Embedded Files
Preview
Preview
In single-variable calculus we used a function's tangent lines to give us information about the graph of the function and to help us approximate the values of the function near the point of tangency (recall that this was referred to as linear approximation in single-variable calculus). In this lesson we'll generalize these concepts to the multivariable context.
In Module 1 we'll discuss how tangent lines generalize to tangent planes in the multivariable context. Then, we'll use this result to generalize the linear approximation approach, yielding what we'll call the tangent plane approximation of a multivariable function.
In Module 2 we'll re-interpret our tangent plane work to extract a new special type of vector: the vector of partial derivatives of the function. We'll call this vector the gradient of the function and explore its connection to the function's level curves.
Finally, in Module 3 we'll generalize the partial derivative concept to the notion of directional derivatives and learn that the gradient plays a crucial part in calculating directional derivatives. We'll also explore some of the applications to optimization associated with directional derivatives.
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 7.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 7 PP.pdfPage updated
Google Sites
Report abuse