Multivariate Calculus - Bruce E. Shapiro

56 LECTURE 8. FUNCTIONS OF TWO VARIABLES
Figure 8.4: Example of a contour plot illustrating the maximum temperature during
the week of July 23, 2006. Rather than annotating the contours the space between
the contours is colored, and the temperature is read by comparison with the legend
on the figure. [Taken from the US NOAA/National Weather Service Climate
Prediction center, as posted at http://www.cpc.ncep.noaa.gov.]
Solution. The level curves for z=0 are the curves
0 = 2 − x − y 2
⇒
⇒
The level curves for z=1 are the curves
y 2 = 2 − x
1 = 2 − x − y 2
⇒
⇒
The level curves for z=-1 are the curves
y = ± √ 2 − x
y 2 = 2 − x − 1 = 1 − x
y = ± √ 1 − x
−1 = 2 − x − y 2
⇒
⇒
y 2 = 2 − x + 1 = 3 − x
y = ± √ 3 − x
The level curves are illustrated in figure 8.5.
Example 8.7 Find a general form for the level curves z = k for the function
z = x2 + y
x + y 2
Revised December 6, 2006. Math 250, Fall 2006