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The image contains exercises related to **finite difference schemes** from a course on numerical methods. These exercises focus on deriving schemes for approximating derivatives with specific accuracy orders using values of the function at different points.

Here’s a breakdown of the problems visible:

### Exercise 1: 1D Schemes
1. **1.1** Derive a finite difference scheme for \( f'(x) \) (the first derivative) with the **highest possible order** using the points \( x, x+h \), and \( x+2h \).
2. **1.2** Derive a scheme for \( f''(x) \) (the second derivative) with the **highest possible order** using the points \( x, x+h \), and \( x+2h \).
3. **1.3** Derive a **symmetric scheme** for \( f'(x) \) with the points \( x+h \) and \( x+2h \).
4. **1.4** Derive finite difference schemes for \( f'(x) \), \( f''(x) \), and \( f^{(4)}(x) \) using the values of the function at \( x-2h, x-h, x, x+h, x+2h \), and \( x+3h \). You are asked to find the **approximation order**.
5. **1.5** Same as 1.4, but using the points \( x+h, x+2h, x+3h \).

### Exercise 2: 2D Taylor Formula (not fully visible)
1. **2.1** Seems to involve deriving finite difference formulas for a function \( f(x, y) \) in two dimensions, involving points within a rectangle.

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### Five related questions:
1. How do you derive a finite difference scheme for the first derivative \( f'(x) \)?
2. What is a symmetric finite difference scheme, and how does it affect error order?
3. How can we increase the accuracy of finite difference schemes by including more points?
4. Why is the spacing \( h \) important in determining the order of accuracy in finite difference methods?
5. How does the inclusion of higher-order derivatives, like \( f^{(4)}(x) \), affect the accuracy and stability of numerical methods?

### Tip:
When deriving finite difference schemes, the **Taylor series expansion** is essential to approximate the function at different points and cancel out lower-order error terms for a higher accuracy scheme.

Exercises from M2 MISCT, Introduction to Finite Differences, focusing on deriving finite difference schemes for approximating derivatives using specific points and obtaining the highest possible order of accuracy.

Explore how to derive finite difference schemes for first and second derivatives using various point sets in 1D and 2D numerical methods. Learn about Taylor series expansions and error approximations to achieve the highest possible order of accuracy.

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