Solução_Calculo_Stewart_6e

F.
168 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
39. The level curves are (y − 2x) 2 = k or y =2x ± √ k,
k ≥ 0, a family of pairs of parallel lines.
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41. The level curves are y − ln x = k or y =lnx + k.
43. The level curves are ye x = k or y = ke −x ,afamilyof
exponential curves.
45. The level curves are y 2 − x 2 = k or y 2 − x 2 = k 2 ,
k ≥ 0. Whenk =0the level curve is the pair of lines
y = ±x. Fork>0, the level curves are hyperbolas with
axis the y-axis.
47. The contour map consists of the level curves k = x 2 +9y 2 ,afamilyof
ellipses with major axis the x-axis. (Or, if k =0, the origin.)
The graph of f (x, y) is the surface z = x 2 +9y 2 , an elliptic paraboloid.
If we visualize lifting each ellipse k = x 2 +9y 2 of the contour map to the plane
z = k, we have horizontal traces that indicate the shape of the graph of f.
49. The isothermals are given by k = 100/(1 + x 2 +2y 2 ) or
x 2 +2y 2 =(100− k)/k [0