COALITION MATH 151, FALL 1997

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

                              Day 5.2


    1.  Activity (preparation for integrated exam):  Exercise 
        2.3.26, p. 131.  Prepare a good team presentation, 
        including graphs and a paragraph.  (15 minutes; provide 
        transparencies, try to put printer output on opaque 
        projector) 
SUCCESSFUL.  OPAQUE PROJECTOR WAS UNAVAILABLE; ONE TEAM 
TRANSFERRED GRAPHS TO TRANSPARENCY, ANOTHER DISPLAYED THEIR MAPLE 
SESSION ON THE COMPUTER PROJECTOR.  TEAMS HAD TROUBLE FINDING 
SOMETHING CONCRETE TO ANSWER THE VERBAL QUESTION AT THE END.

    2.  Trigonometric limits (a critique of the traditional 4)

        A.  Stewart's theorems (1) and (3) merely say that sin 
        and cos are continuous at 0.
SKIPPED THIS REMARK.

        B.  The big one:  (sin x)/x -> 1 as x -> 0.
        Show slide of geometrical proof.  (Rigor belongs in a 
        later course.)

        C.  (cos x - 1)/x -> 0 as x -> 0
        (Follows from B by algebra and trig.)  We already know 
        that cos 0 = 1 and this is a point of continuity, but 
        this says more:  the graph has a horizontal tangent there
        (the function approaches 1 very fast).  Later we will see 
        that

        cos x = 1 - x^2/2 + even smaller terms.

        D.  More complicated limits from the basic one.  Example:

            lim_{t->0} (sin^2 2t)/t^2

        There are at least 2 ways to do this.  (Assuming sin(2t) 
        = 2 sin(t) is not one of them.)
ALSO DID sin(7x)/sin(8x).  FORGOT THE "SECOND WAY" (USING TRIG 
IDENTITY TO GET RID OF THE SQUARE).

    C.  Any more questions before the exam?
NO TIME.