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

188 0 CHAPTER 14 PARTIAL DERIVATIVES
59. z = sin(xy) (a) C (b) ll
Reasons: This function is periodic in both x andy, and the function is the same when x is interchanged withy, so its graph is
symmetric about the plane y = x. ln addition, the function is 0 along tl1e x- andy-axes. These conditions are satisfied only by
Can~ II.
61. z = sin(x- y) (a) F (b) 1
Reasons: This function is periodic in both X andy but is constant along the lines y = X + k, a condition satisfied only
by F and I.
(a) B
(b) VI
Reasons: This function is 0 along the lines x = ±1 andy = ±1. The only contour map in which this could occur is VI. Also
note that the trace in the x z-plane is the parabola z = 1 - x 2 and the trace in the yz-plane is the parabola z = 1 - y 2 , so the
graph is B.
65. k = x + 3y + 5z is a family of parallel plane$ with normal vector (1, 3, 5).
67. Equations for the level surfaces are k = y 2 + z 2 . For k > 0, we have a family of circular cylinders with axis the x-axis and
radius ,fk. When k = 0 the level surface is the x-axis. (fhere are no level surfaces fork < 0.)
69. (a) The graph of g is 'the graph off shifted upward 2 units.
(~)The graph of g is the graph off stretched vertically by a factor of2.
(c) The graph of g i~ the graph off reflected about the xy-plane.
(d) The graph of g(x, y) = - f(x, y) + 2 is the graph off reflected about the xy-plane and then shifted upward 2 units.
71. f(x,y) = 3x - x 4 - 4y 2 - lOxy
20r-----------------.
10
0
y
Three-dimensional view
y
Front view
It does appear that tlle function bas a maximum value, at the higher of the two "hilltops." From the front view graph, the
maximum value appears to be approximately 15. Bolli h illtop~ could be considered local maximum points, as the values off
there are larger than at the neighboring points. There doe~ not appear to be any local minimum point; although tlle valley sh(\pe
between the two peaks looks like a minimum of some kind, some neighboring points have lower function values.
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