AP Calculus AB / Contextual Applications of Differentiation
Lesson
Related rates as a script
Name the equation, differentiate with respect to time, then plug in the instant.
Learning goals
- Set up a related-rates equation from a geometric relation.
- Interpret a derivative with units in a motion or filling story.
Explanation
Related rates problems are chain-rule problems wearing a story. If a radius r and a volume V are tied by a formula, then dV/dt = (dV/dr)(dr/dt). Draw the picture, write the linking equation, mark what is constant, and differentiate with respect to t before you insert the snapshot numbers.
Motion is the cleanest context: velocity is ds/dt, acceleration is dv/dt. A positive velocity with a negative acceleration means the object is still moving in the positive direction but slowing down. Signs are direction, not moral character.
Key terms
- Related rates. A problem in which two or more changing quantities are linked, so their time derivatives are linked.
- Velocity. The derivative of position with respect to time; a signed rate.
Common mistakes
- Plugging in the snapshot radius before differentiating.
- Treating a constant height as if it were changing, or the reverse.
Practice
Original Marlow Works items. Check the answer explanation after you try.
What is lim (x→3) of (x² − 9) / (x − 3)?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
A ladder 10 ft long slides away from a wall. When you write x² + y² = 100, when do you substitute the 6-ft height?
Answer. After differentiating both sides with respect to t, at the instant described.
x and y are functions of time. Substituting too early freezes a variable that is still changing.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source