BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 4

Question 1

In the method of Lagrange multipliers, to optimize f(x,y) subject to g(x,y) = 0, you solve:

Accepted Answer

∇f = λ∇g and g = 0

Answer

The critical points satisfy ∇f = λ∇g (parallel gradients) and the constraint g(x,y) = 0.

Question 2

The convolution (f * g)(t) = ∫₋∞^∞ f(τ)g(t−τ)dτ is useful in applied mathematics primarily because:

Accepted Answer

It transforms to pointwise multiplication under the Fourier transform

Answer

The Convolution Theorem states F{f*g} = F{f}·F{g}, enabling efficient computation via transforms.

Question 3

A Markov chain has transition matrix P = [[0.7,0.3],[0.4,0.6]]. The stationary distribution π satisfies πP = π. What is π₁ (the first component)?

Accepted Answer

4/7

Answer

Solving π₁·0.3 = π₂·0.4 with π₁+π₂=1 gives π₁ = 4/7 ≈ 0.571.

Question 4

The wave equation u_tt = c²u_xx has general solution:

Accepted Answer

u(x,t) = F(x−ct) + G(x+ct)

Answer

D'Alembert's formula gives the general solution as a sum of a left-traveling and a right-traveling wave.

Question 5

The relative error of an approximation x̃ to the true value x is defined as:

Accepted Answer

|x − x̃| / |x|

Answer

Relative error normalizes the absolute error |x − x̃| by the magnitude of the true value |x|.

BMath Bachelor of Mathematics Practice Test