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}

ex 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 ex is the natural logarithmic function
• Notation for the natural log is loge(x) or ln(x)

## 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 = ex

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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