$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\FF}{\mathbb{F}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Study III

Write a study guide for yourself in preparation for the Final Exam.
This is an optional assignment - if you are trying for the “Hold Me Accountable” grading scale (where I weight your exams a bit less in total points, by giving you points for studying/reflecting on them) then you should do this, and turn this in. If you do not want to write a study guide that is fine, it will not hurt you if you are going for the “I got this” grading scheme, where I’ll instead weight your exams higher and not ask for these smaller assignments along the way.

Final Exam Coverage

The exam is 2 hours, and I will write to be slightly shorter than two midterm exams. After consulting with some of you in class we decided on a format: I will give ten problems and you will choose seven of them to complete.

The core of analysis is being able to work sequences and limits in various contexts. We have studied many applications of this general tool, to the study of continuity, differentiability, and integration. The goal of the final is to show that you have mastered this fundamental concept, and have learned how it fits into the wider context. Approximately half of the exam will focus on the basics of sequences and series, and the other half will focus on the applications (continuity, differentiability, integration). Specifically, you can expect 4-5 questions from “The Basics” and 5-6 questions from “Applications” that you will have to choose from.

The Basics I: Sequences

  • Definitions of Supremum and Infimum
  • The completeness axiom
  • Definitions of Convergence, Divergence
  • Proving a sequence converges directly from the definition
  • Limit inequalities and the limit laws
  • The Monotone Convergence Theorem: its proof, and using it to calculate limits.
  • Subsequences: breaking a sequence into subsequences to prove convergence
  • Cauchy sequences: definition + theorem that cauchy is equivalent to convergence

The Basics II: Series

  • Definition of infinite series as limits
  • Examples we can compute directly: telescoping and geometric series
  • Absolute convergence, the Divergence test and alternating series test
  • The comparison theorem, and its consequences: the root and ratio tests.
  • Power series, and using root/ratio test to find convergence

Application I: Continuity

  • Definition in terms of \(\epsilon-\delta\)
  • Proving continuity with sequences
  • The “Continuity Laws”
  • Continuity on Dense Sets
  • Extreme and Intermediate Value Theorems (know their statements and how to use them; I won’t ask about their proofs)

Application II: Differentiability

  • Definition as a limit
  • Using the definition to prove differentiability, and non-differentiability
  • Derivative “Laws” (sums, constant multiples, product, quotient and chain rule)
  • Derivatives, Critical Points, and Extrema (Fermat’s Theorem)
  • Rolles’ Theorem and the Mean Value Theorem
  • Differentiation of Power Series (know how to use the theorem, will not ask about proof)
  • Taylor’s Error Formula (know how to use the theorem, will not ask about its proof)
  • Power Series for the Exponential and Trigonometric Functions

Application II: Integrability

  • The axioms of integration
  • The definitions of upper and lower sums, and upper/lower integrals.
  • Proving a function is not integrable (like the characteristic function of the rationals)
  • Integrability of continuous functions (know how to use the theorem, will not ask about proof)
  • The Fundamental Theorem of Calculus (know how to use the theorem)
  • Basic antiderivatives and methods for antidifferentiation