MATH 163, Spring 2001         Due Date:
Name(s):
Extra Project 3.1: Graphing a Function Using a Table
Objective
To review the graphing of functions by plotting points and "connecting the dots", and to investigate the connection between / and /' as functions.
Narrative
Since fix) = lini/j^o-----------------------, it follows that for small values of h,
h
f {x) „ /(* + hl - /(*).                                               W
Task
a)  Type the command lines below into Maple in the order in which they are listed. These command lines produce a table of values for f(x) = (1 — x2)/(l + x2) and /'(x), x G [—2, 2]
>   # Project  3.1:     Graphing a Function Using a Table
>   restart;
>   f   := x ->   (l-x"2)/(l+x"2);
>   fl   := D(f);
>   a   := -2.0;   b   := 2.0;   N   :=  10;   dx   :=   (b-a)/N;
>   M   := matrix(N+l,3,(Row,Col)->0):
>   M[1,1]   :=   'x';   M[l,2]   :=   'f(x)';   M[l,3]   :=   'fl(x)';
>   Digits   := 2;
>   for i  from  1 to N do
x   := a +   (i-l)*dx:
M[i+l,l]   := x:     M[i+1,2]   :=f(x):     M[i+1,3]   :=fl(x):     od:
>   eval(M);
b)  Continue by typing the following command lines into Maple. These command lines produce an empty graph (on which you will be asked to graph / and /').
>   x   :=   'x';
>   plot(0,x=a..b,y=-l..1);
At this time make a hard-copy of your typed input and Maple's responses. Then, ...
c)  on the graphic you produced in part (b), use the data you produced in part (a) to graph / and /' by hand, each in a different color, and label each graph. (Draw the graphs of / and /' by plotting points and "connecting the dots".)
d)  Illustrate (*) by using the table of values you produced in part (a) to:
1.   approximate /'(l) (you'll be using values in the second column of the table in part (a) to do this), and
2.   compute the absolute value of the difference between this approximation and the actual value of /'(l) (which you can obtain from the third column of the table in part (a)). This is the error in the approximation.