## Supplementary Matlab exercises #1, #2, and #3

Exercise 1

In this exercise, we define  C  to be the plane curve

y2 = x2 - x4,

1. Is the origin a point of  C?
2. Use the Matlab script aPlot to plot  C  in the region  -1.2 < x < 1.2,    -1 < y < 1, in each of the following ways:

• with a 100 by 100 grid;
• with a 51 by 51 grid;
• with a 500 by 500 grid.

3. Which of the plots are more accurate than the others? What features could be contributing to this situation?

Exercise 2

1. Use the Matlab script aPlot to plot the plane curve

y2 = x4 + y4,

in the region  -1 < x < 1,    -1.2 < y < 1.2, in each of the following ways:

• with a 51 by 51 grid;
• with a 501 by 501 grid;
• with a 1000 by 1000 grid.

2. Answer the same questions as in Part c of Exercise 1.

3. (Challenge question) Can you identify what feature makes this curve "worse" than the curve of Exercise 1?

Exercise 3

Consider the following family of plane cubic curves:

y2 = x3 - 3x + c

Plot  6  curves in this family, corresponding to the following parameter values:

1. c = 2
2. c = -2
3. A value of  c slightly above  c = 2
and a value of  c slightly below  c = 2.
4. A value of  c slightly above  c = -2
and a value of  c slightly below  c = -2.

Print a picture that includes all 6 curves, with distinct colors or patterns, and suitable labels.
Alternatively if this is too cluttered, print 2 pictures, each including 3 curves.
Notes:
1. If you don't have access to a color printer, you may print to a file or files (postscript, jpeg, or pdf),
and then e-mail that file {or those files} to me.
2. I usually don't grade on neatness or appearance, but these exercises are a definite exception to that rule.
Indeed, the point here is to make a drawing (or drawings) which will show the viewer what's going on.
If you can draw all 6 curves on 1 sketch, with suitably contrasting colors and separation, that would be wonderful.
Otherwise maybe you can separate them, perhaps into 2 drawings, each with 3 curves.
• If using equally spaced parameter values, then the Matlab script  family  could be useful.
• Otherwise, if adding curves one by one to your sketch, then the Mablab command  hold on  is very useful.
• It's also possible to combine the two previous suggestions, for instance if you're using 2 groups of 3 closely spaced parameter values.
3. There is a feature of the curve corresponding to  c = -2  that Matlab won't show you,
but I may not divulge particulars about this before the due date of the assignment.

Comments and questions to:  roberts@math.umn.edu

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