$($b$)$ Cubic Spline Interpolation

Build a MATLAB function which takes $n$ data points $(x_1,y_1),$dots,$(x_n,y_n)$ as input variables and outputs

a natural cubic spline interpolant $f($in the form of a symbolic function$)$ that fits the data by following the

algorithm presented in class $($the efficient approach that only requires us to solve $n$ equations$).$

i$.$ Use your Cubic Spline Interpolation MATLAB function to produce a natural cubic spline interpolant

$($a symbolic function$)$ that models temperature $T_{\text{S}\text{p}\text{l}\text{i}\text{n}\text{e}}(z)$ as a function of depth $z$ and fits the provided

depth$/$temperature data. You don't need to submit the actual form of the symbolic function $($it will be

ugly$).$ Instead, submit a plot of your spline interpolant $T_{\text{S}\text{p}\text{l}\text{i}\text{n}\text{e}}(z)$ over the interval of depths zin$[0\mathrm{.}1,27\mathrm{.}1].{?}^4$

ii$.$ Submit another plot of the graph of the magnitude of the derivative $|T_{\text{S}\text{p}\text{l}\text{i}\text{n}\text{e}}^{\text{'}}(z)|$ over the interval of depths

zin$[0\mathrm{.}1,27\mathrm{.}1].$ Based on this plot, make a guess for the depth of the thermocline, and then use one of the

open root finding algorithms you built for Question $1($b$)$ to approximate, to $3$ significant figures, the depth

of the thermocline $z_{\text{t}\text{h}\text{e}\text{r}\text{m}\text{o}\text{c}\text{l}\text{i}\text{n}\text{e}}$ according to the temperature distribution $T_{\text{S}\text{p}\text{l}\text{i}\text{n}\text{e}}(z).$ Is this a reasonable

value for $z_{\text{t}\text{h}\text{e}\text{r}\text{m}\text{o}\text{c}\text{l}\text{i}\text{n}\text{e}}?$