Unit 07 · Lesson 03 · Week 19 · Contextual Applications of Derivatives
Related Rates: Geometry and Measurement
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: geometry and measurement through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in related rates: geometry and measurement.
- 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: Geometry and Measurement
Student Worksheet · 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.
A sphere's radius grows at 1 cm/s. Find dV/dt when r=2 cm.
Differentiate x²+y²=100 with respect to time.
Why substitute measurements after differentiating?
What role does a diagram play?
State the unit of dV/dt.
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.