02.42expegraph

# Exponential Graph (Base e)

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

… … $e \approx 2.71828$

**e**is named after Leonhard 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 inverse function to e
^{x}is the natural logarithmic function- Notation for the natural log is
**log**or_{e}(x)**ln(x)**

- Notation for the natural log is

## Exponential Graph (base e)

The graph of $y = e^x$ has the same shape as all other exponential graphs.

- Asymptote: $y = 0$

- y-intercept: $(0,\; 1)$

- 2nd point: $(1, \; e)$

- Domain: $x \in R$

- Range: $y \in R^+$

- Strictly Increasing Graph

## Gradient of y = e^{x}

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.

- The graph of $y = e^x$ is the only exponential graph for which this is true.

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In notation:

… … For $y = e^x$

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… … $\dfrac{dy}{dx}=y$

… … $\dfrac{dy}{dx} = e^x$

OR

… … $\dfrac{d}{dx} \big( e^x \big) = e^x$

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