Principles of Analysis

Final Exam Review Sheet

Saturday, Dec. 14, 8:30 - 11:30 am, Swords 302

The final exam is cumulative, that is, it covers all the material from the first day of class onwards. Approximately 40-50% will cover material since the second exam. This is Sections 4.3, 5.1, 5.2, 5.3, 5.4, 6.1, 6.2 and 6.3.

You should go over homework problems, class notes, previous exams, practice final, etc. Some sample problems from the remaining chapters are listed below. The exam will be designed for 2 hours (twice the length of the mid semester exams) so you should have plenty of time to complete it in the alloted 3 hour time slot. No calculators required or allowed.

EXAM REVIEW SESSION: Thursday 3:00 - 5:00 in SWORDS 302. Please come prepared with specific questions.

The following concepts, definitions, theorems and axioms are the most important material for the exam. This does not mean that other facts, theorems or ideas will not be covered. These concepts and theorems are ones which you may be asked to give a precise definition or statement of. I will expect your answers to be concise and specific, with little margin for error.

Definitions and terminology:

1. Basic Set Theory: null set, union, intersection, complement, A - B, universal set, interval notation
2. Definition of the absolute value function
3. The least upper bound (lub) and greatest lower bound (glb) of a set
4. Countable and uncountable sets
5. The rational and irrational numbers
6. Definition of a convergent sequence (epsilon-n_0 definition)
7. Definition of a subsequence and definition of increasing or decreasing sequences
8. Definition of the limit of a function (epsilon-delta definition)
9. Definition of a continuous function f(x) at x = c
10. Definition of the derivative of f(x) at x = c
11. Upper and lower sums, and upper and lower integrals
12. Definition of an integrable function
13. Definition of a convergent infinite series
14. The Harmonic Series
15. Definition of an alternating series
16. Conditional convergence, absolute convergence
17. Power Series, radius of convergence, interval of convergence
Major Theorems:
1. Principle of Induction
2. The set of real numbers is an uncountable set.
3. Least Upper Bound Axiom (Completeness Axiom)
4. A bounded monotonic sequence converges (MCT)
5. Bolzano-Weierstrass Theorem (BWT)
6. The Big Limit Theorem for sequences or functions (BLT)
7. Intermediate Value Theorem (IVT)
8. Extreme Value Theorem (EVT)
9. Mean Value Theorem (MVT)
10. The Big Integral Theorem (BIT)
11. The Fundamental Theorem of Calculus (FTC)
12. The n-th term test

Problems:

Some problems you should be prepared to do: (from Ch. 5 and 6)

1. Compute the upper and lower sums of a function f(x) given a partition.
2. Use the Fundamental Theorem of Calculus.
3. Determine if a given infinite series converges or diverges. (Know your Tests!)
4. Find the interval of convergence of a given power series.
Proofs:

You will have to do several proofs on the exam. These will be similar to homework problems. In addition, you will be asked to prove at least one of the following major theorems:

1. The square root of 2 is irrational.
2. A bounded monotonic sequence converges.
3. The Product Rule for differentiation
4. The Fundamental Theorem of Calculus

Practice Problems:
Some sample practice problems similar or identical to homework problems are listed below. Solutions are located in the MA242 binder in the Math/Science Library. Note that the solutions from the solution manual are very terse and leave out much of the motivating details I would expect you to include in your proofs.

• Section 4.3 : 9, 11b, 11d, 16, 17
Chapter 5
• Section 5.1 : 2, 3b, 5
• Section 5.2 : 1c, 1d, 3, 5d
• Section 5.3 : 2a, 2d, 4
• Section 5.4 : 1a, 1f, 2c, 4, 5
Chapter 6
• Section 6.1 : 1b, 1g, 1h, 3d, 3f, 8c, 8d, 8f
• Section 6.2 : 1c, 1f, 3, 4d, 6, 12, 13
• Section 6.3 : 1e, 1g, 1h, 2c, 2d, 11