AP Calculus BC / Unit 4: Contextual Applications of Differentiation
4.4 Introduction to Related Rates
Notes
When two quantities are linked by an equation and both change over time, differentiate the equation with respect to $t$ to link their rates.
- Every variable that changes gets a $\displaystyle \frac{d}{dt}$.
- Example: $A=\pi r^2$ gives $\displaystyle \frac{dA}{dt}=2\pi r\frac{dr}{dt}$.
Practice
If $V=s^3$ for a cube, then $\displaystyle \frac{dV}{dt}=$
Chain rule with respect to $t$.
For $x^2+y^2=z^2$, differentiating with respect to $t$ gives:
Each variable gets its own rate.
When should you plug in specific values in a related rates problem?
Plugging in early turns changing quantities into constants.
Ready for more? Take the Unit 4 test.