MATH 450A, Review for the Final Exam

Date and time: Tuesday, Dec 17, 2019, 5:30-7:30pm in CR 5123

Course topics

0. Introduction
   1. Logic and proofs (lecture notes)
   2. Sets and operations on them (lecture notes)
   3. Relations, functions, direct and inverse images (lecture notes)
   4. Composition of functions; one-to-one, onto and inverse functions (lecture notes)
1. The real number system
   1. Axioms of a field (1.1)
   2. The Order Axiom (Axiom I); Inequalities (1.2)
   3. Absolute value (A.1)
   4. Natural numbers and mathematical induction (1.4)
   5. The Completeness Axiom (lecture notes)
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. The Archimedian Principle (2.5)
3. Basic properties of functions on R
   1. The Intermediate-Value Theorem (3.1)
   2. Least upper bound; greatest lower bound (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 Criterion (3.6)
4. Elementary Theory of Differentiation
   1. The derivative in R¹; definitions and basic properties (4.1)
   2. The Rolle and Mean-Value Theorems (4.1)
   3. Inverse functions. Derivative of the inverse function (4.2)
5. Elementary Theory of Integration
   1. The Darboux Integral on R¹ (5.1)
   2. The Riemann Integral (5.2)
   3. The Mean-Value Theorem and the Fundamental Theorem of Calculus (5.1)
6. Infinite Sequences and Infinite Series
   1. Series with Positive Terms. Tests for Convergence (9.1)
   2. Series of Terms of Varying Sign. Power Series (9.2)
   3. Uniform Convergence of Sequences and Series. Weierstrass Test (9.3, 9.4)
   4. Taylor Polynomials and Taylor Series. Lagrange's and Integral Forms of the Remainder (9.4)

Key concepts:

• Axioms of Real Numbers: Field Axioms, Axiom of Inequality and the Completeness Axiom
• The Monotone Convergence Principle (stated as Axiom C in the textbook)
• Induction Principle. Inductive Sets. Proofs by Induction. The Archimedian Principle
• Limits and Continuity. The ε-δ-technique.
• Lower Bound, Upper Bound; Supremum, Infimum; Maximum and Minimum
• Uniform Continuity
• Cauchy Sequence
• Definition of the Derivative. Using the definition to find the derivative of a function defined piecewise
• Inverse Function and its Derivative
• Darboux and Riemann Integrability
• Convergence Tests for Series (Comparison, Limit Comparison, Integral, Ratio, Root, Alternating)
• Absolute and Conditional Convergence
• Interval of Convergence and Radius of Convergence for Power Series
• Pointwise and Uniform Convergence for sequences of functions and for power series. The Weierstass test for Uniform Convergence
• Taylor's Polynomial and Taylor's Series. Formulas for the Remainder.

Key Theorems and Examples:

• The Bolzano-Weierstrass Principle (Theorem 3.10)
• The Extreme Value Theorem (Theorem 3.12)
• The Fundamental Lemma of Differentiation (Theorem 4.8)
• Rolle's and Mean Value Theorems (Theorems 4.11, 12)
• Criterion of Darboux Integrability and Integrability of Continuous and Monotone Functions (Theorem 5.5, Corollary 1, 2)
• Integrability of a Step Function; Non-integrability of the Dirichlet Function (Exercises 10, 11, 5.1)
• Mean Value Theorem for Integrals (Theorem 5.6)
• The Fundamental Theorem of Calculus (Theorems 5.7-8)
• Uniform Limit of Continuous Functions is Continuous (Theorem 9.13)

Problems for Review

You should also go over all questions in the past midterms and review all old homework, completing any work that you missed.

General rules

• All tests are closed books/notes; no electronic devices.
• The duration of the exam is 2 hours.
• Problems will come in the same format as on the quizzes and midterms; you need to show all work necessary to obtain the conclusions.