ALL OF Calculus 2 in 5 minutes
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- čas přidán 6. 12. 2022
- I unfortunately could not finish the whole thing, please forgive me...
However, I may return on this project in the future someday... Wish me luck, I shall try to be a TA next semester :)
Also, this video is not done yet. Expect a re-upload soon
I kinda feel guilty for not finishing in time :(
After all, I put in so much work, only for it not to finish..... That kinda hurts.
Made by Mr-FuzzyPenguin
github.com/Mr-FuzzyPenguin/AM...
Credits:
Thank you to Hillary C. for giving me the materials that were used in the video!
Thanks to Anish for allowing me to create this and share on Discord
Thank you to Bbern for a great calculus semester
Thank you to Linus L. with some of the code and documentation!
Thank you to 3b1b and the Manim Community for all the support, and this wonderful software!
Thank you, for watching!
Life Hack: Watch at 2x speed and learn ALL OF calculus 2 in approximately 2.5 minutes.
Post that video to CZcams
Then watch that at 2 times speed
BAMB!!!! 1.25 minutes
@@peterslattery9581 just download it and speed up x10
@ChannelZero done, thx
i always watch internet in double speed lol
and after 2 to minutes, all data has been lost from the memory 🤣
Big tip: learn all of calculus 4 in 10 minutes by watching the video twice
g e n i us
watch it on 2x speed and learn it in 5 minutes
The best way is to just watch it once on 0.5x speed, that way you get 5x2 minutes of full understanding.
put it on loop to learn all of all of calculus
That's not how _calculus_ works...
This is like calc 2 at my school but without the part where they break your legs and kidnap your family
Damn. You got off easy.
They tore my limbs off and sacrificed me to satan
Instructions unclear, accidentally got a Ph.D in theoretical physics
When I took calc 2 we covered a lot more than this. It was more like:
1. Applications of integrals
2. Integration techniques *(shudder)*
3. Improper integrals & more advanced techniques
4. Integrals in parametric and polar coordinates
5. Series & sequences/everything in this video
I know some places don’t explicitly have “calc 3” but for me it was basically this:
1. R3, space curves, 3D surfaces
2. Vector-valued functions, and the calculus that describes them
3. Higher dimensions, partial derivatives and applications, curl and divergence
4. Multiple integrals in various coordinate systems, and applications
5. Vector fields, line integrals, surface integrals, green’s theorem, stokes theorem
The "(shudder)" was the same reaction I had my friend.
@@outline5975 Definitely one of the worst parts of the math that goes into engineering, but you gain an appreciation for it. Gotta earn your right to use integration tables the way I see it.
@@outline5975 me and my friend were the complete opposite. We loved integration techniques and aced the exam. We then went on to almost fail the exam on series and sequences.
May I know what is the program of Calc 1?
I am just a little bit surprised as the video content is all in my calc 1 course from this semester.
I am studying engineering in Switzerland and the calc1 lecture cover the following subject : Numbers, complex numbers, sequences and series, limites, real function, derivation and integration
Thanks for clarifying that
@@eloidutruy9119 sure. If I remember properly it was something like:
1. Limits
2. Review of slope, derivatives
3. Applications of derivatives
4. Riemann sums, integrals
5. Fundamental theorem of calculus
6. Basic differential equations
thanks for the video, I unfortunately still got 3 1/2 years of school left and am trying to teach calculus myself, your video really helped me get a better overview of what I already can do and what I should learn more in depth, love from germany :)
digga was bist n du für n schwitzer ahahah
welcher jahrgang?
Oh my goodness, 100k views, thank you so much!! I did not expect for this to happen, but I am very thankful for all your support!!
:)
6:26 is that your computer fan in the background? lol.
When I went through Calc 2, the main focus of the class for me was using the techniques of integration to apply integrals to real world scenarios. The series stuff was at the end and kinda pushed to the side. I get that, looking back at the class, this is probably the actual important stuff, but it doesn't really accurately simulate the class
it was probably Calculus 2 for Science and Engineering (thats what its called at my uni) its similar but has physics applications, while regular calc 1 and 2 focus on more of the theory
Took Calc II at UCONN summer session. (maybe don't do that) way too many years before seeing this ridiculously cool video. Looking forward to more, Fuzzy!
Thank you for the video! It helped clarify and visualize most of the stuff I learned during the first year of university (that being calc II)
thank you, Stanley! I hope your hard work pays off! :DD
Thank you, I hope so too!
