Calculus / Unit 5: Applications of Derivatives I
5.3 Related Rates Basics
Notes
In a related rates problem, several quantities change with time and are linked by an equation. Differentiate that equation with respect to $t$.
- Steps: draw/describe, write an equation linking the variables, differentiate with respect to $t$ (chain rule), then substitute known values.
- Example: a circle has $A = \pi r^2$, so $\displaystyle \frac{dA}{dt} = 2\pi r \frac{dr}{dt}$.
- Substitute the instant's numbers after differentiating, not before.
Practice
The radius of a circle grows at 2 cm/s. How fast is the area growing when $r = 5$ cm?
$\displaystyle \frac{dA}{dt} = 2\pi r\frac{dr}{dt} = 2\pi(5)(2) = 20\pi$.
A square's side grows at 3 cm/s. How fast is its area growing when the side is 4 cm?
$A = s^2$, so $\displaystyle \frac{dA}{dt} = 2s\frac{ds}{dt} = 2(4)(3) = 24$.
When should you substitute the values that hold at one specific instant?
If you plug in constants first, the changing quantity looks constant and its derivative is lost.
Ready for more? Take the Unit 5 test.