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

164 D CHAPTER 13 VECTOR FUNCTIONS
for x > - ~ ln. 2, the maximum curvature is attained at the point (-~ ln 2, e(- In 2 ) 1 2 ) = (- ~ ln 2, ~).
Since lim e"'(1 + e 2 x)- 3 1 2 = 0, ~~:(x) approaches 0 as x---+ oo.
:.~: - oo
33. (a) C appears to be changing direction more quickly at P than Q, so we would expect the curvature to be great~r
,
at P.
(b) First we sketch approximate osculating circles at P and Q. Using the
axes scale as a guide, we measure the radius of the osculating circle
at P to be approximately 0.8 units, thus p = .!. =>
K,
~ 1.3. Similar!y,'we estimate the radius of the
p .8
"'= .!. ~ - 0
1
c
osculating circle at Q to be 1.4 units, son, = .!. ~ ....!...
4 ~ p 1.
0.7.
X
j6x-' 1 1 6
[1 + (-2x-3)z]3/2 = :z;4 (1 + 4x- 6)3/ 2'
The appearance of the two humps in this graph is perhaps a little surprising, but it is
explained by the fact that y = x - 2 increases asymptotically at the origin from both
-1
directions, and so its graph has very little bend there. [Note th~t r;,(O) is undefined.]
37. r(t) = (tet, e- 1 ,../2t) => .r '(t) = ((t+l)et,-e-'·,../2), r"(t) = ((t+2)et,e-t, O). Then
r' (t) x r" (t) = ( -.J2e-t, ../2(t + 2)e 1 , 2t + 3), · Jr ' (t) x r " (t)l = y'2e- 21 + 2(t + 2) 2 e 2 t + (2t + 3)2,
l r'(t)J = . l(t + 1 )2e2t + e zt + 2 , ( ) lr'(t) x r "(t)l y'2e- 2 t +'2(t + 2)2e 2 t + (2t + 3)2
v and "' t = lr'(t)13 = ((t + 1)2e2t + e-2t + 2]3/2
We plot the space curve and its curvature function for - 5 :::; t :::; 5 below.
K(l)
0.6
y
- 5
5 I
From the graph of r;,(t) we see that curvature is maximized fort = 0, so the curve bends most sharply at the point (0, 1, 0).
The curve bends more gradually as we move away from this point, becoming almost linear. This is reflected in the curvature
graph, where n,(t) becomes nearly 0 as /tl increases.
39. Notice that the curve b has two inflection points at which the graph appears alrnos,t straight. We would expect the curvature to
be 0 or nearly 0 at these values, bu't the curve a isn't near 0 there. Th~ s, a must be the graph of y = f ( x) rather than the graph
of curvature, and b is the graph .of y = K-( x).
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