# Exponential Growth

## Key Questions

• For graphing an exponential function, basically three things have to be observed. First is the end behavior of the function. One should be able to figure out what happens when x goes to +infy and what happens when x goes to - infy. Next is the point at which the graph would crosss y -axis, that is when x=0. To sketch the general out line, a few points can by worked out by assigning different values to x. which can be plotted.

A function, whose derivative is proportional to itself, is an exponential.

#### Explanation:

If a function has a derivative proportional to itself, and the proportionality factor is real, then the function grows or decays exponentially.

$y$ is an exponential growth function of $x$ if $y = a \cdot {b}^{x}$ for some $a > 0$ and some $b > 1$. This is often written as $y = a \cdot {e}^{k \cdot x}$, where $e = 2.718281828459 \ldots$ and ${e}^{k} = b$ (so $k = \ln \left(b\right) > 0$).

#### Explanation:

This is the definition of what an exponential growth function is.

On the other hand, if $0 < b < 1$, then the function is called an exponential decay function (and $k = \ln \left(b\right) < 0$).

You should graph examples of functions like these on your calculator to see what their graphs look like.

The number $e = 2.718281828459 \ldots$ is a "special" number in mathematics (special like $\pi$ is special). When it's used as the base of an exponential function in calculus, the resulting calculations you can do with it are simpler than they would be if you used some other base.

The function $\ln \left(x\right) = {\log}_{e} \left(x\right)$ is called the "natural logarithm" and the same comments apply to it.