Lesson 15: Taylor Series
Preview
Lesson Preview
We now know how to approximate a function by a polynomial—we use the function’s nth degree Taylor polynomial—and calculate the maximum error involved in doing so (via Taylor’s Theorem). Can we reduce that error to zero? That is, can we express a function as an “infinite degree” Taylor polynomial? In this lesson we’ll do just that. In Module 1 we’ll introduce Taylor series, and then begin calculating and exploring them. Then, in Module 2 we’ll revisit Taylor’s Theorem and use it to develop a method for determining when Taylor series converge to the function used to calculate them.
Learn
The lesson notes below contain a learning plan with three stages -- Learn, Reflect, and Practice -- and guidance for what to do within each stage. Some tips for you as you work through this resource, and those that it points to:
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
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Video 1 (Example 15.2)
Video 2 (Example 15.3)
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
Work through the practice problems suggested below to see how much of this lesson you've understood.
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