r(sin θ − θ cos θ).
|cos π − 1|
(2 − 2cosπ) 3/2 = |−1 − 1|
[2 − 2(−1)] 3/2 = 1 4 .
10.3 Polar Coordinates
1. (a) 2, π 3
By adding 2π to π 3 , we obtain the point 2, 7π 3
. The direction
opposite π 3 is 4π 3 ,so −2, 4π 3
requirement.
is a point that satisfies the r