BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 5 — Questions and Answers
Question 1: The Runge-Kutta 4th-order (RK4) method achieves local truncation error of order:
- O(h⁵) (Correct answer)
- O(h⁴)
- O(h²)
- O(h³)
Correct answer: O(h⁵)
RK4 has local truncation error O(h⁵) and global error O(h⁴), making it far more accurate than Euler's method.
Question 2: Green's theorem relates a line integral around a simple closed curve C to a double integral over the region D it encloses. It states ∮_C (P dx + Q dy) =
- ∬_D (∂Q/∂x − ∂P/∂y) dA (Correct answer)
- ∬_D (∂P/∂x + ∂Q/∂y) dA
- ∬_D (P + Q) dA
- ∬_D (∂P/∂y − ∂Q/∂x) dA
Correct answer: ∬_D (∂Q/∂x − ∂P/∂y) dA
Green's theorem converts the line integral to ∬_D (∂Q/∂x − ∂P/∂y) dA over the enclosed region.
Question 3: For an underdamped second-order system with damping ratio ζ < 1, the step response exhibits:
- Oscillatory overshoot that decays exponentially (Correct answer)
- Monotone rise to steady state with no overshoot
- Sustained oscillations that never decay
- Immediate jump to the final value
Correct answer: Oscillatory overshoot that decays exponentially
Underdamped systems have complex conjugate poles, producing oscillatory behavior with amplitude decaying as e^(−ζωₙt).
Question 4: The partial fraction decomposition of 1/[(s+1)(s+2)] is:
- 1/(s+1) − 1/(s+2) (Correct answer)
- 1/(s+1) + 1/(s+2)
- −1/(s+1) + 1/(s+2)
- 2/(s+1) − 1/(s+2)
Correct answer: 1/(s+1) − 1/(s+2)
Setting 1 = A(s+2) + B(s+1) gives A = 1 at s = −1 and B = −1 at s = −2.
Question 5: In probability, the Central Limit Theorem states that the sample mean X̄ of n i.i.d. variables with mean μ and variance σ² is approximately distributed as:
- N(μ, σ²/n) for large n (Correct answer)
- N(0, 1) for all n
- N(μ, σ²) for all n
- N(nμ, nσ²) for large n
Correct answer: N(μ, σ²/n) for large n
The CLT states X̄ ~ N(μ, σ²/n) approximately for large n, with variance shrinking as the sample grows.
Question 6: A vector field F is conservative if and only if:
- curl F = 0 on a simply connected domain (Correct answer)
- div F = 0
- F is bounded
- F has constant magnitude
Correct answer: curl F = 0 on a simply connected domain
On a simply connected domain, F is conservative (F = ∇φ) if and only if curl F = 0.
Question 7: Using the trapezoidal rule to approximate ∫₀¹ eˣ dx with n = 2 subintervals (h = 0.5), the estimate is:
- (0.5/2)[e⁰ + 2e^0.5 + e¹] (Correct answer)
- (0.5)[e⁰ + e¹]
- (0.5/3)[e⁰ + 4e^0.5 + e¹]
- (0.5)[e^0.5 + e¹]
Correct answer: (0.5/2)[e⁰ + 2e^0.5 + e¹]
The trapezoidal rule with n=2 gives (h/2)[f(0)+2f(0.5)+f(1)] = (0.25)[1 + 2e^0.5 + e], approximately 1.7539.
BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 5
The Runge-Kutta 4th-order (RK4) method achieves local truncation error of order: