COALITION MATH 151, FALL 1997

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

                             Day 14.2


1.  List of things to look for when sketching a curve.  Offer to 
explain (in due time) any that need explanation (cf. 3 below).

2.  Activity -- Example 1:  f(x) = - x^3 + 3x.
    Alternate 1-minute brainstorming with demands for suggestions 
    on what to do next.  Build sign table for f, f', f".  Sketch.

3.  Concavity and the second derivative

    A.  Geometrical definition of concavity

    B.  Second derivative as the favorite function of incumbent 
    politicians.  (Things are still getting worse, but not as 
    fast as they were.) 

    C.  Second-derivative test for extrema

    D.  Inflection points

GOT THIS FAR.  POSTPONED E TO DAY 15_1.
    E.  Why study concavity?  (Why eat broccoli?)  Challenge:
    Prove that if f"(x) = (x^2 + 1)f(x) and f(0) = 0, then
    f(x) is not zero for any other value of x.