Develop a computer program that uses the Newton$-$Raphson method to calculate the roots of a given function. Then use the program to calculate the positive root of each of the following functions.

a$.f(x)=0\mathrm{.}51x-sin(x)$

b$.f(x)=x^{10}-1$

For each case, obtain and present the following results.

Plot of the function over an appropriate range to show the root.

Tabulate the results for the first $50$ iterations showing

i $($iteration no$.)$

$x(i)$

$\%$ relative error.

Computer$-$generated plot of the root estimate as a function of iteration number.

Computer$-$generated plot of the $\%$ relative error as a function of iteration number.

Explain the reason for the dramatic difference in convergence rate for the two cases.

Program listing $($so the professor or assistant can duplicate results$-$if needed$).$

For case $($a$)$ use an initial guess of $x=1\mathrm{.}5$ and for Case $($b$)$ use an initial guess of $x=0\mathrm{.}5.$