COALITION MATH 152, SPRING 1998

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

                             Day 26.T


    1.  The usual paper shuffling:  Old homework, CAPA, revised 
    new homework, lab sheet if available.  

    2.  Announce freshman math contest tomorrow.

    3.  Congratulations on good project proposal presentations

    4.  While passing out transparencies, quickly review MVT and 
    Taylor's theorem and quickly indicate how one is an extension 
    of the other (point 8 of previous day).

    5.  Planck radiation law activity (point 4 of previous day).
    Point out that geometric series is the key!  Find two terms 
    in each series, the first of which is the historic limiting 
    formula.  Odd teams look at lambda large, even at lambda 
    small.
REASONABLY SUCCESSFUL.  MUCH HANDHOLDING NEEDED.

    6.  Perturbation theory:  Solve [epsilon]x^2 + 2x + 1 = 0
    to first order (second order if time allows).  Check against 
    exact solution and note that one root was lost because it 
    does not have the assumed form.

    7.  Be prepared for air drag problem in lab next week!

    8.  4 ways of finding the Maclaurin series of 1/(5 - 3x).
    (Try to get students to propose them.)

        A.  Taylor's formula

        B.  Long division (see p. 662)

        C.  Geometric series

        D.  Recursion relation
POSTPONED -- GOT TO IT A WEEK LATER.
        
    9.  Background for the rigorous theory of limits consists of
    (a) logic (quantifiers) and (b) inequalities (describing 
    intervals).  The latter will be left to CAPA and recitation 
    (consult Appendix A).  Suggestions for studying limits and 
    logic:

        A.  Read Sec. 1.4 superficially, with special attention 
        to the definition of "limit" on p. 71.   Also, review 
        definition of "continuity" on p. 80.

        B.  Read "Logic" handout.

        C.  STUDY Sec. 1.4.

        D.  Don't get up tight over all this.
GOT THIS FAR (? -- BECAUSE OF PRESSURE OF BUSINESS, I GOT 2 WEEKS 
BEHIND IN MAINTAINING THIS DIARY, AND DON'T REMEMBER EXACTLY WHAT 
HAPPENED WHEN DURING WEEKS 26 AND 27.)

   10.  Logic example:  Everybody loves a lover.  Does this mean:

        A.  \forall x \exists y [y is a lover AND x loves y]

        B.  \exists y \forall x [y is a lover AND x loves y]

        C.  Neither of these?

        I suggest:  "If you're a lover, everybody else will love 
        you."

        D.  \forall y [y is a lover => \forall x (x loves y)]

        But why stop there?

        E.  \forall y [\exists z (y loves z) => \forall x (x 
        loves y)]