>tfw finally thought I understood what derivative and integral

Once you "get it" then dynamic systems will never look the same, "seeing" calculus is the sci-pill.

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>>8364349

>"seeing"

It's just like those fucking magic eye pictures, isn't it? I could never get those to work REEEEEEE

>>8364337
>derivative functions
A derivative function of f is a function that maps the independent variable to the slopes of the tangent lines on f. An integral function of f is a function that maps the independent variable to the enclosed areas between f and the x axis. What is so confusing about this?

>>8364532

>on an intuitive level

You probably don't know what intuiting is like because you're autistic.

>>8364532

Not OP but I was confused as to why the integral is the area of the graph. I think I see what he means by "intuity" because with simpler operations like adding, the "opposite" would be subtracting. It just seems weird the opposite of a derivative (slope) is the area under the graph.

>>8364337
>PROBABLYANALGEBRAMISTAKE

>>8364535
You can consider an integral to be the limit of sum of areas. So when you integral a function, you're taking the sum of a bunch of areal values of that function, hence, it gives you the area under that curve, right? But then, if you differentiate the integral, you're taking the instantaneous rate of change of the area, which is the original function, hence, the two cancel each other out.

You could also consider the (technically incorrect but still intuitive) argument, that given a function f, and its derivative df/dx, then you integrate it with respect to x, you get integral(df/dx *dx), so you "cancel" the "dx"s (this totally isn't true, mind you, I'm just trying to give you an understanding), you're left with the integral of df, which are tiny bits of the original function f, so if you integrate all those tiny bits of f, you get f.

>>8364533
Autists are the only ones who intuit. NTs can only muster a pale imitation.

>>8364535
If you look up Euler's method, it is pretty clear that making an integral function of a derivative function of f just produces the function f (up to a constant, at least). Additionally, if you look up the fundamental theorem of calculus, it is clear that making a derivative function of an integral function of f just produces f again (as the area of the integral function bounded by f, the x axis, the line x = x_0 and the line x = x_0 + h, where h is a very small positive number, is approximentally f(x_0)*(x_0 + h - x_0) = f(x_0)*h. During differentiation, you take this number, and divide it by (x_0 + h - x_0) = h, so the derivative function of the integral function of f at the point x_0 would just be f(x_0)).

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>>8364337
The first plots the slope of the function while the second plots the area under the function (between the function and 0).

Do you need an ms paint drawing?

>>8364545
Probably anal gebra mistake?

>>8364704
>>8364532
>regurgitating = intuition

>>8364349
>tfw you realize it's all about solutions to DEs
>tfw you realize that it's all linear transformations and algebra
>tfw analysis gets cucked by algebra constantly
why even live

>>8364349
Abstract algebra is the sci pill.
Calculus is just harder elementary algebra

Genius here. You just gotta visualize it

>>8365752
what he said

Read this book for intuitive knowledge.
https://mitpress.mit.edu/sites/default/files/titles/free_download/9780262526548_Art_of_Insight.pdf

>>8365754

I know, the problem is I don't know what to visualize. I can see subtraction and addition as mental images that make sense, but nothing for these.

>>8364337
A derivative is just an equation for slope at a point.

An integral is just an equation for the summation of values over specific portion of a function.

>>8365858

>intuitive

Why are there so many autists on /sci/?

>>8365864
How is that not intuitive?

You can derive calculus yourself from those basic starting points.

>>8365865

You just repeated their literal definition, that's the opposite of giving someonen an intuitive understanding.

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>>8365877
Try drawing it out, the literal definition is incredibly intuitive

>>8365877
But.. I mean.. what else is there for something so simple.

Is there an "intuitive" definition for adding, or does the literal one there suffice?

>>8364337
The derivative of a function at a given point is the local "growth rate" of that function at this given point.

When you have the antiderivative of a given function, by definition, its "local growth rate" is equal to the "height" of the given function. It obviously justify nothing but for me it makes sense that the antiderivative is used to calculate the area under the curve as really, "the growth rate is the height".

It's only my way of seeing it maybe it's just shit.

>>8365903

>Is there an "intuitive" definition for adding

Yes. Adding is iterated incrementing in the same way multiplication is iterated adding and exponentiation is iterated multiplying.

Adding tells you you many times to increment given a starting number.

>>8365910

At the very least you actually understand what intuiting is which puts you ahead of most everyone else in this thread. Cool ideas.

>>8365911
Try doing this for the reals and you're going to quickly lose any idea of "intuitive".

