Math 3

Winter 2004

Section 2.1, Tangent and velocity

We show the graphs of a function f and some secants.

> f:=x->x^3/3: x0:=1:y0:=f(x0): printf(" x1 f(x1) m=(f(x1)-f(x0))/(x1-x0) \n\n"):
for n from 0 to 4 do
x1 := x0+2^(-n): m[n] := (f(x1)-f(x0))/(x1-x0): printf("%15.10f %15.10f %15.10f\n",x1,f(x1),m[n]);
end do:

>

        x1            f(x1)      m=(f(x1)-f(x0))/(x1-x0) 

   2.0000000000    2.6666666670    2.3333333330

   1.5000000000    1.1250000000    1.5833333330

   1.2500000000     .6510416667    1.2708333330

   1.1250000000     .4746093750    1.1302083330

   1.0625000000     .3998209635    1.0638020830

A graph is always good:

> plot([f(x),m[0]*(x-x0)+y0,m[1]*(x-x0)+y0,m[2]*(x-x0)+y0,m[3]*(x-x0)+y0],x=0.7..2,y=-1..3, color=[black,blue,red,green,yellow]);

>