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Summarize Calculus

Anonymous Mon 30 Aug 08:18:38 2021 No.13580134 View Reply Original Report

Quoted By: >>13580308 >>13580354 >>13580616 >>13580839 >>13581969 >>13582267 >>13590333 >>13590493

I want to summarize calculus with this thread.
My calc is rusty and my integration fucking sucks (seriously, what is up with integration being so hard?) so I wanted to make sure I have it down and maybe get some help from you guys in case I missed anything. I know there are other fags here who barely have Algebra and Trig down so I wanted to dump notes here because I know they'll be useful.

Anonymous

Anonymous Mon 30 Aug 2021 08:20:29 No.13580141 Report

Quoted By: >>13580161 >>13580268 >>13580276 >>13582167 >>13590983

The xkcd guy is such a gay midwit

Anonymous

Anonymous Mon 30 Aug 2021 08:23:12 No.13580151 Report

Quoted By:

Oh additional comment before I start. This thread is meant to summarize how to DO calculus so proofs aren't really important here (to utter a mathematical heresy).
Ok now I'll start.

Anonymous

Anonymous Mon 30 Aug 2021 08:26:58 No.13580161 Report

Quoted By:

>>13580141
true but this particular comic is ok in a “funny cause its true xD” kind of way. he’s done much worse

Anonymous

Anonymous Mon 30 Aug 2021 09:10:10 No.13580268 Report

Quoted By: >>13582904

>>13580141

I met the guy once ca. ~2008

IIRC he was majoring in Mathematics

Anonymous

Anonymous Mon 30 Aug 2021 09:12:23 No.13580276 Report

Quoted By: >>13580288

>>13580141
Hes also a simp

Anonymous

Anonymous Mon 30 Aug 2021 09:16:33 No.13580288 Report

Quoted By: >>13586051

>>13580276
QRD?

Anonymous

Anonymous Mon 30 Aug 2021 09:22:52 No.13580308 Report

Quoted By: >>13581972

>>13580134
Integration is an art unto itself. Check out redpenblackpen on YT is my only advice

Anonymous

Anonymous Mon 30 Aug 2021 09:37:30 No.13580354 Report

Quoted By:

>>13580134
Differentiation:
Take two points close to x
Calculate slope (y2-y1)/(x2-x1)

Integration:
Use for loop to calculate trapezoid area

Sets

Sets Mon 30 Aug 2021 10:40:04 No.13580496 Report

Quoted By:

A really quick introduction to sets is usually given so I'll start there too. Without being rigorous this is what you need to know:
A set is a collection of unique things; the things that make up a set are known as it's elements or it's members. The members of a set can be any kind of object (e.g. lines, numbers, variables, etc.). A set is defined when we know what is in the set and what is not in the set.
One way of defining what is in a set is to list all the objects in the set between curly brackets like so: $\{1,2,3\}$ ; sometimes when this is not possible because of the length of the set (or maybe for a more practical reason, if too much space would be taken up listing the entire set), and if the pattern for what elements are apart of the set and which are not is obvious, the non-rigorous notation of ellipses is used so that when we write $\{1,2,3,\dots\}$ it's obvious (though maybe not exactly certain) that what is meant is the set of all natural numbers and when we write $\{2,3,5,7,\dots\}$ it's obvious to every mathematician that what we are listing is the set of all prime numbers.
Because the members of a set must be unique, things like repetition are not recognized; so for a collection like $\{1,1,2,2,3\}$ the set $\{1,2,3\}$ contains all the items of said collection. Like any other mathematical object sets can be represented by a variable and by convention the variable is usually a capital letter (e.g. $A, B, C, S, X, \textrm{etc.}$ ). To say that $2$ is a member of the set $\{1,2,3\}$ we write $2\in\{1,2,3\}$ ; likewise to say that x is a member of the set S we write $x\in S$ .

Sets

Sets Mon 30 Aug 2021 10:41:37 No.13580498 Report

Quoted By:

btw feel free to add anything, just remember the topic is calculus not sets
$\textrm{Equality of Sets}$
Two set are said to be equal if they share all their members. To represent equality we write $A=B$ , and because of the definition we know this means that every member of the set A is a member of the set B and vice versa.
Because of this definition of equality order does not matter in defining a set because if we take a set and reorder some of it's members we would still have the same members and no new ones and this means that a statement like $\{1,2,3\}=\{2,3,1\}$ is valid.
$\textrm{Subsets and Supersets}$
A subset is a set contained in another set, that is to say, if A is a subset of B then all the elements of A are members of B; a superset is set which contains another set (e.g. B in the previous example).
For example, $\{1,2,3\}$ is a set that contains the set $\{1\}$ and the set $\{1,2\}$ and even the set $\{1,2,3\}$ itself--a subset which is specified to not be equal to it's superset is called a proper or strict subset.
To denote that A is a subset of B we write $A\subset B$ and to denote that B is a superset of A we write $B\supset A$ .
It should be noted that the usages of these symbols are not uniform and some people use $\subset$ to denote a subset while others use it to specifically denote a proper subset (likewise for $\supset$ ); the symbol $\subseteq$ exist to emphasize that the subset can be equal to the equal superset while the symbol $\subsetneq$ exists to specifically to represent strict subsets (likewise for $\supset$ ).

