Day 3 in Mat329

• Reminder: stow those small, cellular rectangles, but bring out your laptops.
• No quiz today, but in general we'll have quizzes on Fridays.
• Roll

Just FYI, regarding Snow's data,

• a recent paper (2003) about the cholera case
• John Snow's case data (time series) from 1854.

I'd like you to work in pairs -- hopefully we'll have enough laptops and copies of Mathematica.

• Who's never used Mathematica, or doesn't feel comfortable with it?
• Who's a Mathematica wizard (or at least has a good start!)?

1. Try to recreate graphs on p. 915, 59-64, to confirm your thinking about which is which. For these, you will need to create models of the functions. This isn't easy! Good luck.... In the end you can take a look at mine, but try your own first.
2. We could look at p. 913, #34-36.
3. We could look at other homework problems, which are due Tuesday, and attempt to use Mathematica to work on them.

   For homework you should use Mathematica in any way you can that doesn't violate the spirit of the exercises. So some exercises are clearly supposed to be done by hand -- and remember that for the exams you won't have Mathematica or your graphing calculator. So use Mathematica as a tool.

4. Joey asked about "sketching" yesterday, and I suggested using Mathematica graphics as a model -- just as an artist might. You can teach yourself to draw better. Exercise your ability to draw.
5. We could move ahead into 14.2!

• As usual we start with the technical definition of a limit in the univariate case (found in section 1.7 of your book, p. 73). Let's take a look at that.

• We then ask "how must we modify this for multivariate functions?"

• In the univariate case, how can a limit fail to exist?
  • Single holes are no problem;
  • Jumps are bad (including infinite ones);
  • Functions may fail to "settle down": sin(1/x) around x=0.

• The same sorts of problems will occur for multivariate functions. But, ironically, limits may exist along particular lines of approach but not agree. This is something new and different. Let's take a look at p. 917, Tables 1 and 2.

  Example 2, p. 919 is also interesting. It might be interesting to look at this example in Mathematica.

• The generalization of the univariate limit of section 1.7 is on p. 917, and well illustrated on p. 918.
  • The difference is in the domain, as usual. Instead of a segment surrounding a, we require a circle surrounding (a,b); in general, a ball surrounding the point in the domain at which we desire the limit, if it exists.

  • Issued a challenge about getting within a tiny value (epsilon) of the limiting value L, we must be able to find a ball small enough (as indicated by radius ) that function values within that "delta-neighborhood" are within of L.