Euler’s number, e, has a range of applications and is used primarily in exponential functions. The graph of y=e^{x} has a gradient equal to the function, and is therefore **the** exponential function.

## Exponential functions

Any function in the form f(x)=a^{x} are known as exponential functions. They all pass through the point (0,1) as any number to the power of 0 will be 1. Each of these graphs has a gradient function, a way to find the gradient at any point along the line – it will be proportional to the function itself. E.g:

You can see from the table that a value of a between 2 and 3 will create a gradient function equal to the function itself. This value is an irrational number which is approximately 2.718 and called e. When the graph of y=e^{x }is differentiated or integrated, it keeps the same value.

The graph of y=e^{x } looks like:

## Uses of exponentials

The graph of y=e^{x } can be used in a range of applications, the most common being modelling population growth.

#### Capacitors

It can also be used to describe the charge and discharge of capacitors. The potential difference across the capacitor during discharge can be described using the equation:

where** V** is the pd at time** t**, **V _{o}** is the initial pd for a capacitor with capacitance

**C**in a circuit with resistance

**R**. As it is e to the power of a negative, the graph have its maximum value at the lowest value of t.

#### Radioactive decay

Radioactive nuclei have a property known as half life, the time taken for the activity of the sample to decrease by half. This remains constant throughout decay and leads to an exponential pattern of decay occurring. The number of undecayed nuclei remaining in the sample can be calculated using the equation:

This is where **N** is the remaining nuclei at time** t**, **N _{0}** is the initial number of nuclei and

**λ**is the decay constant of the sample.