BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 3

Question 1

Which numerical method has quadratic convergence for finding roots of a smooth function?

Accepted Answer

Newton-Raphson method

Answer

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:

Accepted Answer

u_xx = 0, so u is linear in x

Answer

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:

Accepted Answer

ad - bc

Answer

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?

Accepted Answer

f and ∂f/∂y are continuous in a rectangle containing (x₀, y₀)

Answer

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):

Accepted Answer

1 Hz and 3 Hz

Answer

cos(2πft) has frequency f; here 2π·1 gives f=1 Hz and 2π·3 gives f=3 Hz.

BMath Bachelor of Mathematics Practice Test