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

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73.<br />

10-------------<br />

5<br />

0<br />

- 5<br />

y<br />

SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES D 189<br />

x+y<br />

f ( x, y) = - 2<br />

-- 2<br />

• As both x and y become large, the function values<br />

X +y<br />

appear to approach 0, regardless of which direction is consider<strong>ed</strong>. As<br />

(x, y) approaches the ~rigin, the graph exhibits asymptotic behavior.<br />

From some directions, f(x, y)--+ oo, while in others f(x, y) --+ -oo.<br />

(These are the vertical spikes visible in the graph.) If the graph is<br />

examin<strong>ed</strong> carefully, however, one can see that f(x, y) approaches 0<br />

along the line y = - x.<br />

75. f(x, y) = ecx 2 +v 2 • First, if c = 0, the graph is the cylindrical surface<br />

z = eY 2 (whose level curves are parallel lines). When c > 0, the vertical trace<br />

above they-axis remains fix<strong>ed</strong> whi.le the sides of the surface in the x-direction<br />

"curl" upward, giving the graph a shape resembling an elliptic paraboloid. The<br />

z<br />

level curves of the surface are ellipses center<strong>ed</strong> at the origin.<br />

c=O<br />

For 0 < c < 1, the ellipses have major axis the x-axis and the eccentricity increases as c --+ 0.<br />

y<br />

c = 0.5 (level curves in increments of 1)<br />

-1.2<br />

For c = 1 the level curves are circles center<strong>ed</strong> at the origin.<br />

-1.2<br />

c = 1 (level curves in increments of 1)<br />

[continu<strong>ed</strong>)<br />

© 2012 Cengage Le:uning. All Rights Rcscn·al. Mlly not be scann<strong>ed</strong>, copi<strong>ed</strong>, or duplicotcd 1<br />

or post<strong>ed</strong> to o publicly oot

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