Day 20 in MAT229

Today:

• Announcements:
  • Your exams are returned. The results were not what I would have liked.
    • Exam 1
    • Exam 1 key
    • Some comments:
      • Show Work!
      • There was a fair amount of logarithm abuse -- properties of logs are pretty rigid -- no wiggle room!
      • Integrals require differentials. The integral
        

        says "add up an infinite number of infinitely small chunks to get something finite (if you can)." If you add up an infinite number of finite chunks you're likely to get something infinite. We only add up infinitesimals. That's why it's so distressing for me to see someone drop a "dx".

        

        is usually thought of a height () times a width ().

      • Too many struggled with
        • position, velocity, and acceleration: the most important threesome of calculus -- function, first derivative, second derivative.

        • computing distance travelled from velocity.

        • the equation of a tangent line. The most important thing is that the function be linear -- it's graph is a line.

    • All that being said, I was able to put a key together based on your collective work -- that is, someone was able to do essentially every part.

    You will have until Friday to correct your exams, and get them back to me. Return your original exam, as well as a new copy of the exam with all problems repaired. You will earn back 50% of the lost points by doing so.

  • You have an assignment due today....

• Numerical (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).

    

    

  • Setting up your calculator for numerical integration....
    • Midpoint
    • LRR
    • RRR
    • Trapezoidal
    • Simpson's rule

  • Here is a hand out which emphasizes the error term message. Let's take a look at this handout:
    • Compute estimates on the sheet by hand.
    • Check calculations on the calculator.
    • 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?

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

• Improper Integration (section 7.8)