Consider the function $f(x)=xtan(x).$ Now answer the following:

$($a$)(2$ marks$)$ Evaluate the numerical derivative of $f(x)$ at $x=1\mathrm{.}0$ with step size $h=0\mathrm{.}2$ using the

forward and central difference methods up to $4$ significant figures.

$($b$)(4$ marks$)$ Compute the upper bound of the truncation error of $f(x)$ at $x=1\mathrm{.}0$ using $h=0\mathrm{.}2$ for

the backward and central difference methods up to $7$ significant figures.

$($c$)(4$ marks$)$ Deduce an expression for $D_h^1$ from $D_h$ by replacing h with $(4$h$/3)$ using the

Richardson extrapolation method.

$(3+2$ marks$)$ The following Data set is generated by the function $f(x)=2cos(x)-x+x^{2}sin(x).$

Based on the above data, compute $f(2\mathrm{.}5)$ using the Central Difference method, and also

calculate the relative error. Use $5$ significant figures.

Consider the function $f(x)=7x^4-4e^{-5x}$ Now answer the following:

a$)(3$ marks$)$ Compute $D_{0\mathrm{.}2}^{(1)}$ at $x=3\mathrm{.}4$ using Richardson extrapolation method up to $4$

significant figures.

b$)(2$ marks$)$ Compute $D_{0\mathrm{.}2}^{(2)}$ at $x=3\mathrm{.}4$ using Richardson extrapolation method up to $4$

significant figures.