৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

Central difference is more accurate than forward/backward and has error of order h², while forward/backward have error O(h).

Simpson’s 1/3 rule works only if n is even, because it groups subintervals in pairs.

.Simpson’s 1/3 rule is derived by fitting a parabola (quadratic polynomial) through 3 points: f(x₀),f(x₁),f(x₂).
That’s why it is more accurate than trapezoidal rule, which only fits a straight line.

Trapezoidal rule error: O(h²).
Midpoint rule error: O(h²).
Simpson’s 1/3 rule error: O(h⁴) → much smaller error.
Simpson’s 3/8 rule also has O(h⁴), but in practice 1/3 rule is often simpler and slightly better.

Trapezoidal gave 0.375 (approx).
Simpson’s 1/3 gave 0.333 (exact).
Simpson’s rule is exact for polynomials up to degree 3.

Trapezoidal rule assumes the area under the curve is made up of trapeziums formed by straight line joining successive points.

Trapezoidal Rule → exact for degree ≤1
Midpoint Rule → exact for degree ≤1
Simpson’s 1/3 Rule → exact for degree ≤3
Simpson’s 3/8 Rule → exact for degree ≤3

Exactness:

Simpson’s 3/8 Rule is exact for cubic polynomials (degree ≤3), if the number of intervals is a multiple of 3.
That’s why in the MCQ about “exact for all polynomials up to degree 3,” 3/8 Rule was the correct choice.

General Accuracy (for any smooth function, same step size h):
Simpson’s 1/3 Rule is fourth-order accurate (error ∝ h⁴) and generally more efficient for most applications.
Simpson’s 3/8 Rule is also fourth-order accurate, but requires more points per segment (4 points), so it’s slightly less convenient.
Conclusion:

If the question asks “best accuracy for the same step size”, Simpson’s 1/3 Rule  is usually preferred.
If the question asks “exact for cubic polynomials”, Simpson’s 3/8 Rule is safer.

Heun’s method improves Euler by averaging the slope at the current point and the predicted next point.
This gives second-order accuracy (O(h²)), better than Euler.

Euler is a first-order method, meaning global error is proportional to h (O(h)). Reducing h improves accuracy linearly.

Steps A–C are part of RK4 slope calculations.
RK4 has fourth-order accuracy (O(h⁴)) → highly accurate for moderate h.

Multistep methods like Adams-Bashforth use previous computed points to predict y_n+1, increasing efficiency for long computations.
Single-step methods (Euler, Heun, RK4) only use the current point.

Smaller h gives better slope approximation → lower truncation error.
However, too small h may increase round-off error in computer calculations.

An IVP specifies both the derivative and initial condition.
Only option B provides y(x₀​)=y₀​, allowing step-by-step numerical solution.

IVP requires first derivative and initial value. Option B specifies y(0)=0. Option D is second-order, but only one initial condition is given → incomplete for an IVP.

It uses linear interpolation between points (endpoints + 2*sum of interior points).

RK4 global error ∝ h4, very accurate even for moderate $.$

Exact value, since Simpson’s 1/3 rule is exact for cubic polynomials

(exact value, since Simpson’s 3/8 rule is exact for cubic polynomials