Question $2.$ Interpolation of Temperature Data

In this question $($Question $2)$ we will use interpolation and root finding algorithms to analyze heat transfer in an

environmental application. Lakes in temperate climates can become thermally stratified during the summer: a layer

of warm buoyant water near the surface, called the epilimnion, overlies a layer of colder denser water near the bottom,

called the hypolimnion. The thermocline is a thin horizontal layer that separates the epilimnion and hypolimnion.

Quantifying thermal stratification by approximating the depth of the thermocline can be important when estimating

the impacts of pollution on a lake.

The following data represent measurements of the temperature $T($in $C)$ at various depths $z($in $m)$ measured

from a single location on a stratified lake in a temperate region similar to Guelph, $O{N}^1$ :

${?}^1$ This is not real data, but the values selected are close to values from a real data set drawn from a lake in the Midwestern United

States.

For the purpose of this assignment, for a given temperature distribution $T(z)$ for depths zin$[0,27\mathrm{.}2],$ we define the

depth of the thermocline $z_{\text{t}\text{h}\text{e}\text{r}\text{m}\text{o}\text{c}\text{l}\text{i}\text{n}\text{e}}$ as the depth $z$ where the magnitude of the derivative $|T^{\text{'}}(z)|$ is maximized

over the interval zin$[0,27\mathrm{.}2].{?}^2$

$($a$)$ Newton Polynomial 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 polynomial interpolant $f$ of degree $n-1($in the form of a symbolic function$)$ by following Newton's divided

difference algorithm. Then:

i$.$ Use your Newton Interpolation MATLAB function to produce a polynomial interpolant $($a symbolic func$-$

tion$)$ that models temperature $T_{\text{N}\text{e}\text{w}\text{t}\text{o}\text{n}}(z)$ 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 polynomial interpolant $T_{\text{N}\text{e}\text{w}\text{t}\text{o}\text{n}}(z)$ over the interval of depths zin$[0,27\mathrm{.}2].{?}^3$

ii$.$ Submit another plot of the graph of the magnitude of the derivative $|T_{\text{N}\text{e}\text{w}\text{t}\text{o}\text{n}}^{\text{'}}(z)|$ over the interval of

depths zin$[0,27\mathrm{.}2].$ Based on this plot, what would you discern is 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 $T_{\text{N}\text{e}\text{w}\text{t}\text{o}\text{n}}(z)?$ Is this a reasonable value for $z_{\text{t}\text{h}\text{e}\text{r}\text{m}\text{o}\text{c}\text{l}\text{i}\text{n}\text{e}}?$