notes/uni/mmme/1026_maths_for_engineering/calculus.md

---
author: Alvie Rahman
date: \today
title: MMME1026 // Calculus
tags: [ uni, nottingham, mechanical, engineering, mmme1026, maths, calculus ]
---

# Calculus of One Variable Functions

## Key Terms

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<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
$f(x)$.

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<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 $g(x) = 0$.

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<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*.


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## Important Functions

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<summary>

### Exponential Functions

$$f(x) = e^x = \exp x$$

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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 $x$.

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<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:
  - $\frac{\mathrm{d}}{\mathrm{d}x} \sinh x = \cosh x$
  - $\frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x$
  - $\frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}$

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<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 $x > 0$.

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<summary>

### Implicit Functions

An implicit function takes the form

$$f(x, y) = 0$$

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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.

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