Unit 4: Contextual Applications of Differentiation
1. Related rates (free preview)
Related rates questions connect how fast one quantity changes to how fast another does, through an equation linking them. The recipe never changes:
Write an equation relating the quantities. Differentiate both sides with respect to time $t$ (every variable picks up a $\frac{d}{dt}$ by the chain rule). Substitute the known values and rates. Solve for the unknown rate. Substitute only after differentiating, never before.
Worked example
A circle's radius grows at $2$ cm/s. How fast is the area growing when $r = 5$?
- Relation: $A = \pi r^2$.
- Differentiate in $t$: $\dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt}$.
- Substitute: $2\pi(5)(2) = 20\pi$ cm²/s.
Common mistake: plugging in the specific value ($r = 5$) before differentiating. That freezes the variable and kills the rate. Differentiate first, substitute last.
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The rest of Unit 4 is in the full course
The exam's most word-problem-heavy unit, fully worked:
- Motion: velocity, acceleration, speed, and direction changes
- The full related-rates playbook: ladders, cones, shadows, angles
- Linearization and tangent-line approximation (with over/under analysis)
- L'Hospital's Rule, including when it silently does not apply
- 45 exam-style questions with verified step-by-step solutions
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