Calculus II Quiz 2 Apr. 9, 2009 Name: Student ID number:

Calculus II Quiz 2 Apr. 9, 2009

Name: Student ID number:

1. (10 pts) Compute the Taylor polynomial of degree 2 about a = 0 for f (x) = 1

1 + x.

2. (10 pts) Solve dy

dx = 3y, with y(0) =−5, i.e. y = −5 when x = 0.

3. Suppose that dy

dx = y(y− 1)

(a) (10 pts) Find the equilibria of the differential equation.

(b) (10 pts) Discuss the stability of the equilibria (by graph or eigenvalue)

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Name: Student ID number:

4. Given a matrix A =

[ 4 2

−3 −1 ]

(a) (20 pts) Find the eigenvalues and eigenvectors of A

(b) (10 pts) Let v₁, v₂ be two eigenvectors found in part (4a). Find A¹⁰v₁ and A¹⁰v₂ without using a calculator.

5. (10 pts) Show that lim

(x,y)→(0,0)

2xy

x²+ y² does not exist.

6. (20 pts) Given f (x) = 2x− 1

xy, find ∂f

∂x and ∂f

∂y.

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