A function f(x), that is smooth and convex-shaped between interval (x_l , s_u) is shown in the figure. This function is observed at odd number of regularly spaced points. If the area under the function is computed numerically, then . 

Consider the equation $${{du} \over {dt}} = 3{t^2} + 1$$ with $$u=0$$ at $$t=0.$$ This is numerically solved by using the forward Euler method with a step size. $$\,\Delta t = 2.$$ The absolute error in the solution at the end of the first time step is __________

Newton-Raphson method is to be used to find root of equation $$\,3x - {e^x} + \sin \,x = 0.\,\,$$ If the initial trial value for the root is taken as $$0.333,$$ the next approximation for the root would be _________ (note: answer up to three decimal)

The integral $$\,\int_{{x_1}}^{{x_2}} {{x^2}dx\,\,} $$ with $${x_2} > {x_1} > 0$$ is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $${\rm I}$$ is the exact value of the integral obtained analytically and $$J$$ is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?