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CHAPTER 41
Fig. 41-21
41.38
Describe the level curves of f(x, y) = y - x
2 .
This is a family of parabolas y = x
2 + k, all opening upwards and having the y-axis as axis of symmetry.
There is one level curve through every point. See Fig. 41-22.
Fig. 41-22
Fig. 41-23
41.39
Describe the level curves of f(x, y) = ylx.
This is the family of all "punctured" lines y = kx through the origin, but With the origin excluded. There
is a level curve through every point not on the y-axis. See Fig. 41-23.
41.40
Describe the level curves of f(x, y) = 1 Ixy,
These are the rectangular hyperbolas xy = k (k>0) and xy = k ()t<0); see Fig. 41-24. There is a
level curve through every point not on a coordinate axis.
Fig. 41-24
Fig. 41-25
Describe the level curves of f(x, y) -
The level curves are y + x = k ^0. These are the parallel lines of slope —1 and nonnegative y-intercept
(Fig. 41-25). There is a level curve through every point on or above the line y = —x.
41.41
41.42
Describe the level curves of f(x, y) =
The level curves are the curves 1 - y — x
2 = k >0, or y = -x
1 + I — k = -x
2 + c, where c£l. This
is a family of parabolas with the y-axis as axis of symmetry, opening downward, and with vertex at (0, c); see Fig.
41-26. There is a level curve through every point on or below the parabola y = -x
2 + 1.