Common data set for Problems $1$ and $2$

The temperature in a region varies as in the following table:

$\backslash$table$[[$Depth$,z(m),0,0\mathrm{.}5,1\mathrm{.}0,1\mathrm{.}5,2\mathrm{.}0,2\mathrm{.}5,3\mathrm{.}0],[$Temperature $(C),70,68,55,22,13,11,10]]$

Problem $1($Interpolation$)$: find the temperature $0\mathrm{.}2$ m above and $0\mathrm{.}2$ m below the thermocline depth.

$T(z_0+0\mathrm{.}2m)=$

$T(z_0-0\mathrm{.}2m)=$

Use a second order interpolation polynomial of the form $y(x)=a_0+a_{1}x+a_2{x}^2.$ Provide details about the chosen points from the dataset to solve for $a_0,a_1,a_2$ along with the asked values of temperature.

Problem $2($Numerical differentiation$)$: At the depth of the thermocline $(z=z_0),$ compute the value of the heat flux J as described by Fourier's law:

$J=-k\frac{dT}{dz}$

$k=0\mathrm{.}6\frac{W}{mK}$

Ensure that you maintain the correct SI unit for all involved quantities. Use first$-$order central differencing to fourth order accuracy. State the formula that you select to use $($no need to derive$).$