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).
	                       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.

1. Where SHOULD the error tend as h → 0 ?
  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 ?

3. 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

4. Which formula performs better ? in what sense ?

Submit the following, in a plain text file Lab4.txt
  Lab 4: NAME , date
  *************************************************
  a. Answers to 2 for Forward FD:
  =================================================
  b. Answers to 2 for Centered FD:
  =================================================
  c. Answers to 4.
  *************************************************
  d. your code for centered FD (cleaned up!)