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

CHAPTER 13 REVIEW D 173
43. The tangential component of a is the length of the projection of a onto T, so we sketch
the scalar projection of a in the tangential direction to the curve and estimate its length to
be 4.5 (using the fact that a has length 10 as a guide). Similarly, the normal component of
a is the length of the projection of a onto N, so we sketch the scalar projection of a in the
normal direction to the curve and estimate its lengttJ to be 9.0. Thus aT ::::::: 4.5 cmf s 2 and
X
45. _If the engines are turned off at timet, then the spacecraft will continue to travel in the direction ofv(t), so we need at such
that for some scalars > 0, r(t) + s v (t) = (6, 4, 9). v(t) = r ' (t) = i + ~ j + (t 2
~ 1
) 2
k =>
4 8(3-t)t
so 7- t2 + 1 + (t2 + 1)2 = 9 ¢}
24t- 12t 2 - 4
- -;-:-;;:--:--::-.;--:- = 2 ¢} t 4 + 8t 2 - 12t + 3 = 0.
(t2 + 1)2
·a is easily seen that t = 1 is a root of this polynomial. Al~o 2 + ln 1 + 3 ~ 1 = 4, sot= 1 is the desired solution.
13 Review
CONCEPT CHECK
1. A vector function is a function whose domain is a set of real numbers and whose range is a set of vectors. To find the derivative
or integral, we can differentiate or integrate each component of the vector function.
2. The tip of the moving vector r (t) of a continuous vector function traces out a space curve.
3. The tangent vector to a smooth curve at a point P with position vector r(t) is the vector r ' (t). The tangent line at p ' is the line
through P parnllel to the tangent vector r ' (t). The unit tangent vector is T (t) = ,:m,.
4. (a) (a)- (f) See Theorem 13.2.3.
5. Use Formula 1 3.3.2, or equivalently, 1 3.3.3.
6. (a) TI1e curvature of a curve is"' = I~~ I where Tis the unit tangent vector.
I I
(b) K.(t) = . T r '(t)
'(t)
(c) K.(t) = lr' (t ) X r" (t)l
[r'(t)[ 3 _ if"(x) l ·
(d) K-(x)- [1 + (f'(x))2j3/2
T'(t)
7. (a) The unit normal vector: N (t) = IT'(t)l' The binormal vector: B (t) = T (t) x N (t).
(b) See the discussion preceding Example 7 in Section 13.3.
8. (a) Ifr(t) is the position vector of the particle on the space curve, the velocity v (t) = r'(t), the speed is given by lv (t )i,
and the acceleration a (t) = v ' (t) = r" (t).
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