COALITION MATH 151, FALL 1997

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

                              Day 7.2


    1.  Implicit differentiation:  4 examples.

        A.  Redo the focal length (or parallel resistors) problem 
        from Sec. 2.3 homework. 

        B.  Finish the proof of the power rule. Example:  Find 
        d/dx x^{2/3} by differentiating 

            y^3 = x^2.

        C.  Find d/dx ln(x) by differentiating

            e^y = x.

        Note that this fills a hole in the power rule for 
        antidifferentiation.

        D.  x^3 + y^3 + xy^2 + x^2y - 25x - 25y =0

            i.  Discuss the need to draw a "window" to get a 
            graph segment that defines a function.

            ii.  Calculate formula for y'.

            iii.  IF we know a point on the graph, then the 
            formula gives the derivative of the implicit function 
            at that point.  Example: (3,4) 

            iv.  Other examples:  (3,-3) and (3,-4).  Lesson:  
            WHICH function we are differentiating is not 
            determined until we plug in a value for y.

            v.  The secret of constructing this example:  
            equation factors into

            (x^2 + y^2 -25)(x+y) = 0,

            whose graph is obviously the union of a circle and a 
            line.

            vi.  What will happen if you try to use the formula 
            for y' at

                a.  a point of vertical tangency:  x = 5 ?

                b.  a point of intersection:  x = 5/sqrt{2} ?

    2.  Announce problems for related rates jigsaw:  Subtract 10 
    from your team number and prepare corresponding problem from 
    Sec. 2.8.
FINISHED (PARTLY BY STARTING TOO EARLY).  ALSO DID THE POSTPONED 
POINT 2 OF DAY 7.1 (AMBIGUITY OF THE POLAR ANGLE).