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

174 D CHAPTER 13 VECTOR FUNCTIONS
(b) a = a7:T +aNN where aT = v' and aN = 1w 2 .
9. See the statement of Kepler's Laws on page 892 [ET 868].
TRUE-FALSE QUIZ
1. True. If we reparametrize the curve by replacing u = t 3 , we have r(u) = u i + 2u ~ + 3uk, which is a line through the origin
with direction vector i + 2 j + 3 k.
3. False. The vector function represents a line, but the line does not pass through the origin; the x-component is 0 only for t ;, 0
which co~esponds to the point {0, 3, 0) not {0, 0, 0).
d
5. False. By Formula 5 ofTheorem 13.2.3, -d (u(t) x v(t)] = u '(t ) x v (t) + u (t) x v'(t).
. t
7. False. K- is the magnitude of the rate of change of the unit tangent vector T with respect to arc length s, not with respect tot.
9. True. At an inflection point where f is twice continuously differentiable we must have !" ( x) = 0, and by Equation 13.3.11,
the curvature is 0 there.
11. False. If r(t) is the position of a moving particle at timet and lr(t)l = 1 then the particle lies on the unit circle or the unit
sphere, but this does not mean that the speed lr'(t)l must be constant. As a counterexample, let r (t) = (t, Jl=t2), then
r'(t) = ( 1, -t/J1 - t 2 ) and lr(t)l = Jt 2 + 1 - t 2 = 1 but lr'(t)l = yf1 + t 2 /(1- t 2 ) = 1 /~ which is not
constant.
13. True. See the discussion preceding Example 7 in Section 13.3.
EXERCISES
1. (a) The corresponding parametric equations for the curve are x = t,
y = cos 1rt, z = sin -rrt. Since y 2 + z 2 = 1, the curve is contained in a
I
circular cylinder with axis the x-axis. Since x = t, the curve is a helix.
(b) r(t) = t i + cos rij +sin -rrt k =>
r'(t) = i--rrsin 1rt j + -rrcos -rrt k ~
X
r" ( t) = --rr 2 cos 1rt j - 1r 2 siri 7rt k
3. The projection of the curve C of interSection onto the xy-plane is the circle x 2 + y 2 = 16, z = 0. So we can write
x = 4 cost, y = 4 sin t, 0 S t S 21r. From the equation of the plane, w~ have z = 5 -
x = 5 - 4 cos t, so parametric
equations for Care x = 4 cost, y = 4 sin t, z = 5 - 4 cost, 0 S t S 2-rr, and the corresponding vector function is
r(t) = 4 cost i + 4 sin t j + (5 - 4 cost) k , 0 S t S 21r.
© 2012 Ccngngc Leoming. All Rights Rcsem:d . Mny nol be scanned, copied, orduplicoled, or posted loa publicly IICccssible wt:bsilc, .in -.'hole or in part.