Properties of the Exponential Function - Maple Help
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Exploring properties of the Exponential function

Main Concept

The graph of the exponential function fx=ⅇx has a very interesting property. If you draw a vertical line (green in the graph to the right) from a point x0,ⅇx0 on the graph down to the point x0,0 on the x-axis, then draw another line 1 unit to the left (red), to the point x01,0, and then finally complete the triangle by drawing the line through x01,0 and x0,ⅇx0 (magenta), this final line will just touch the graph of ⅇx at this latter point without passing through the graph; that is, this line is tangent to the graph of ⅇx at the point x0,ⅇx0. This property (with the base of the triangle having length 1) and the specification that the function has value 1 at x=0 completely and uniquely determine the exponential function ⅇx.


The slope of this (magenta) line tells you how fast the function is growing. So this property of the exponential function can be summarized this way: At every point, how big the exponential function is and how fast it is growing are the same.

Click or drag on the graph to see a demonstration of this property.


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