Q$1)$ Polynomial fitting is widely used in engineering. One application is fitting a polynomial to

noisy experimental data. To practice that you can consider the following scenario.

An object is left to fall freely from a certain elevation. During the fall a sensor mounted on the

object records the time of the fall and the elevation of the object. However, there is always a

noise in the measured data which needs to be considered when evaluating the data.

We can generate this kind of data in Matlab in the following way.

$[20]$ a$)$ Write a Matlab script that evaluates the $"f"$ function for every element of the $t$ array and

plots t vs$.$ yMeasured. $($yMeasured $=f(t))$

$[50]$ b$)$ Suppose we wish to approximate this data $($generated by the Matlab function above$)$ by

a $2$nd order polynomial. Find by hand the least square fit to this data. $($Carry out all the

calculations by hand and show them clearly on paper.$)$

$[30]$ c$)$ Write a Matlab script that calculates the $2^{nd}$ order polynomial fit for the same data and

have both the real data $($yMeasured$)$ and the fitted polynomial data$($yFit$)$ in the same plot.

Please use the "polyfit" function for fitting and "polyval" function for the evaluation of the

polynomial fit. $$ Check the lecture notes for the use or refer to Matlab help.