Question [3]-Least Square Fitting Method Often an exact solution to a problem in applied mathematics is difficult to obtain. However, it is usually just as useful to find arbitrarily close approximations to a solution. In particular, finding "linear approximations" is a potent technique in applied mathematics. One basic case is the situation where a system of linear equations has no solution, and it is desirable to find a "best approximation" to a solution to the system. This method is called "least squares approximation of data". Hint - Let the system of equations $A_{m\ast n}{x}_{n\ast 1}=b_{m\ast 1}$ is inconsistent system. - Then, any solution $[\tilde{\mathbf{x}}]$ to the normal equations $\left({\mathbf{A}}^{\mathrm{T}}\mathbf{A}\right)\tilde{\mathbf{x}}=\left({\mathbf{A}}^{\mathrm{T}}\mathbf{b}\right)$ is a best approximation in the sense that $\Vert b-A\tilde{\mathrm{x}}\Vert$ is the minimum value of $\Vert b-A\mathrm{x}\Vert$ as $[\mathrm{X}]$ ranges over all columns in ${\mathbb{R}}^n$. Example To measure the take-off performance of an airplane, the horizontal position of the plane was measured every second, from $t=0$ to $t=12$. The positions (in feet) were:

$\mathrm{t}$	0	1	2	3	4	5	6	7	8	9	10	11	12
$\mathrm{y}(\mathrm{t})$	0	8.8	29.9	62	104.7	159.1	222	294.5	380.4	471.1	571.7	686.8	809.2

- Find the least-squares cubic curve $y(t)={\beta }_0+{\beta }_1\ast t+{\beta }_2\ast t^2+{\beta }_3\ast t^3$ for these data. - Use the result of part (a) to estimate the velocity of the plane when $t=4.5$ seconds Hint: Writing a code that applies least square fitting algorithm will be a bonus.