Example 1: Forward Difference Approximation Consider the function f(x) = x². We want to estimate the derivative f'(2) using the forward difference approximation with a step size of h = 0.1.

Using the forward difference approximation formula:

f'(x) ≈ (f(x + h) – f(x)) / h

At x = 2, we have:

f'(2) ≈ (f(2 + 0.1) – f(2)) / h ≈ (f(2.1) – f(2)) / h ≈ ((2.1)² – (2)²) / h ≈ (4.41 – 4) / h ≈ 0.41 / h ≈ 4.1

Therefore, the forward difference approximation gives an estimated derivative of f'(2) ≈ 4.1.

Example 2: Backward Difference Approximation Consider the same function f(x) = x². Now we want to estimate the derivative f'(2) using the backward difference approximation with a step size of h = 0.1.

Using the backward difference approximation formula:

f'(x) ≈ (f(x) – f(x – h)) / h

At x = 2, we have:

f'(2) ≈ (f(2) – f(2 – 0.1)) / h ≈ (f(2) – f(1.9)) / h ≈ (2² – (1.9)²) / h ≈ (4 – 3.61) / h ≈ 0.39 / h ≈ 3.9

Therefore, the backward difference approximation gives an estimated derivative of f'(2) ≈ 3.9.

Example 3: Central Difference Approximation Again, consider the function f(x) = x². We will estimate the derivative f'(2) using the central difference approximation with a step size of h = 0.1.

Using the central difference approximation formula:

f'(x) ≈ (f(x + h) – f(x – h)) / (2h)

At x = 2, we have:

f'(2) ≈ (f(2 + 0.1) – f(2 – 0.1)) / (2h) ≈ (f(2.1) – f(1.9)) / (2h) ≈ ((2.1)² – (1.9)²) / (2h) ≈ (4.41 – 3.61) / (2h) ≈ 0.8 / (2h) ≈ 4

Therefore, the central difference approximation gives an estimated derivative of f'(2) ≈ 4.

I apologize for any confusion caused by the previous response. I hope this clarifies the equations and provides accurate examples of the forward, backward, and central difference approximations for estimating derivatives.