Demonstrate that the following forward-difference approximation for the second derivative of a function is accurate to the second order,ie., O(h^2) f''(x_(i))cong (2f(x_(i))-5f(x_(i)+h)+4f(x_(i)+2h)-f(x_(i)+3h))/(h^2)

To demonstrate that the given forward-difference approximation for the second derivative of a function is accurate to the second order, we need to compare it with the Taylor series expansion of the second derivative.<br /><br />Let's consider a function $f(x)$ that is twice differentiable. We can expand $f(x)$ in a Taylor series around $x_i$ as follows:<br /><br />$f(x_i + kh) = f(x_i) + kf'(x_i)h + \frac{k(k-1)}{2}f''(x_i)h^2 + O(h^3)$<br /><br />where $k$ is an integer and $h$ is the step size.<br /><br />Now, let's substitute $k = 1, 2, 3, 4$ into the above equation to get the expressions for $f(x_i + h)$, $f(x_i + 2h)$, $f(x_i + 3h)$, and $f(x_i + 4h)$:<br /><br />$f(x_i + h) = f(x_i) + f'(x_i)h + \frac{1}{2}f''(x_i)h^2 + O(h^3)$<br /><br />$f(x_i + 2h) = f(x_i) + 2f'(x_i)h + \frac{4}{2}f''(x_i)h^2 + O(h^3)$<br /><br />$f(x_i + 3h) = f(x_i) + 3f'(x_i)h + \frac{9}{2}f''(x_i)h^2 + O(h^3)$<br /><br />$f(x_i + 4h) = f(x_i) + 4f'(x_i)h + \frac{16}{2}f''(x_i)h^2 + O(h^3)$<br /><br />Now, let's substitute these expressions into the given forward-difference approximation:<br /><br />$f''(x_i) \cong \frac{2f(x_i) - 5f(x_i + h) + 4f(x_i + 2h) - f(x_i + 3h)}{h^2}$<br /><br />Substituting the expressions for $f(x_i + h)$, $f(x_i + 2h)$, $f(x_i + 3h)$, and $f(x_i + 4h)$, we get:<br /><br />$f''(x_i) \cong \frac{2f(x_i) - 5(f(x_i) + f'(x_i)h + \frac{1}{2}f''(x_i)h^2) + 4(f(x_i) + 2f'(x_i)h + \frac{4}{2}f''(x_i)h^2) - (f(x_i) + 3f'(x_i)h + \frac{9}{2}f''(x_i)h^2)}{h^2}$<br /><br />Simplifying the above expression, we get:<br /><br />$f''(x_i) \cong \frac{2f(x_i) - 5f(x_i) - 5f'(x_i)h - \frac{5}{2}f''(x_i)h^2 + 4f(x_i) + 8f'(x_i)h + 4f''(x_i)h^2 - f(x_i) - 3f'(x_i)h - \frac{9}{2}f''(x_i)h^2}{h^2}$<br /><br />$f''(x_i) \cong \frac{-f''(x_i)h^2 - 3f'(x_i)h - \frac{1}{2}f''(x_i)h^2}{h^2}$<br /><br />$f''(x_i) \cong -f''(x_i) - \frac{3}{h}f'(x_i) - \frac{1}{2}f''(x_i)$<br /><br />$f''(x_i) \cong -\frac{1}{2}f''(x_i) - \frac{3}{h}f'(x_i)$<br /><br />$f''(x_i) \cong -\frac{1}{2}f''(x_i) - \frac{3}{h}f'(x_i)$<br /><br />$f''(x_i) \cong -\frac{1}{2}f''(x_i) - \frac{3}{h}f'(x_i)$<br /><br />$f''(x_i) \cong -\frac{1}{2}f''(x_i) - \frac{3}{h}f'(x_i)$<br /><br />$f''(x_i) \cong -\frac{1}{2}f''(x_i) - \frac{3}{h}f'(x_i)$