Why is calculus so ... EASY ?
Calculus made easy, the Mathologer way :)
00:00 Intro
00:49 Calculus made easy. Silvanus P. Thompson comes alive
03:12 Part 1: Car calculus
12:05 Part 2: Differential calculus, elementary functions
19:08 Part 3: Integral calculus
27:21 Part 4: Leibniz magic notation
30:02 Animations: product rule
31:43 quotient rule
32:18 powers of x
33:10 sum rule
33:52 chain rule
34:54 exponential functions
35:30 natural logarithm
35:56 sine
36:32 Leibniz notation in action
36:43 Creepy animations of Thompson and Leibniz
37:00 Thank you!
Online version of Silvanus P. Thompson's book "Calculus made easy" at Project Gutenberg:
www.gutenberg.org/ebooks/33283
There is also a version of this book annotated by the great Martin Gardner. That's the one to get if you after a hardcopy.
www.amazon.com/Calculus-Made-...
Paranormal distribution maths t-shirt:
tinyurl.com/2p6x5jre for other versions of the same idea google "paranormal distribution math t-shirt"
Creepy animations: www.myheritage.com/deep-nosta...
Music: Morning mandolin by Chris Haugen and Game changer by ikoliks.
Thank you very much to Eduardo Ochs for his subtitles in Brazilian Portuguese.
Burkard
Calculus is incredibly easy and trivial if you already know calculus
As easy as the derivate of e^x
@@plaierdifortnaiti9955 On the same level as an integral from 0 to 0
as easy as proving fermats last theorem, walk in the park.
@@plaierdifortnaiti9955 is this e^x? Im not sure if I remember my calculus well, so I need a refresh.
@@kosherre6243 yeah
Looking back on calculus, most of the things I actually had issues with were not the core concepts, but in fact was my ability to perform algebra without making small mistakes, remembering and applying trigonometric identities, and getting used to new notation. To anyone going through Calculus I urge you not to stress too much about it, just do your best it comes in time!
Literally me
I had this issue as well when I took it…except also adding in not understanding the core concepts.
When the prof was giving lessons I could always do the work on the board... but when I went to take a test it's like everything had changed and I had learned nothing.
@JordonPatrickMears11211988 guys everyone makes small mistakes even scientists, they only difference is they have the opportunity and time to correct themselves. Your taught to triple check your math for a reason. They didn't teach you how to check it just for shits and giggles they want you to succeed. Check your fucking work.
I am in calculus right now trying to refine my foundational knowledge because of these exact reasons.
"Calculus is easy, if you are me." - Gottfried Wilhelm Leibniz
Really ?🤔
The concepts of calculus are easy, and so is making programs to do it. The hard part of calculus is how they teach it in school. They want you to solve it with all of the rules to memorize. But memorizing all those rules- and the exact situations in which to use them- is the difficult part. The ideas of differentiation and integration can frankly be understood by anyone who can understand the area of a circle and how to graph a line; in other words, a late elementary school student or older. But for me, calculus was the first math class where suddenly there was no ability to look at a problem and know immediately how to solve it; you had to try different things on the same problem until it worked. And that does make it more difficult than any previous math class. Granted, it really doesn't have to be that way. Teachers could teach it differently and you wouldn't have that problem.
@natalieeuley1734 - After watching this video, I am righteously indignant that my calculus profs in college didn't teach it this way. I guess they needed to justify spending three months three times a week going over various derivatives and integrals. For Pete's sake, after watching this, I've taken a line equations and integrated it into the area of a triangle, derived it back to a line equation. My calculus teachers NEVER explained it to me that way, i.e. founded on something I already know from high school geometry.
👏🏽👏🏽👏🏽👏🏽👏🏽👏🏽
@@timrogers2638It's not for Pete's sake it's for PITY's sake!
@@roberttelarket4934"for pete's sake" is an idiom originating around the early 20th century that i can safely say is commonly used and understood in the UK at the very least
@@-YELDAH: It's for the sake of PITY not for the sake of someone by the name of PETE! (Yes I know it's common to not to use pity and has become an alternate way.
It is easy. The harder part is learning all the prerequisite material you need to know to start to learn calculus. But if you know algebra, trig, and geometry really well, calculus is incredibly easy.
mastering the basics!
@@peamutbubber Exactly. I had an 8th grade education up until I was 30. I then went to community college and took Algebra I & II, Trig, and pre-calc. Transferred to a university and took Calc 1, 2, & 3, Diff. Eq, Linear Algebra, prob & stats, and a bunch of other math courses like Linear System Theory. Because I took all those basics as an adult and all these classes more or less one after the other, I did very well. The basics are very important.
Trig is a nightmare.
@@toby9999 Noooo! Trig is awesome. It's my favorite math subject.
@@TheBabelCorner Yeah, I just bought some books on that, and abstract algebra. Haven't started yet. Pretty sure it's what will lead us to the new age.
