Table 6-1. Some Formulas for Computing Derivatives (For tables with equally spaced entries) First derivative, using two points: Forward difference y i = y i + 1 -y i h Central difference y i = y i + 1 -y i-1 2h Backward difference y i = y i -y i – 1 h First derivative, using three points: Forward difference y i = -y i + 2 +4y i+1 -3y i 2h First derivative, using four points: Central difference y i = -y i + 2 +8y i+1 -8y i-1 +y i-2 12h Second derivative, using three points: y i ^ * = y i + 1 -2y i+1 +y i h^ 2 Forward difference y i = y i + 1 -2y i *y i-1 h^ 2 Central difference vec y i = y i -2y i – 1 +y i-1 h^ 2 Backward difference Second derivative, using four points: Forward difference y i ^ * = 2y_{i} -5y i+1 +4y i+1 -y i+3 h^ 2 Second derivative, using five points: Central difference y i ^ = -y i + 2 +16y i+1 -30y i +16y i-1 -y i-2 12h^ 2 Third derivative, using four points Forward difference y i ^ — = y i + 1 -3y i+1 +3y i+1 -y i h^ 3
The Correct Answer and Explanation is :
Corrected Formulas for Computing Derivatives
- First Derivative (Using Two Points):
- Forward Difference:
[
f'(x_i) = \frac{y_{i+1} – y_i}{h}
] - Central Difference:
[
f'(x_i) = \frac{y_{i+1} – y_{i-1}}{2h}
] - Backward Difference:
[
f'(x_i) = \frac{y_i – y_{i-1}}{h}
]
- First Derivative (Using Three Points):
- Forward Difference:
[
f'(x_i) = \frac{-y_{i+2} + 4y_{i+1} – 3y_i}{2h}
]
- First Derivative (Using Four Points):
- Central Difference:
[
f'(x_i) = \frac{-y_{i+2} + 8y_{i+1} – 8y_{i-1} + y_{i-2}}{12h}
]
- Second Derivative (Using Three Points):
- Central Difference:
[
f”(x_i) = \frac{y_{i+1} – 2y_i + y_{i-1}}{h^2}
] - Forward Difference:
[
f”(x_i) = \frac{y_{i+2} – 2y_{i+1} + y_i}{h^2}
] - Backward Difference:
[
f”(x_i) = \frac{y_i – 2y_{i-1} + y_{i-2}}{h^2}
]
- Second Derivative (Using Five Points):
- Central Difference:
[
f”(x_i) = \frac{-y_{i+2} + 16y_{i+1} – 30y_i + 16y_{i-1} – y_{i-2}}{12h^2}
]
- Third Derivative (Using Four Points):
- Forward Difference:
[
f”'(x_i) = \frac{-y_{i+3} + 3y_{i+2} – 3y_{i+1} + y_i}{h^3}
]
Explanation (300 Words)
The numerical differentiation formulas estimate derivatives using discrete data points, assuming equal spacing ( h ). These methods are essential in scenarios where the function is not given analytically, such as experimental data or simulations.
- First Derivatives:
The forward difference uses the next point ( y_{i+1} ) relative to ( y_i ), while the backward difference uses the prior point ( y_{i-1} ). The central difference is often more accurate as it averages forward and backward approximations, using both ( y_{i+1} ) and ( y_{i-1} ). - Higher Accuracy:
For increased precision, formulas use additional points (e.g., three or four points). These incorporate weights to improve accuracy while maintaining the same order of ( h ). - Second Derivatives:
Similar principles apply to the second derivative, but it measures curvature. The central difference formula is symmetric, making it more robust against errors in data compared to forward or backward approximations. - Higher Derivatives:
Third and higher-order derivatives are calculated similarly but involve more points and higher powers of ( h ), which can amplify numerical errors.
Each formula balances complexity, accuracy, and computational effort. Central differences generally offer better accuracy but require data on both sides of the target point.