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

184 0 CHAPTER 14 PARTIAL DERIVATIVES
9. (a) g(2, - 1) = cos(2 + 2( - 1)) = cos(O) = 1
(b) x + 2y is defined for all choices of values for x andy and the cosine function is defined for all input values, so the domain
of g is JR 2 .
·(c) The range of the cosine function is [- 1, 1] and x + 2y generates all possible input values for the cosine function, so the
range ofcos(x + 2y) is (-1, 1].
(b) ft, .,fij, vz are defined o~ ly when x ~ 0, y ~ 0, z ~ 0, and ln(4 - x 2 - y 2 - z 2 ) is defined when
4 - x 2 - y 2 - z 2 > 0 0, or ix 2 + y 2 < 1. So the domain off
is { (x, y) j ix 2 + y 2 < 1 }, the interior of an ellipse.
y
, ,.,~,..- - - - - -- ..........,
,,~ X
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17. J1 - x 2 is defined only when 1 - x 2 ~ 0, or
x 2 ::; 1 ¢:}
-1 ::; x ::; .1, and \./1 - y 2 is defined
only when 1 - y 2 ~ 0, or y 2 ::; 1 -¢* -1 ::; y::; 1.
Thus the domain off is
{(x,y)l-1::;x::;1, - 1 ::; y ::; 1}.
19. Jy - x 2 is defi~ed only when y - x 2 ~ 0, or y ~ x 2 .
ln addition, f is not defined if 1 - x 2 = 0