The central difference approximation to the gradient of a function fat the point xi, which is uniformly sampled with spacing Δxis:

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The central difference approximation to the gradient of a function fat the point xi, which is uniformly sampled with spacing Δxis:

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The question asks for the central difference approximation to the gradient (derivative) of a function at the point i, given uniform spacing Δx between samples. 
Option 1: fi+1 − fi−1 over 2Δx. This matches the standard central difference formula: (f(x_{i+1}) − f(x_{i−1})) / (2Δx), which is known to provide a good approximation to f'(x_i) with error O(Δx^2). This is the correct form.
Option 2: fi − fi−1 over Δx. This is the backward difference approximation, which uses only the current and previous values: (f(x_i) − f(x_{i−1})) / Δx. While it estimates the derivative, it is not the central difference and generally has higher truncation error than the central difference.
Option 3: fi+1 − fi over Δx. This is the forward difference approximation, using the current and next values: (f(x_{i+1}) − f(x_i)) / Δx. Like the backward difference, it is not the central difference and also has higher truncation error compared to the central form.
In summary, the central difference formula requires the two neighboring points on either side of x_i and combines them as (f(x_{i+1}) − f(x_{i−1})) / (2Δx). The other two options represent first-order forward or backward difference schemes, which are not the central difference and generally less accurate for the same grid spacing.

AMME2000 BMET2960 BMET9960 (ND) Week 2 Pre-work

The central difference approximation to the gradient of a function fat the point xi, which is uniformly sampled with spacing Δxis:

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