Solução_Calculo_Stewart_6e

F.
158 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
8. (a) If r(t) is the position vector of the particle on the space curve, the velocity v(t) =r 0 (t), the speed is given by |v(t)|,
and the acceleration a(t) =v 0 (t) =r 00 (t).
(b) a = a T T + a N N where a T = v 0 and a N = κv 2 .
9. See the statement of Kepler’s Laws on page 880 [ET page 844].
1. True. If we reparametrize the curve by replacing u = t 3 ,wehaver(u) =u i +2u j +3u k, which is a line through the origin
with direction vector i +2j +3k.
3. False. By Formula 5 of Theorem 14.2.3[ ET 13.2.3],
d
dt [u(t) × v(t)] = u0 (t) × v(t)+u(t) × v 0 (t).
5. False. κ is the magnitude of the rate of change of the unit tangent vector T with respect to arc length s,notwithrespecttot.
7. True. At an inflection point where f is twice continuously differentiable we must have f 00 (x) =0, and by Equation 14.3.11
[ET 13.3.11], the curvature is 0 there.
9. False. If r(t) is the position of a moving particle at time t and |r(t)| =1then the particle lies on the unit circle or the unit
sphere, but this does not mean that the speed |r 0 (t)| must be constant. As a counterexample, let r(t) = t, √ 1 − t 2 ,then
r 0 (t) = 1, −t/ √ 1 − t 2 and |r(t)| = √ t 2 +1− t 2 =1but |r 0 (t)| = 1+t 2 /(1 − t 2 )=1/ √ 1 − t 2 which is not
constant.
11. True. See the discussion preceding Example 7 in Section 14.3 [ ET 13.3].
1. (a) The corresponding parametric equations for the curve are x = t,
y =cosπt, z =sinπt. Sincey 2 + z 2 =1, the curve is contained in a
circular cylinder with axis the x-axis. Since x = t, the curve is a helix.
(b) r(t) =t i +cosπt j +sinπt k ⇒
r 0 (t) =i − π sin πt j + π cos πt k ⇒
r 00 (t) =−π 2 cos πt j − π 2 sin πt k
3. The projection of the curve C of intersection onto the xy-plane is the circle x 2 + y 2 =16,z =0.Sowecanwrite
x =4cost, y =4sint, 0 ≤ t ≤ 2π. From the equation of the plane, we have z =5− x =5− 4cost, so parametric
equations for C are x =4cost, y =4sint, z =5− 4cost, 0 ≤ t ≤ 2π, and the corresponding vector function is
r(t) =4cost i +4sint j +(5− 4cost) k, 0 ≤ t ≤ 2π.