1. **Stating the problem:** We are given the function $f(x) = e^x$, a step size $h = 0.5$, and the value $f(0) = 1$. We want to approximate the derivative of $f(x)$ at $x=0$ using finite difference methods. 2. **Formula used:** The finite difference methods approximate derivatives using function values at discrete points. The common formulas are: - Forward difference: $$f'(x) \approx \frac{f(x+h) - f(x)}{h}$$ - Backward difference: $$f'(x) \approx \frac{f(x) - f(x-h)}{h}$$ - Central difference: $$f'(x) \approx \frac{f(x+h) - f(x-h)}{2h}$$ 3. **Calculate function values:** - $f(0) = e^0 = 1$ - $f(0.5) = e^{0.5} = e^{\frac{1}{2}}$ - $f(-0.5) = e^{-0.5} = e^{-\frac{1}{2}}$ 4. **Apply forward difference:** $$f'(0) \approx \frac{f(0.5) - f(0)}{0.5} = \frac{e^{0.5} - 1}{0.5}$$ 5. **Apply backward difference:** $$f'(0) \approx \frac{f(0) - f(-0.5)}{0.5} = \frac{1 - e^{-0.5}}{0.5}$$ 6. **Apply central difference:** $$f'(0) \approx \frac{f(0.5) - f(-0.5)}{2 \times 0.5} = \frac{e^{0.5} - e^{-0.5}}{1} = e^{0.5} - e^{-0.5}$$ 7. **Evaluate numerical values:** - $e^{0.5} \approx 1.64872$ - $e^{-0.5} \approx 0.60653$ 8. **Calculate approximations:** - Forward difference: $$\frac{1.64872 - 1}{0.5} = \frac{0.64872}{0.5} = 1.29744$$ - Backward difference: $$\frac{1 - 0.60653}{0.5} = \frac{0.39347}{0.5} = 0.78694$$ - Central difference: $$1.64872 - 0.60653 = 1.04219$$ 9. **Exact derivative:** The exact derivative of $f(x) = e^x$ at $x=0$ is $f'(0) = e^0 = 1$. 10. **Conclusion:** The central difference method gives a better approximation ($1.04219$) compared to forward ($1.29744$) and backward ($0.78694$) differences for this step size. **Final answer:** - Forward difference approximation: $1.29744$ - Backward difference approximation: $0.78694$ - Central difference approximation: $1.04219$