Part I $(55pt)$ The coefficient of thermal expansion $()$ describes how the size of an object changes with a change in temperature $(T),$ and it is essential for the analysis of convection caused by thermal buoyancy. $$ can be calculated by using the temperature$-$dependence of density $()$ or specific volume $(v)$ as described below.

$=\frac{1}{v}(\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{T}}{)}_p=\frac{1}{\frac{1}{}}(\frac{\text{d}\text{e}\text{l}(\frac{1}{})}{\text{d}\text{e}\text{l}\text{T}}{)}_p=-\frac{1}{}(\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{T}}{)}_p$

Therefore, the differentiation of density is needed to find $.$ Using the provided table below $($water density with respect to $T[1])$ and numerical differentiation, you will compute the thermal expansion coefficient. Neglecting all other variables, $=()$ more than six significant figures for $.$ Note that the unit of $K^{-1}$ is the same as the unit $C^{-1}$

$\backslash$table$[[$T$,C,,k\frac{g}{m^3},$T$,C,,k\frac{g}{m^3},$T$,C,,k\frac{g}{m^3}$