Question $1$

$50$ marks

Consider the problem of interpolating the data

$\backslash$table$[[k,0,1,2,3],[x_k,-1\mathrm{.}20,0\mathrm{.}500,2\mathrm{.}20,3\mathrm{.}10],[y_k,-0\mathrm{.}400,1\mathrm{.}20,5\mathrm{.}50,10\mathrm{.}2]]$

using a polynomial interpolant $P_3(x)$

$($a$)$ Write out the interpolating conditions that $P_3(x)$ satisfies. Follow the recipe on slide $11$ of lecture $14.$

$($b$)$ Construct the linear system for the interpolation data in the table above.

$($c$)$ Solve the linear system for the interpolation coefficients $($you may use NumpPy$).$

$($d$)$ Produce a plot that shows the interpolation data in the table with points as well as the graph of the interpolant. Make sure the axes are labeled and the scale and limits of the axes are appropriate $($extending slightly to the left of $x_0$ and to the right of $x_3).$ Include the figure in your LaTeX submission and add a caption that explains the figure in one or two sentences.