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

8 8
47. x 2 + 2y 2 + 3z 2 = 1 =? - (x 2 + 2y 2 + 3z 2 ) = - (1)
ax
ox
8z - 2x x 8 ( 2 2 2) 8 ( )
- 8
= - = - - 3
, and - 8
x + 2y + 3z = - 0
1
x 6z z y y
SECTION 14.3 PARTIAL DERIVATIVES D 199
8z
8z
2x + 0 + 6z ox = 0 =? 6z ax = - 2x =?
8z
0 + 4y + 6z 8y = 0 =?
8z
6z - 0
= -4y =?
y
49. ez = xyz => ez '- 8z = y (
X -
8z + Z · 1 )
=?
ax ax
• 8z 8z yz
(e- - xy) - 8
= yz, so - 8
= -_--.
x x e- - x:y
_ 8z 8z
e- - - xy - = yz =>
OX ox
8z xz
ay = e=- xy
=>
• 8z
8z
e- - xy - = xz , so
8y
e· oy - xy oy = xz => ( - ) 8z
51. (a) z = f(x ) + g(y) => ~: = /'(x), ~ = ,q'(y)
. 8z df QU df I 1
(b) z = f(x + y). Let u = x + y. Then ox = du ox = du (1) = f (u) = f (x + y),
. az = df au= df (1) = J'(u) = f'(x + y).
8y du 8y du
53. f(x , y) = x 3 y 5 + 2x 4 y =? f,(x, y) = 3x 2 y 5 + Bx 3 y, fu(x, y) = 5x 3 y 4 + 2x 4 . Then l rr::c (x, y) = 6xy 5 + 24x 2 y,
f xu(x, y) = 15x 2 y 4 + 8x 3 , fux(x, y) = 15x 2 y 4 + Bx 3 , and f uu (x, y) = 20x 3 y 3 •
55. w = Ju2 + v2
(u
uv ( 1) ( 2 2) - 3/ 2 ( ) . uv
u + v2)3 2' .
2 + v2)3/ 2 ' W uu = v -2 u +v 2u = ( 2 I
1 · ~ - v · ~(u 2 + v 2 ) - 1 1 2 (2v) Juz + v1 - v 2 j .,Ju2 + v 2 u 2 + v 2 - v 2 u 2
Wuv = ( Ju2 + v2)2 ' = u2 + v2 = (u2 + v2)3/2 = (u2 + v2)3/2.
x + y
57. z = arctan-- =?
1 - xy
1 (1)(1-xy)-(x+y)(- y)_ l+y 2 _ 1+y 2
Zx = 1 + (..:.±1L..) 2 . (1 - x y)2 - (1 - x y)2 + (x +y)2 - 1 + x2 + y2 + x2y2
1- xy
1 + y 2 1
= (1 + x2)(1 + y2) = 1 + x2 '
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