One of the first exams I had in physics the teacher gave us velocity over time graphs and we had to “be the car” and move in distance over time. Now that I’m 63 and still remember this tells me it was one of the best learning tasks I ever had.
We are learning this now!
207076
I forgot everything and I'm still a working Engineer.
@@TC-hh8dc And how do you do at work?
@@elfoyadordeperrosxd1882 He probably fell victim to the usual lack of higher-position jobs, and ended up in a lower-position job, than his education actually merits; where he doesn’t need this stuff, in his day-to-day work 🤔.
Calculus makes things easier once you know it. Learning integration is a perfect example. First we were taught to integrate using infinite rectangles , trapezoids , etc. It was tricky to find the correct formula and take the limit. However, once we were taught anti derivatives , it became much easier.
Yes, everything builds upon the previous information taught. I am extremely grateful to have been able to take Calculus in highschool, so many interesting concepts! I liked it so much, I even bought a book to read alongside.
Would it be a reasonable analogy to say learning an arithmetic operation like the way one can learn a useful thing such as multiplication methods(I’ve seen substantially different methods in different countries) vs the inverse (or reverse operating of Division…again plenty of ways of doing details, but they all seem harder to master…..why? Well, Let’s ask mr. Owl
4:45
What are you talking about? Integrals are anti-derivatives.
@@mariag2916 riemann sums
Silvanus Thompson’s book “Calculus Made Easy” sparked my interest in higher math when I was younger and definitely influenced me into becoming a math major, absolute gem of a book every calc student needs a copy
I wish he'd also written "Algebra Made Easy".
In high school, I was surprised by how by far the hardest parts of Calc I and II simply involved a lot of steps of algebra. Things like partial fraction decomposition are a major pain, but actually integrating the resulting rational functions was very straightforward--once you did the necessary algebra (completing the square, etc.). Then in Calc III, I found it was much the same. Vector algebra is obnoxious, but the calc part really isn't so bad. I will say though that it gets much worse. Nonlinear differential equations are way harder than anything you have to deal with in a high school algebra class. I'd sooner factor ten solvable quintics than stare at a system of nonlinear PDEs until my brain melts.
Numerical methods baby, numerical methods.
@@topilinkala1594 Blessed be double precision
And I found probability with continuous variables to be much easier than discrete probability.
I mean if non-linear PDEs were easy, I know several people who’d be out of a job haha
@@CousinoMacul For sure, it's not even close. Check out the problem of finding the distribution of the maximum of some iid discrete random variables. Then compare it to the absolutely continuous version.
I found calculus to be really easy when I first learned it, but it was always the algebra that held me back. Just as they say, people take calculus to finally fail algebra.
Currently Restudying Algebra. I'm a 3rd Year Electrical Engineering Student. I passed Calculus subjects like Differential, Integral and Differential Equation. Id say that I understood them without being aware that im also learning algebra, the knowledge gap in that subject. But still I want to learn it in an active way not just because I solved higher math. Professors don't really explain where that formula comes from or what it means. They just prioritize the process of solving and application of it. Id say if I take a BS in Physics again maybe I got to know it more deeply.
If you did not pass or did not learn Algebra well, then you are not ready for calculus or any higher math
@@kierpaolodesepeda4428If you did not pass or did not learn Algebra well, then you are not ready for calculus or any higher math
Then you did not understand Calculus if you had problems Wirth Algebra. If you can't do Algebra you can't do Calculus either. Algebraic manipulations or mechanization is the foundation stone to do Calculus and higher math. What you said is like saying that you found University or College easier than elementary school
@@jesusandrade1378 I mean it worked out for me, considering that I’m currently working on my masters in physics.
When I started learning Calculus in High School, I began to realise that everything I had learned in maths before then from simple arithmetic, geometry, algebra and trigonometry was leading up to it. Does this mean that one reason we learn how to add and subtract is so that we can eventually do Calculus?
I think it's more accurate to say that later maths, when discovered, were based on existing maths. So all math is built from the foundations. As you work through the foundations as a student, you get the tools required to start the higher level stuff; hence why calculus uses so many skills you built.
Arithmetic was taught to the masses so that citizens could challenge the authority of the Church. King Henry 8th of England thought enlightened citizens would see that the Church was corrupt when they were educated, then follow his own great reasoning rather than religious dogma. At the time the Church of England was as powerful as its King.
@@stephenkane9630 Henry VIII’s “own great reasoning.” That’s an interesting point of view. I would be curious to know how you reached such a conclusion?
I think it would be more beneficial to introduce the concept of Calculus immediately. So that students understand what and why they are learning algebra and trig.
I really don’t think that maths as a subject revolves around calculus. It is one area but there are many other areas that are not linked to calculus. Showing that there is no general quintic formula uses Galois theory, and this uses no calculus at all.
