MATH 350, Review for the Final Exam

Test topics

1. The real number system
   1. Sets and functions (1.1)
   2. Properties of the real numbers as an ordered field (1.2)
   3. Supremum and infimum of a set. The completeness axiom (1.3)
   4. Cardinality. Countable and uncountable sets (1.3)

2. Sequences of real numbers
   1. Sequences and limits (2.1)
   2. Subsequences, subsequential limits (2.2)
   3. Limit points of sets of real numbers; the Bolzano-Weierstrass principle (2.3)
   4. The upper and lower limits (2.3)
   5. Cauchy sequences (2.3)

3. Topology of the real numbers
1. Open and closed sets (3.1)
2. Compactness (3.1)
4. Limits and continuity
1. Limit of a function (4.1)
2. Definition of continuity. Continuity on a set (4.1)
3. Properties of continuous functions (4.1)
4. Uniform continuity (4.1)
5. Differentiation
1. The derivative of a function (5.1)
2. Mean value theorems (5.2)
3. Taylor's formula (5.2)
6. Integration
1. The Darboux integral (6.1, to page 140: the end of proof of Theorem 6-7)
2. Properties of the integral (6.2, to page 154)

Important definitions/facts

• Functions; direct and inverse images
• One-to-one, onto and invertible functions
• Axioms of a field, axioms of order; the completeness axiom
• Bounded sets; upper and lower bounds; supremum and infimum
• The Archimedian principle
• Cardinality, countable sets
• Sequence; limit of a sequence
• Subsequences; subsequential limits. Upper and lower limits
• Monotone increasing and decreasing sequences
• Cauchy sequences
• Open and closed sets; compact sets
• Limit points, isolated points, interior points, boundary points, closure
• The Heine-Borel theorem
• Limits, continuity
• Global and local extrema
• Intermediate value property
• Uniformly continous functions
• Definitions of the derivative
• Generalized mean value thorem
• Taylor's formula and Lagrange's form of the remainder
• Darboux integral. Definition using the upper and lower sums
• Riemann integral. The equivalence between the Darboux and Riemann integrals

Theorems that you need to know how to prove

• Properties of supremums and infimums (Theorems 6.1, 6.2, 6.12)
• Every convergent sequence is bounded (Theorem 2.3)
• Arithmetic operations and limits (Theorem 2.4)
• Comparison theorems for limits (Theorem 2.5)
• Every bounded monotone sequence is convergent (Theorem 2.6)
• Principle of nested intervals (Theorem 2.7)
• The Bolzano-Weierstrass principle and its corollaries (Theorems 2.12, 2.14, Corollary 2.14)
• Cauchy sequences: (2.3: problems 13, 14)
• Limit points, boundary points and their properties (Theorem 3.6 and corollaries)
• Algebraic operations on continous functions(Theorem 4.3)
• A function continuous on a compact interval is uniformly continuous (Theorem 4.10)
• A continous function defined on a compact interval is bounded (Theorem 4.7 and Corollaries)
• Extreme value theorem (Corollary 4.7(b))
• Intermediate value theorem (Theorem 4.9 and Corollary 4.9)
• Differentiable function is continuous (Theorem 5.2)
• Rolle's theorem, Mean-value theorem (Theorems 5.7, 5.8)
• Properties of upper and lower sums (Theorem 6.3)
• ε-criterion for integrability (Theorem 6.4)
• Every continous function is Darboux integrable (Theorem 6.7)
• Properties of integrable functions (Theorems 6.13, 6.14)

General rules

• All tests are closed books/notes; graphing calculators, cell phones or laptops are not permitted. A calculator will not be 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 the expectation is that you write a clear and concise proof for the statement given in the question.