COALITION MATH 151, FALL 1997

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

                         Days 3.1 and 3.2


    1.  Questions?  

    2.  Continuity and limits (intuitive minilecture)

        A.  The precise definition of a limit is the deepest 
        concept in calculus.  We postpone it to a later semester.  
        Therefore, we can't really prove theorems about limits.
        Intuitively, it seems better to talk about continuity 
        first.
GOT ONLY TO MIDDLE OF B ON DAY 3.1, BECAUSE MOST OF DAY WAS SPENT 
FINISHING 2.2.  CONTINUED NEXT DAY.

        B.  Intuitively, a function is continuous if its graph 
        can be drawn without lifting the pen from the paper. 
        [draw example]  How might this fail to be true? [sketch 
        x/|x|, 1/x^2, x/x; mention (sin x)/x to show why the last 
        type is not trivial]  Remark: All these examples are 
        continuous EVERYWHERE EXCEPT at x = 0. 

        C.  A function is continuous if nearby inputs yield 
        nearby outputs:  If x is close to a, then f(x) is close 
        to f(a).  A function that violates this condition is very 
        inconvenient for an engineer!  Imagine having to aim a 
        gun so that it will shoot a bullet through a hole no 
        bigger than the bullet itself (x = angle, f(x) = place 
        where bullet lands).

        D.  Intuitive definition:  lim_{x->a} f(x) = L  means 
        that f(x) can be guaranteed to be arbitrarily close to L 
        by choosing x sufficiently close to a. [show drawing of 
        Tate's drill press, and an example where it does not 
        close]  Precise definition is in Sec. 1.4.

        E.  Definition:  f is continuous at a means that
                
                 lim_{x->a} f(x) = f(a).

        This combines 3 assertions:
          i) f(a) is defined.
         ii) lim_{x->a} f(x) exists.
        iii) These two numbers are equal. 
        [show slide and relate to previous examples]

        F.  "Limit" is the fundamental concept of calculus.  
        Everything else is defined in terms of it:
            continuity
            derivative
            integral
            sum of infinite series

        G.  Limit theorems (Sec. 1.3) tell us how to calculate 
        limits.  E.g., "lim" can be pushed inside sums; quotients 
        OK unless denominator's limit is 0.  More generally, the 
        operational significance of continuity is that "lim" can 
        be pushed through a continuous function: 

                lim_{x->a} f(g(x)) = f(lim_{x->a} g(x))
    
        if f is continuous (and the inner limit exists and is in 
        the domain of f).

    3.  Activity:  Subtract 6 from your team number and answer 
    the corresponding problem on p. 68 (Exercises 1.3.15ff).
    (Call up a decent number of these; collect the papers and 
    grade on existence.)
THIS WAS VERY SUCCESSFUL ON DAY 3.2.  21 MINUTES WERE AVAILABLE, 
AND AFTER 6 MINUTES OF TEAM WORK, MORE THAN HALF OF THE 20 TEAMS 
WERE ABLE TO PRESENT SOLUTIONS (ALL BASICALLY CORRECT).