Awesome video. I learned late in life that this kind of math isn't something I'm naturally bad at, just something that requires more effort on my part than, say, writing an essay on Wittgenstein's late period thought. But then again, calculus is something that requires a lot of effort for MOST people. Anyway, it's great to have resources like this, which are obviously the product of a great deal of passionate labor on the part of Mathologer.
When you mentioned _Calculus Made Easy_ I thought, "Hey, I have that book!" and ran to the bookshelf to retrieve it. As it turns out, no I don't. I have a book called _Calculus the Easy Way_ by Douglas Downing of Yale University, © 1982. It's a fun little book wherein the protagonist is involved in a shipwreck and washes ashore in the land of Carmorra where he, in essence, helps its denizens invent calculus in order to answer burning questions involving the speeds of trains, the areas of fields, the simple harmonic motion of a spring-powered chicken scaring machine, etc.
Have to to check out that book you mention there. Sounds like a lot of fun :)
Funny story!
I have that book!
Google has it in its repertoire.
It sounds really good. I hope u tell the truth.
Pro tip: if you are manipulating equations without regard for units, you are doing mathematics. If instead you DO consider units, you are probably doing physics, engineering, or some other useful thing ;-)
You know Calculus is difficult when someone writes a whole book about how easy it is.
I'm deeply moved by this class! Your passion for teaching shines through, and it's impossible not to be inspired by your enthusiasm.
I always liked to describe differentiation as just a bunch of rules you have to apply and it's usually straight forward how to do it. Integration on the other hand consists of either knowing the answer or trying to manipulate the function until you do.... with the optional third step of giving up and looking it up on a table. Also I like that the music got way more epic as soon as you got to the chain rule.
The chain rule definitely deserves epic music!
I agree with you about Integration. I just put my head down for the entire summer after I completed Calc II, and I solved well over a thousand problems from many different texts. Practice, practice, practice...
Calculus is only hard until the point where it clicks in your head and then you feel: "How could I not understand it?" Tangent 🤣: Archimedes was really close. I think he makes for an incredibly plausible "what if?" scenario. What if he discovered calculus? How much would it change the world?
Archimedes video is in the pipeline :)
How far could he get without algebra, or even a good grasp on real numbers?
@@Mathologer Fantastic to hear it. On subject of "easy" books. I have Polish book from 1946, written by one of the few survivors of Polish School of Mathematics. It is on Complex numbers. In less then 30 pages it takes you from "what is complex number" to "calculus on complex numbers". It is incredibly easy to follow and it's free. The only problem is that it is in Polish.
@@Mathologer I want to contact with you send me email..
It would make the Greeks more misogynistic. Technological progress would get delayed even more because people would think that they already reached the technological celling.
Having always been a touch afraid of calculus this video is a revelation. Thanks for framing this in such a straightforward way!
best explanation ever on a math topic from KZhead I've ever found, I'm currently learning cuadratic functions and this video was really easy to understand!
Nice explanation of calculus. However, I would have preferred the masterclass on Galois theory you promised three years ago 🙂
I hope he'll cover Inverse Galois Theory in that, too :>
I also want complex calculus made easy.
Lmao
@@InXLsisDeo jezuz I haven't even thought of that being a possibility
Hard calculus made easy!
It always irks me when people teach the subtraction rule and the quotient rule as separate unique rules. The subtraction rule is the addition rule and the constant multiple rule combined. After all, f-g is simply f+c*g where c=-1. Same rules, no need to make a special subtraction rule. The quotient rule can be explained by the product and chain rules combined. f/g = f*h (product) where h = g^-1 (chain). I just dislike having to learn special cases when they're not at all special cases.
you mean you don't want to learn the special rule for ((f(g) * h + i * j(k))/(l + m(n)))' ? 🙃
@@JNCressey I do not lol. I've also found today, looking closely at the sine/cosine sum-of-angles rules that it's not 4 rules, it's actually just 2: sin(a+-b) = sin(a)cos(b)+-cos(a)*sin(b) cos(a+-b) = cos(a)cos(b)-+sin(a)sin(b) And the double angle rules come from those rules just with the understanding that b = a. So that's 2 rules to capture 6. Matt from stand-up maths recently did a video about the nice values for 30-45-60 angles and said remembering it as sqrt(1)/2, sqrt(2)/2 and sqrt(3)/2 is mathematically wrong. To that I say pfoey because the point of the tool isn't pure mathematics, it is to _remember_ the actual mathematics, which can be deduced from the uglier (but easy to remember) form.
I’m fairly certain if you just use logarithms, addition rule and the chain rule, you can do without the power rule, product rule, quotient rule, exponent rule, but the expressions get quite unwieldy very fast
I think the reason why the quotient rule is treated separately is probably because of how or rather, when it gets teached. When I got taught calculus (Rhineland Palatinate, Germany), I learned the chain rule first before going over to the product rule while the quotient rule is, as you described, simply applying the product and chain rule together. However, it's also common to learn the chain rule _after_ the quotient rule (something which I learned from bprp when he accidentally included a chain rule question in a test before it was introduced) which makes deriving the latter more difficult if you can't use the chain rule. One thing what can be said for sure is that the quotient rule for integrals is never taught since it is so situational, it may never come up in practise.
