MATH 350, Review for the Final Exam

Date and time: Tuesday, July 7, 2015, 12:00-1:40pm in LO 1326.

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; Lecture Notes: properties of sup and inf)
   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. Elementary Theory of 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)
   3. Inverse functions. Derivative of the inverse function (4.2)
   4. Properties of Elementary Functions and their derivatives (Lecture notes)

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)

Important definitions/facts/techniques

• Axioms of real numbers
• The Archimedian principle
• Bernoulli inequality; limits of geometric sequences (Homework 5)
• Bounded sets; upper and lower bounds; maximum and minimum; supremum and infimum; supremum and infimum of functions
• Infinite sequences; limits of sequences; subsequences
• Monotone (nondecreasing, increasing, nonincreasing and decreasing) sequences and functions.
• Cauchy property
• Limits, continuity; ε-δ-technique (and its variations for sequences, infinite limits and limits at infinity)
• Uniformly continous functions
• Definition of the derivative
• L'Hopital's rule
• Power functions with rational exponents (Homework 5, lecture notes on the webpage)
• Exponential function: definition and properties (Lecture notes on the webpage)
• Exponential function: derivative and the number e (Lecture notes on the webpage)
• Logarithmic functions and their derivatives (Lecture notes on the webpage)
• Trigonometric functions and their derivatives (Lecture notes on the webpage)
• Upper and lower sums; upper and lower Darboux integrals. Criterion of integrability
• Riemann integral. The equivalence between the Darboux and Riemann integrals

Theorems to know with proofs:

• The Bolzano-Weierstrass theorem (Theorem 3.10)
• Boundedness and Exteme Value Theorems (Theorem 3.11, 12)
• The chain rule and its partial converse (Theorems 4.8-4.9, problem 10)
• Rolle's and mean-value theorems (Theorems 4.11, 12, 14)
• Properties of integrals (Theorem 5.3)
• Criterion of integrability (Theorem 5.5)
• Mean-value theorem for integrals (Theorem 5.6)
• The fundamental theorem of calculus (Theorems 5.7-8)
• Proofs for limits of sequences aⁿ, aⁿ/n^m, a^1/n with a > 1; bⁿ, nbⁿ, b^1/n with 0 < b < 1.

Key types of questions (the list is not all-inclusive):

• Formulate and prove one of the theorems from the list "with proofs" (see above)
• Formal logic, sets, functions, images and inverse images (review problems in Homework 1)
• Properties of supremums and infimums/examples (review homework/midterm questions, work through problems in the last three assignments)
• Find the limit of a sequence, using the techniques discussed in class/included in homework; Find a limit related to the number e (work through lecture notes posted on the webpage)
• Cauchy sequences: determine whether or not a sequence is Cauchy; use definition or the Cauchy Criterion
• Uniform continuity: find out whether or not a function is uniformly continuous (review problems in the homework)
• Use ε-δ-technique to prove continuity or to find a limit
• Find a derivative of a function using the definition (integer powers, roots, simple rational functions, exponential function)
• A question related to properties of power functions and the definitions of exponential/logarithmic functions (work through problems in Homework 5 and lecture notes posted on the webpage)
• Prove a property of upper/lower integrals (part of Theorems 5.3-5)
• A problem on integration (work through problems in the last two assignments)

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 1 hours 40 minutes.
• 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.