Determine average rates of change using difference quotients
Represent the derivative of a function as the limit of a difference quotient
I can distinguish between the average and instantaneous rate of change.
I can evaluate the limit of a difference quotient.
Quick Lesson Plan
This lesson is a huge first step to understanding the definition of the derivative, which drives the first semester’s worth of content in AP Calc (AB). Students go through the process of calculating average rates of change over intervals that get smaller and smaller, in the hopes of estimating Felix Baumgartner’s exact velocity 30 seconds after jumping out his helium balloon.
This lesson is longer than others and requires strategic pausing. We suggest having students work in groups on questions 1-3 (the front page), then pausing as a class to debrief the front page, adding the margin notes shown in the answer key. This is critical for getting students to start to see the structure of our new notation for a difference quotient and to ultimately arrive at this idea of “h”. The color-coding is intentional and strategically placed to get students to see the key aspects of the definition of the derivative.
We suggest pausing again after question 6, filling in the margin notes, and discussing answers. Be sure to get students to point out how in 3a and b as well as 4a and b, the points chosen are the same distance away, just coming from the left and the right. After question 6, students should be thinking about how small the interval really needs to be to get a good estimate. Suggest that we could have asked them to find the average speed between t=29.5 and t=30 and then t=30 and t=30.5, or between t=29.9 and t=30 and t=30 and t=30.1. The length of the interval is arbitrary. If something is arbitrary, we can give it a variable name. Let’s call it h. Then allow students to finish 7 and 8. Reveal at the end that even though Baumgartner did not go supersonic by t=30, he did go supersonic at t=34.
This is a FULL lesson so be sure to plan accordingly. If you can not finish in one day, we highly recommend that you complete the experience and debrief portion all on the first day.
We feel strongly about finishing the experience and debrief in one day, so we are going to discuss our Unit 1 test results on day two of this lesson. The ideas in today’s topic require continuous development, thoughtful homework and then a revisit on the next day. This is a big idea!! Allow students time to process this!!
Although students will not be asked to find a derivative using the formal definition of a derivative, they WILL be asked to recognize multiple forms of a difference quotient in a limit problem. Many multiple choice questions are simplified by recognizing that the complicated limit is really just asking for a derivative. In subsequent Unit 2 lessons, the Check Your Understanding problems will incorporate some practice recognizing the definition of a derivative within limit problems.
The transition from a concrete average rate of change with known values to the more abstract notation with a’s, x’s, and h’s can be a challenge for students. As often as possible, refer students back to concrete examples and show what is taking the place of those values. The color coding of the margin notes should help students start to see the structure of a difference quotient.
Sometimes having two different definitions for the derivative can be tricky for students. Remind them that conceptually, they are getting at the same idea, but using a slightly different notation. I tell my students that it’s like the difference between saying someone’s car is getting closer to your own as you sit still (x approaching a), and saying the distance between your two cars is getting very small (h approaching 0). We purposefully draw two different diagrams in the Important Ideas box to help students see both representations.