@@adityaattri5414 That's how I explain it, typically.
This video helped me so much, and really helped me enjoy calculus (and mathematics in general) much more. Your reasoning made understanding the notation and mathematical processes much easier. I couldn't thank you enough Mathologer!
Very nicely explained, calculus has always been my favourite, now you just showed another interesting way to learn it, great effort!
Only 20 years too late! This is the calculus lesson I wish I had while I was an engineering student. Very well done!
"A Tour of the Calculus" (Berlinski) was successful in terms of sales, but a pompous flop in terms of making calculus accessible. Martin Gardner prepared an edition of Silvanus Thompson's "Calculus Made Easy" that includes some additional chapters. Steven Strogatz's "Infinite Powers" is a fascinating contemporary take on how calculus works and what it can do. Highly recommended. I also took a crack at it in "A Stroll through Calculus," which is subtitled "A Guide for the Merely Curious" because it keeps the math as elementary as possible. (It's been used as a textbook for non-STEM majors who need a math class.)
I can't recall what famous mathematician once had this quote (in German): "Ableiten ist Handwerk, aber Integrieren ist eine Kunst". In english like: "Taking a derivate is a craft, but integration is art." When you know the rules, you can take the derivative of any function, no matter how complicate it is. But integration can be a pain in the butt. Without the help of substitution tables, I was quite busted during my studies at university when it came to quotient of functions. Thanks Burkard for this video (as always)!
Probably Gauss or Leibniz
Thank you for this kind of video, it really helps to drive home some of the concepts that are often explained in such convoluted terms
Yet again, you manage in 30 mins to better explain something than my maths teachers could over a year. Bravo, sir!
still dont get it xD
Doing seperation of variables in Diff Eqs was the first time it really hit me how nice Leibniz notation is. It's really just the chain rule, but still, super nice.
Wonderful lectures on Calculus! This is the best video I have ever seen on the topic of calculus!👍👍
Thank yyou for your clarity and humour .It made remembering Calculus so much fun. Bless you.
Never commented here before... Burkard, you seem like the coolest person! Loved every one of your videos that I have watched. I wish I had the internet when I was a kid. Learning math with you as a kid would have been so much simpler and so much more fun. Thanks for everything! You rock!
:)
When I was at school, learning maths or any other subject was not for fun; we had sport for that.
@@thomaskember3412 I found school to be like that for most of my peers: sports was more their thing than the other subjects. I didn't particularly enjoy sports, my physique wasn't really made for sports, at least I thought so then. I found nearly all the other subjects to be very interesting though. Sadly, not all my teachers were willing and/or able to present their subjects in an interesting way. Unlike @Mathologer, who keeps me interested with every video!
@@Mathologer BALANCED inertia/INERTIAL RESISTANCE is fundamental (ON BALANCE), AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE consistent WITH F=ma AND WHAT IS E=MC2; AS the rotation of WHAT IS THE MOON matches the revolution; AS WHAT IS E=MC2 is taken directly from F=ma. (c squared CLEARLY represents a dimension of SPACE ON BALANCE.) Consider TIME AND time dilation ON BALANCE. Great. The stars AND PLANETS are POINTS in the night sky ON BALANCE. “Mass"/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent with/as what is BALANCED electromagnetic/gravitational force/ENERGY, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE). Great. Indeed, consider what is the fully illuminated AND setting/WHITE MOON ON BALANCE !!! Consider what is THE EYE ON BALANCE !!! c squared CLEARLY (and necessarily) represents a dimension of SPACE ON BALANCE. Now, consider what is the TRANSLUCENT AND BLUE sky ON BALANCE !!! Indeed, notice what is the orange AND setting Sun ON BALANCE !!! WHAT IS E=MC2 is taken directly from F=ma, AS the rotation of WHAT IS THE MOON matches the revolution; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE !!! Think. Great. By Frank Martin DiMeglio
@@SvenBerkvensMatthijsse BALANCED inertia/INERTIAL RESISTANCE is fundamental (ON BALANCE), AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE consistent WITH F=ma AND WHAT IS E=MC2; AS the rotation of WHAT IS THE MOON matches the revolution; AS WHAT IS E=MC2 is taken directly from F=ma. (c squared CLEARLY represents a dimension of SPACE ON BALANCE.) Consider TIME AND time dilation ON BALANCE. Great. The stars AND PLANETS are POINTS in the night sky ON BALANCE. “Mass"/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent with/as what is BALANCED electromagnetic/gravitational force/ENERGY, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE). Great. Indeed, consider what is the fully illuminated AND setting/WHITE MOON ON BALANCE !!! Consider what is THE EYE ON BALANCE !!! c squared CLEARLY (and necessarily) represents a dimension of SPACE ON BALANCE. Now, consider what is the TRANSLUCENT AND BLUE sky ON BALANCE !!! Indeed, notice what is the orange AND setting Sun ON BALANCE !!! WHAT IS E=MC2 is taken directly from F=ma, AS the rotation of WHAT IS THE MOON matches the revolution; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE !!! Think. Great. By Frank Martin DiMeglio
Welcome back i dont know why should i watch and rewatch your videos over and over again ??????? Thanks much prof .
