Have you been following my lessons in order?
If so, then you've heard me say (many times):
The special property of the natural exponential function discussed in this section will give you new appreciation for this fact!
The $y$-value of a Point
on the Graph of the Natural Exponential Function
tells you the Slope of the Tangent Line
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Usually, calculus is needed to find slopes of tangent lines.
But... |
 For the curve $\,y = {\text{e}}^x\,$: $y$-value of point $=$ slope of tangent line |
The sketches below show just how simple things are for the natural exponential function!
If you know the $\,y\,$-value of a point, then you know the slope of the tangent line at that point.
If you know the slope of the tangent line at a point, then you know the $\,y\,$-value of the point.
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At this point, the $\,y\,$-values are changing $\,\color{red}{5}\,$ times faster than the $\,x\,$-values. |
At this point, the $\,y\,$-values are changing $\,\color{red}{10}\,$ times faster than the $\,x\,$-values. |
At this point, the $\,y\,$-values are changing $\,\color{red}{20}\,$ times faster than the $\,x\,$-values. |
At this point, the $\,y\,$-values are changing $\,\color{red}{30}\,$ times faster than the $\,x\,$-values. |
| If $\,x\,$ changes by $\,0.1\,,$ we expect $\,y\,$ to change by about $\,5(0.1) = 0.5\,.$ |
If $\,x\,$ changes by $\,0.1\,,$ we expect $\,y\,$ to change by about $\,10(0.1) = 1\,.$ |
If $\,x\,$ changes by $\,0.1\,,$ we expect $\,y\,$ to change by about $\,20(0.1) = 2\,.$ |
If $\,x\,$ changes by $\,0.1\,,$ we expect $\,y\,$ to change by about $\,30(0.1) = 3\,.$ |
Think about this.
When (say) $\,x = 10\,,$ then $\,y = {\text{e}}^{10} \approx 22{,}026\,.$
At this point, the slope of the tangent line is about $\,22{,}026\,.$
At this point, the
instantaneous rate of change is about $\,22{,}026\,.$
At this point, the outputs are changing about $\,22{,}026\,$ times as fast as the inputs!
Wow!
And this is only when $\,x\,$ is $\,10\,$!
Exponential functions get big fast!
For Those Who Know Some Calculus and Differential Equations
The derivative of $\,f(x) = {\text{e}}^x\,$ is $\,f'(x) = {\text{e}}^x\,.$
More generally, the derivative of $\,f(x) = K{\text{e}}^x\,$ is $\,f'(x) = K{\text{e}}^x\,.$
Notice that the derivative of the function is the same as the original function.
The $\,y\,$-values of points (given by the original function) and the slopes of tangent lines (given by the derivative) are the same.
We could ask:
What functions are the same as their derivatives?
Equivalently, we could ask: What are the solutions of the differential equation $\,y = y\,'\,$?
Head up to WolframAlpha and type in:
solve y = y'
The family of solutions is $\,y = K{\text{e}}^x\,$ (one for each real number $\,K\,$).
are the ONLY functions
where finding slopes of tangent lines
is as easy as
reading off the $\,y\,$-values of points!
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
simple versus compound interest