COALITION MATH 152, SPRING 1998

              (S. A. Fulling, assisted by Vera Rice)

                             Day 25.R


    1.  Review Taylor's formula and the list of Maclaurin series.
    
    2.  Example of algebraic manipulation:  Find the Maclaurin 
    series of e^{-x} \sqrt{4 + x}.  Note: High-order terms are 
    like [less] significant digits. 

    3.  TEAM RAT (5 min):  Suppose f(x) = 1 + 3x + x^2 + ....
    Find first 3 terms in Taylor expansion of e^x f(x). HINTS:

        A.  "It's just a big polynomial ..."

        B.  Remember the significant figures analogy.

    [ANSWER:  1 + 4x + (9/2) x^2 + ...]

    4.  Density of black body radiation.  After brief discussion, 
    a TEAM EXERCISE:  Odd teams derive Rayleigh-Jeans for large 
    lambda, even teams derive Wien's Law for small lambda.
I POSTPONED THIS, SO THAT THERE WOULD NOT BE TWO ACTIVITIES IN 
RAPID SUCCESSION.

    5.  Theory break:  Careful statement of Taylor's Theorem with 
    Remainder.  To get a context for what this is all about ...
    [Condense the following as time constraints dictate.]

    6.  Mean Value Theorem:  Statement, disclaimer of importance 
    of numerical value, graphical proof, [insert team exercise 7],
    use in proving uniqueness of antiderivatives mod constants.

    7.  TEAM EXERCISE (no plastic):  Sketch a graph of a function 
    that violates the MVT by violating one or the other of the 
    hypotheses:

        A. not differentiable in the interior

        B. not continuous at an endpoint

    (Each team should produce one example of each type.)
GOT THIS FAR.  WILL DO POINTS 4 AND 8 NEXT TIME.

    8.  How the MVT generalizes to Taylor's theorem

        A.  Contrast definition of continuity, MVT, and best 
        linear approximation.

        B.  Get an infinite alternating hierarchy of 
        approximation theorems, demanding increasing 
        differentiability.