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Preview
In this lesson we’ll start the transition to studying multivariable functions, focusing first on visualizing them and learning how to generalize the notions of limit and continuity to the multivariable context.
In Module 1 we’ll introduce the relevant terminology and introduce the notion of level curves, which in the simplest case will help us use 2D thinking to visualize 3D graphs of functions.
In Module 2 we’ll generalize the notion of a limit to the setting of two-variable functions. We’ll discover one added complication—an infinite number of directions of approach in the limit (as opposed to just the “left-hand” and “right-hand” directions in single-variable calculus)—but develop techniques to deal with this.
Finally, in Module 3 we’ll generalize the notion of continuity to the setting of two-variable functions. We’ll see that most of the results we already know about continuity carry over into this multivariable context (e.g., “multivariable polynomials” are continuous).
Review
Review
Learn
Learn
Work through the lesson notes below, consulting the videos below it when you get to the "See Class Notes" boxes. For your records, the annotated lesson notes are below the videos. Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson Notes
Lesson 5.pdfModule 1 Video
Module 1 Video
Module 2 Video
Module 2 Video
Module 3 Video
Module 3 Video
Reflect
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 5 PP.pdfPage updated
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