11 Exponential functions
Exponential functions crop up in applied mathematics everywhere. This section looks at these important functions, so important that, Professor Albert Bartlett said the following about them in this lecture Arithmetic, Population and Energy.
The greatest shortcoming of the human race is our inability to understand the exponential function.
11.1 Getting to know exponential functions
An exponential function comes in the for, \(y=a^x\). They can increase incredibly fast. Take for example \(y=2^x\)
| \(x\) | \(y=2^x\) |
|---|---|
| \(-2\) | \(y = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}\) |
| \(-1\) | \(y = 2^{-1} = \frac{1}{2^1} = \frac{1}{2}\) |
| \(0\) | \(y = 2^{0} = 1\) |
| \(1\) | \(y = 2^{1} = 2\) |
| \(2\) | \(y = 2^{2} = 4\) |
| \(3\) | \(y = 2^{3} = 8\) |
| \(4\) | \(y = 2^{4} = 16\) |
Plotting these points give a graph that looks like:
Notice the following key points about the graph.
- The graph quickly increases.
- It crosses the \(y\) axis at \(1\) (all exponential graphs do this).
- It never goes under the \(x\) axis.
11.2 The exponential function
There is one exponential function that is so important that it is called the exponential function. It is written as \(y=e^x\) where \(e\) is an irrational number (an infinitely long decimal number that doesn’t repeat itself, \(/pi\) is an irrational number too). The value of \(e\) is:
\[ e = 2.71828182845904523536028747135266249775724709369995... \] ish.
The reason why it is special is that when \(y=e^x\), the derivative is itself, that is \(\frac{dy}{dx}=e^x\). Below is a graph of \(y=a^x\) (solid red line) and it derivative (dashed blue line), you can open it up and change the value of \(a\) from \(2\) to \(4\). \(a\) is set to \(2\) to begin with, notice how the derivative is beneath the curve \(y=a^x\). When \(a\) is increased the derivative moves above \(y=a^x\). The point where the two curves overlap is when \(a=e\).
If \(y=e^x\) then \(\frac{dy}{dx}=e^x\).
11.3 Differentiating \(e^x\)
The rule for differentiating \(e^x\) is if \(y = ke^{ax}\) then \(\frac{dy}{dx}= ake^{ax}\).
Use that rule to try the following questions.