Solve the equation $$x = 10\,\cos \,\left( x \right)$$ using the Newton-Raphson method. The initial guess is $$x = {\pi \over 4}.$$ The value of the predicted root after the first iteration, up to second decimal, is _____________.

The error in numerically computing the integral $$\,\int\limits_0^\pi {\left( {\sin \,x + \cos \,x} \right)dx\,\,\,} $$ using the trapezoidal rule with three intervals of equal length between $$0$$ and $$\pi $$ is _________.

The values of function $$(x)$$ at $$5$$ discrete points are given below:

Using Trapezodial rule with step size of $$0.1,$$ the value of $$\,\int\limits_0^{0.4} {f\left( x \right)\,dx\,\,} $$ is __________.

Simpson's $${1 \over 3}$$ rule is used to integrate the function $$f\left( x \right) = {3 \over 5}{x^2} + {9 \over 5}\,\,$$ between $$x=0$$ and $$x=1$$ using the least number of equal sub-intervals. The value of the integral is __________.