Introduction to Analysis -- Fall 2011

Information about the final exam

Our final exam will be Monday, 12/19, 5:30 - 8 PM.   Note that the exam starts earlier than our normal class time.  I plan to be on campus starting at 3pm for any last minute consultation.

Everyone should again bring his or her portfolio to the exam for me to review.  You should also remember to pick it up after the exam as you leave.

What to know for the exam.  The exam will cover  sections 14, 15, 17, 18, 19, 20, 23, 24, 25, 28, 29 upto but not including theorem 29.8.  However, the following sections will receive less emphasis on the exam: 15, 20, 28, and 29.  Those were discussed more brieffly in class than the others (and in the case of 20, essentially not at all).  You should be prepared to state definitions and theorems from those sections, and perform routine computations or discuss simple examples.  In sections 14 and 15 I will not ask you to do the type of problem covered in calculus 2 to determine whether a specifc series converges.  You should know what the convergence tests are and have a general idea of how they are proved.  You should also be able to apply them  in a theoretical way -- for example, apply the ratio test to prove that a power series converges absolutely in the interior of the interval bounded by + the radiis of convergence.  From the last two sections we covered, I am more likely to ask you to prove the results in the section than to ask you to apply those results.  In particular, you should know the basic ideas of the proofs of Fermat's theorem (29.1), Rolle's Theorem, the Mean Value Theorem, and the corollaries on pp 216-217.

As on the other exams, you should know the statements of definitions and theorems, and be prepared to both explain what the theorems mean and provide proofs, except in cases where the proofs are very complicated.  Also be prepared to apply theorems to prove new results.  As one way of demonstrating your understanding of definitions, be able to verify that they hold (or do not hold) in specific examples, or to make up examples of your own.   For example, you  should be able to determine if functions are uniformly continuous, or if sequences and series of functions are uniformly convergent or uniformly Cauchy, or to give examples of functions/sequences/series that do or do not have these properties.  And in all of these, you are expected to be able to substantiate your conclusions.