Final answer:
The centered divided difference formula for approximating the first derivative can be derived using Taylor series. The error term is O(h^2), where h is the interval between the points used for approximation.
Step-by-step explanation:
To derive the centered divided difference formula for approximating the first derivative using Taylor series, we start with the Taylor series expansion of the function at a specific point.
Let's assume we have a function f(x) and we want to approximate its first derivative f'(x) at a point a. The Taylor series expansion of f(x) around a is:
f(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + (f'''(a)/3!)(x-a)^3 + ...
If we truncate the series after the second term, we get:
f(x) ≈ f(a) + f'(a)(x-a)
To approximate f'(a), we rearrange the equation:
f'(a) ≈ (f(x) - f(a))/(x-a)
This derived formula is known as the centered divided difference formula for approximating the first derivative.
Now, let's discuss the error term in Big O notation. The error term for the centered divided difference formula is O(h^2), where h is the interval between the points used for approximation. This means that as h becomes smaller, the error decreases quadratically.