Unit 07 · Lesson 04 · Week 20 · Contextual Applications of Derivatives
Related Rates: Volume and Flow
Related-rates problems connect quantities that change together. An equation describes the geometric or physical relationship; differentiating with respect to time turns that relation into a rate equation. This lesson emphasizes related rates: volume and flow through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in related rates: volume and flow.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do derivatives describe motion, related quantities, sensitivity, and local approximation in authentic contexts?
Visual model
Related-rates problems connect quantities that change together. An equation describes the geometric or physical relationship; differentiating with respect to time turns that relation into a rate equation.
\frac{d}{dt}F(x,y)=0\frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt}
Worked reasoning
Misconception check: Substituting fixed measurements before differentiating can erase the variable whose rate is needed.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Related Rates: Volume and Flow
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
A circle's radius grows at 2 cm/s. Find dA/dt when r=3 cm.
Answer: 12π cm²/s
Reasoning: dA/dt=2πr dr/dt=2π·3·2.
Reasoning: dA/dt=2πr dr/dt=2π·3·2.
A sphere's radius grows at 1 cm/s. Find dV/dt when r=2 cm.
Answer: 16π cm³/s
Reasoning: dV/dt=4πr²dr/dt.
Reasoning: dV/dt=4πr²dr/dt.
Differentiate x²+y²=100 with respect to time.
Answer: 2x dx/dt+2y dy/dt=0
Reasoning: Both coordinates vary with time.
Reasoning: Both coordinates vary with time.
Why substitute measurements after differentiating?
Answer: To preserve the changing variables and their rates.
Reasoning: Early substitution can turn variables into constants.
Reasoning: Early substitution can turn variables into constants.
What role does a diagram play?
Answer: It defines quantities and their relationship.
Reasoning: The rate equation comes from a correct model.
Reasoning: The rate equation comes from a correct model.
State the unit of dV/dt.
Answer: cubic units per unit time
Reasoning: Volume changes over time.
Reasoning: Volume changes over time.