BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 3 — Questions and Answers
Question 1: Which numerical method has quadratic convergence for finding roots of a smooth function?
- Newton-Raphson method (Correct answer)
- Bisection method
- Fixed-point iteration
- Secant method
Correct answer: Newton-Raphson method
Newton-Raphson converges quadratically near a simple root because the error satisfies eₙ₊₁ ≈ C·eₙ².
Question 2: For the heat equation u_t = α²u_xx, the steady-state solution (u_t = 0) satisfies:
- u_xx = 0, so u is linear in x (Correct answer)
- u = 0 everywhere
- u is exponential in x
- u is sinusoidal in x
Correct answer: u_xx = 0, so u is linear in x
Setting u_t = 0 gives u_xx = 0, whose general solution is u(x) = Ax + B, a linear function.
Question 3: The determinant of a 2×2 matrix [[a,b],[c,d]] equals:
- ad - bc (Correct answer)
- ac - bd
- ad + bc
- ab - cd
Correct answer: ad - bc
By definition, det([[a,b],[c,d]]) = ad - bc.
Question 4: Which condition guarantees a unique solution to the initial value problem y' = f(x,y), y(x₀) = y₀?
- f and ∂f/∂y are continuous in a rectangle containing (x₀, y₀) (Correct answer)
- f is bounded on all of ℝ²
- f is a polynomial in y
- y₀ = 0
Correct answer: f and ∂f/∂y are continuous in a rectangle containing (x₀, y₀)
The Picard–Lindelöf theorem requires f to be continuous and Lipschitz in y, which continuity of ∂f/∂y ensures locally.
Question 5: A signal f(t) = cos(2πt) + cos(6πt) contains frequencies (in Hz):
- 1 Hz and 3 Hz (Correct answer)
- 2π Hz and 6π Hz
- 1/2 Hz and 1/3 Hz
- 2 Hz and 6 Hz
Correct answer: 1 Hz and 3 Hz
cos(2πft) has frequency f; here 2π·1 gives f=1 Hz and 2π·3 gives f=3 Hz.
Question 6: The eigenvalues of the matrix [[3,1],[0,3]] are:
- Both equal to 3 (repeated) (Correct answer)
- 1 and 3
- 0 and 3
- 3 and -3
Correct answer: Both equal to 3 (repeated)
An upper-triangular matrix has eigenvalues equal to its diagonal entries, both of which are 3.
Question 7: Using Simpson's 1/3 rule with n = 2 subintervals to approximate ∫₀² x² dx, the result is:
- 8/3 (Correct answer)
- 2
- 4
- 3
Correct answer: 8/3
Simpson's rule gives (h/3)[f(0)+4f(1)+f(2)] = (1/3)[0+4+4] = 8/3, which is exact here.
BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 3
Which numerical method has quadratic convergence for finding roots of a smooth function?