That's what I've always asked myself, I can't believe how amazingly simple calculus is. Truly wonderful.
Great, as always! Loved the animations of Thompson and Leibnitz and the metal soundtrack of the end (though I've always been a fan of the usual wistful and nostalgic guitar theme)--it was distracting, but worth the distraction!
I always go back to time, distance, speed, and acceleration whenever I need to get an intuition about differentials, it's the simplest indeed, and it's so natural to all of us
I'll be honest, I don't know why, but the second problem of integration, about there not being elementary antiderivatives of all elementary functions, just crushed me psychologically when I was learning calculus. I know there are ways around the problem, and I memorized them enough to do decently well in calc, but somehow the trial and error nature of it just lost some of the luster off something I previously quite enjoyed, and turned me off of pursuing any higher forms of calculus. To this day I can handle most high school math up to that point with only minimal references, but all the various methods of integration slipped away from me like water once the class was over.
From what i know, the only integration methods are substitution and integration by parts. The difficult part is to make very nonobvious transformations to get difficult integrals to yield to those methods, and those transformations are so specific to each individual difficult integral that i wouldn't call them methods
Just ignore integration then. I prefer combinatorics myself.
Just reading the words “elementary anti derivatives of all elementary functions” hurt my brain. I’ve found a free version of the book mentioned. Hopefully it makes more sense to me than this video.
@@MekazaBitrusty one step at a time man, and this video really isn't anything more than a refresher for basic calculus. Its only good if you already knew everything the video covers.
Well if you think about it you just don't know what C is. Yes it is sometime feels like flying blind. Uncertainty is a bitch.
65yo, haven't used calculus in over 40 years, still remember the heart of calculus and find it easy to follow this wonderful presentation. I see calculus as "depowering" or "powering" operations. Exponents become addition and back again. I imagine this ranking of increasingly more powerful operations, and calculus as the rules for going up and down in effect. Very similar to the way explained here by graphs and the table of elementary operations. Go up for slope, go down for area. From the first it seemed so intuitive back when I was young. Then again, so did dancing molecules which led to my career in medicine and biochemistry as an expert in single carbon metabolism ( the dance of the B vitamins). Mathologer is a wonderful channel because he loves this. There's an inherent beauty that simplicity brings; but you must first love knowing for the sake if knowing, not some other goal. Sorry, an old man reminiscing of when he still had a functional mind here. Soon again, I will think.
That's great. Thanks for sharing :)
I remember how hard it was learning calculus in university. 25 years later, I went back for a Master's in Engineering and realized how amazingly simple and easy it was.
my high school had a copy of that same book, Thompson's "Calculus Made Easy". i remember using it to teach myself calculus before i ever took a class because i wanted to help my girlfriend at the time with her homework lol
Amazing! Even if i still remember the calculus and derivative rules, your video makes it easy: the car analogy is pretty much spot on! The end animations are the top cherry! "Simple isn't it?" 😁 Thank you!
This video helped me a lot! I already studied calculus, but I didn't understand most things I was doing, and this video was a savior for me! Awesome work!!!
Wow. That's the best intuitive explanation of calculus I found. Thanks a lot 💙
Its so nice to watch a simplified version of maths. This should be played on the very 1st calculus lesson in highschool. During my classes they just have us formulas and didn't give much explanation, but on uni on mathematical analysis we went thoroughly through all the proofs. None of those 2 approaches are good for people which have never heard what calculus is, so i don't understand why on most highschools they either don't explain it or prove it in a way that highschoolers have no chance of understanding.
just starting to really enjoy maths and discovered it as one of my passions, (doing hours everyday), your videos are very fun and useful! much appreciated!
Great :)
Because of your videos I learned moving numbers from some place to another by my mind and solve equations, so thanks for teaching me the magic of mathematics.❤️
12th grader here, this is the best educational video I have ever watched. I can not tell you how clear this has made everything to me, I just started calculus, but it feels like I have been doing it for months now. Genuinely thank you.
My favourite extension of calculus is the vector calculus. I studied it in the first year of my graduation. It's been over a decade since - how time flies!!!!
Thanks for the video! One more bump you might want to mention is that we're assuming continuous functions and sometimes assuming continuous derivatives. Works great for the Mathologermobile, which I assume can't teleport or go from 0 to 60 in zero seconds.
Lot's more bumps once you seriously start looking for them :) Always a balancing act to figure out what to say and what not to say in these videos.
