## Math 355: Introduction to Analysis Study Guide for the Final Exam

The final exam is on Monday, May 7, from 9:00am to 12:00pm, in MERR 4 (note that this is not our usual classroom).

Reading/exam period office hours are Thursday 2:00-3:30pm and Friday 2:00-3:00pm.

Daniella Bennet will hold drop-in hours 2pm-5pm at the Q-Center during reading week and an extended session specifically for Analysis on Sunday (May 6), 7pm - 9pm

The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

The final exam is cumulative, so it is important that you study the material from earlier chapters as well. Please be sure to review homework and example problems for the chapters given below. If you would like to see additional examples and practice problems, I recommend looking at the textbook How to think about analysis by Lara Alcock, which is on reserve for this course in the Science Library.

Another good source of practice problems comes from old final exams and the department comprehensive exams. You may also find it useful to look over the summary of topics covered in the Analysis section of the comprehensive exam.

6.3: This section extends the ideas of the previous theorem to differentiability (not just continuity). Be familiar with all theorems in this section.

6.4-6.5: These sections deal with series of functions, and in particular power series, and combine our previous results about series of real numbers and sequences of functions. You must be familiar with all theorems and proofs in these sections, except for Abel's Lemma and Abel's Theorem, which you are not required to know for this exam.

6.6: Be familiar with the general examples of Taylor series and standard substitution techniques used as given in p. 197-200 (before Lagrange's Remainder Theorem).

7.2-7.4: Know what we mean by partition, upper sum, lower sum, upper integral, lower integral, and Riemann integrability. Know all the theorems and proof techniques used in these sections.

7.5: Know the statement and proofs of both parts of the Fundamental Theorem of Calculus. Pay careful attention to how part (i) is mainly a computational statement, but part (ii) asserts continuity and differentiability in certain circumstances.