Day 28 in MAT229

Last time

Next time

Announcements:
- I have graded the Canvas part of the exams; I'll have a look at the IMath portion next, and Prof. Wilkinson informs me that there were a fair number of issues. So hopefully your score on that might rise.
- We're in week 8, covering a little more on improper integrals, and then getting a start into the world of Taylor polynomials.
  We love polynomials because they're so simple; we use polynomials to approximate functions for the same reason that we use rectangles to approximate general areas -- because we they're simple!
  But that's for next time.
Today: A lab giving us one more crack at improper integrals.
1. Improper integrals have some sort of infinity causing us trouble: either an $\infty$ staring us in the face, e.g. \[ \int_{1}^{\infty}\frac{1}{x^2}\ dx \] or cleverly concealed inside the integrand (which blows up to $\infty$ at an endpoint, which is the case here, or at some point on the interior of the interval of integration): \[ \int_{0}^{1}\frac{1}{\sqrt{x}}\ dx \]
2. So we still can't (yet) do anything about that integral running over an infinite interval, but we can handle that second problem numerically: of an integrand with a vertical asymptote at an endpoint.
  But we'll see that only the midpoint is helpful in this case.
3. Lab 8 (.nb) (a pdf version)
4. Recording of Tuesday's lab
5. Notes from Tuesday's lab
Links/Notes:
- Caesar Bao recommends the following commands to auto-save your Notebook -- to help you recover more easily from a crash:
```
SetOptions[$FrontEndSession, NotebookAutoSave -> True]
NotebookSave[]
```

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