a) A cubic polynomial $y = f(x)$ satisfies the following data: \begin…

Previous year question for Numerical Methods & Transform Calculus.

Question

a) A cubic polynomial $y = f(x)$ satisfies the following data:

\begin{tabular}{c|ccccc}

$x$ & 0 & 1 & 3 & 4 \\

\hline

$f(x)$ & 1 & 4 & 40 & 85

\end{tabular}

Determine the function $f(x)$ and hence find $f''(2)$.

b) Evaluate $\int\limits_{0}^{1} \sqrt{(1-x^2)} dx$ by Trapezoidal rule, taking $h = 0.1$.

c) Define the operators $\Delta, \nabla, \mu, \delta, E$ and $E^{-1}$. Also show that $\mu\delta = \sinh(hD)$.