Back to Courses

1 Scalar Functions & Limits

Scalar Functions

Definition: A function consists of 3 things:
1. A set $X$ called the domain.
2. A set $Y$ called the co-domain.
3. A "rule" $F$ which given any $x\in X$ produces $F(x)\in Y$. We write this as $F: X\rightarrow Y$.
Definition: A scalar function, $F: \mathbb{R}^n\rightarrow\mathbb{R}$ is a function with domain $X\subseteq\mathbb{R}^n$ and co-domain $Y\subseteq\mathbb{R}$.
e.g. $f(x,y)=16-x^2-4y^2$: Domain$=\mathbb{R}^2$, Co-domain$=\{p\in\mathbb{R} | p\leq 16\}$.
Definition: The level curves of $f(x,y)$ are the curves in $\mathbb{R}^2$ of the form $f(x,y)=k$ where $k\in\mathbb{R}$. e.g. $16-x^2-4y^2=k\rightarrow x^2+4y^2=16-k$, which is an ellipse.
Definition: The cross sections of $f(x,y)$ are the curves in $\mathbb{R}^2$ given by $z=f(c,y)$ or $z=f(x,d)$ where $c,d\in\mathbb{R}$.
Tip: If $f(x,y)=g(x)+h(y)$, use cross-sections.

Limits

Definition: The limit, L, of $f(x)$ as $x$ approaches $a$ is defined as:
$$\forall\epsilon >0\text{, }\exists\delta > 0\text{ such that if }0<|x-a|<\delta\text{, then }|f(x)-L|<\epsilon$$
When $f$ is univariate, there are only 2 ways to approach the limit, i.e. from the left and right.
Given $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and $\vec{a}\in\mathbb{R}^2$, let $\vec{x}=(x,y)$ and $\vec{a}=(a,b)$, then the norm of $\vec{x}$ and $\vec{a}$ is $$||\vec{x}-\vec{a}||=\sqrt{(x-a)^2+(y-b)^2}$$ This is the distance between $\vec{x}$ and $\vec{a}$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9862058162689209, "perplexity": 199.1995175294554}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267157574.27/warc/CC-MAIN-20180921225800-20180922010200-00293.warc.gz"}