Day 36 in Mat229

• The exam
• Key
• Top score: 91.5
• Median: 73.5
• Comments:
  • On #1 I had assumed that you'd all use the averages for two of the methods, when you could -- i.e. trap is the average of rrr and lrr; and simp is the weighted average of midpoint and trap.

    I seesawed a bit between two graphics for the plot on page 1: do I include the \(x\)-axis or not? It's a little deceptive when doing area calculations if you don't include that axis; but I wanted to show the concavity as well as possible, to help you assess whether midpoint and trapezoidal should be under or over-estimates.

  • I'll make a few comments on a copy of the test for the rest of the problems.

• if the terms of a series don't converge to 0 as they traipse off to infinity, then the series must diverge.

  That's equivalent to the statement that if the series converges, then its terms must converge to 0.

• The converse of that statement is not true: if the terms of a series converge to 0 as they traipse off to infinity, it doesn't mean that the series must converge.

  The harmonic series is a good example; it diverges, even as its terms go to zero: \[ \sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2}+\frac{1}{3}+\frac{1}{4} + \ldots \]

Thus you're confronting two somewhat unfamiliar animals simultaneously. What ties them together is this idea of running off to infinity, and trying to figure out how to successfully add up an infinite amount of stuff.

Great thought experiments, but perhaps intellectually troubling. That's perhaps why calculus is difficult: there are lots of intellectually troubling things going on. You don't want to turn it into abstract nonsense, however: if you have any questions, please pop by for an office visit!

1. Integral Test (.nb)
2. Integral Test (.pdf)