- Math Club -- coming up....
- Sehnert Banquet and Lecture, October 12th (Monday evening, 6:15). This year we will have Frank Morgan ("Mr. Bubbles!").
- May I have your agreements?
- You have an assignment of problems: see the
- Schedule and Assignments page to keep track of these.
- Derivatives and integrals inverse processes?
- Via the fundamental theorem, as you may recall:
- Via the fundamental theorem, as you may recall:
- Problem:
- to estimate a root of the equation
- Solution?
- Calculator Demo
- to estimate a root of the equation
- What's integration all about? Well, we usually start by going about calculating (signed) areas:
"dA" is called an "infinitesimal" -- it's a tiny chunk of area -- tinier than anything you know!
- There are various schemes for numerical integration:
- Left Rectangular
- Right Rectangular
- Midpoint
- Trapezoidal
- Simpson's rule.
- Three of the rules are so called "rectangle rules" (LRR, RRR, Midpoint Rule);
but as the image to the right (above) and the graphical insight on p. 415 show, we can think of the Midpoint rule as being a "Tangent rule":
- The Trapezoidal rule is really just the average of the LRR and RRRs. This gives rise to an important observation, which I want to encourage you to think about, and here it is:
When you have two estimates, you have a third (their arithmetic average, in this case).
- Time now for a hand out which emphasizes this message.
- Simpson's rule is a "blend" (or weighted average) of the Midpoint and Trapezoidal rules, which
perfectly balances the errors of the two to generate a better rule:
Now we can go further:
When you have two estimates, you have infinitely more (their weighted arithmetic averages).
- Examples:
- #9, p. 421
- #20, p. 421
- Links:
- Here's a nice summary of the issue, focused on the use of Mathematica to illustrate them. Interesting even if you don't use Mathematica.