Day 24 in MAT229

Today:

• Announcements:
  • Exam 2: next week, 3/5: it will cover chapter 7, specifically.

  • HW4 is due tomorrow.

  • Let's have a look at a few things in our numerical integration worksheet.

    Let me start by pointing out that I mislead you in showing that 2nd derivative plot: how do we know it's wrong? The good news is that, in terms of bounds, it was pretty darned close!

    The stories the problems tell:

    • First set of problems: with equal $n$, midpoint is about twice as good as trapezoidal.

    • Second set: with different $n$, more $n$ trap beats few $n$ mid. Both get within the tolerance required, by a lot. The bound is conservative.

    • Third set: by bounding the fourth derivative, you find that you need only $n=6$ rectangles for Simpson's rule. So to compute $S_6$, you'll need $T_3$ and $M_3$: \[ S_6=\frac{2M_3+T_3}{3} \] Some of you just ran out and grabbed some musty old $M$ and $T$s (specifically those from the previous set of problems), and stuffed them in there. In fact, the $M$ and $T$s are very specifically related to the number of rectangles used. You must use a "matched pair" of $M$ and $T$s.

• Today: Section 11.1: Infinite Sequences
  • The opening page of Chapter 11 in our text offers some exciting prospects. Anything wrapped up with infinity is exciting!

    "No one will drive us from the paradise which Cantor created for us." -- David Hilbert

  • We'll operate off of Wilkinson's Sequences page.

    One of the features that I illustrate with these notes is that of creating a function which serves to implement an algorithm.

    I also illustrate a recursive (self-referential) algorithm, for computing square roots.