Math 302

Spring 2010

Prof. Donnay


Class Materials and Assignments





Class Material




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.




Ch. 28

Sup metric satisfies 3 properties of metric. Analysis concepts defined on a metric space.

Week 1 quiz review.

 Quiz #1 is available on shelf outside my office – hand in on class on Wed.

HW #1-2 for Wed 1/27 and Fri 1/29





NO CLASS  as Prof Donnay is out of town.






Function spaces, Cn[a,b], example of C1 but not C2 function; Map/Function between function spaces (Differentiation, Integration).

Start of Differentiation Ch. 14.






More on positivity property of metric spaces. In a metric space, if a sequence converges, the limit is unique and the sequence is bounded.

Wk 2 quiz review

Quiz 2 avaiable outside my office; due by 4pm on Monday

Hw #3-4 for Wed Feb 3 and Fri Feb 5.





Local extrema Theorem, Rolle’s Theorem,

Mean Value Theorem Ch 14


7 W



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


8 F



Differentiable -> Continuous

Intro to Uniform Continuity Ch. 11

Wk 3 quiz review

Hw #5-6 for Wed Feb 10 and Fri Feb 12. Material from Stewart on Riemann Sums: 1, 2, 3

9 M



Uniform Continuity (Cont fn on Compact Set)

Intro to integration via Riemann Sums Ch. 15


10 F



More on (Riemann) integration; existence of non-integrable function. Riemann Sum Worksheet.

Wk 4 quiz review

HW #6-7

11 M



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


12 W



Fundamental  Theorem of Calculus Ch. 16


13 F



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

Wk 5 quiz review

HW #8-9

14/ M 2/22


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

Mathematica illustration of sequences 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 #10-11 .Slight change in HW. Do the revised HW assignment instead:

HW Revised #10-11.

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




 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

Wk 6-7 quiz Review

22 F 3/19


Review of Sequences of functions for exam.

Updated list of proofs that might be on exam.


23 M 3/22


Instructions for Midterm.


24 W



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 “the same”. 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



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.

HW #14, due Friday 4/16

32 M 4/12


Introduction to Series (Ch. 19): what does

1-1+1 -1 + … equal?


32 W 4/14


Geometric Series


33 F 4/17


Nth term test, Harmonic Series, Integral test

HW #15, due Friday 4/23

34 M 4/20


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


35 W 4/22


Cauchy convergence criteria, Series of functions, Weierstrass M-Test (Ch. 21)


36 F 4/24


Presentation 1: Bifurcations and Dynamical Systems.

-           Choices for Final Work

-           Information on writing paper.

HW #16, due Friday 4/30.