Interpreting the Meaning of the Derivative in Context (Topic 4.1)

Unit 4 - Day 1

​Learning Objectives​
  • Interpret the meaning of a derivative in context.

​Success Criteria
  • I can interpret a derivative as an instantaneous rate of change.

  • I can state the units of f’(x) by dividing the unit for f(x) by the unit for x. 

  • I can write a sentence interpreting the derivative value at a certain point in context.

Quick Lesson Plan
Activity: A Summer Day of Calculus

     

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Lesson Handout

Answer Key

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Overview

Unit 4 moves students into new applications of derivatives. First and second derivatives are utilized to justify behaviors of functions (increasing and decreasing), locate important points (relative and global extrema as well as points of inflection) (all in Unit 5) and relate the behaviors of the independent and dependent variables (in related rates problems). To begin Unit 4, the investigation and practice problems for Topic 4.1 focus on accurate labeling and precise interpretations of derivatives in a given context. Several components are required to fully and correctly interpret a derivative and students will practice writing those components in a fully formed sentence.

Teaching Tips

One of the most important aspects of the derivative is its ability to describe function behavior at a point. While Units 2 and 3 encouraged students to perceive derivative values as the slope of a tangent line, Unit 4 extends the interpretation of a derivative to include context. Emphasize early that successive derivatives add a layer to the label since each derivative is comparing the prior function to the independent variable.
Have students invent scenarios with humorous or ridiculous units: Perhaps B(c) represents the number of broken legs supported by c crutches. Or M(t) is the number of messages sent by t thumbs. Or A(p) is the minutes of agony created by p calculus problems. Students can write their own equations, find derivatives with correct labels and then interpret their derivatives in context.

Exam Insights

Interpreting a derivative (and later, integrals) in context is a staple of the AP Calculus Test. Students are graded not only on their ability to correctly write a derivative expression and find a numeric value at a given point (time), but also on their ability to communicate the correct interpretation. Free response questions on this topic from past AP Tests are abundant. For recent examples, see the questions and scoring rubrics listed here: 2019 AB 1 and 4; 2018 AB 4; 2017 AB 2; 2016 AB 1, 2, and 5

Student Misconceptions

Most students (physics students particularly) are familiar with common units on acceleration: meters per second squared. Without working through the derivative process, however, few can actually explain why these units are used for acceleration. Work through several examples where position, s(t), is measured in meters and time, t, is measured in seconds. The first derivative is easily understood to be velocity where v(t) = s’(t) and v(t) will be measured in meters per second. The first derivative adds a layer to the label: meters has become meters per second. The second derivative is now called acceleration where a(t) = v’(t) = s”(t) and a(t) will be measured in meters per second per second. The second derivative adds a layer to the label because acceleration describes the change in meters per second (which is velocity) per second. This commonly heard, but commonly misunderstood, unit is meters per second squared.
Have some fun with this idea! Students are often entertained by the higher-order derivatives of position. After acceleration we have jerk, snap, crackle and pop! Have them determine the labels on pop and perhaps even interpret this derivative.

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