Consider the atmospheric temperature data in table 1 to estimate the temperature at altitude z = 12,000 m. Use natural cubic splines to interpolate. You can use MATLAB to solve the linear system of equations for the coefficients of the splines as discussed in class.

Table 1:
knots index, i altitude (m) temperature (◦C)
1 9000 -43.42
2 10000 -49.90
3 15000 -56.50
4 20000 -56.50
Table 1: Atmospheric temperature. The data is also stored in air temperature.mat.
Source: https://www.engineeringtoolbox.com/standard-atmosphere-d_604.html.

Problem 2:
Repeat problem 2, but use quadratic spline. For a quadratic spline, the piecewise function takes the form: si(x) = ai + bi(x − xi) + ci(x − xi)2 where ai, bi, and ci are coefficients to be determined. Start with the continuity of si and s′i at the interior knots. Apply clamped condition which sets the first derivative at the last knot to be zero. Clearly state any additional assumptions that are used to construct the splines