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The Fundamental Theorem of Calculus and Accumulation Functions (Topic 6.4)

Unit 6 - Day 7

​Learning Objectives​
  • Represent accumulation functions using definite integrals.

  • Find derivatives of accumulation functions.

​Success Criteria
  • I can interpret the meaning of the independent and dependent variables of accumulation functions.

  • I can apply the FTC to find the derivative of an accumulation function.

Quick Lesson Plan
Activity: Under Cover



Lesson Handout

Answer Key


Today, students will discover part of the Fundamental Theorem of Calculus:  that the derivative of an accumulation function is the integrand function. A rate of change function measuring umbrellas per hour is used to review accumulation functions as students estimate the number of umbrellas produced by a struggling umbrella company. Question 3 of the activity leads students to introduce x as the independent variable of the accumulation function. The derivative of this accumulation function is then easily connected to the integrand (rate of change) function. 

Teaching Tips

Encourage students to use only one geometric region for questions 1b and 2b; it is more important to have enough time to complete the Check Your Understanding section than to make an accurate approximation in 1b and 2b. We strongly recommend a thoughtful discussion/debrief of all the CYU exercises as question 3 ties together cleverly conceptual thinking about the FTC with students’ analytic work.

Exam Insights

The AP Test assesses student knowledge of Topic 6.4 on both the multiple-choice and free response sections. Typically, the analytic skill of writing the derivative of an accumulation function is found on the multiple-choice section. Interpreting the derivative of an accumulation function (use of correct units, interpreting numerical solutions in context) is FRQ material.

Student Misconceptions

Notation around the derivative of an accumulation function can be confusing and frustrating for calculus students. Using the context of umbrella production allows students to attach units and meaning to integrand (rate of change function), integral (accumulation function), and the derivative of an integral (back to the rate of change function). Clarify also that the rate of change function (the graph of the integrand) does not use the same independent variable as the accumulation function. Again, using context whenever possible is tremendously helpful for students.

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