I'm starting calc II tomorrow and this is a great quick look at the new material! Thank you!
Was this accurate to what your class covered?
@@dac7294 Yes
how did it go for you?
Awesome vid bro!!!
My class mostly focused on integrals and series, that was like 85% the rest was Differential Equations, Vectors in R2, and R3 , Polar coordinates and area using integration, and lastly calculating volume for shapes using integration
Beautiful!
I first watched this video as a bright eyed and bushy tailed algebra 2 student understanding nothing. Now, as my calculus 2 class wraps up, I finally understand every concept in this video to at least a fundamental level.
Truly a fantastic feeling after so many watches
Cheers
Excellent video
watching this re-kindled my PTSD.
Gonna flex a bit but I was in this guys class. Bbern for life
This guy was a legend for posting all the zoom vids. Carried my grade and was the most polite dude I've ever seen
I cant wait till I have to answer these but on a timed exam
@fuzzyPemguinAMS I want to make math videos too, can you tell me where you got your typesetting? Its very 3b1b (edit I learned, Manim!)
Nice overview of like 1/6 of calc 2
What’s the program used to make these kind of videos? Great work!! Really enjoyed
They are created with "Manim" library (or Manim-Community)
wait holy crap im in calc 3 now watching this for review and i just realized we were in the same calc 2 class last year with bbern
👋
Can you make Calc 1? Can't start 2 before finishing 1 can you
What I learned from this: Smart guys don’t have good microphones
I will be taking Calc 2 final exam this Saturday 😂
In my university calc II is quite different:
(That's the First half)
.anologue calc I theorem applied to multivarible function
.theory of mesure, mostly Lebesgue
.Fourier series
.integrals
Watching this as a pre calc student in high school what am i looking at
This is just sequences and series though.
The rest is applications/optimization, continuity/differentiability, integration techniques (parts, trig substitution, etc), solids of revolution, arc length, and parametric/polar coordinates which is 80-90% of the course.
Thx man gotta calc bc final tomorrow and it’s getting late.
Cool video
Out of curiosity, why is this about Sequences and Series? I know you said you didn’t finish making the video, but why did you decide that you were going to start the video with Sequences and Series? Did your Calc 2 class start with Sequences and Series? I’m curious because I have yet to hear of a Calc 2 class like that. The earliest I’ve ever heard of people putting Sequences and Series is right after all the integration techniques (so like about 2/5s into the course)
Anyway I think it’s a cool approach to teach Sequences and Series by starting with Taylor/McLaurin Series which is rarely done. And you’ve got some awesome visuals in here
Why are you trying to keep it under 5 minutes? Why not shoot for something more reasonable like 20 minutes. How are you going to do it under 5 minutes if you do this video again? How can you shave off any time? Actually you could sensibly shave off time in the beginning by skipping the explanation of polynomials. But 5 minutes is still too much of a constraint. Why not aim for 20
Really cool video!
It is mostly about sequences and series as that was what my Calculus 2 course started off with! I hadn't finished making the rest of the video such as integration techniques and other topics. I really only had (about) one week to learn how to program and use Manim and object-oriented programming in Python :/
Also I was making this while at University so it was significantly more challenging as I had other courses. I was making this video during finals week, as I wanted a way to share some medium of sharing my study material with my peers. This video was it :)
To be quite honest, I don't really know why I chose 5 minutes! I kind of wanted a more brief way for my peers to study, not something complete and comprehensive... Also due to my time constraints (as I had other finals to study for) I hadn't finished it sadly. I am considering a more comprehensive 10 minute re-upload soon! Your feedback is valuable for me to make the next re-upload the best it can be, thank you!
Oh wow that’s really cool that your class started with Sequences and Series
Yeah I can tell videos like these take a long time to make
I’m sure if you make a complete version it will be really good 👍
Which shoftwear you use bro for this awesome animetion...?
ManimCE. It has been put in the description!
Alternative title: All of calculus 2 in approximately 0 min (corrected to the nearest 100 mins)
Is this a world record calc 2 speedrun?
HUGE
Amazing editing! What software are you using?
Probably 3blue1brown's library
It's created by the "Manim" library (or Manim-Community)
What is your major?
