04Cexpegraph

= Exponential Graphs (Base e) =


 * Euler's number, **//e,//** is an irrational number.


 * e ≈ 2.71828


 * e is named after Leonard Euler a Swiss mathematician (18th Century)
 * a more exact value of e can be calculated in several ways (see here) {not part of course}
 * //e// x is a very important function in mathematics and is called the **natural exponential function**.
 * The index laws apply to //e// as with any other base.
 * The opposite function to //e// x is the **natural logarithmic function**.
 * Notation for the natural log is **//log// e (x)** or **//ln//(x)**.

The graph of y = e x follows the same rules as all other exponential graphs.

Asymptote: y = 0

y-intercept: (0, 1)

2nd point: (1, e)

Domain: x Î R

Range: y Î R +

Strictly Increasing Graph

The most interesting thing about the graph of y = e x, is that for every point (x, y), the __gradient__ is the same as the y-coordinate. This can be seen in this activity. The graph of y = e x is the only exponential graph for which this is true.

In notation: math \\ \text{For } y = e^x \qquad. \\ . \\ \Rightarrow \quad \dfrac{dy}{dx}=y \\ \\ \Rightarrow \quad \dfrac{dy}{dx} = e^x math

OR

math \Rightarrow \quad \dfrac{d}{dx} \big( e^x \big) = e^x \qquad. math Go to top of page flat

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