COALITION MATH 152, SPRING 1998

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

                             Day 17.R


    1.  Invite teams to write up their max/min problems for 
    extra credit (5 pts if Web-ready, 3 pts if photocopying is 
    needed).  Deadline of next Thursday (a week before the test).  
    Poll on whether to work in temp teams (problems automatically 
    assigned) or new teams (duplications likely).  Report which
    problems were most often mentioned in reports.

    2.  Newton's method (following prepared transparencies)

        A.  The problem of root-finding and the ideas of sliding 
        down the tangent and of iterating.

        B.  Use differentials to derive the formula.

        C.  Contrast Newton's method and differential 
        approximation for finding square roots; in Newton, "f" is 
        not the square root function!

        D.  Quick description of the secant method (ref. Acton, 
        Numerical Methods that Work).

    3.  Inverse functions and their derivatives

        A.  Volume and radius of a sphere (Maple demo, roughly 
        following lab by Yasskin and Fulling).  Show the cube 
        function on a transparency and invert it.  Discuss 
        "standard" variables (x -> y always) vs. "physical" 
        variables (which retain their names after inversion).

        B.  (continuing the demo)  Calculate and plot tangent 
        lines, note slope relationship, end with generic formula 
        for derivative of inverse.

        C.  More examples of inverses (prepared transparencies)

            i.  exp and log

           ii.  square root, demonstrating the complications:

                a.  Inverse may be undefined somewhere.

                b.  Inverse may be multivalued elsewhere; 
                convention needed to define a FUNCTION.

          iii.  arcsin (demonstrating same points)

ALL THIS WENT WELL.