Math 242 - Fall 2016 - Final Exam Information

Math 242: Principles of Analysis
Fall 2016
Final Exam Information .

Department of Mathematics and Computer Science
College of the Holy Cross

Date, Time and Location

Wednesday, December 14, from 11:30am to 2:00pm in Swords 321

General Information

The final exam will be comprehensive and cover all material we have studied throughout the semester. The test will be closed book, closed notes and no calculators will be allowed. Phones and all other electronic devices should be turned off for the duration of the test.

Important Definitions/Theorems/Axioms

You are expected to know all material covered on the three midterm exams. Consult the following review pages for information about those topics:

In addition, the exam will cover material in sections 5.1, 5.2, 5.3, 5.4, 6.3, and 6.4 that we have done since the last midterm. Below is a complete list of definitions and theorems that you should know. Know precise statements of the ones in bold.

Chapter 1: The Real Numbers

Principle of Mathematical Induction
Definition 1.5.2
Definition 1.5.3
Least Upper Bound Axiom
Theorem 1.5.13 (Archimedian Property)
Theorem 1.5.14
Corollary 1.5.15
Theorem 1.5.17
Corollary 1.5.18

Chapter 2: Sequences

Definition 2.1.4 (Convergence of a Sequence)
Definition 2.2.2
Theorem 2.2.4
Theorem 2.2.5
Theorem 2.2.7
Theorem 2.2.9
Definition 2.3.1
Theorem 2.3.3 (Monotone Convergence Theorem)
Definition 2.5.1
Theorem 2.5.8
Theorem 2.5.12 (Bolzano-Weierstrass Theorem)
Definition 2.6.1 (Cauchy sequence)
Theorem 2.6.4
Definition of a contractive sequence.
Contractive sequences are Cauchy and therefore converge.
Definition 2.7.1 (infinite limits)
Theorem 2.7.5

Chapter 3: Functions and Continuity

Definition 3.1.1 (Limit of a Function)
Theorem 3.2.3
Theorem 3.2.4
Theorem 3.2.8
Theorem 3.2.9 (Squeeze Theorem)
Definition 3.3.2
Theorem 3.3.4
Definition 3.4.1 (Continuity)
Theorem 3.4.10
Theorem 3.4.11
Theorem 3.4.13
Theorem 3.4.16
Theorem 3.5.2 (Intermediate Value Theorem)
Definition 3.5.7
Theorem 3.5.9
Theorem 3.5.10 (Extreme Value Theorem)
Definition 3.6.1
Theorem 3.6.8

Chapter 4: The Derivative

Definition 4.1.1 (Definition of the Derivative)
Theorem 4.1.8
Theorem 4.2.1
Theorem 4.2.5
Theorem 4.3.3
Theorem 4.3.9 (Rolle's Theorem)
Theorem 4.3.11 (Mean Value Theorem)
Definition 4.3.14
Theorem 4.3.15
Theorem 4.4.3
Theorem 4.4.4

Chapter 5: The Integral

Definition 5.1.1
Definition 5.1.4
Definition 5.1.11
Definition 5.1.13 (Integrable)
Theorem 5.1.16
Theorem 5.1.18
Theorem 5.2.3
Theorem 5.2.5
Theorem 5.2.8
Theorem 5.2.9
Theorem 5.2.11
Theorem 5.3.5, Corollary 5.3.6 and Corollary 5.3.7
Theorem 5.4.1 (Fundamental Theorem of Calculus)

Chapter 6: Infinite Series

Definition 6.1.1
Theorem 6.1.3
Theorem 6.1.5
The Cauchy Condensation Test
Theorem 6.1.7
Theorem 6.1.11
Corollary 6.1.12
Theorem 6.2.2
Corollary 6.2.3
Theorem 6.2.5
Theorem 6.2.9
The Root Test
Definition 6.3.1
Theorem 6.3.3
Corollary 6.3.5
Theorem 6.3.9
Corollary 6.3.11
Corollary 6.3.12
Corollary 6.3.15
Definition 6.4.1
Theorem 6.4.3
Example 6.4.4, parts (a) and (b)

Finally, I will ask you to write a complete proof of one or two of the following theorems:

Theorem 2.3.3
Theorem 2.6.4
Theorem 3.2.3
Theorem 3.5.2
Theorem 4.1.8
Theorem 4.3.9 and 4.3.11
Theorem 5.3.3

Math 242: Principles of Analysis Fall 2016 Final Exam Information .