Use the following data for problems $9-12$

Carbon is diffused into a steel bearing to case$-$harden it$.$ The carbon concentration was measured at various depths and

is shown below.

Use Newton's method to approximate the concentration of carbon at a depth of $0\mathrm{.}15mm$ with a $1^{st}$ order

interpolating polynomial.

Use Newton's method to approximate the concentration of carbon at a depth of $0\mathrm{.}15mm$ with a $2^{nd}$ order

interpolating polynomial.

Use Newton's method to approximate the concentration of carbon at a depth of $0\mathrm{.}15mm$ with a $3^{rd}$ order

interpolating polynomial.

Use linear algebra to find the coefficients of the $3^{rd}$ order polynomial that passes through the data points at

depths of $0,0\mathrm{.}1,0\mathrm{.}35,$ and $0\mathrm{.}45.$ Use MATLAB to solve the system of equations. Plot the complete data set and

the fit line on the same graph. Include axis labels and a legend.