COALITION MATH 151, FALL 1997

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

                              Day 4.2


    1.  Antiderivatives, velocity, and acceleration.

        A.  Speedometer-odometer paradigm.

        B.  Activity:  Polynomial example.  State problem in 
        mathematical and then in physical terms.  Emphasize that 
        at^2/2 + ... applies only to CONSTANT acceleration. 

        C.  State formulas for derivatives of sine and cosine.
    
        D.  Activity:  Problems with cosine as antiderivative.  
        State problem in mathematical and then in physical terms.  
        Emphasize that the constant of integration is not always 
        the initial value. 

        E.  Mention that e^{-x^2} and (sin x)/x do not have 
        elementary antiderivatives.
THIS WENT WELL, BUT CONSUMED THE ENTIRE PERIOD.  (TIME AT 
BEGINNING HAD TO BE SPENT RESOLVING CAPA TROUBLES.)  THE 
EMPHASIZED ERRORS DID NOT APPEAR IN EITHER MATHEMATICAL OR 
PHYSICAL FORMULATION (UNLIKE LAST YEAR'S TESTS).  BOTH TEAMS 
DOING THE INITIAL MATHEMATICAL PROBLEMS HAD SURPRISING TECHNICAL 
DIFFICULTY, BUT MOST OTHER TEAMS SEEMED QUITE COMPETENT.
    POSTPONED THE FOLLOWING TO NEXT DAY.

    2.  Derivatives and antiderivatives in Maple.

        A.  Explore  D, diff, unapply, Limit, value, D@@2.

        B.  Explore  int, Int, value.  (Cf. 1E.)

        C.  Direction fields:
            with(DEtools);
            dfieldplot(diff(y(x),x)=<expression in x>, y(x), 
                -1..3, y=-1..4);
            (These ranges are good for sqrt(1+x^3) - x, 
            corresponding to Stewart ed. 3 p. 247, Fig. 4.)
SUCCESSFUL, FOLLOWING THIS SCRIPT:

> f := 3^x;
> D(f);
> diff(f);
> diff(f,x);
> g:=unapply(f,x);
> h:= x-> 3^x;
> i:=unapply(3^x,x);
> D(g);
> Limit( (g(x+h) - g(x))/h, h=0);
> value(");
> Delta_f := subs(x=x+h, f) - f;
> D@@2(h);
> (D@@2)(h);
> Int(x^2, x);
> value(");
> int(x^2, x);
> (D@@(-1))(g);
> int(f,x);
> int(g,x);
> int(g(x), x);
> k := unapply(", x);
> Int(exp(-x^2), x);
> value(");
> Int(sin(x)/x, x);
> value(");
> help(Si);
> help(erf);
> plot(erf);
> plot(Si);
> with(DEtools);
> dfieldplot( diff(y(x),x) = x, y(x), -10..10, y=-10..10);
> dfieldplot(diff(y(x),x) = sqrt(1+x^3) -x, y(x), -1..3, 
     y=-1..4); #Stewart p. 257 Fig. 4
> dfieldplot(diff(y(x),x) = exp(-x^2), y(x), -2..2, y= -2..2);
>