Dear Professor Polster, I wish Your videos became a part of mandatory study materials at tech. universities. Sincerely, I learned more and many mathematical concepts "clicked into the right place" within my mind while watching Your videos. I have a PhD. in electrical engineering where I worked with Markov chains and complex calculus on a daily basis. Had I had the chance to have such a quality study materials 15 years ago, I feel could have learned 10 times more in the limited timespan. Your visual proofs and way of calculation helped me to perform quick calculations by just manipulating symbols in my imagination rather than writing everything down. One big thank You, since You have the talent not only to understand the topic, but also explain it as real παιδαγωγός.
This was epic. I want to get a good book with plenty to practice and teach myself calculus. I think this is very useful!
Imo you only really need the chain rule to rule them all. The product rule *is* the chain rule for multiple variables. And all others can be derived from it and might sometimes be trivially easy Like this: d/dt f(x(t), y(t)) = df/dx dx/dt + df/dy dy/dt That's the multivariate chain rule, and it's extremely close to the product rule. If you simply take f: KxK -> K to be the product on K, you get: d/dt x(t) * y(t) = y dx/dt + x dy/dt which is literally just the product rule. If you take it to be the sum instead, you get the sum rule, etc. And the quotient rule is one I never ever use: Just transform to powers and apply the product rule! (Which nets you the power rule too)
Calculus is so beautiful and elegant theory....and also it is really easy if learned from the correct teacher or book
Great. Has you got any book recommendation? In positive case, please share it.
@@jullyanolino I've got a recommendation in the video :) Actually, there is a version of this book I recommend annotated by the great Martin Gardner. That's the one to get :)
Wait, are you implying someone standing at the front of the class exuding hot air about gibberish slapped on a whiteboard isn't teaching?
@@Mathologerwhat do you think of the book by William Anthony Granville ? And what about the calculus book by Stephen Banach ?
at 22:17. Thank you. That is the best description of +C I have encountered. Well done and appreciated. :)
Far removed from mathematical studies (6 years since undergrad) … but this might be the best way to describe derivatives and integrals to quite literally anyone and I love it!
Thompson's book is superb. It's like being led through the steps of calculus by a friendly Victorian uncle
Done. This motivated me 100times to excel at calculus. I'm in 9th standard and that is surely old enough to learn about this. I am sure I will give effort and completely understand the formulas, rules and everything of calculus. Thank you Sir! Funfact : calculus is one of the most scoring parts in maths. If your brain refuses to understand the calculus things, take a small break and think of professor Calculus from Tintin. Also you control your brain, it doesn't control you. Cheers 🍻 bye
No please don't study calculus, it will make me have competition in future
Great video. Animation are very nice and they help a lot. You have putted so much work into this video and I thank you for that.
The book in the description actually makes the learning process much better 🔥🔥
00:00 Intro 00:49 Calculus made easy. Silvanus P. Thompson comes alive 03:12 Part 1: Car calculus 12:05 Part 2: Differential calculus, elementary functions 19:08 Part 3: Integral calculus 27:21 Part 4: Leibniz magic notation 30:02 Animations: product rule 31:43 quotient rule 32:18 powers of x 33:10 sum rule 33:52 chain rule 34:54 exponential functions 35:30 natural logarithm 35:56 sine 36:32 Leibniz notation in action 36:43 Creepy animations of Thompson and Leibniz 37:00 Thank you!
it's already in the video timeline
Math was very challenging until grade 10, honestly, in my case, I understood math very quickly afterward. Calculus was the easiest subject during university. Now after 10 years I still remember these rules. Math is all about understanding concepts and visualization of problems in your head, most students I came across learn math by doing examples and memorizing types of problems, at first glance, it seems the correct approach, but in reality, when the problem is slightly changed they struggle to solve it.
Anyway, the more techniques you know, of integration or otherwise, the bigger and better your toolbox becomes, and your field or view expands to solve a wider variety of problems, you identify special cases, limit cases, power series, special functions , etc.
This was a great watch. What makes calculus hard is having to use big words to describe very simple things we all understand. If you put your foot on the accelerator and speed up 10 km/h/s after one second you're traveling 10 km/h, after 2 -> 20 km/h after 10 seconds -> 100 km/h. If you then drive 100 km/h for two hour you traveled 200 km All math my 12 year old can do. The problem is when you try to explain *how* that works. Start using big words like "curves", "slopes" and "areas" and abstract concepts of difference in time dx/dy and peoples' brains start leaking out their ears. Math as it taught from a young age is all about combing numbers to get some number as a result and what matters in getting a correct result. It took me forever to wrap my head around the concept that solving for dx/dy does not mean putting an actual specific value in and getting a result, it's a way to generally define what happens for **any** value you could put in.
Fractional Calculus would make for a great series of videos. I happened upon it once and the next day a math professor came in near the end of my physics class. I asked him about it and he had never heard of it. Soon after he gave me and my physics professor a short series of lectures on it. I've always regretted not recording those.
