Consider the function: f$($x$)=1/2($x$-2)^(2)*[2$pi$(0\mathrm{.}55)^(2)]$

Consider the interval: $0<=$ x $<=4$

Carry out the following tasks

Task $1$:

Compute nx $=41$ equidistantly distributed discretized points along x$$and the values in an array named x

Task $2$:

Compute the function values f$($x$)$for each value of x and store the results in an array named f

Task $3$:

Compute the points at which the function values should be interpolated as the midpoints between the values stored in x$,$ store the values in the array x$\_$int

Task $4$:

Interpolate the function values at all x$\_$int locations using the following methods:

linear interpolation, store the results in f$\_$int$\_$linear $($hint: interp$1$ function$)$

cubic spline interpolation: store the results in f$\_$int$\_$spline $($hint: interp$1$ function$)$

lagrangian interpolation: store the results in f$\_$int$\_$lagrange $($hint: use the provided int$\_$lagrange$$function$)$

Task $5$:

Compute the exact values at the interpolation locations x$\_$int and store the results in f$\_$int$\_$exact

Task $6$:

Compute the maximum error between the numerically interpolated values and the exact values for each of the methods and store the results in

e$\_$linear

e$\_$spline

e$\_$lagrangian

Task $7$:

Plot the results in $3$ subplots arranged in $3$ lines and $1$ column

subplot $1$:

plot f$\_$vis vs x$\_$vis as solid black line with line width $3$

plot f$\_$int$\_$linear vs$\_$x$\_$int as a solid red line with circles marking the points, use also line width $3$

subplot $2$:

plot f$\_$vis vs x$\_$vis as solid black line with line width $3$

plot f$\_$int$\_$spline vs$\_$x$\_$int as a solid blue line with circles marking the points, use also line width $3$

subplot $3$:

plot f$\_$vis vs x$\_$vis as solid black line with line width $3$

plot f$\_$int$\_$lagrange vs$\_$x$\_$int as a solid green line with circles marking the points, use also line width $3$

MATLAB Code