Rates of Change

Assume $x(t)$ is a displacement, then $x'(t)$ or $\dfrac{dx}{dt}$ is the instantaneous rate of change in displacement with respect to time, which is velocity.
Example where quantities vary with time, or with respect to some other value.
  • temperature changes
  • height of the surface of water in a pond
  • speed of a car
$\dfrac{dy}{dx}$ gives the rate of change in $y$ with respect to $x$.
We can therefore use the derivative of a function to inform us the rate at which something is happening.

Example 1

The number of people in a colony is approximated by the formula $P(t)=30000+72t^2-t^4$ where $t$ is the time in years after the year 2000.

(a) Find the number of people in the colony at the year 2000.
(b) Find the number of people in the colony at the year 2008
(c) Find the rate at which the number of people in the colony is changing when $t=2$.
(d) Describe the significance of the result of part (c).
(e) Find the rate at which the number of people in the colony is changing when $t=8$.
(f) Describe the significance of the result of part (e).