Sets

Sets Mon 30 Aug 2021 10:42:38 No.13580499 Report

Quoted By:

$\textrm{Intersections}$
Given two sets, the intersection set is the set of elements that exist in both sets. For example, the intersection of the set $\{f,a,g,g,o,t\}$ and $\{o,p\}$ is the set $\{o\}$ because the element o is common to both sets. To denote the intersection of A and B we write $A\cap B$ .
$\textrm{Union}$
Given two sets, the union set is the set that contains all the elements of A and all the elements of B and only those elements. For example, for the sets {1,2,3} and {2,4,6} we have the union set $\{1,2,3,4,6\}$ . To denote the union of A and B we write $A\cup B$ .

Anonymous

Anonymous Mon 30 Aug 2021 11:40:38 No.13580616 Report

Quoted By:

>>13580134
there should be a grad school version of this comic where the integration side is "function is measurable so I can integrate it" and the differentiation side is all these crazy weak derivatives, sobolev embedding tricks or difference quotient estimates and regularity theory in order to be able to differentiate a function

Sets

Sets Mon 30 Aug 2021 13:06:19 No.13580817 Report

Quoted By:

$\mathbb{N}$ is the set of natural numbers. These are the set of positive whole numbers (e.g. $1,2,3,4,\dots$ ) which sometimes includes and sometimes doesn't include the number $0$ .
$\mathbb{Z}$ is the set of integers. This is the set of whole numbers, positive and negative, and it always includes $0$ .
$\mathbb{Q}$ is the set of rational numbers. This is the set of numbers that can be represented as a ratio of integers (i.e. all numbers $\frac{p}{q}, \textrm{where }p,q\in\mathbb{Z}\textrm{ and }q\neq0$ ).
$\mathbb{R}$ is the set of real numbers. This is the set of numbers that can be represented via an infinite decimal expansion (e.g. $0.\overline{99},2.\overline{00},0.123456789101112\dots$ ).
$\mathbb{I}$ is the set of imaginary numbers. This is the set of numbers which are multiples of $i$ (where $i=\sqrt{-1}$ ).
$\mathbb{C}$ is the set of complex numbers. This is the set of numbers which can be represented as the sum of some real number and a multiple of $i$ (that is, a number that can be represented in the form $a+bi$ ).

Anonymous

Anonymous Mon 30 Aug 2021 13:15:36 No.13580839 Report

Quoted By:

>>13580134
Just practice dude? The only way to get good at calculus. I banged out piles of written exercises and notes. You can check pauls online math notes

Intervals

Intervals Mon 30 Aug 2021 14:00:08 No.13580970 Report

Quoted By:

An interval is a set that contains all real numbers between two given numbers. It can be thought of then as a line segment of the real number line whose endpoints are the two numbers given. To define an interval we need to first define whether either endpoint is included in the interval.
If we had an interval between $0$ and $10$ for example, we might choose to include $0$ but not $10$ and the set would look like $\{0,\frac{1}{8},\frac{1}{4},\frac{1}{2},1,2,\dots,e,\dots,3,\pi,4,5,6,7,8,9,9.5,9.9,9.99,9.999,9.9999,9.999989,\dots\}$ .
If we have an interval from $a$ to $b$ these are the possibilities.
>if both a and b are included then the interval is called closed and is denoted $[a,b]$ .
>if both a and b are not included then the interval is called open and is denoted $(a,b)$ .
>if a is included but b is not then the interval is called half-closed or half-open and is denoted $[a,b)$ .
>if a is not included but b is then the interval is called half-closed or half-open and is denoted $(a,b]$ .
The aforementioned possibilities assumed that both a and b where finite quantities; suppose one of the endpoints was $\pm\infty$ then it is always denoted as an open endpoint (i.e. $\pm\infty$ is never included).
By geometric analogy, an interval of the form $(-\infty, b]$ or $(-\infty, b)$ or $(a, \infty)$ or $[a, \infty)$ is called a ray.

Intervals

Intervals Mon 30 Aug 2021 14:06:23 No.13580988 Report

Quoted By: >>13581010

Nonstandard notation sometimes uses
> ] for (
and
> [ for )
The following forms are used in the following cases then
> (a,b)
]a,b[
> [a,b)
[a,b[
> (a,b]
]a,b]

Anonymous

Anonymous Mon 30 Aug 2021 14:12:17 No.13581010 Report

Quoted By: >>13582044

>>13580988
It's standard.

Sequences

Sequences Mon 30 Aug 2021 18:28:45 No.13581941 Report

Quoted By:

A sequence is a collection of mathematical objects which allows repetitions and where order matters. It, like a set, is usually denoted as a list of elements enclosed in either curly brackets or, unlike sets, in parentheses.
Therefor the sequence $(1,2,3)$ is neither the same as $(1,2,2,3)$ nor $(3,2,1)$ .
Because order matters so does position, and the position of an element in a sequence is called it's index or rank; indexes are numbered sequentially with natural numbers, the first is either the 0th or 1st index depending on convention, but usually the 1st. The nth index of a sequence is represented with a subscript $n$ like so $a_n$ , where a represents some element of the sequence.
Besides listing it's elements we can define a sequence like so: $(a_n) = f(n)$
or likewise: $(f(n))_{n=a}^{b}$ where a is the upperbound of the sequence and b is the lower bond of the sequence.

