There are options for executing both single and multiple cells.
#MAX MATLAB CODE#
After that, use the entries on the Cell Menu or the icons on the toolbar to execute the code in the cells provided in the file. The file maxmin.m is designed to be run in "cell mode." Open the file maxmin.m in the Matlab editor, then enable cell mode from the Cell Menu.
#MAX MATLAB DOWNLOAD#
Download the file to a directory or folder on your system. You can download the Matlab file at the following link. Matlab FilesĪlthough the following file features advanced use of Matlab, we include it here for those interested in discovering how we generated the images for this activity. The results in Table 1 are identical to the results found using Figure 9.
Note how the graphical visualization and analytical solution complement one another. First, declare symbolic variables `x` and `y`, then compute `f`. This requires that we find the partial derivative, a simple task using Matlab's Symbolic Toolbox. The first step of the anayltical approach is to find the critical values of the function `f(x,y)=x^3+y^3+3x^2-3y^2-8`. This means that we are constantly walking downhill as we approach the point `(0,2)`, making `(0,2)` a local minimum point.
This means that we are constantly walking uphill as we approach the point `(-2,0)`, making `(-2,0)` a local maximum point. Note that the gradient vectors all point inward near the point `(-2,0)`, regardless of approach.That's because your are going uphill as you approach the saddle point, but fall away downhill in another direction. The gradient vectors "flow" toward the saddle point in one direction, only to "flow" away in another direction. Note the action of the gradient field near the suspected saddle points at `(0,0)` and `(-2,2)`.Recall that the gradient vectors point in the direction of greatest increase of the function i.e., they point "uphill.".Wonderful! Here are some important observations regarding Figure 9. Similar comments are in order for the local maximum that is marked in Figure 1. That is why this point is a local minimum, and not an absolute minimum. However, in its immediate locale or neighborhood, it is a lowest point. In the case of the local minimum, note that it is not the absolute lowest point on the surface, because there are other points on the surface that are lower still. In Figure 1, we've marked a local minimum and a local maximum on the surface. A local maximum is a point on the surface that is the highet point in its immediate neighborhood. We will investigate three types of extrema:Ī local minimum is a point on a surface that is the lowest point in its immediate neighborhood. In this activity, we will apply those visualizations to help determine extrema of multivariable functions of the form `f:R^2\to R`.
In the activities Contour Maps in Matlab and The Gradient in Matlab, we developed visualizations of level curves and the gradient field.
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