Exponents

A fundamental concept in Calculus.

When dealing with exponents, smart people defined the number $e$.

Definition of e

The slope of the function $e^t$ at any point $t$ is $e$.

Consider the function $a^t$. An alternative (and often preferable) way of writing that is $e^{ct}$, where $c$ is $ln(a)$.

Formula for Exponents:

Formula:

$P(t)=P_0{e}^{kt}$
Where:

• $P(t)$ is the value at time $t$,
• $P_0$ is the initial starting value,
• $k$ is the growth constant.

If the growth constant is $ln(a)$, then the function is multiplying by $a$ each iteration. For example, if it’s $ln(2)$, the function is constantly doubling.

Also a note:

Note

When a variable’s Rate of Change is directly proportional to itself, the variable is an exponential.