I have a degree in Math. As most degrees needing math, there were Calc 1, 2, and 3. There is usually another one after that. After finishing those, I remember taking a Math 300, Into to Calc. This class made all the rest of the calc classes easy/obvious, I never did figure out if the 300 class was so enlightening since I had the other classes or if I could have taken the class earlier and the other classes would have been easier.
In high school I was always in the advanced classes but that meant I got into calculus with zero precalc. Got a math major but focused on computer science so forgot all my calculus (not quite 100%, but most of it). I keep going back about every decade and relearning calculus I used to ace. I found an old assignment where I did about 20 pages of proof. Other than recognizing my own handwriting, I had no clue how I did it. It's not like riding a bike
I love this! Why burden the new student of calculus with all the limit and epsilon-delta proofs before you even start to learn about derivatives and integrals. Here, you just jump right into the fun, powerful stuff and make it easy to understand. Once the student has a grasp of what you can do with calculus, then maybe dig into it a little deeper with the formal proofs involving limits, etc. Learning is so much easier when you first understand why you'd want to learn whatever the subject is. This video does that!
Speaking as an outsider, I have always got the impression that the teaching of mathematics is very tradition-strong. This is ironic, since (for example) modern views of real numbers include those from Cauchy, who invented them to teach basic calculus rigorously. I've always thought that a book of "stuff we invented or discovered because we had to teach basic classes" would be a neat book for inculcating humility in academics who dislike the "service classes". (Another example, this time from chemistry if anyone cares: the theoretical justification for looking for noble (then "inert") gas compounds came out of someone seeing by chance that the first ionization energies of oxygen gas and xenon were about the same, and compounds like O2PtF6 were known, so ...)
Yeah, basically what Newton and Leibniz and the next few generations were doing before Cauchy and Weierstrauss et al came along and decided there needed to be more rigor. You do need to have some conceptual idea of limits to understand what a derivative is, but we don’t need much if any of the algebraic and analytical formality that we often shove on students in the first couple weeks. Had this discussion about AP Calc recently. The AP exam doesn’t do a whole lot algebraicly with limits (no epsilon delta proofs at all) and where this some algebra, it’s probably a limit definition of the derivative in disguise or it is a L’Hospital rule question. But teachers typically stick to what they know.
Love this summary! Thank you so much!
This is a beautiful and very informative video. Thank you for this
I agree. It is also a very "attractive" type of math. What I mean is that once you get it over with, by completing the classes, you kinda want to go back and continue doing calculus. Its not the same with linear algebra or other maths. At least thats my opinion on cal.
Maybe you could do a follow-up video about the Risch algorithm to find anti-derivatives for elementary functions. Would be interesting to get an understanding how that works in principle.
Would be nice but probably a bit too fiddly unless one focusses on the functions generated by a smaller set of atoms :)
I need a Gravol for limits. But when you freeze time to a dot....how many times was your answer off .0001 to the book? You did the same math? Order of opperations. And write at the top of each page y=f(x).
Wow the inertia of a dot.
But NONE of this math actually work in the real world. Why? because IF you accept Einstein's theories, (I dont) then you cant get to even square one, where if you plot a simple constant acceleration over time, from zero to light speed, then you cant get a straight line which is necessary and logical if classical Physics is true, but in Einstein's stuffed up physics, the straight line showing velocity is not a constant slope at all. Because as velocity increases the acceleration slows, as he claimed that you cant get to light speed ever, so that acceleration has to stop totally as you approach light speed.. and it does this trick according to Einstein, by having Time slow to stopped, and distances shrink. So you can't have a classical Plot anymore under these conditions. And as modern confused Physicists demand, Einstein is correct and classical Physics is wrong. (not only wrong at high speeds, its necessarily wrong at ANY speed, the correct equations are not Newtonian, but Relativity equations are only able to produce the exact correct result.) So the equation d=v.t is useless, as Time is not a constant, it varies with motion. And so Distance is also unable to be calculated by this equation, it too shrinks with velocity, which we can only measure using Time and Distance, which shrink anyway! Classical physics equations can only be applicable if you reject Einstein's Physics, you should not be allowed to cling to Newton and Galileo and still claim that Einstein is correct, They are totally incompatible in every regard, least of which is the foundation of Newtons Physics requires a stable Time, Distance and Mass universe, but Einstein's universe is based on the exact opposite.
@@everythingisalllies2141 Everything is skewed. Sorry boss how many decimal points again....wuh? Round down? WTF? What if the radius of a circle is a limit. Unattainable.
I'm thoroughly enjoying a reentry to calc. Still catching up with the language, so it's a bit bumpy. So the first part was, as always, a challenge. The last section is an eye opener. When these solutions are presented in that way it all becomes very clear. Thank you, please continue!
I've been teaching myself calculus, and I was very surprised how easy it is. I started in August, only having taken algebra 1 and sort of teaching myself algebra 2/basic trig. It is so interesting.
