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Preview
This lesson generalizes the vector/parametric view of lines in 3D (from the previous lesson) to develop the vector/parametric view of curves in 3D. Mirroring what we did in the previous lesson, we will come to view a curve in 3D as the output of a vector of parametric functions as the parameter ranges over some interval. We’ll call these vector-valued functions in this lesson and explore their calculus.
In Module 1 we’ll define vector-valued functions and learn how to visualize them.
In Module 2 we’ll discuss the calculus of vector-valued functions, including their limits, derivatives, and integrals.
Finally, in Module 3 we’ll define the tangent vector and tangent line to a vector-valued function, generalizing the analogous notions we are familiar with from single-variable calculus.
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 4.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 4 PP.pdfPage updated
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