Math 242 Principles of Analysis

Exam #2

Thursday, November 19, Stein 223, 1:00 - 1:50 pm

The second exam covers Sections 2.5 through 2.7 and Chapters 3 and 4. You should go over homework problems (as well as the partial solutions), quizzes and your class notes. The exam will be designed to take the full class period (45-50 minutes). No calculators are required or allowed.

Exam Review Session: We will review for the exam during Tuesday's class on Nov. 17. Please come prepared with specific questions. Some practice problems are available here.

The following concepts, definitions, theorems and examples are crucial material for the exam. As with the quizzes, you may be asked to give a precise definition or carefully state a particular theorem. I will expect your answers to be concise and specific, with little margin for error. You may also be asked some short answer questions similar to those on quizzes.

Definitions, terminology, important examples:

1. Subsequence, monotone sequence, Cauchy sequence
2. Infinite series, partial sums, convergence, conditional versus absolute convergence, geometric series, p-series
3. Point-Set Topology: Cantor set, open and closed sets, closure of a set, limit points, isolated points, bounded, compact, connected, separated sets
4. Definition of a limit (for BOTH sequences and functions), definition of continuity, bounded function
5. Harmonic Series, Dirichlet's function (and modified versions)
Theorems:
1. Convergence Tests for Infinite Series: nth-term test, comparison test, ratio test, alternating series test, etc.
2. The Big Limit Theorems for sequences, functions and series (BLT)
3. Bolzano-Weierstrass Theorem (BWT), Cauchy Criterion (CC)
4. Facts about open and closed sets: arbitrary union of open sets is open, finite intersection of open sets is open, complement of an open set is closed, etc.
5. Heine-Borel Theorem (A set is compact if and only if it is closed and bounded.)
6. Sequence characterization of functional limits, divergence criterion for limits, 4 ways to describe continuity, discontinuity criterion, algebraic continuity theorem, composition of continuous functions is continuous
7. Continuous functions take compact sets to compact sets and connected sets to connected sets.
8. Squeeze Theorem
9. Intermediate Value Theorem (IVT)
10. Extreme Value Theorem (EVT)

Proofs:

You will have to do a couple of proofs on the exam. These will be similar to homework problems. Two types of problems likely to be asked on the exam include:

1. Using the epsilon-delta definition to verify a limit of a function.
2. Determining whether a function is continuous at a point and proving your claim.