/*  Approximate the derivative f'(x) using a second order approximation
    of the form R(h) = c1 D(h) + c2 D(2h) where

         D(h) = (f(x+h) - f(x))/h

    and constants c1 and c2 are given by Richardson extrapolation. */

#include <stdio.h>
#include <math.h>

double f(double x){
    return 1/(x*x-x+3);
}
double df(double x){
    double t=x*x-x+3;
    return -(2*x-1)/(t*t);
}
double D(double x,double h){
    return (f(x+h)-f(x))/h;
}
double R(double x,double h){
    /* Please fill in this function */
}

#define N 8 
double y[N]= {
    4.44444444444444e-02, 2.22222222222222e-02,
    7.12788259958069e-03, 1.90869551612988e-03,
    4.83780672011547e-04, 1.21029365004299e-04,
    3.02168640078926e-05, 7.54583882628390e-06 };

int main(){
    printf("This is problem 3 on exam part 2.\n");
    printf("%21s %21s %21s\n","h","R_h","Error");
    double eold=1,x=1,dfx=df(x),h=1,flag=0;
    for(int n=0;n<N;n++){
        double t=R(x,h),e=fabs(t-dfx);
        if(fabs(e-y[n])>y[n]*1e-6) flag=1;
        printf("%21.14e %21.14e %21.14e\n",h,t,e);
        h/=2;
    }
    if(!flag){
        printf("Congratulations!  Your answer is correct!\n");
    } else {
        printf("Please try again.  Your answer is incorrect.\n");
    }
    return 0;
}