The schedule below lists the planned activities for this course. See the course syllabus for additional course information. Typos are possible. You are responsible for tracking university closures, final exam schedules, etc. through the NC State University Academic Calendar.
Due dates for assignments should be confirmed through Moodle. Always check the assignments themselves for specific dates. These due dates are tentative based on the start of the semester; dates may change on Moodle without changing on this schedule.
This week we cover the basics of logic and proofs and introduce sets of real numbers. The textbook in this course is not required; however, if you have it, note the section numbers for each lecture (we begin with a slightly different ordering of topics).
Major concepts: Introduction to real numbers. Logic and proof writing. Sets and functions, bounded sets (supremum and infimum) and the Nested Interval Property.
This week is all about sets, defined last week in the second lecture. We will cover Sections 1.3 and some additional material on open and closed sets.
Major concepts: Cardinality and density. The basic topology of open and closed sets.
We finish our unit on sets with a look at a very interesting example: the famous Cantor set. This set is not covered in the book--you should learn the material covered in lecture. You will not be held accountable for the additional material beyond the scope of the lecture/quiz/homework, but you may wish to read more on those topics in the posted online textbook. Next up, we begin the second unit of the course, on sequences.
Major concepts: The Cantor Set and a quick look at dimension, sequences and convergence, limit theorems.
This week we finish up the required material on sequences. If you want to take sequences further, there are two optional Student Choice lectures on sequence topics.
Major concepts: Cauchy sequences, subsequences, the Monotone Subsequence Theorem, the Monotone Convergence Theorem, the Bolzano-Weierstrass Theorem.
This week we begin working with functions, beginning with the concept of a continuous function.
Major concepts: the definition of continuity, and the knowledge we get when a continuous mapping is over [a, b].
We begin Riemann integration with partitions and lower/upper sums; integrability.
This week we discuss integrating discontinuous functions using the notion of measure, define improper integration, and then start differentiability.
The Mean Value Theorem (+preliminary results), and the Fundamental Theorem of Calculus (FTC).
We wrap up Unit 2 with Inverse Functions, and then the rest of the week dedicated to Midterm 2!
This week, we begin looking at sequences of functions and related topics (convergence to a limit function, types of convergence). Please utilize this Desmos illustrator: Sequence of Functions (Desmos Demonstration).
This week, we continue studying convergence for sequences of functions, including describing convergence using norms.
Major concepts: limit Theorems for sequences of functions. The Sup Norm. Metric spaces and examples.
This week we begin series, which consists of material you may recall from a second semester calculus course. We will of course go further than a regular calculus course. Note that a lot of results about series follow from results about sequences. This observation stems from the fact that a series converges if and only if its sequence of "partial sums" converges. Make sure throughout this chapter that you distinguish working with a series vs. working with a sequence.
Major concepts: Contraction mapping. Series of real constants.
The Weierstrass M-test for series of functions. Power series and why they are useful.
Taylor series (coefficients). A look at Fourier series. (There is no written homework pertaining to Fourier Series, but there is a Moodle quiz due the following week.)
Wrap up, e.g. with a Student Choice activity.