Consider the differential equation given below. Using the Euler method with the step size (h) of 0.5 , the value of $y$ at $x=1.0$ is equal to _________ (rounded off to 1 decimal place).

$$ \frac{d y}{d x}=y+2 x-x^2 ; y(0)=1 \quad(0 \leq x<\infty) $$

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)