Anonymous

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Anonymous Mon 30 Aug 2021 18:34:38 No.13581969 Report

Quoted By:

>>13580134
>I want to summarize calculus with this thread.
total thing on outside equals total change on inside

Anonymous

Anonymous Mon 30 Aug 2021 18:34:56 No.13581972 Report

Quoted By:

>>13580308
This guy’s channel is really nice. I watched a bunch of his videos while taking cal 2. He’s also got this natural goofiness that’s highly entertaining.

Anonymous

Anonymous Mon 30 Aug 2021 18:50:02 No.13582044 Report

Quoted By:

>>13581010
Show me a math paper written in the past 10 years that uses this notation.

Anonymous

Anonymous Mon 30 Aug 2021 19:18:00 No.13582167 Report

Quoted By: >>13582175 >>13582226

>>13580141
well he's not wrong, integration is much more computationally complex than differentiation and you need to git gud at applying a lot more different heuristics to be reasonably proficient at doing it by hand, unaided.

Anonymous

Anonymous Mon 30 Aug 2021 19:19:39 No.13582175 Report

Quoted By: >>13582370

>>13582167
only a vanishingly small number of integrals are exactly solvable in closed form. it's a lot easier to numerically compute integrals than derivatives.

Anonymous

Anonymous Mon 30 Aug 2021 19:36:19 No.13582226 Report

Quoted By:

>>13582167
Of course it is, the class of integrable functions is far, far larger and less nicer than that of differentiable functions. For starters, every monotone function is integrable.

Anonymous

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Anonymous Mon 30 Aug 2021 19:46:15 No.13582267 Report

Quoted By:

>>13580134
x = np.linspace(a, b, 10**6)
f(x).mean() / (b - a)

Simple as.

Anonymous

Anonymous Mon 30 Aug 2021 20:13:44 No.13582370 Report

Quoted By: >>13583035 >>13583102

>>13582175
What's so hard about computing (f(x+h)-f(x))/h for small h?

Anonymous

Anonymous Mon 30 Aug 2021 22:30:53 No.13582904 Report

Quoted By:

>>13580268
>IIRC he was majoring in Mathematics
So hes a midwit

Anonymous

Anonymous Mon 30 Aug 2021 23:00:47 No.13583035 Report

Quoted By: >>13583042

>>13582370
it's really easy to right numerically unstable ode and pde solvers

Anonymous

Anonymous Mon 30 Aug 2021 23:01:59 No.13583042 Report

Quoted By:

>>13583035
That's numeric integration not differentiation.

Anonymous

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Anonymous Mon 30 Aug 2021 23:18:29 No.13583102 Report

Quoted By:

>>13582370
Integration is continuous for the uniform norm and differentiation isn't.
This is important for entirely practical reasons: when you measure something with a ruler or a thermometer you usually have an uniform bound on the error.

Anonymous

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Anonymous Tue 31 Aug 2021 12:15:09 No.13585289 Report

Quoted By:

bump

Anonymous

Anonymous Tue 31 Aug 2021 12:24:18 No.13585314 Report

Quoted By:

The difference between differentiation and integration is the same as the difference between P and NP.
Which is the same as the difference between checking that a solution to a sudoku is correct and finding a solution.

Anonymous

Anonymous Tue 31 Aug 2021 16:07:09 No.13586051 Report

Quoted By:

>>13580288
he has a gf

Anonymous

Anonymous Wed 01 Sep 2021 02:03:51 No.13588698 Report

Quoted By:

Fuck

Anonymous

Anonymous Wed 01 Sep 2021 10:18:28 No.13590333 Report

Quoted By: >>13590418

>>13580134
>>13580134
Do I need this to be able to do things with numerical methods?

Anonymous

Anonymous Wed 01 Sep 2021 10:41:25 No.13590418 Report

Quoted By:

>>13590333
Need to, no. You can brute force most integrals numerically. You may be able to optimize by substitution or i.b.p but typically its not required.

Anonymous

Anonymous Wed 01 Sep 2021 10:59:31 No.13590470 Report

Quoted By: >>13590496

>numerical methods
>indefinite integrals block your path

Anonymous

Anonymous Wed 01 Sep 2021 11:06:38 No.13590493 Report

Quoted By:

>>13580134
Limits of error

Anonymous

Anonymous Wed 01 Sep 2021 11:07:49 No.13590496 Report

Quoted By:

>>13590470
>numerical methods
>indefinite integrals
no problem here

Anonymous

Anonymous Wed 01 Sep 2021 14:28:09 No.13590983 Report

Quoted By:

>>13580141
fpbp
I wish this shit would stay on reddit where it belongs

Anonymous

Anonymous Thu 02 Sep 2021 10:24:58 No.13594355 Report

Quoted By:

its just probabilities under a graph

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