Question

Which one of the following statements about the central difference and averaging operators is correct?

• A. The central difference operator is used for approximating the second derivative, while the averaging operator is used for approximating the first derivative.
• B. Both the operators approximate the derivative with the same order of accuracy.
• C. The central difference operator is more accurate than the averaging operator for approximating the derivative.
• D. The averaging operator is used for approximating the second derivative, while the central difference operator approximates the first derivative.

Hint:

Taylor-expand both operators; only the scaled central difference converges to a derivative.

Answer 1

The Correct Option is C

Step 1: Recall the definitions. The central difference operator is\[\delta f(x)=f\left(x+\frac{h}{2}\right)-f\left(x-\frac{h}{2}\right)\]and the averaging (mean) operator is\[\mu f(x)=\frac{1}{2}\left[f\left(x+\frac{h}{2}\right)+f\left(x-\frac{h}{2}\right)\right]\]

Step 2: Expand both by Taylor series about $x$.\[\delta f(x)=hf'(x)+\frac{h^3}{24}f'''(x)+\cdots \ \Rightarrow\ \frac{\delta f(x)}{h}=f'(x)+O(h^2)\]\[\mu f(x)=f(x)+\frac{h^2}{8}f''(x)+\cdots\]

Step 3: The scaled central difference $\delta f(x)/h$ genuinely converges to the first derivative $f'(x)$ with error $O(h^2)$, a standard and accurate way to approximate a derivative. The averaging operator, however, converges to the function value $f(x)$ itself, with a curvature correction, not to any derivative, so it does not serve as a derivative approximation at all. Hence, for approximating a derivative, the central difference operator is the more accurate and appropriate choice.

\[\boxed{\text{The central difference operator is more accurate than the averaging operator for approximating the derivative.}}\]