Write a well commented MATLAB program to calculate the $1$st $,2$nd $,$ and $3$rd derivative of

an equation using forward finite difference, backward finite difference and central finite

difference methods with an error of the second order O$($h$2$

$)$ for forward and backward finite

different methods and an error of $4$th order for central finite method O$($h$4$

$).$

In your program give the user the capability to insert the step size and the value at which

the derivative is to be calculated. Write the equation in a separate $.$m file as function.

$5$

$2-$ Use your program to find the $1$st$,2$nd$,$ and $3$rd derivatives of the following equation $($f$($x$)=$

x$4+2$x $-$

$2$

$)$ at x$=1$ using two different values for the step size, the first value is h$1=0\mathrm{.}25$

and the second one is h$2=0\mathrm{.}01.$

Solution $1$ at h$1=0\mathrm{.}25$:

First$\_$D$\_$Forward$=$

Second$\_$D$\_$Forward $=$

Third$\_$D$\_$Forward $=1.$

First$\_$D$\_$Backward$=$

Second$\_$D$\_$Backward $=$

Third$\_$D$\_$Backward $=$

First$\_$D$\_$Central$=$

Second$\_$D$\_$Central $=$

Third$\_$D$\_$Central $=$

Solution $2$ at h$2=0\mathrm{.}01$:

First$\_$D$\_$Forward$=$

Second$\_$D$\_$Forward $=$

Third$\_$D$\_$Forward $=1.$

First$\_$D$\_$Backward$=$

Second$\_$D$\_$Backward $=$

Third$\_$D$\_$Backward $=$

First$\_$D$\_$Central$=$

Second$\_$D$\_$Central $=$

Third$\_$D$\_$Central $=$

$6$

Discussion and Analysis

$1.$ Calculate the relative error in the answers for all methods found using h$=0\mathrm{.}25$ assuming

that the answers found using step size h$=0\mathrm{.}01$ is the right one