Newton-Raphson method for solving algebraic equations is based on

The exact solution of $\int_0^4 \frac{d x}{1+x}$ is represented as $n$.

If $m$ represents numerically evaluated value of the above integral using Trapezoidal rule by considering four equal subintervals in the range of $x$, then ( $m-n$ ) is

$$ \text { The values of a function } f \text { obtained for different values of } x \text { are shown in the table below. } $$

$$ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline f(x) & 0.9 & 2.0 & 1.5 & 1.8 & 0.4 \\ \hline \end{array} $$

$$ \text { Using Simpson's one-third rule, } $$

$$ \int_0^1 f(x) d x \approx $$__________[Rounded off to 2 decimal places]

In order to numerically solve the ordinary differential equation dy/dt = -y for t > 0, with an initial condition y(0) = 1, the following scheme is employed:

$\frac{y_{n+1} - y_{n}}{\Delta t} = -\frac{1}{2}(y_{n+1} + y_{n}).$

Here, $\Delta t$ is the time step and $y_n = y(n\Delta t)$ for $n = 0, 1, 2, \ldots.$ This numerical scheme will yield a solution with non-physical oscillations for $\Delta t > h.$ The value of h is