MATH 350, Review for the Final Exam

Date and time: Wednesday May 18, 2011, 5:30-7:30pm in CR 5117.

Course topics

0. Introduction
   1. Sets and operations on them (Kirkwood 1-1; lecture notes)
   2. Relations, functions, direct and inverse images (Kirkwood, 1-1; lecture notes)
   3. Logic and proofs (lecture notes)
   4. Composition of functions; one-to-one, onto and inverse functions (Kirkwood, 1-1; lecture notes)

1. The real number system
   1. Axioms of a field (1.1)
   2. The order axiom; inequalities (1.2)
   3. Absolute value (A.1)
   4. Completeness axiom (lecture notes)
   5. Natural numbers and mathematical induction (1.4)

2. Continuity and limits
   1. Continuity and limits. Definitions and examples (2.1)
   2. Properties of limits (2.2)
   3. One-sided limits (2.3)
   4. Limits at infinity. Infinite limits (2.4)
   5. Limits of Sequences (2.5)
   6. The Completeness Axiom and the Archimedian Principle (2.5 and the lecture notes)

3. Properties of Continuous Functions
   1. The Intermediate-Value Theorem (3.1)
   2. Least Upper Bounds, Greatest Lower Bounds (3.2)
   3. The Bolzano-Weierstrass Theorem (3.3)
   4. The Boundedness and Extreme-Value Theorems (3.4)
   5. Uniform Continuity (3.5)
   6. The Cauchy Principle (3.6)

4. Differentiation
   1. The derivative; definitions and basic properties (4.1)
   2. The Rolle Theorem and the Mean-Value Theorem. L'Hopital's rule (4.1)

5. Integration
1. The Darboux integral for functions on R¹ (5.1)
2. Properties of the integral (5.1, Thm. 5.3 and the Corollary)
3. The Riemann integral (5.2)
4. Mean-value theorem for integrals and the fundamental theorem of calculus (5.1)
6. Taylor's formula
1. Taylor's polynomial. Lagrange's form of the remainder (lecture notes)
2. The integral form of the remainder (lecture notes)
3. Taylor expansions of elementary functions (lecture notes)

Important definitions/facts

• Axioms of real numbers
• The Archimedian principle
• Bounded sets; upper and lower bounds; maximum and minimum; supremum and infimum
• Infinite sequences; limits of sequences; subsequences
• Monotone (nondecreasing, increasing, nonincreasing and decreasing) sequences and functions.
• Cauchy property
• Limits, continuity
• Uniformly continous functions
• Definition of the derivative
• L'Hopital's rule
• Upper and lower sums. Darboux integrals
• Riemann integral. The equivalence between the Darboux and Riemann integrals
• Taylor's polynomial; Taylor's formula; remainder
• o(f(x)) and O(f(x)) symbols

Theorems which you need to know with proofs:

• The monotone convergence principle (based on Axiom C, see lecture notes)
• Theorems about limits (Theorems 2.1-2.10)
• The Bolzano-Weierstrass theorem (Theorem 3.10)
• A subsequence of a convergent sequence is convergent (Theorem 3.10')
• Intermediate value theorem (Theorem 3.3)
• Boundedness and Exteme Value Theorems (Theorem 3.11, 12)
• The Cauchy criterion (Theorem 3.14)
• Properties of derivatives (Theorems 4.1-4.7)
• Fundamental lemma of differentiation; the chain rule (Theorems 4.8-4.9)
• Rolle's and mean-value theorems (Theorems 4.11, 12, 14)
• Properties of integrals (Theorem 5.3)
• Criterion of integrability (Theorem 5.5)
• Integrability of continuous functions (Corollary 1 to Thm 5.5)
• Mean-value theorem for integrals (Theorem 5.6)
• The fundamental theorem of calculus (Theorems 5.7-8)

General rules

• All tests are closed books/notes; electronic devices are not permitted, with the exception of a basic scientific calculator (TI 30X IIS or similar). A calculator is not required to solve the exam problems.
• The length of the exam is 2 hours.
• Problems will come in the same format as on the quizzes and midterms; you will be asked to present a complete solution to each problem in a readable format.