Given the following data.

$\backslash$table$[[x,1\mathrm{.}5,1\mathrm{.}6,1\mathrm{.}7,1\mathrm{.}8,1\mathrm{.}9,2\mathrm{.}0],[f(x),1\mathrm{.}00,2\mathrm{.}35,3\mathrm{.}12,5\mathrm{.}62,7\mathrm{.}70,9\mathrm{.}50]]$

Estimate $f^{\text{'}}(1\mathrm{.}5)$ and $f^{\text{'}}(2)$ using the most suitable $3-$point finite difference formulae with $h=0\mathrm{.}1.$

$[6$ marks$]$

$5.$ Consider the integral

${\displaystyle \underset{0}{\overset{1}{}}}(2+\frac{1}{3}sin2x{)}^{3}dx$

Approximate the above integral using trapezoidal rule with $h=0\mathrm{.}1.$

$[6$ marks$]$

$6.$ Evaluate the integral

${\displaystyle \underset{0}{\overset{3}{}}}\frac{e^{x}sinx}{1+x^2}dx$

using $3-$point Gauss quadrature formula.

$[7$ marks$]$