How does the average rate of change between two points lead to the derivative at a single point?
By the end of Unit 1, students can calculate limits and distinguish between average and instantaneous rates of change. Unit 2 brings those ideas together: as one point on a graph approaches another, students use a limit to calculate the instantaneous rate of change at a single point. Then they see how finding that rate at different inputs gives rise to a new function: the derivative.
The videos below are part of the complete AP Calculus AB/BC Animated Curriculum. As graphs, labels, and equations appear one step at a time, students can follow the reasoning that connects one idea to the next. The animation carries the visual sequence, leaving you free to watch your students, notice when something isn’t clicking, and respond in the moment.
As I get ready to start Unit 2 with my own students, I thought I’d share the first three animated lessons from this unit with our community. Start with both parts of Lesson 2.1 below. If you’d like to continue, use one of the buttons below to request Lessons 2.2 and 2.3—four more videos—along with guided notes for all three lessons.
How can we calculate the rate of change at a single point?
Students already know how to find an average rate of change between two points and estimate an instantaneous rate from a graph. In Part 1 of this lesson, they watch one point approach the other and see how a limit gives the exact instantaneous rate. The focus is on understanding where the formula comes from. In Part 2 of this lesson, they put both rate-of-change formulas to work, using the matching guided notes to think through each example.
đź’ˇ Ready to put this idea to work?
Watch Part 2 of Lesson 2.1 below. If you’d like to continue with Lessons 2.2 and 2.3, use the button to request four more videos and guided notes for all three lessons.
In Part 2, students put the ideas from the first video to work. Two examples—one involving submarine depths and one involving customers at a bagel shop—give them practice calculating average rates of change and using limits to find or represent instantaneous rates of change.
I typically pause when each question appears, giving students time to work through it and discuss their thinking before revealing the animated solution. Press play to see how you might use this lesson with your own students.
Want to use these lessons with your own students? The matching guided notes include the key ideas and give students space to work through each example before the animated solution appears.
Use the button below to request Lessons 2.2 and 2.3—four more videos—along with guided notes for all three lessons.
I hope these two parts of Lesson 2.1 help your students see how average rates of change and limits connect to the rate at a single point. If you try the lesson with your class, I’d love to hear what clicked and where students still had questions.
This lesson is part of the complete AP Calculus AB/BC Animated Curriculum. If you’d like to continue, use the button below to request Lessons 2.2 and 2.3 and guided notes for all three lessons.