Relative Growth Rate, Using Derivatives

The Law of Natural Growth is useful in determining the rate at which quantities grow or decay at in proportion to their size.

To understand where this formula comes from, take the equation y=f(t) where t is the number of bacteria at time t. If this is true, the you can expect a growth rate proportional to the population at f(t) where $f'\left(t\right)=k\cdot f\left(t\right)$ predicting what will happen to the population over time. This gives us the theorem pictured above.

$$\text{Tell a Joke Day}$$

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Today (August 16) is National Tell a Joke Day!

There are three functions hanging out on a street corner. Diverging functions; the worst of the worst: $3x$, $x^2$, and $e^x$. They're clearly up to no good. Word on the street is that they're not abiding by the PEMDAS law. Terrible stuff.

Anyways, $3x$ spots something in the distance and turns to $x^2$. "Oh no man, look down the street, there's a differential operator cop walking right towards us! If we don't get out of here, she's going to differentiate us down to a constant!"

But $e^x$ isn't worried. "Calm down, I ain't scared of no police. I'm $e^x$, and she's on our turf. I'll take care of this."

So $e^x$ walks right up to the differential operator cop and says, "It looks to me like you might be lost my friend. Let me introduce myself, I'm $e^x$. Now, what in Euler's good name are you doing on my turf? You best turn around before things get messy."

The differential operator is unworried as she looks $e^x$ up and down. She smiles and says, "Hey $e^x$, let me introduce myself. I'm the differential operator . . . with respect to $y$."