Hopefully taking calc 2 next semester so this is super helpful
Just a heads up, it seems his class is pretty unique, most calc 2 courses focus on integration techniques. I highly recommend math is fun, it's a website that really helped me get thru my course
@@davidawakim5473 oh awesome man! I’ll give it a try, thanks for the suggestion 👍
Fuzzy Penguin maths
What is this course, I am currently taking calculus 1 and it is infinitely more formal and rigorous, this shit is just surface level no way this is the only thing that is taught
What’s after Calc 3? Is it differential equations or linear algebra?
Depending on your field of study, differential equations, linear algebra, intro to proofs, probability and statistics, number theory, real analysis and complex analysis are some of the courses you can study after calculus 3 (though calc 3 is not necessarily a prereq. for all of them and the order in which some of these courses & calc3 can be taken is interchangeable).
Everything Lukas said is right if you’re going into undergrad math/physics typically. If you’re going into a more engineering/sciences discipline, colleges typically will just want you doing Linear Algebra, some form of differential equations (ODE, PDE, etc), and probability statistics.
Even then, you learn most of the useful stuff from these classes in your own major’s classes anyway.
Alright, thanks everyone. I'm going into computer engineering, so I believe I'll eventually come into contact with linear algebra and beyond.
We did all of that and more in calc 1
I'm studying engineering in Italy and all this things are part of the first math course which is analysis 1
But this is what we covered in Calculus 1 though.
semester vs quarter system unis organize curriculum differently
*integrable
ngl I came here thinking it’d be a speed math joke but I’m scared to take 2nd year Calc I’m out
bro I can pc worked so hard I can hear it's fan in earphones
3:08 I've taken Calc 2. You definitely don't need to memorize this, especially when you can just look it up or derive it yourself.
to state the obvious, not every calc 2 course is identical. your experience is not some universal calc 2 experience, so it's strange that you feel qualified to make such a universal statement
@@matta5749 I should clarify: I meant for the real world after taking the course
Quick question why is there no + C at 3:24?
Whoops, thank you for pointing it out! It seems I did forget that!!
Thanks man i joind a competition in my uni didnt know what it was about turns out its about calc 2 (im a first year studying math001)
Note: yes im speedrunning calc 2
Bets are open on how many questions i get right👇(we have 20 on the exam)
If you re-upload this, I hope you correct the _multiple_ errors? Many are "only" sloppy wording - but that alone is already really bad when one talks about maths!
0:55 "using only arithmetic and x raised to the power of a positive integer"
Last that I looked, arithmetic includes division. So 1/x² would be a polynomial, according to this definition! Then you go on to list 1/x² as an "invalid polynomial", contradicting your own definition! Why don't you use the correct (and simpler!) definition that a polynomial is a sum of multiples of integer powers of x?
1:10: ,,algebraic equations''
What you then show are not "equations", but "expressions"! (And the statement here _yet again_ contradicts your own definition what a polynomial is!)
1:40: Not really an error here. But I'd say it is _really_ a bad idea to talk about a power series _before_ one talks about regular series, the crucial topic of convergence etc.
3:05: Again, these are not equations, but expressions or functions.
3:20: The word "integrateable" is not wrong, but quite unusual. The word "integrable" is _much_ more common.
textranch.com/298279/integrateable/or/integrable/
3:25: Here you play quite loose with the concept of "integrable". What this word means is _not_ that one can write down a primitive function, but that the Riemann sum exists! (or alternatively, the Lebesgue sum etc.)
3:50: One does not divide the Taylor series by the new power, but one does divide _each term_ of the Taylor series by the new power! (And BTW, it's customary to write names with an uppercase letter: Taylor, not taylor.)
3:55 The word "solve" is unusual here, usually one would say something like "calculate" or "evaluate". One only talks about "solving" when one has an equation and is looking for the solution(s).
4:05: Shouldn't one first talk about the convergence and divergence of sequences before one goes on to series? (Or was that topic covered in Calc 1 already?)
4:10 and 4:30 and 6:13: Again, calculate or evaluate, not solve.
4:35: This is _not_ a function, but a series! Additionally, divergence does _not_ simply mean that it goes to infinity or minus infinity, it simply means that the series has no limit! As you yourself say at 5:05, hence contradicting yourself!
4:48: Writing n on the right hand side is simply wrong! That only would be right if you write a limit n to infinity in front of the expression on the right hand side!
5:00: The sum on the right hand side has to start with +1, not with -1. Because the series starts at n = 0, and obviously, (-1) to the power of 0 is +1.
5:10: Not really a new topic. You already wrote down an interval of convergence (-1 < r < 1) at 4:35. That the variable was called r there, not x, changes nothing about that.
