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The second derivative of a function f is computed using the fourth-order Central Divided Difference method with a step length h. The CORRECT expression for the second derivative is
- 1/(12h²) [−f_(i+2) + 16 f_(i+1) − 30 f_i + 16 f_(i−1) − f_(i−2)]
- 1/(12h²) [f_(i+2) + 16 f_(i+1) − 30 f_i + 16 f_(i−1) − f_(i−2)]
- 1/(12h²) [−f_(i+2) + 16 f_(i+1) − 30 f_i + 16 f_(i−1) + f_(i−2)]
- 1/(12h²) [−f_(i+2) − 16 f_(i+1) + 30 f_i − 16 f_(i−1) − f_(i−2)]
Correct answer: 1/(12h²) [−f_(i+2) + 16 f_(i+1) − 30 f_i + 16 f_(i−1) − f_(i−2)]
Solution
The correct expression accurately represents the second derivative using the fourth-order Central Divided Difference method by appropriately weighing the function values at points around the target point, ensuring that the contributions from points further away are balanced to achieve higher accuracy.
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