COALITION MATH 151, FALL 1997

        Group II (S. A. Fulling, assisted by Cary Lasher)

                              Day 9.3


BEGAN WITH AN "ATTITUDE RAT" AND PEP TALK; ANNOUNCED RESULTS OF 
THE QUIZ THAT CONSTITUTED THE CURVE ON EXAM 2.
THEN FINISHED THE NUMERICAL INTEGRATION LECTURE ON ERROR BOUNDS. 

    1.  Miscellaneous properties of the definite integral
    (Refer to pp. 277 and 279 of ed. 3 for the whole list.)
POSTPONED THIS TO NEXT DAY.
        
        A.  Integral from b to a = - in from a to b.

        B.  Integral from a to b = int from a to c + int from c 
        to b. (Remark that proof needs Riemann's messy 
        definition.) 

        C.  Linearity.

        D.  f >= g implies int of f >= int of g.  (This and E 
        assume a < b!)

        E.  Upper and lower bounds:  (b-a) Max f and (b-a) Min f.

        F.  Mean value theorem for integrals (by picture only).

    2.  The fundamental theorem(s) of calculus

        A.  FT I:  Differentiation undoes integration.

            1.  Proof by picture.

            2.  Examples (with class participation)
THIS WAS UNUSUALLY SUCCESSFUL:  3 DIFFERENT STUDENTS (NOT ALL 
BEST IN CLASS) SUPPLIED CORRECT ANSWERS TO THE SOCRATIC PROBLEMS.
                i.  a power (so theorem can be verified)

                ii.  1/t (so theorem can be verified by those who 
                know logs)

                iii.  something with a compound limit (requiring 
                chain rule) 

        B.  FT II:  Integration undoes differentiation, up to a 
        constant.

            1.  Remarks on practical importance.

            2.  Proof from FT I and the lemma:

        C.  Theorem:  If two functions have the same derivative, 
        they differ only by a constant (if the domain is a single 
        interval).

            1.  Promise proof from the mean value theorem later 
            (or reverse development following Tucker).

            2.  Demonstrate need for the parenthetical hypothesis 
            by the example of derivative x/\sqrt{x^2 -1} with 
            disconnected domain.