**Exercise 1**

- Is the origin a point of
*C*? - 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.

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

**Exercise 2**

- Use the Matlab script aPlot to plot the plane curve

*y*^{2}=*x*^{4}+*y*^{4},

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.

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

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

**Exercise 3**

Consider the following family of plane cubic curves:

*y*^{2} = *x*^{3} - 3*x* + *c*

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

*c*= 2*c*= -2- A value of
*c*slightly above*c*= 2

and a value of*c*slightly below*c*= 2. - 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.

if this is too cluttered, print 2 pictures, each including 3 curves.*Alternatively**Notes***:**- 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. - I
grade on neatness or appearance, but these exercises*usually don't*to that rule.*are a definite exception*

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
parameter values, then the Matlab script*equally spaced*`family`could be useful. - Otherwise, if adding curves one by one to your sketch, then the Mablab
command
`hold on`is very useful. for instance if you're using 2 groups of 3 closely spaced parameter values.*It's also possible to combine the two previous suggestions,*

- If using
- 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.

- If you don't have access to
a color printer, you may print to a file or files (postscript, jpeg, or pdf),

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

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