Lesson 12: Related Rates

Preview

Here's the on-ramp for this lesson.


What Will We Learn?

  • This lesson is dedicated to the topic of Related Rates. As the name implies, this part of calculus is about studying how two rates -- derivatives -- are related to each other. For example, the volume V of a sphere of radius r is V(r)=(4/3)*(pi)*r^3, and therefore if the radius changes in time -- example: it grows linearly with time -- then we'd expect the volume to also change in time. In this lesson we'll learn how to quantify this intuition, and how to apply it to other contexts.


Why Do We Need to Learn This?

  • Related Rates is the first topic where we'll start delving deeper into mathematical modeling -- the process of converting a real-world problem in mathematics, solving it, and them interpreting the solution in real-world terms. As such, this lesson builds the foundation we'll need to later discuss the most powerful application of calculus to real-world contexts -- optimization.

Review

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

Lesson 12.pdf

Video 1 (Example 4.1)

Video 2 (Example 4.2)

Video 3 (Example 4.3)

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.

Lesson 12 PP.pdf