BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 4 — Questions and Answers
Question 1: In the method of Lagrange multipliers, to optimize f(x,y) subject to g(x,y) = 0, you solve:
- ∇f = λ∇g and g = 0 (Correct answer)
- ∇f = 0 and ∇g = 0
- f = λg
- ∇(f - g) = 0
Correct answer: ∇f = λ∇g and g = 0
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:
- It transforms to pointwise multiplication under the Fourier transform (Correct answer)
- It always produces a smoother function
- It computes the cross-correlation of two signals
- It solves boundary value problems directly
Correct answer: It transforms to pointwise multiplication under the Fourier transform
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)?
- 4/7 (Correct answer)
- 0.7
- 0.4
- 1/2
Correct answer: 4/7
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:
- u(x,t) = F(x−ct) + G(x+ct) (Correct answer)
- u(x,t) = e^(−c²t)sin(x)
- u(x,t) = F(x)G(t)
- u(x,t) = ct·x
Correct answer: u(x,t) = F(x−ct) + G(x+ct)
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:
- |x − x̃| / |x| (Correct answer)
- |x − x̃|
- x̃ / x
- (x − x̃)² / x²
Correct answer: |x − x̃| / |x|
Relative error normalizes the absolute error |x − x̃| by the magnitude of the true value |x|.
Question 6: Which of the following is a second-order linear ODE with constant coefficients?
- y'' + 3y' + 2y = 0 (Correct answer)
- y'' + y·y' = 0
- y'' = y²
- (y')² + y = 0
Correct answer: y'' + 3y' + 2y = 0
The equation y'' + 3y' + 2y = 0 is linear with constant coefficients 1, 3, and 2.
Question 7: In numerical linear algebra, the condition number of a matrix A is defined as:
- ‖A‖·‖A⁻¹‖ (Correct answer)
- det(A)
- trace(A)/n
- ‖A‖/‖A⁻¹‖
Correct answer: ‖A‖·‖A⁻¹‖
κ(A) = ‖A‖·‖A⁻¹‖ measures how much the output can change relative to a small change in input.
BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 4
In the method of Lagrange multipliers, to optimize f(x,y) subject to g(x,y) = 0, you solve: