Principles of Analysis

Exam #2 (Fall 2002)

Wednesday, November 13, In Class

The second exam covers Sections 2.2, 2.3, 2.4, 2.5, 3.1, 3.2, 3.3, 3.4, 3.5, 4.1 and 4.2. You should go over homework problems and class notes. The exam will be designed to take the full class period (45-50 minutes). No calculators required or allowed.

The following concepts, definitions, theorems and axioms are crucial material for the exam. You WILL BE ASKED to give a precise definition or carefully state a particular theorem. (eg. Give the definition of an increasing sequence.) I will expect your answers to be concise and specific, with little margin for error.

Definitions and terminology:
Be sure you know the definitions and understand the following concepts thoroughly.

1. Definition of limit (for BOTH sequences and functions), deleted neighborhood
2. Left and right-hand limits, limits involving infinity
3. Subsequence, increasing and decreasing sequences (and functions), recursive sequences
4. Definition of continuity, removable discontinuity
5. Bounded sequence, bounded function
6. Definition of the derivative
Major Theorems:
1. Least Upper Bound Axiom (this is still very important)
2. The Big Limit Theorem for sequences AND functions (BLT)
3. A bounded monotonic sequence converges (MCT)
4. Bolzano-Weierstrass Theorem (BWT)
5. Intermediate Value Theorem (IVT)
6. Extreme Value Theorem (EVT)
7. Differentiation Theorems (product rule, quotient rule, chain rule, etc.)

Problems:

You will have to do a couple of proofs on the exam. These will be similar to homework problems. You may also be asked to compute something, such as a limit of a sequence or function. Types of problems most likely to be asked on the exam include:

1. Using the epsilon-delta definition to prove a limit exists.
2. Determining where and why a function is continuous or differentiable using the definitions.
3. Showing you have a solid understanding of the absolute value function.

Practice Problems:
Some sample practice problems similar or identical to homework problems are listed below. Solutions will be located in a 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.

Chapter 2

• Section 2.2 : 1b, 1d, 5a, 6, 8
• Section 2.3 : 2c, 5
• Section 2.4 : 5a, 5d, 7d, 7e, 7f
• Section 2.5 : 1a, 1b, 6, 12
Chapter 3
• Section 3.1 : 2d, 2h, 2p, 3, 6
• Section 3.2 : 7
• Section 3.3 : 3d, 3e, 3f, 5b, 5h, 6
• Section 3.4 : 1a, 1d, 1f, 1i, 2, 11c, 11d
• Section 3.5 : 5, 8, 13,
Chapter 4
• Section 4.1 : 1b, 1c, 5a, 11b, 18a, 18d
• Section 4.2 : 2, 5