5:33 and 6:44: Not the function converges, but the _series_ converges _to_ the function.
5:40 and 6:00: The graph also does not converge. Again, it's the _series_ that converges.
6:40: Where do you get the r = 0 from?!? The series here is _not_ a power series, it makes no sense to talk about r here! Do you want to _compare_ the series with a power series, i. e. find an upper bound?! Then you should say that explicitly!
Good catch! Though, 1/x^2 is invalid due to it having a negative exponent, but thanks for letting me know for the error, I should definitely be more clear! I think a better wording I should write is:
"Using only addition or subtraction and multiple terms of x raised to the power of a positive integer"
I know you mentioned there are multiple errors, if you spot any more of them, please let me know! I am still thinking about potentially making an updated version soon!
@@fuzzypenguinams3207 "Though, 1/x^2 is invalid due to it having a negative exponent"
Yes, one can write it using a negative exponent, and then it is invalid. However, one can also write it using division and a _positive_ exponent (1/x²), and then it would be valid. Hence there is even an internal contradiction in your definition.
"I know you mentioned there are multiple errors, if you spot any more of them, please let me know!"
Huh? Did you miss 90% of my comment? There already is a long list.
@@bjornfeuerbacher5514 Whoops, sorry! didn't see the "Read more" button for some reason! Thank you so much for the comprehensive list!
tryhard
Some good feedback here. But I don’t think it’s inappropriate to discuss power series before looking at convergence.
I'm a community college student and
I just finished my Calc 2 course and I ended up with a 93.17%. Should I be very proud of myself?
Absolutely, well done! Some people struggle with Calculus, and 93.15% is an insanely high grade! (Maybe even higher than mine)!
@@fuzzypenguinams3207 Actually, I got 93.17% . Thank you for your comment!
Hi fuzzypenguin!
Helloo!!
i have a calculus bernhardt teacher at my highschool LOL
It’s a bit too fast especially at 2x speed.
We did all that and much more in calc 1 xddd
Me too… The video was (and is) still helpful nonetheless
how did u do the animations
Hi! I did the animations using Manim (the community edition, with Cairo as the rendering backend). The animations was entirely programmatically created in python, but since I wrote the code really late at night, a lot of the code is... unreadable (to say the least). But I hope this video inspired you to try and make your own animation!
@@fuzzypenguinams3207 I wanna start doing it but like I feel like it’s hard to learn
@@AstroMo Hi again! It seems intimidating to learn, but I'd like to tell you that I learned how to make this animation within a single week (starting from complete zero knowledge)! I encourage you to try and learn Manim, and there even is a Discord if you have questions, and on top of that, it has highly detailed documentation that you can find on the documentation page. Don't be intimidated to learn it, it certainly is not hard to learn! From Transform(), to Square(), to Text(), you'll find many different functions of Manim that you will definitely find useful for any animation you can think of!
@@fuzzypenguinams3207 Alr man thanks I appreciate it
Nice vid but i wished you could talk louder as i had to use headphones despite being at max vol
Wow this blew up!
Starts from 00:36
Great video but that's not how computers compute transcendental functions
대략적으로라는건 굉장히 위험한 멘트 아니가요 이 메롱판에서
Where is All of Calculus 1?
at epfl all of that is just two/three chapters among dozens in analysis 1 😂
i literallyl have my speakers turned up to max and can barely hear you
I personally prefer the Bugatti series
Haha, I love that joke, I thought the exact same thing when I heard of "MacLaurin"!
3:12. i think u meant e^x & not just e
Whoops! Looks like the same issue is present in cos, I forgot to write cos(x). I also unfortunately did not do the Taylor Series proof for cos(x), e^(x) or a/(1-x) because I didn't have enough time to put it in a mere 5 minute video :)
Thanks, let me know if there are any other errors! I am considering a re-upload soon!
i am currently 11/16 weeks in my calc 2 semester and we have learnt absolutely none of this??????????
How bout now?
There is no need to try to memorize the formulas, I've just known them by heart ever since Algebra 2
Congrats! Even I haven't memorized those formulas yet :P
I know that there will be some people out there that haven't memorized it or known it, so it might be good just to add it (just in case)!
But thank you for the suggestion nonetheless, I am trying to be a TA next semester, this will be very valuable feedback prior to my TA experience!
@@fuzzypenguinams3207 Are you an undergrad?
