Level Curves and Partial Derivatives. Contour Plots and Level Curves

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 that
fit on a piece of paper and tell you about what the real world looks like.

So what a
contour plot looks like is something like this.
[Graph] show all the points where
f(x,y) = some fixed constant,
chosen at regular
intervals
Figure 1 shows a plot of the function f(x, y) as determined by values on the axes x
and y. The function has been graphed as a series of curves corresponding to
different values of x and y.

For example, one curve might correspond to points where
f 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 space
above that. It's like a map telling you how high things are. And what you would want
to do with the function is really to be able to tell quickly what's the value at a given
point.
Well, let's say that we want to determine the function f(x) for a specific value of x.

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We
know that f(x) is somewhere between 1 and 2. Actually it's much faster to use the
table of values than to graph the function because reading the values directly from
the table requires less time than plotting them, measuring their coordinates, and
calculating their slopes.
The function f of (x, y) is periodic with period 2 if it equals some fixed values at
regular intervals. Iypically, 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 a
sequence of positive integers. We can then construct the following sequence: O
minus 1 (or minus 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 horizontally
produces 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 some
sort of blob out there. And if we slice it by the plane z equals 1, then we will get a
level 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 the
resulting level curves on the same map. We then squish all the contour maps
together and use that as our final product.
[Graph]
Each line of the graph represents a line on which you could traverse a mountain
without ever going up or down. If you want to cross a mountain and stay at the same
elevation, then you should just walk along one of these lines.