Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2022
Sporkz
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33
Level curves and partial derivatives. Contour plots and level curves The contour plot is another way to look at functions of two variables. It gives a slightly different view of the surface than the standard plot, and it's useful for visualizing regions where the function has extreme values. A contour plot is a way of representing the function of two variables by using a map.The same way that when you walk around, you have actual geographical maps thatfit on a piece of paper and tell you about what the real world looks like. So what acontour plot looks like is something like this. Figure 1 shows a plot of the function f(x, y) as determined by values on the axes xand y. The function has been graphed as a series of curves corresponding todifferent values of x and y. For example, one curve might correspond to points wheref equals 1; another curve might correspond to points where f equals 2; and so on.When you see a graph like this, it's supposed to look like the function sits in spaceabove that. It's like a map telling you how high things are. And what you would wantto do with the function is really to be able to tell quickly what's the value at a givenpoint. Well, let's say that we want to determine the function f(x) for a specific value of x. Weknow that f(x) is somewhere between 1 and 2. Actually it's much faster to use thetable of values than to graph the function because reading the values directly fromthe table requires less time than plotting them, measuring their coordinates, andcalculating their slopes. The function f of (x, y) is periodic with period 2 if it equals some fixed values atregular intervals. Typically, these fixed values are chosen at regular intervals.Consider the following example. Let's say we choose the integers 1, 2, 3, and 0 for asequence of positive integers. We can then construct the following sequence: 0
minus 1 (or minus 1) = -1; 1 minus 2 (or minus 2) = -3; 2 minus 3 (or minus 3) = -5.This sequence may be used to demonstrate how cutting a graph horizontallyproduces new graphs that are similar to the original. So horizontal planes have equations of the form z equals some constant, z equals 0,z equals 1, z equals 2, and so on. So maybe the curve of my function will be somesort of blob out there. And if we slice it by the plane z equals 1, then we will get alevel curve that is points where f(x, y) = 1. And so that is called a level curve of f(x). To create a contour map, we repeat the process for each value of z and plot theresulting level curves on the same map. We then squish all the contour mapstogether and use that as our final product. Each line of the graph represents a line on which you could traverse a mountainwithout ever going up or down. If you want to cross a mountain and stay at the sameelevation, then you should just walk along one of these lines.
Visualizing Functions: Contour Plots and Level Curves
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