$$(5/1000)P(t+h)h < P(t) - P(t+h) < (5/1000)P(t)h$$
$P(t)$ stands for the # of gallons of pollutants in a cistern at time $t$ minutes after a spill.
5 is the rate the cistern fills with gallons of clean water per minute
1000 is the size in gallons of the cistern
I understand that since $P(t)$ is the initial amount of pollutants in the cistern, $P(t+h)$ is the amount of pollutants at a later time and therefore would be less. $P(t)$ - $P(t+h)$ is the actual amount of pollutants at a given time after the spill.
One thing I don't understand is why $h$ is being multiplied at the left and right equations. Also, why it is beneficial to set up an equality like this.
How does one get the derivative $P'(t) = -(5/1000)P(t)$?
P.S.: (This is my first exposure to differential equations/integrals) I may have trouble understanding terminology. Please explain gently so I can follow THANKS!