Math 242 Principles of Analysis

Final Exam

Tuesday, Dec. 15, 8:30 - 11:30 am, Stein 217

The final exam is cumulative, that is, it covers all the material from the first day of class onwards. Approximately one-third will cover material since the second exam. This is Chapters 5 and 7.

You should go over homework problems (as well as the partial solutions), quizzes and your class notes. The exam will be designed for 2 hours (twice the length of the midterm exams) so you should have plenty of time to complete it in the alloted 3 hour time slot. No calculators required or allowed.

Exam Review Session: Saturday, Dec. 12, 12:00 - 1:30 in Swords 328. Please come prepared with specific questions. Some practice problems are available here.

The following concepts, definitions, theorems and examples are crucial material for the exam. As with the quizzes, you may be asked to give a precise definition or carefully state a particular theorem. I will expect your answers to be concise and specific, with little margin for error. You may also be asked some short answer questions similar to those on quizzes.

Definitions, terminology and examples:

  1. Basic Set Theory: empty set, union, intersection, complement, A - B, universal set, subset, def. of when two sets are equal, De Morgan's Laws
  2. Natural numbers N, integers Z, rationals Q, irrationals I, reals R
  3. Logic: negation, truth tables, "or" versus "and", implications, converse, contrapositive, if and only if
  4. Function, domain, range, image, pre-image, f(A), f^{-1}(B), one-to-one, onto, bijection, inverse
  5. Bounds: maximum, minimum, upper and lower bounds, the supremum and infimum of a set
  6. Cardinality: infinite sets, A ~ B, countable sets, uncountable sets
  7. Sequences: convergent sequence, divergent sequence, increasing, decreasing and monotone sequences, recursive sequences, Cauchy sequence, Fibonacci sequence, triangle inequality, bounded sequence
  8. Infinite series, partial sums, convergence, conditional versus absolute convergence, geometric series, p-series
  9. Point-Set Topology: Cantor set, open and closed sets, closure of a set, limit points, isolated points, bounded sets, compact sets, connected sets, separated sets
  10. Definition of a limit (for BOTH sequences and functions), definition of continuity, bounded function
  11. Harmonic Series, Dirichlet's function (and modified versions)
  12. Definition of the derivative of f(x) at c
  13. Integration: partitions, refinements, upper and lower sums, upper and lower integrals, integrable function, interchanging the limits of integration
Axioms and Theorems:
  1. Fundamental Theorem of Arithmetic, Square root of a prime is irrational
  2. Principle of Induction
  3. Axiom of Completeness, The Sup (and Inf) Lemma
  4. Nested Interval Property
  5. Archimedean Property, The rationals and irrationals are each dense in R.
  6. Theorems on Countable sets (eg. the rationals are countable, the reals are uncountable)
  7. Theorems on Sequences: The Big Limit Theorem, Every convergent sequence is bounded, Order Limit Theorem, Monotone Convergence Theorem (MCT), the Squeeze Theorem
  8. Convergence Tests for Infinite Series: nth-term test, comparison test, ratio test, alternating series test, etc.
  9. The Big Limit Theorems for sequences, functions and series (BLT)
  10. Bolzano-Weierstrass Theorem (BWT), Cauchy Criterion (CC)
  11. Facts about open and closed sets: Arbitrary union of open sets is open, Finite intersection of open sets is open, Complement of an open set is closed, etc.
  12. Heine-Borel Theorem (A set is compact if and only if it is closed and bounded.)
  13. Connected sets in R are intervals.
  14. Sequence characterization of functional limits, Divergence criterion for limits, 4 ways to describe continuity, Discontinuity criterion, Algebraic continuity theorem, Composition of continuous functions is continuous
  15. Continuous functions take compact sets to compact sets and connected sets to connected sets.
  16. Squeeze Theorem for sequences and functions
  17. Intermediate Value Theorem (IVT)
  18. Extreme Value Theorem (EVT)
  19. Differentiability implies continuity, Standard differentiation rules (sum, product, quotient and chain rules)
  20. Interior Extremum Theorem, Darboux's Theorem
  21. Mean Value Theorem (MVT)
  22. L'Hopital's Rule
  23. Lemmas about upper and lower sums: Adding points to a partition increases the lower sum and decreases the upper sum, etc.
  24. Criteria for integrability (Thm. 7.2.8) and its corollary involving sequences of partitions, Continuous functions are integrable
  25. Properties of the integral (Thm. 7.4.1 and Thm. 7.4.2)
  26. The Fundamental Theorem of Calculus (FTC)

Proofs:

You will have to do several proofs on the exam. These will be similar to homework problems. In addition, you will be asked to prove at least one of the following major theorems:

  1. The square root of 2 is irrational.
  2. The real numbers are an uncountable set.
  3. The Fundamental Theorem of Calculus