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Rates of Change in Applied Contexts Other than Motion (Topic 4.3)

Unit 4 - Day 4

​Learning Objectives​
  • Interpret rates of change in applied contexts.

​Success Criteria
  • I can identify Rate In—Rate Out problems.

  • I can calculate derivatives and interpret their meaning in context.

Quick Lesson Plan
Activity: How Many Shoppers on Black Friday?



Lesson Handout

Answer Key


In the previous two activities (4.1 and 4.2), students used derivatives to solve real world problems involving rates of change.  Topic 4.3 continues the application of derivatives to investigate rates of change in varied contexts.  Emphasize daily that differentiation is the common underlying structure for solving such a wide variety of problems and the application of differentiation in context requires a solid understanding of labels and units of measure. Determining units on equation values or derivative values while reading the stem of a question will go a long way toward student understanding of the task ahead of them as well as the meaning of their solution. 

Teaching Tips

Remind students of the importance of their work from Units 2 and 3: concepts and skills introduced earlier in the course are now used to describe real world situations. Determining the correct mathematical procedure from a verbal description can be challenging for students. We encourage explicit modeling of identifying and labeling key information from the text (function values at a point in time vs. derivative values), translating phrases into calculus notation (for example, “rate of change” indicates a derivative), and determining what quantities are changing (as well as the correct units for describing that change). Reading aloud the question stem, and sharing your thinking as you begin work on a problem, allows them to hear your reasoning and models how they should approach their work.

The Activity questions and the Check Your Understanding! problem set have been designed to provide different contexts with similar solving procedures. After working through these examples, students should be more familiar with the presentation of AP “story problems” and more intentional in their problem-solving decisions. Continue to remind students of the structure for labels on derivatives: each successive derivative adds a layer to the label!

Exam Insights

Problems from Topics 4.3 appear on the FRQ section of the AP Calculus Test. Students have had to navigate questions about escalators, bananas, and trees in the past few years. Take time to look through the student samples provided by College Board to familiarize yourself with common missteps. Labels often earn points on their own merit so the importance of this part of a student’s solution cannot be trivialized.

For interesting FRQs, see the following:
2019 AB #1 (fish), #2 (ubiquitous particles), #4 (water)
2018 AB #1 (escalator), #2 (ubiquitous particles), #4 (trees)
2017 AB #1 (water), #2 (bananas), #4 (potatoes), #5 (ubiquitous particle)
2014 AB #1 (decomposing grass), #4 (two trains)
2013 AB #1 (unprocessed gravel), #3 (coffee)

Student Misconceptions

After working numerous motion problems, students may adopt the belief that every first derivative represents velocity and every second derivative represents acceleration. Interpreting the problem stem is imperative so that our students are not waxing about velocity when the context of the problem involves the rate at which bananas disappear from the grocery shelf or the rate at which people enter or leave an amusement park. Context matters: labels are important and have meaning. Student solutions must reflect not only correct mathematics but a correct understanding of what their solution describes.

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