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.  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
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a. Forward FD: