1. Using Taylor series expansion derives the error term for the following formulas:

• f'(x) ≈ 1/2h (-f(x + 2h) + 4f (x + h) - 3f(x)))
• f"(x) ≈ 1/h²(f (x) - 2f (x + h) + f (x + 2h))

2. Use method of undetermined coefficients to derive second order scheme for ∂u/∂x using three points in the following way:

u'(kΔx) ≈ c₁u(kΔx) + c₂u((k - 1) Δx) + c₃u((k - 2) Δx).

3. Use method of undetermined coefficients to show that it is impossible to approximate u_xx, at point kΔx to the third or higher order using only the points kΔx, (k - 1) Δx and (k + 1) Δx.

4. Consider the following diffusion problem:

• Construct implicit in time finite-difference approximation of the problem (1) with Δx = 1/4 and Δt = 1/10. Use three different approximations for Neumann boundary condition:

- First order

- Second order using "ghost" point approach

- Approximation derived in problem 2.

• Build matrix A and right hand side vector F corresponding to the each of the three approximate problems.