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=30and 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 ?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 ?
