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GATE 2024 CE (CE2) – Question 13

Engineering Mathematics · Numerical Methods: Error analysis, algebraic equations, interpolation, differentiation, integration, ODEs · 1 mark · Multiple choice

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. $\frac{1}{12h^2}[-f_{i+2} + 16f_{i+1} - 30f_i + 16f_{i-1} - f_{i-2}]$
  2. $\frac{1}{12h^2}[f_{i+2} + 16f_{i+1} - 30f_i + 16f_{i-1} - f_{i-2}]$
  3. $\frac{1}{12h^2}[-f_{i+2} + 16f_{i+1} - 30f_i + 16f_{i-1} + f_{i-2}]$
  4. $\frac{1}{12h^2}[-f_{i+2} - 16f_{i+1} + 30f_i - 16f_{i-1} - f_{i-2}]$

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Correct answer: (A) $\frac{1}{12h^2}[-f_{i+2} + 16f_{i+1} - 30f_i + 16f_{i-1} - f_{i-2}]$

Explanation

The fourth-order central formula for the second derivative uses the weights $-1, 16, -30, 16, -1$ on $f_{i+2}, f_{i+1}, f_i, f_{i-1}, f_{i-2}$, all divided by $12h^2$. The weights sum to zero, as they must for a derivative. Only option A has the signs right.