Numerical Analysis Techniques in Engineering Mathematics

A: A: Bisection’s only guarantee is the sign change. Lose that and you’re just chopping space. B: Speed isn’t the core risk. Validity is. C: Secant uses slope approximation, not interval bracketing. D: Round-off isn’t the driver at this stage.

A: A: Simpson’s is fourth-order. h halves, error scales with h^4. That’s 2^4. B: That’s trapezoidal thinking. C: That’s second-order logic leaking in. D: Polynomial exactness doesn’t mean zero improvement for smooth functions.

A: A: Secant slope uses f(x1)−f(x0) over x1−x0. Zero denominator. Hard stop. B: There’s no slope to slow down with. C: No derivative is computed anywhere. D: No interval logic exists here.

A: A: Divergence can land you anywhere. Downstream assumes validity. B: Conservatism isn’t guaranteed. C: That behavior has to be coded. It isn’t inherent. D: Accuracy is already gone.

A: A: Higher frequency means more curvature per step. Linear segments don’t capture it. B: Damping isn’t accuracy. C: Trapezoidal is only exact for linear functions. D: Cancellation isn’t guaranteed unless symmetry is perfect.

A: A: Each iteration halves the interval. 2^10 ≈ 1000. That matches 1e-3. B: That barely shrinks anything. C: You’re confusing tolerance with iteration count. D: That overshoots by an order of magnitude.

A: A: Composite Simpson’s needs pairs of intervals. Odd count breaks the pattern. B: It’s not a small degradation. It’s misuse. C: That behavior isn’t implicit unless coded. D: The interior weighting is already wrong.

A: A: Without a cap, divergence can propagate indefinitely. B: Precision doesn’t fix bad math. C: Initial guesses help but don’t guarantee convergence. D: Smoothing hides the problem.

A: A: Flat slope kills Newton steps. Bracketing restores guaranteed convergence. B: Tolerance doesn’t fix slope pathology. C: More iterations of a failing method don’t help. D: Linearization near zero slope is ill-conditioned.