From 42bc5f626d585bbf8f481a6da5045d9f929c7395 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Thu, 2 Dec 2021 16:10:52 +0000 Subject: [PATCH] Add notes on calculus 1a,1b, 2a --- .../calculus.md | 137 +++++++++ .../images/Sinh_cosh_tanh.svg | 265 ++++++++++++++++++ 2 files changed, 402 insertions(+) create mode 100755 mechanical/mmme1026_maths_for_engineering/calculus.md create mode 100644 mechanical/mmme1026_maths_for_engineering/images/Sinh_cosh_tanh.svg diff --git a/mechanical/mmme1026_maths_for_engineering/calculus.md b/mechanical/mmme1026_maths_for_engineering/calculus.md new file mode 100755 index 0000000..ff04acc --- /dev/null +++ b/mechanical/mmme1026_maths_for_engineering/calculus.md @@ -0,0 +1,137 @@ +--- +author: Alvie Rahman +date: \today +title: MMME1026 // Calculus +tags: [ uni, nottingham, mechanical, engineering, mmme1026, maths, calculus ] +--- + +# Calculus of One Variable Functions + +## Key Terms + +
+ + +### Function + +A function is a rule that assigns a **unique** value $f(x)$ to each value $x$ in a given *domain*. + + + +The set of value taken by $f(x)$ when $x$ takes all possible value in the domain is the *range* of +$f(x)$. + +
+ +
+ +### Rational Functions + +A function of the type + +$$ \frac{f(x)}{g(x)} $$ + + + +where $f$ and $g$ are polynomials, is called a rational function. + +Its range has to exclude all those values of $x$ where $g(x) = 0$. + + +
+ +### 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: + +$$f^{-1}(y) = f^{-1}(f(x)) = x$$ + +### 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 $a$. + +#### 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*. + +## Important Functions + +### Exponential Function + +$$f(x) = e^x = \exp x$$ + +
+ +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 $x$. + +### Hyperbolic Functions (sinh and cosh) + +The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defined by: + +$$\sinh x = \frac 1 2 (e^x - e^{-x}) \text{ and } \cosh x = \frac 1 2 (e^x + e^{-x})$$ +$$\tanh = \frac{\sinh x}{\cosh x}$$ + +![[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: + - $\frac{\mathrm{d}}{\mathrm{d}x} \sinh x = \cosh x$ + - $\frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x$ + - $\frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}$ + +## Natural Logarithm + +$$\ln{e^y} = \ln{\exp y} = y$$ + +Since the exponential of any real number is positive, the domain of $\ln$ is $x > 0$. + +## Implicit Functions + +An implicit function takes the form + +$$f(x, y) = 0$$ + +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. diff --git a/mechanical/mmme1026_maths_for_engineering/images/Sinh_cosh_tanh.svg b/mechanical/mmme1026_maths_for_engineering/images/Sinh_cosh_tanh.svg new file mode 100644 index 0000000..8915483 --- /dev/null +++ b/mechanical/mmme1026_maths_for_engineering/images/Sinh_cosh_tanh.svg @@ -0,0 +1,265 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file