MATH 1280 HOMEWORK 3 SOLUTIONS
Section 3.4
Problem 2: Collect the bifurcation information for x = rx sinh x.
Solution: Fixed points occur at intersections of rx and sinh x. The three qualitatively dierent cases are
r < 1, r = 1, r > 1. Cartoons of rx and
Exercises
April 21, 2010
Exercise 1. Let f (x, y) = ln(x2 + y 2 + 1).
(1) Find all local minima and maxima of f on its domain.
(2) Find all global minima and maxima of f on its domain.
(3) Find the tangent plane to the surface z = f (x, y) at (x0 , y0 ) =
SYLLABUS
MATH 1280, Sect. 2
Spring 2010
1. General information
Lectures: MTWF 10:45am-11:35am in ST 205
Instructor: Emanuele Macr`
Email: macri(AT)math(DOT)utah(DOT)edu
Phone: (801) 581 6898
Oce: JWB 317
Oce hours: TF 9:30am-10:30am
On-line information
SYLLABUS
MATH 1280, Sect. 1
Spring 2010
1. General information
Lectures: MTWF 08:35am-09:25am in ST 205
Instructor: Emanuele Macr`
Email: macri(AT)math(DOT)utah(DOT)edu
Phone: (801) 581 6898
Oce: JWB 317
Oce hours: TF 9:30am-10:30am
On-line information
MATH 1280 HOMEWORK 6 SOLUTIONS
Section 6.1
Problem 4: Find and classify the xed points. Sketch the nullclines, the vector eld, and a plausible phase
portrait. x = y, y = x(1 + y) 1.
Solution: There is a single xed point at (1, 0). The Jacobian is given by
Math 1280 notes,5, Lyapunov functions (9.6)
In this section we return to the question of whether there is an E function for
a system
x0 = F (x; y)
(1)
y 0 = G (x; y)
and what to do when there is not. (I using F and G instead of f and g because
m
the text