>>8365911
>Adding tells you you many times to increment given a starting number.
>Derivative tells you the slope at a point
>Integral tells you the area under the curve

Spot the hardest definition

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aah, I remember this from high school, here's a diagram I quickly drew up OP, hope its help

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>>8365990

>derivation
>not differentiation

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>>8366041
sorry

integral = area under graph
derivative = slope

>>8368145
t. high schooler enrolled in calc 1

>>8368173
t. uppity undergrad

>>8368145
>have to define coordinate system
>have to define metric
>all for the measly sake of a simple integral

Get your pleb ass out of here.

I'd just like to add that I've always kind of known what the derivative and integral are... but now that I'm in calc 3 I'm starting to understand the actual literal definition of a derivative and what integration actually is (Has nothing to do with the class I am just catching on to this stuff now)

This thread makes me question if most of you are idiots because you are unable to understand this on an intuitive level, or if I'm the idiot, failing to understand the depth of this topic.

>>8368301
>/sci/
>maybe gorilla posters have a point

>>8368301
Honestly, the very concept of "intuitiveness" in math is lost on me. I'm not a mathematician though, so it matters little.

>>8365822
This. Sanjoy Mahajan is one bad mothercalculer.

>>8368396
It just means phrasing something such that a child could understand it.

>>8364532
>undergrad intuition

>>8368756

No, that's not at all what it means.

Intuitive understanding is when you get why something is the way it is and works the way it does as opposed to just knowing the calculation steps or wikipedia definition like most everyone else in this thread has tried giving as an answer.

Non-intuitive knowledge is taking lines of code off of stackoverflow and plugging them into your program without really understanding how they work while intuitive knowledge is where you know what each part of the code does and why and won't be surprised by its behavior when it's used in a different context.

People's varying levels of understanding of pi and e are also good examples of this dichotomy. If your understanding of pi is that it's 3.14159... and your understanding of e is that it's 2.71828... then you don't really have an intuition for them. If on the other hand you can explain them coherently in terms of cyclicism and growth respectively then you're actually intuiting them.

>>8368811
Yes, that's definitely a part of it too. The reason I emphasized explaining it simply is that often people think they know why but they are relying on a deep web of theoretical material they have acquired and only think they understand.

>>8364576

shut up autist

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>>8364576

>Autists... intuit
>Can't figure out how basic social cues and body language work despite being immersed in them on a daily basis

>>8365911
>Adding is iterated incrementing in the same way multiplication is iterated adding
>this circular definition
>intuitive

>>8370185

It's not circular since you already have a working definition of addition and use it all the time. This is the *intuitive* explanation for it that lets you understand what addition really means beyond what you know about it as a memorized calculation, and it's given premised on the knowledge that you already use addition and multiplication but just don't have an intuitive understanding yet.

>>8370199
>This is the *intuitive* explanation for it that lets you understand what addition really means beyond what you know about it as a memorized calculation
I don't see in any way how this "intuitive" explanation gives more insight over the traditional one

>>8370217

It explains where addition comes from in the way everyone learns where multiplication and exponentiation come from. It's also useful to understand addition in terms of iterated incrementing if you ever want to write a programming language and prove it's turing complete by implementing the primitive recursive functions. Doing so requires creating numbers, incrementation, and addition from scratch using recursive functions.

>>8370227
>It explains where addition comes from in the way everyone learns where multiplication and exponentiation come from.
It doesn't explains where it comes from, is just a comparision.

>It's also useful to understand addition in terms of iterated incrementing

yeah if you are just working with integers

>>8370269

>if you are just working with integers

How does iterated incrementing not apply to addition with decimal numbers?

>>8370269
>>8370273

Also that is where it comes from, as in if you implement it from scratch that's how you would do it, by creating a function that iterates the successor (i.e. incrementing) function.

>>8370273
what does it means to iterate a number e or pi times?

>>8370341

If you're talking about *adding* e or pi to a number, you would still use iterated incrementing, only you would work with decimal approximations of e and pi.

>>8366069
Is English not your first language mate? It is "Differentiation" and "Derivative", not "Derivation" or "Differentiative".

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>>8370183
>implying autists aren't the ones who focus on how many breaths the girl they're talking to is making and completely ignore their surroundings and then respond with "y-you too"

>>8364337
>>8364533
intuitive level mathematics is a meme

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>>8370622

>he doesn't intuitively understand his maths

How's that cave treating you?

>>8364337

If f(x) is real valued, then its derivative is also a real valued function but its integral is an R-torsor.