---
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 Functions
$$f(x) = e^x = \exp x$$
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:
$$\sinh x = \frac 1 2 (e^x - e^{-x}) \text{ and } \cosh x = \frac 1 2 (e^x + e^{-x})$$
$$\tanh = \frac{\sinh x}{\cosh x}$$
](./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.