Day 26 in Mat229

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Announcements
- As a scientist, I feel compelled to share this editorial from the preeminent scientific journal Nature (25 February 2025):
  Trump 2.0: an assault on science anywhere is an assault on science everywhere: US President Donald Trump is taking a wrecking ball to science and to international institutions. The global research community must take a stand against these attacks.
- Your quiz today is over numerical integration.
- We've gotten a little bit behind: I want to finish off improper integrals today (comparisons). Next Monday we will move on to Sequences: Vol 2, Sec 5.1: Sequences.
  You had a fair share of exercises assigned for this section. Any questions on them?
- Lab tomorrow will be over applied numerical integration, with data.
Today:
- Last time we worked through some examples from our improper integral handout. (Here's the Mathematica version if you wish.)
- We were on the verge of talking about comparisons. For that, we turn to an additional handout (and here's the Mathematica version).
  - For comparisons, we need to keep in mind this fundamental relationship:
    
    1 1
    
    >
    
    small BIG
    
    So
    - decreasing the magnitude of the denominator increases the magnitude of the quantity;
    - increasing the magnitude of the denominator decreases the magnitude of the quantity.
    It's easy to get confused.
  - Notice that the final problems ask you to approximate the integrals (if they exist) using the midpoint method. This method is an "open" method (meaning it doesn't tred on the endpoints, which are problems because of what we call a singularity at an endpoint -- the integrand blows up there.
    - Can you integrate this analytically?
    - Can you bound the second derivative, so that you can find the number of rectangles needed to achieve a certain tolerance?
- Quiz
Links:
- Wolfram Alpha
- Mathematica on-line is an option, if you are at a computer without Mathematica installed. I've tried it and it worked pretty well!

Website maintained by Andy Long. Comments appreciated.

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