M371 - Alexiades
                      Lab 4
        Finite Difference approximation to derivatives
Write a code to approximate the derivative of a function  f(x) 
using  forward  finite difference quotient

	              f( x + h ) - f( x )
	    f'(x)  ≈  -------------------     (for small h).

For the function  f(x) = sin(x), at x=1 ,  compute the FD quotients for
	  h = 1/2k, k=5,...,N,  with N=30
and compare with the exact derivative  cos(x).
Output  k , h , error.  Where SHOULD the error tend as h  0 ?

1. Look at the numbers. Does the error behave as expected ? 
   Output to a file "out" (or to arrays in matlab), and plot it 
	[ gnuplot>  plot "out" u 2:3 with lines ]
   Which direction is the curve traversed, left to right or right 
   to left ?  Look at the numbers.  h is decreasing 
   exponentially, so the points pile up on the vertical axis. The 
   plot is poorely scaled.  To see what's happening, use logarithmic
   scale, i.e. output  k , log(h) , log(error) and replot.

2. What is the minimum error ?  at what k ?
   Why does the error get worse for smaller h ?

Repeat, using  centered finite differences
[copy your code to a another file and modify it]

	            f( x + h ) - f( x - h )
	    f'(x) ≈ -----------------------     (for small h).
	                    2 h 

3. Which formula performs better ?  in what sense ?

Submit the following, in a plain text file lab4.txt

 Lab 4: NAME , date
 a. Forward FD:
    Answers to 2.
 b. Centered FD:
    Answers to 2.
 c. Answers to 3.
 d. your code for centered FD (cleaned up!)