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Q$1.(20$ points$)$ Suppose we want to approximate the function $f(x)=x^{2}ln(x).$ What is the $3$rd Taylor polynomial $p_3(x)$ of $f(x)$ at $x=1?$

Q$2$ points$)$ Compute the root of $f(x)=x^3-3.$

$($a$)$ Starting with the interval $1,2,$ apply two iterations of the bisection method. What exactly does it provide? What is the final resulting approximation of $\sqrt[3]{3}.$

$($b$)$ Starting with the initial approximation $1,$ apply two iterations of the Newton method. Write down the absolute and relative errors.Q$3.(20$ points$)$ Eatimate the number of iterations needed to achieve an approximation with accuracy ${10}^{-4}$ to the solution of $f(x)=x^3+4x^2+4x-4$ lying in the interval $\{$:$|0,1\}$ yping, the bisection method.

Q$4.$ Consider the follow table of values for an unknown function $f(x).$

$\backslash$table$[[x,f(x)$