Then you must carry them into middle-age. ;P
I set my volume to max and can still barely hear you
Theres no bass in the sounds. Sounds kinda hollow
I didn't have a very good mic at the time ;-;
This is real analysis, not calc 2, right?
This is all necessary for real analysis but is calc 2
These are not the topics most students cover in calculus 2. This is more like an introductory real analysis course.
Good video but did not include ALL of Calculus 2.
Cosx2 is just sinx2/2
Can you clarify with powers? But thank you for your input!
Did you mean:
cos(x^2) = 0.5*sin(x^2)
I tried graphing the functions and Desmos but I couldn't verify this, but I'm sure you might have some reasonable Mathematics proof backing with the trig identities that proves this behavior. Keep me posted!
@@fuzzypenguinams3207 The integral of cos(x^2) is (sin(x^2))/2. Reverse chain rule.
Sure it is a nice video, but this was only Calculus 1... Calculus 2 is (mostly) about multidimensional calculus...
It depends on the school you go to. Some schools split single variable calculus into two courses, Calc 1 and Calc 2 and have multivariable calculus as Calc 3. Other schools only have two calculus courses, a single variable course Calc 1 and a multivariable course Calc 2.
calculus 2 is still singlevariable iirc you just go more in depth
" if you are hearing this message then I'm dead" type vibes.
Nope, I'm still here! I'm kind of busy these days thinking of the next video!
Someone made a video "ALL of Calculus 3 in 8 minutes", and posted a link to this video so people could find the previous episode in this meta series.. So now you (or someone) needs to pass the torch on, and post the link to "ALL of Calculus 1 in X minutes"
I was actually thinking about doing that!! That is crazy!! I will go and check out the Calculus 3 video, I am so glad that this video inspired others!
this stressed me out
good video, could be louder -- I have my volume all the way up and still it could be louder
The integral of cos(x^2) is (sin(x^2))/2
Somewhat unfortunate that there is not significant emphasis on the proofs of these statements. They are just being given without context.
Such is the spirit of calc 2 in North America
It’s a 5 minute video… If you’re looking for the reasoning behind the theorems and formulae, you should probably just take a calculus class
2:45 Nonsense. If a function can be written in terms of Euler's formula then it is integratable . Any trigonometric identity can be written in terms of Euler's formula. Again, mathematicians have polluted rational mathematics (as in sane) with this nonsense. You have got to clean this up!!
This is not helpful at all. I don’t care what anyone says. No one will come out of this video and say, “I get it now”. 😂
Obviously 5 minutes is a very short time to completely cover everything from calculus 2. However I think he did a very good job with the animations and explanations. This video can also help you visualize everything which would be very helpful for visual learners
This is generally for review if you already know it.
And that’s another video in the ocean of videos spreading lies that the most common way a computer finds trig values is with polynomial expansions
You're correct. After looking into it a bit more (as it is part of my major) it turns out CORDIC is MUCH more efficient than using polynomial expansions. However, at the time, I have not learned bitwise operations, nor did I know about how computers ACTUALLY do math. Oh well, we grow and learn. Thanks for pointing it out!
@@fuzzypenguinams3207 What’s crazy is my literal calculus course in high school claimed polynomials were how all calculators do it, and I privately told my teacher later that that’s really not as true as it seems, and the CORDIC algorithm is much more efficient, and he was completely taken aback. This is a really weird lie and I wonder what made it spread so hard. Maybe because it’s such a quick thing to say “So why do we need these polynomials? Because calculators duhhhh”
You could have added more if you didn’t spend time by apologizing about explaining an old ass concept.
Sorry :(
It's due to my perfectionist-natured self, I wanted to clarify to the one (or two) of my friends that knew about the development of this video. I wanted to explain to them why I didn't finish the full video...
But yes, I do believe that given a month (or so), I would have been able to complete the video. Do note, I am a full-time college student and I didn't have that much free time in my hands.
(edit): Thank you for your suggestion however, I know from much feedback, that the apology in the beginning of the video is certainly unnecessary. I will now seriously consider removing it.
Most important convergence lifehack:
Σaᵢ converges ⇔ Σ2ⁱa₂ᵢ converges
At least if the terms are positive and decreasing.
It is a challenge to come up with something positive and decreasing where this doesn't work, because the growth should be slower than log(log(log(n))).
Just to be clear, you are allowed to have as many logs as you want.
In this case, the terms would have been
1/(n log(n) log(log(n)) ).