BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 5

Question 1

The Runge-Kutta 4th-order (RK4) method achieves local truncation error of order:

Accepted Answer

O(h⁵)

Answer

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) =

Accepted Answer

∬_D (∂Q/∂x − ∂P/∂y) dA

Answer

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:

Accepted Answer

Oscillatory overshoot that decays exponentially

Answer

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:

Accepted Answer

1/(s+1) − 1/(s+2)

Answer

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:

Accepted Answer

N(μ, σ²/n) for large n

Answer

The CLT states X̄ ~ N(μ, σ²/n) approximately for large n, with variance shrinking as the sample grows.

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 5