Embedded Files
Preview
Preview
Here's the on-ramp for this lesson.
What Will We Learn?
In this lesson we'll learn about an additional interpretation associated with the derivative -- the tangent line to a graph at a point is the "linearization" of the function near that point. Visually, this means that if you zoom into the point of tangency you'll see the graph of the function start to look more and more like the graph of the tangent line (Figure 4.4 on pg. 90 in Calculus Simplified illustrates this). This insight will then help us understand how the sign of the derivative -- positive or negative -- gives us information about whether the function is increasing or decreasing near the point of tangency. Filling in the details of that connection will take up the last chunk of this lesson.
Why Do We Need to Learn This?
This lesson marks our journey towards that powerful application of calculus I mentioned in the previous lesson: optimization. Optimization is about maximizing and minimizing functions. And one tool we'll use in searching for the extrema of a function is the Increasing/Decreasing Test, which we cover in this lesson. Separate from the optimization tie-in, the concepts we'll learn in this lesson will help us approximate functions and function values (example: we'll be able to approximate the square root of 3 without a calculator) and also sketch the graph of a function, again without a calculator.
Review
Review
Learn
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 Notes
Lesson 13.pdfVideo 1 (Example 4.6)
Video 1 (Example 4.6)
Video 2 (Example 4.7)
Video 2 (Example 4.7)
Video 3 (Example 4.8)
Video 3 (Example 4.8)
Video 4 (Example 4.11)
Video 4 (Example 4.11)
Video 5 (Example 4.12)
Video 5 (Example 4.12)
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 13 PP.pdfPage updated
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