fix header levels
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@ -95,16 +95,16 @@ If the two limits are equal, we simply refer to the *limit*.
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## Important Functions
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<details>
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<summary>
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### Exponential Function
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$$f(x) = e^x = \exp x$$
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<details>
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<summary>
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It can also be written as an infinite series:
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</summary>
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It can also be written as an infinite series:
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$$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$
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</details>
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@ -116,6 +116,11 @@ The two important limits to know are:
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Note that $e^x > 0$ for all real values of $x$.
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</details>
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<details>
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<summary>
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### Hyperbolic Functions (sinh and cosh)
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The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defined by:
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@ -123,6 +128,8 @@ The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defin
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$$\sinh x = \frac 1 2 (e^x - e^{-x}) \text{ and } \cosh x = \frac 1 2 (e^x + e^{-x})$$
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$$\tanh = \frac{\sinh x}{\cosh x}$$
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</summary>
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![[Fylwind at English Wikipedia, Public domain, via Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Sinh_cosh_tanh.svg)](./images/Sinh_cosh_tanh.svg)
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Some key facts about these functions:
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@ -136,17 +143,33 @@ Some key facts about these functions:
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- $\frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x$
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- $\frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}$
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## Natural Logarithm
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</details>
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<details>
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<summary>
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### Natural Logarithm
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$$\ln{e^y} = \ln{\exp y} = y$$
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</summary>
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Since the exponential of any real number is positive, the domain of $\ln$ is $x > 0$.
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## Implicit Functions
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</details>
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<details>
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<summary>
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### Implicit Functions
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An implicit function takes the form
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$$f(x, y) = 0$$
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</summary>
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To draw the curve of an implicit function you have to rewrite it in the form $y = f(x)$.
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There may be more than one $y$ value for each $x$ value.
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</details>
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