Math 164 - quiz two state carefully a theorem that

Question

Quiz two

1. Use n = 1 iterations of Runge-Kutta to obtain an approximate solution vector (ω₀, ω₁) of the first order IVP y' = y for t  ∈ (0,1), y(0) = 1. What exact number does ω₁ approximate?

2. State carefully a theorem that guarantees that Newton's method will converge to a zero of a real valued function of one real variable.

3. Write down a system of two first order ODE that has precisely two equilibrium solutions.

4. Find all eigenvalues of

	1	0	0
A =	0	3	1
	0	1	3