This project is under construction
The due date for project 3 is 11 Feb

General Rules for Maple projects.

1. Your name must appear on all printed pages as Maple input.
2. Assignments must be stabled (not paper clipped).
3. Maple output can be confusing if the worksheet is not executed
from top to bottom. Make sure you to execute the the worksheet
from top to bottom before printing.
4. Some Maple programs use too big font for printing. Be sure the
font size is reasonable.
5. The student version of Maple has a "smartplot", do NOT use this.
6. Include your name in the title of any plot.

Hints:

Project 3. (This is not a group project)
Plotting functions and derivatives on the same graph.
#A. Expressions vs Functions. 
g:=x^2; is an expression and not a function. g(6) outputs x(6)^2
f:=x->x^2; is a function. f(6) outputs 36. F is a function, f(x) is an
expression. To turn an expression into a function, use unapply.
h:=unapply(g,x) makes h into a function.
#B. derivatives. "diff". The usage is diff( expression, variable )
diff(g,x) yields 2x. diff(f(x),x) yields 2x, diff(f,x) yields 0. The
output of diff is an expression. To make a functional dirivative
fprime:=unapply(diff(f(x),x),x));
#C. "Diff" is a "pretty printer". It does nothing, but prints well.
Diff(g,x); will print partial x^2 over partial x. (a type of derivative
known as partial derivative. 
#D. "D" the operator D acts on functions and gives functions. D(f) is
the function x->2x. So D(f)(23) is 46, and D(f)(x) is 2x.
#E. Numerical approximations to the derivative. Classically
f'(x) = limit (f(x+h)-f(x))/h as h->0. So (f(x+h) - f(x))/h is the
easiest approximation and as h ->0 is gets closer and closer to f'.
However the symmetric average (f(x+h)-f(x-h))/2h is significantly
better in practice and is what is commonly used in graphing calculators.
#F. Our own deltaf
deltaf(x,h):=(x,h)->(f(x+h)-f(x-h))/(2*h);
This is function of two variables. 
deltaf(x,1.0) is the difference when h = 1.0 and
deltaf(x,0.0001) is the difference when h = 0.0001
#G. 2nd dervative. Maple uses diff(f(x),x,x); for the 2nd derivative. There
are other methods including diff(diff(f(x),x),x) and diff(f(x),x$2). Others
ways using D work too.

Problems:
For problems 1-3 f is sin(x) and deltaf is like 6 above.
#1 Plot f(x) and deltaf(x,h) on the same plot for h = 0.001 for
2 periods of f(x).
#2 plot deltaf(x,h) and diff(f(x),x) or the same graph for h=1.0
#3 plot deltaf(x,h) and diff(f(x),x) or the same graph for h=.1
#4 Find a table for f(x)=sin(x)/x and
x = 1, 0.1, 0.001, ... 0.0000000001 = 1e-10 with the seq command
(hint x co-ordinate is  10^(-i) for i=0..10)
#5 Plot f(x)=4^x and its diff on the same grap from -2..2
#6 Plot f(x)=ln(x) and its diff on the same grap from 1/2..10
#7 Plot f(x)=x(x-1)(x+1) and its diff on the same graph from -2..2
#8 Plot f(x)=x(x-1)(x+1) and its 2nd diff on the same graph from -2..2
#9 Plot f(x)=x(x-1)(x+1)(x-3/2)(x+3/2) and its diff on the same graph
from -2..2 with y restricted between -10 and 10.  Find the roots
of the derivative to at least 4 decimal places.
#10 Plot f(x)=x(x-1)(x+1)(x-3/2)(x+3/2) and its 2nd diff on the same
graph from -2..2 with y restricted between -10 and 10. Find the roots
of the 2nd derivative to at least 4 decimal places.
#9 Plot