1. **Problem Statement:** Explain the finite differences method with an example. 2. **What is the Finite Differences Method?** The finite differences method is a numerical technique used to approximate derivatives of functions by using values of the function at discrete points. 3. **Formula and Explanation:** The first finite difference of a function $f$ at point $x$ with step size $h$ is given by: $$\Delta f(x) = f(x+h) - f(x)$$ This approximates the derivative $f'(x)$ as: $$f'(x) \approx \frac{\Delta f(x)}{h} = \frac{f(x+h) - f(x)}{h}$$ 4. **Example:** Suppose we want to approximate the derivative of $f(x) = x^2$ at $x=2$ using $h=0.1$. Calculate: $$f(2) = 2^2 = 4$$ $$f(2+0.1) = f(2.1) = (2.1)^2 = 4.41$$ Then the finite difference approximation is: $$f'(2) \approx \frac{f(2.1) - f(2)}{0.1} = \frac{4.41 - 4}{0.1} = \frac{0.41}{0.1} = 4.1$$ 5. **Compare with exact derivative:** The exact derivative of $f(x) = x^2$ is: $$f'(x) = 2x$$ At $x=2$, $f'(2) = 4$. 6. **Interpretation:** The finite difference method gives an approximate derivative of $4.1$, which is close to the exact value $4$. The accuracy depends on the choice of $h$; smaller $h$ generally gives better approximations. This method is widely used in numerical analysis and computational applications where analytical derivatives are difficult to obtain.