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fix header levels

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Akbar Rahman 2021-12-02 16:36:15 +00:00
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mechanical/mmme1026_maths_for_engineering/calculus.md
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@ -95,16 +95,16 @@ If the two limits are equal, we simply refer to the *limit*.
## Important Functions ## Important Functions
<details>
<summary>
### Exponential Function ### Exponential Function
$$f(x) = e^x = \exp x$$ $$f(x) = e^x = \exp x$$
<details>
<summary>
It can also be written as an infinite series:
</summary> </summary>
It can also be written as an infinite series:
$$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$ $$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$
</details> </details>
@ -116,6 +116,11 @@ The two important limits to know are:
Note that $e^x > 0$ for all real values of $x$. Note that $e^x > 0$ for all real values of $x$.
</details>
<details>
<summary>
### Hyperbolic Functions (sinh and cosh) ### Hyperbolic Functions (sinh and cosh)
The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defined by: The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defined by:
@ -123,6 +128,8 @@ The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defin
$$\sinh x = \frac 1 2 (e^x - e^{-x}) \text{ and } \cosh x = \frac 1 2 (e^x + e^{-x})$$ $$\sinh x = \frac 1 2 (e^x - e^{-x}) \text{ and } \cosh x = \frac 1 2 (e^x + e^{-x})$$
$$\tanh = \frac{\sinh x}{\cosh x}$$ $$\tanh = \frac{\sinh x}{\cosh x}$$
</summary>
![[Fylwind at English Wikipedia, Public domain, via Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Sinh_cosh_tanh.svg)](./images/Sinh_cosh_tanh.svg) ![[Fylwind at English Wikipedia, Public domain, via Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Sinh_cosh_tanh.svg)](./images/Sinh_cosh_tanh.svg)
Some key facts about these functions: Some key facts about these functions:
@ -136,17 +143,33 @@ Some key facts about these functions:
- $\frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x$ - $\frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x$
- $\frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}$ - $\frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}$
## Natural Logarithm </details>
<details>
<summary>
### Natural Logarithm
$$\ln{e^y} = \ln{\exp y} = y$$ $$\ln{e^y} = \ln{\exp y} = y$$
</summary>
Since the exponential of any real number is positive, the domain of $\ln$ is $x > 0$. Since the exponential of any real number is positive, the domain of $\ln$ is $x > 0$.
## Implicit Functions </details>
<details>
<summary>
### Implicit Functions
An implicit function takes the form An implicit function takes the form
$$f(x, y) = 0$$ $$f(x, y) = 0$$
</summary>
To draw the curve of an implicit function you have to rewrite it in the form $y = f(x)$. To draw the curve of an implicit function you have to rewrite it in the form $y = f(x)$.
There may be more than one $y$ value for each $x$ value. There may be more than one $y$ value for each $x$ value.
</details>