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

SECTION 13.1 VECTOR FUNCTIONS AND SPACE CURVES D 155
47. For the particles to collide, we require r1 ( t) = r 2 ( t) ¢:> ( t 2 , 7t - 12, t 2 ) = .( 4t - 3, t2, 5t - 6). Equating components
gives t2 = 4t- 3, 7t - 12 = t2, and t 2 = 5t- 6. From the first equation, t 2 - 4t + 3 = 0 ¢:> (t- 3)(t -1) = 0 sot = 1
or t = 3. t = 1 does not satisfy the other two equations, butt = 3 does. The piuticles collide when t = 3, at the
point (9, 9, 9).
49. Let u (t) = (u1(t), u2(t), u3(t)) and v (t) = (v1 (t), V2(t), va(t)). In each part of this problem the basic procedure is to use
Equation l and then analyze the individual component functions using the limit properties we have already developed for
real-valued functions.
(a) lim u (t) + lim v(t) = I lim u1 (t), lim u2(t), lim ua(t)) + I Jim V1 (t), lim v2(t), lim v3(t)) and the limits of these
t-+a. t--+a \ t -+a t-+a t--+a \t--+a t--+a t-a.
component functions must each exist since the vector functions both possess limits as. t -+ a. Then adding the two vectors
and using the addition property oflimits for real-valued functions, we have that
lim u(t) + lim v (t) = I Jim u1 (t) + lim v1 (t), lim u2(t) + lim v2(t), lim ua(t) + lim v3 (t))
t --..o. t -+a \t--+a
= lim (u1(t) +VI (t), U2(t) + V2(t), 'U3(t) + V3(t))
t-+a .
[using (I) backward]
= lim (u(t ) + v(t)]
t-a
(b) lim cu(t) = lim (cu1 (t), cu2(t), C'U3(t)) = I lim cu1(t), lim cu2(t) , lim cua(t))
t -+a t --+ a \t- a t -+a t -+a
= I c lim u1(t), c lim u~ (t), c lim u3(t)) = c I lim u1 (t), lim u 2 (t), lim ua(t))
\ t-a t -+ a t -+ a \t__,. a. t-+a t-+a
= c lim (u1 (t), u2(t), u3(t)) = c lim u (t)
t--+a
t-a
(c) lim u(t) · lim v(t) = I lim u1(t), lim u2(t), lim u3(t)) ·I lim v1(t), lim v 2 (t), lim v3(t))
t-+a t--+a \t-a t -+a t-a \t--. n. · t-+a t - a.
= [lim u1(t)] [u~ v1(t)] + [lim u2(t)] [lim v2(t)] _;_ .[lim us (t)] [urn v3(t)]
t --+a t -+a t -+ a t --+a t -+a t-+a
@) 20 12 Ccngagc Learning. All Rights Rcscn·cd. May not be: 5Callflcd, copied. or duplicalcd, or posted too publicly accessible website, in whole or in part.