MATH 350, Review for Midterm Test 2

6/26/2015

Test topics

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

List of Theorems to know with proofs:

• Nested intervals principle
• The Intermediate Value Theorem
• Properties of supremums/infimums (Corollary to Thm. 3.5; see Lecture Notes)
• A function continuous on a closed interval is bounded
• Extreme Value Theorem
• The Cauchy Criterion
• Derivatives of sums/products/quotients. The Chain Rule and its converse
• The inverse function theorem, for increasing and decreasing functions
• Rolle's Theorem, the Mean Value and Generalized Mean Value Theorems
• L'Hopital's Rule "0/0"

Important definitions/facts/techniques:

• Limit of a sequence; techniques of computing limits of sequences (Lecture Notes)
• Sequential definition of limit (Lecture Notes)
• Monotone sequences; Monotone Convergence Principle
• Upper bounds, lower bounds; supremums and infimums; find sup/inf of a set or a function
• Sequences aⁿ, aⁿ/n^m, a^1/n with a > 1; bⁿ, nbⁿ, b^1/n with 0 < b < 1; Bernoulli inequality
• Subsequences. The Bolzano-Weierstrass theorem
• Definition of uniform continuity; techniques to show that functions are / are not uniformly continuous
• A function continuous on a closed interval is uniformly continuous
• Applying the Cauchy criterion for partial sums of series: problems 1-3 and 7 in 3.7
• Definition of the derivative; differentiability; left and right-sided derivatives
• The fundamental lemma of differentiation
• Properties of derivatives; the chain rule and its partial converse
• Condition for local extremum (Theorem 4.10)
• Monotonicity conditions for differentiable functions (Theorem 4.13)
• Definitions and properties of power functions f(x)=x^p for rational exponents
• Definitions and properties of the exponential and logarithmic functions for real x
• The definition of the number e as a limit; derivatives of the exponential functions
• Examples of functions xⁿsin(1/x), xⁿcos(1/x) for Continuity, Intermediate Value Theorem, Differentiability and Rolle's Theorem

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

• Prove one of the theorems from the list "with proofs"
• Find a supremum or infimum or prove a property of supremums and infimums
• Find the limit of a sequence, using the techniques discussed in class/included in homework; Find a limit related to the number e
• 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
• 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
• A question related to Problems 15 or 25 in 4.1.

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 60 minutes.
• Problems will come in the same format as on the quizzes: you will need to write a concise proof to answer each question.