Lesson 3: 4-3 Rates of Change in Applied Contexts Other Than Motion
Duration of Days: 1
Lesson Objective
Apply the concept of a derivative to model rates of change in biological systems (e.g., population growth, bacteria spread).
Use derivatives to calculate marginal cost, marginal revenue, and marginal profit in economic contexts.
Analyze volumetric flow rates (e.g., water entering or leaving a container).
Formulate and solve equations involving temperature change (Newton’s Law of Cooling contexts) and other physical phenomena.
Interpret the first and second derivatives to describe the rate at which a rate is changing (e.g., the rate of growth of a population is slowing down).
How does "marginal cost" relate to the slope of a total cost function?
If a rate of change is positive but the second derivative is negative, what does that imply about the future state of the system?
How can we distinguish between the amount of a substance and the rate at which that substance is being added?
Why is the derivative an essential tool for predicting trends in business and medicine?
Marginal Cost / Revenue / Profit
Population Dynamics
Volumetric Flow Rate
Rate of Change of a Rate
Amount vs. Rate
Net Change
Input/Output Rates
Proportionality Constant
Topic 4.3: Rates of Change in Applied Contexts Other Than Motion.
Standard CHA-3.C: Solve problems involving rates of change in applied contexts.
Mathematical Practice 1.A: Identify the question to be answered or problem to be solved.
Description:
This lesson takes the skills from 4.1 (interpreting) and 4.2 (calculating) and applies them to diverse scenarios. Students often work with "In/Out" problems, such as water being pumped into a tank at a rate, while leaking out at a rate.
Purpose: The AP Exam rarely limits calculus to motion. Students must be comfortable seeing "prime" notation applied to variables like P(Population), T (Temperature), or C (Cost). This section builds the flexibility needed for the "Variety of Contexts" FRQs.
DOK 1 (Recall): Differentiating a given cost function to find the marginal cost at a specific production level.
DOK 2 (Skill/Concept): Identifying whether the amount of a substance in a system is increasing or decreasing by comparing "Rate In" vs. "Rate Out."
DOK 3 (Strategic Thinking): Explaining the implications of a second derivative in a business context (e.g., "The profit is increasing, but at a decreasing rate").
Scaffolding (Support):Contextual Variables Table: Provide a reference sheet that maps standard variables to their context.
The "Net Change" Formula: Use a visual graphic showing: Net Rate = Rate IN - Rate OUT.
Extension (Challenge):Optimization Preview: Challenge students to find the production level where Marginal Revenue equals Marginal Cost and explain why that might be important for a business.
Linked Contexts: Provide a problem where the rate of change of one variable depends on another non-time variable (e.g., the rate of change of a medication's effectiveness relative to the dosage).
College Board AP Classroom Assessments