MATH 350, Review for Midterm Test 2

Test topics

2. Continuity and limits
   1. Limits of Sequences (2.5)
   2. 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. 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)

Important definitions/facts

• Limit of a sequence
• Monotone sequences; monotone convergence principle (with proof)
• The Archimedian principle
• Nested intervals principle; the intermediate-value theorem ( with proofs)
• Least upper bound (supremum) and greatest lower bound (infimum). Properties of supremums/infimums (Corollary to Thm. 3.5, with proofs)
• Subsequences. The Bolzano-Weierstrass theorem (without proof)
• The boundedness and extreme value theorems (with proofs)
• Definition of uniform continuity and techniques on showing that functions are / are not uniformly continuous
• The Cauchy criterion (with proof)
• Definition of the derivative, and the list of properties (with proofs)
• The fundamental lemma of differentiation and the chain rule, with proofs
• The first derivative test (Theorem 4.10) with proof
• Rolle's theorem and mean-value theorems, with proofs
• L'Hopital's rule (without proof)

Types of questions that will be on the test:

• Prove one of the theorems listed above
• Limits of sequences; Cauchy sequences
• Subsequences; application of the Bolzano-Weierstrass theorem
• Uniform continuity
• Example on differentiability/non-differentiability
• Applications of Rolle's and mean-value theorems

General rules

• All tests are closed books/notes; graphing calculators, cell phones or other electronic devices are not permitted. A basic scientific calculator (example: TI 30X IIS or similar) may be helpful but it is not required.
• The duration of the test is 1 hr 15 minutes.
• Problems will come in the same format as on the quizzes: you will need to write a concise proof to answer each question.