Why this chapter matters
Calculus is most useful when derivative values answer real questions about speed, flow, growth, and changing geometry.
What you will learn
- Interpret derivative signs and magnitudes in verbal and physical contexts.
- Solve related-rates problems by linking variables and differentiating over time.
- Use position, velocity, and acceleration relationships in motion models.
Lessons in this chapter
- Rates in context and unitsTranslate derivative statements into precise real-world meaning.
- Related rates setupBuild a geometric or physical equation before differentiating with respect to time. Read the full guide →
- Linearization and local approximationUse tangent lines to estimate nearby values and changes.
- Motion along a lineConnect derivatives of position to velocity and acceleration decisions.
Study task
A spherical balloon radius increases at 0.2 cm/s when r = 10 cm. Find the rate of volume change and state units clearly.
Chapter checkpoint
If s(t) = t^3 - 6t^2 + 9t, what is the velocity at t = 1?
v(t) = s'(t) = 3t^2 - 12t + 9, so v(1) = 0.