Contextual Applications of Differentiation
Core ideas
This domain translates derivatives into rates. Interpret $\frac{ds}{dt}$ as velocity and $\frac{dv}{dt}$ as acceleration, with sign indicating direction. Related rates problems connect quantities whose rates are linked by an equation; differentiate that equation with respect to time and substitute known values at the instant of interest.
Motion analysis uses position, velocity, and acceleration together: speeding up when velocity and acceleration share a sign, slowing down when they differ. Straight-line motion questions often ask for when a particle changes direction (velocity zero and sign change) or is at rest. Linear approximation and differentials estimate changes: $dy\approx f'(x),dx$.
Traps
- In related rates, differentiating before substituting values; substitute only after differentiating, and distinguish constants from variables.
- Confusing speed (magnitude) with velocity (signed).
- Forgetting units or that a rate is negative when a quantity decreases.
- Assuming a particle reverses direction at every zero of velocity; a sign change is required.
Worked example
A balloon's radius grows at $2$ cm/s. Volume $V=\frac{4}{3}\pi r^3$. Then $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$. When $r=5$, $\frac{dV}{dt}=4\pi(25)(2)=200\pi$ cm³/s. The key is differentiating the relation first, then substituting the instantaneous radius.