Math 302
Spring
2010
Prof.
Donnay
Class
Materials and Assignments
Class#, Date 
Section 
Class
Material 
Assignment 
1 W
1/20 
a.
Ch. 27 
b.
Metric Spaces: Taxi Cab metric, C[0,1]. 
Due
Friday 1/22: Prove that Taxi Cab Metric satisfies 3 properties of metric.
Complete review
worksheet of Analysis definitions. 
2 F
1/22 
Ch.
28 
Sup
metric satisfies 3 properties of metric. Analysis concepts defined on a
metric space. 
Quiz #1 is available on shelf outside
my office – hand in on class on Wed. HW #12 for Wed 1/27 and Fri 1/29 
3
M
1/25 

NO
CLASS as Prof Donnay is out of
town. 

4 W
1/27 

Function
spaces, C^{n}[a,b], example of C^{1} but not C^{2}
function; Map/Function between function spaces (Differentiation,
Integration). Start
of Differentiation Ch. 14. 

5 F
1/29 

More
on positivity property of metric spaces. In a metric space, if a sequence
converges, the limit is unique and the sequence is bounded. 
Quiz
2 avaiable outside my office; due by 4pm on Monday Hw #34 for Wed Feb 3 and Fri Feb 5. 
6 M 2/1 

Local extrema Theorem, RolleÕs Theorem, Mean Value Theorem Ch 14 

7 W 2/3 

Implications of Mean Value Theorem, relation to linear approximation and then to Taylor Polynomials, 

8 F 2/5 

Differentiable > Continuous Intro to Uniform Continuity Ch. 11 
Hw #56 for Wed Feb 10 and Fri Feb 12. Material from Stewart on Riemann Sums: 1, 2, 3 
9 M 2/8 

Uniform Continuity (Cont fn on Compact Set) Intro to integration via Riemann Sums Ch. 15 

10 F 2/12 

More on (Riemann) integration; existence of nonintegrable function. Riemann Sum Worksheet. 

11 M 2/15 

Proof of Thm: Continuous fn on compact set is integrable. 

12 W 2/17 

Fundamental Theorem of Calculus Ch. 16 

13 F 2/19 

Numerical Riemann Integration with Mathematica; Tayor polynomials, sequences of functions (Ch 17). Mathematica handout 

14/ M 2/22 

Pointwise and uniform limits of functions (Ch 17). Introductory worksheet on limits of functions. 

15 W 2/24 

HW review: uniform continuity via Mean Value Theorem; existence of Riemann Integral for a function that is discontinuous 

16 M 3/1 

Uniform convergence of continuous functions and associated theorems (Ch. 17) 
HW #1011 .Slight change in HW. Do the revised HW assignment instead: 
18 W 3/3 

Uniform convergence allows interchange of limit and integral. 

19 F 3/5 

Uniform convergence is the same as convergence in the sup metric of the metric space C[a,b]. Mathematica and the command Manipulate (Mathematica file) , (pdf file)to visualize a sequence of functions converging to a limit function. 

20 M 3/15 

Projects: information and list of potential projects.
Information about test prep Dynamical Systems via Staircase method, plots of various f(x) functions. 
Choice of group projects: due Monday 3/22 
21 W 3/17 

Relation of staircase diagram to hopping diagram. Classifying dynamics of linear functions. Varying the initial conditions. 
Quiz 7 : due Friday 3/19 
22 F 3/19 

Review of Sequences of functions for exam. Updated list of proofs that might be on exam. 

23 M 3/22 



24 W 3/24 

Parameter space diagram for the linear maps: m values in R. Various regions grouped by type of dynamics. 

25 F 3/26 

Special types: m = +1, m =  1, m = 0. Bifurcation/tipping point – small change in parameter value can lead to major change in the dynamical. Stable systems – can change parameter and the dynamics do not change. 

26 M 3/28 

Topologically conjugacy – rigoruous definition of Òthe sameÓ. Equivalence relation. 

27 W 3/30 

Invariants of conjugacy. If two systems are the top. conj., then this property must be the same for both systems. A way to determine that two systems are not the same. 

28 F 4/2 

Local attracting and repelling points. Finding multiple fixed points and characterizing the type. 

29 M 4/5 

Periodic points, attracting/repelling periodic pts; Carrying capacity, change of variables for quadratic map 

30 W 4/7 

Base 3 expansions 

31 F 4/9 

Cantor Set and Base 3; Cantor set is uncountable. 

32 M 4/12 

Introduction to Series (Ch. 19): what does 11+1 1 + É equal? 

32 W 4/14 

Geometric Series 

33 F 4/17 

Nth term test, Harmonic Series, Integral test 

34 M 4/20 

Comparision Test, Absolutely convergent, Conditionally convergent (Ch. 20) 

35 W 4/22 

Cauchy convergence criteria, Series of functions, Weierstrass MTest (Ch. 21) 

36 F 4/24 

Presentation 1: Bifurcations and Dynamical Systems. 
HW #16, due Friday 4/30. 



































































