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

SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES 0 191
c = - 5, z = 2 c = - 10 c= - 2
When -2 < c:::; 0, z ~ 0 for all x andy. Ifx andy have the same sign, then
x 2 + y 2 + cxy ~ x 2 + 1i - 2xy = (x-y) 2 ~ 0. IJthey have opposite signs, then cxy ~ 0. The intersection with the
surface and the plane z = k > 0 is an ellipse (see graph below). The intersection with the s urface and the planes x = 0 and
y = 0 are parabolas z = y 2 and z = x 2 respectively, so the surface is an elliptic paraboloid.
When c > 0 the graphs have the same shape, but are reflected in the plane x = 0, because
x 2 + y 2 + cxy = ( - x) 2 + y 2 + (- c)( - x)y. That is, the value of z is the same for c at (x, y) as it is for-cat ( -x, y).
c = -1, z = 2' c = O c = 10
So the surface is an elliptic paraboloid for 0 < c < 2, a parabolic cylinder for c = 2, and a-hyperbolic paraboloid for c > 2.
79. (a) P = bLo: K 1 - o: =? p = bL"' J