Day 18 in MAT229

Today:

• Announcements:
  • Your exam tomorrow will cover the material we covered in chapter 6 as well as section 7.1.

  • Today we have a review, which will consist of many questions from you!

  • Your 7.1 homework is due today. Questions concerning this homework are welcome for the review. Graded
    • 2
    • 20
    • 46

  • Your 6.8 homework is returned: graded
    • 6
      • many of you used an determinate form!
      • Don't drop limits
      • Don't abuse equality
      • Chad's idea....

    • 20 -- simplify....
    • 28
      • must use L'Hopital twice....
      • Don't differentiate the quotient -- differentiate numerator and denominator separately.

• If there is any time left after the review, we will continue with section 7.7: approximate integration.
  • 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 ("vanishingly small")!

    What's numerical integration all about? We do pretty much the same thing, only we have

    

    where the are small, but not vanishingly small.

  • There are various schemes for numerical integration:
    • Left Rectangular
    • Right Rectangular
    • Midpoint
    • Trapezoidal
    • Simpson's rule.
    Let's see what unifies and what distinguishes them.

  • 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 below 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).

    

  • 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:
    

    Notice that the number of subintervals in Simpson's rule must be even.

    Now we can go further:

    When you have two estimates, you have infinitely many more.

    (their weighted arithmetic averages).

    

    

  • Here is a hand out which emphasizes this message. Let's take a look at this:
    • Check midpoint calculation
    • Let's do the calculations by the formulas, for
      • Trapezoid
      • Simpson's
    • Check that we get the same things using the averages
    • Calculate the true value of the integral, and compare to the approximations. What's surprising?

  • Setting up your calculator for numerical integration....

  • Examples:
    • #29, p. 541
    • Others?

  • 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.