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uni/mmme/1026_maths_for_engineering/calculus.md

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+---
+author: Alvie Rahman
+date: \today
+title: MMME1026 // Calculus
+tags: [ uni, nottingham, mechanical, engineering, mmme1026, maths, calculus ]
+---
+
+# Calculus of One Variable Functions
+
+## Key Terms
+
+<details>
+<summary>
+
+### Function
+
+A function is a rule that assigns a **unique** value $f(x)$ to each value $x$ in a given *domain*.
+
+</summary>
+
+The set of value taken by $f(x)$ when $x$ takes all possible value in the domain is the *range* of
+$f(x)$.
+
+</details>
+
+<details>
+<summary>
+
+### Rational Functions
+
+A function of the type
+
+$$ \frac{f(x)}{g(x)} $$
+
+</summary>
+
+where $f$ and $g$ are polynomials, is called a rational function.
+
+Its range has to exclude all those values of $x$ where $g(x) = 0$.
+
+</details>
+
+<details>
+<summary>
+
+### Inverse Functions
+
+Consider the function $f(x) = y$.
+If $f$ is such that for each $y$ in the range there is exactly one $x$ in the domain,
+we can define the inverse $f^{-1}$ as:
+
+$$f^{-1}(y) = f^{-1}(f(x)) = x$$
+
+</summary>
+</details>
+
+<details>
+<summary>
+
+### Limits
+
+Consider the following:
+
+$$f(x) = \frac{\sin x}{x}$$
+
+The value of the function can be easily calculated when $x \neq 0$, but when $x=0$, we get the
+expression $\frac{\sin 0 }{0}$.
+However, when we evaluate $f(x)$ for values that approach 0, those values of $f(x)$ approach 1.
+
+This suggests defining the limit of a function
+
+$$\lim_{x \rightarrow a} f(x)$$
+
+to be the limiting value, if it exists, of $f(x)$ as $x$ gets approaches $a$.
+
+</summary>
+
+#### Limits from Above and Below
+
+Sometimes approaching 0 with small positive values of $x$ gives you a different limit from
+approaching with small negative values of $x$.
+
+The limit you get from approaching 0 with positive values is known as the limit from above:
+
+$$\lim_{x \rightarrow a^+} f(x)$$
+
+and with negative values is known as the limit from below:
+
+$$\lim_{x \rightarrow a^-} f(x)$$
+
+If the two limits are equal, we simply refer to the *limit*.
+
+
+</details>
+
+## Important Functions
+
+<details>
+<summary>
+
+### Exponential Functions
+
+$$f(x) = e^x = \exp x$$
+
+</summary>
+
+It can also be written as an infinite series:
+
+$$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$
+
+The two important limits to know are:
+
+- as $x \rightarrow + \infty$, $\exp x \rightarrow +\infty$ ($e^x \rightarrow +\infty$)
+- as $x \rightarrow -\infty$, $\exp x \rightarrow 0$ ($e^x \rightarrow 0$)
+
+Note that $e^x > 0$ for all real values of $x$.
+
+</details>
+
+<details>
+<summary>
+
+### Hyperbolic Functions (sinh and cosh)
+
+The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defined by:
+
+$$\sinh x = \frac 1 2 (e^x - e^{-x}) \text{ and } \cosh x = \frac 1 2 (e^x + e^{-x})$$
+$$\tanh = \frac{\sinh x}{\cosh x}$$
+
+</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)
+
+Some key facts about these functions:
+
+- $\cosh$ has even symmetry and $\sinh$ and $\tanh$ have odd symmetry
+- as $x \rightarrow + \infty$, $\cosh x \rightarrow +\infty$ and $\sinh x \rightarrow +\infty$
+- $\cosh^2x - \sinh^2x = 1$
+- $\tanh$'s limits are -1 and +1
+- Derivatives:
+  - $\frac{\mathrm{d}}{\mathrm{d}x} \sinh x = \cosh x$
+  - $\frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x$
+  - $\frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}$
+
+</details>
+
+<details>
+<summary>
+
+### Natural Logarithm
+
+$$\ln{e^y} = \ln{\exp y} = y$$
+
+</summary>
+
+Since the exponential of any real number is positive, the domain of $\ln$ is $x > 0$.
+
+</details>
+
+<details>
+<summary>
+
+### Implicit Functions
+
+An implicit function takes the form
+
+$$f(x, y) = 0$$
+
+</summary>
+
+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.
+
+</details>