I found that running the segment starting at 16:58 at 1/4 speed several times and saying the process as it occurred Very helpful. (But don't forget to MUTE it. The music will kill you.)
Personally I find the limit definition of calculus very unwieldy and intuitive. Working with Leibniz notation like this is always much preferred in my eye. Turning your differential operations into statements about the convergence of sequences doesn't seem like a particularly natural step and is one that I don't tend to see people do when working with calculus in physics or dealing with them numerically. In physics and while working with numerical methods I feel like what we do most resembles non-standard calculus, especially ERNA and ERNA^A. Sam Sanders 2010 article "More infinity for a better finitism" gives a really nice write up of this. Being able to manipulate infinitesimals symbolically without fearing that one of your implied sequences stops converging is great peace of mind. In particular their example of pushing sums through integrals without fear (as long as your result is sensible) is one of those things I bet thousands of people do without thinking about whether it is justified.
I’m with you!!!
Non-standard analysis should really be standard
Definitely this is the way I prefer to introduce calculus :)
I did learn starting from limits. That was interesting.
@@Mathologer Do you teach the hyperreals in analysis 1, or do you teach infinitessimals more as thinking aid?
One of the best explanations I've seen
Amazing video! A fresh and new take on Calculus logic for me. It really helps!
Oh, this reminds me of large parts of my first calculus class, in particular when my teacher stated that calculous was easy, it's the algebra you have to go through to DO calculus that's hard. Which is true to an extent. It too me a long time to realize that d/dx meant "As X changes". I was a lot happier when I figured that out. I don't know if I'd have been less intimidated if I'd known at the time that if you have a number of datapoints, you can figure a formula that fits them and then to calculus to THAT. Ah, well. Don't forget that the Cs cancel! But your teacher will be mad if you don't put them in! I wonder if calculus can be used to prove that the distrebution of numbers in pi is normal. I got lost in analytical geometry. I understood the bits but putting them together was beyond me. And ultimately, I was more interested in humanities. But I did do very well on the math portion of my GRE's. :) I wonder what the Austrian Autobahn's like.
Fermat invented it first, Newton's fluxions make the most physical sense. Leibnitz's notation is way the best.
The best of all possible worlds!
@@gordonglenn2089 Shut it, Pangloss. :)
- This is a lovely video. The heavy lifting is done by all the properties of the limit (in a mathematical sense), which are disguised in the clever notation. - 35:10 e = lim (1 + x)^(1/x) as x goes to 0. Swap the usage of lim and write e = (1 + dx)^(1/dx) instead (abusing/exploiting the notation). Raise both terms to the power of dx, then subtract 1: dx = e^dx -1. Finally, divide by dx to get 1 = (e^dx-1)/dx - By using the concept of the car as a _vehicle_ for the explanation (pun intended), some hard issues are avoided, such as "can I differentiate any function at any point?" or "can I integrate *any* function on any interval?" - We Spanish speakers have access to a very neat opening interrogation symbol, "¿". This is very handy, given that you know that the phrase is an interrogation from the very beginning. - The rule for the derivative of the inverse function is graphically very intuitive after noticing that the plot of the inverse function is the same as the original function but mirrored about the line y=x. Hence, the slope of the inverse function is the (algebraic) inverse of the slope of the original. I'm sure Mathologer is aware of this fact, and probably decided to leave this property out for the sake of the total length of the video. - 27:22 That math joke is brilliant. Here's a similar one I found elsewhere: take the equation x^2 = 25. Of course (?) you can cancel each number 2 appearing on both sides, which leaves x = 5, the correct solution for the original equation. I wonder if there are more of these... - 32:15 Fun fact: the quotient rule for h = f/g can also be obtained by applying the product rule to h = f * (1/g) first and then using the chain rule with the functions 1/x and g. - 33:17 The rule (x^n)' = n x^(n-1) is the same for ANY value of n, not only positive integers, but for any real number. But the proof is more involved and I would start by writing x = e^(ln(x)). I liked the music of the final part!
Awesome presentation on the beauty of calculus. This is how calculus should be introduced.
Ah yes, the subject that made me rip out my scalp
I always struggled with algebra but actually found integral calculus to be very easy.
I almost screamed when you mentioned "Calculus made easy" in the video! It was my first exposure to calculus as well and it is a fantastic book
Vielen Dank für das ausführliche und gut durchdachte Video👍
Thank you so so much
11:20 "None of what I said so far is really terrifying" Are you sure?!?! You just talked about suddenly stopping and going backwards on the Germany Autobahn, that _is_ terrifying.
:)
Beauty. Loved the simplified explanation
well explained thank you , this was a great refresher for the basic concepts
Being honest...I rarely make it to the end of a Mathologer video. In this case I've surprised myself by not only making it to the end, but in a seemingly too short amount of time. No video stretches like a Mathologer video.
Love the kick ass music at the end
Great :)
You still didn't integrate imple
Great presentation, thanks , keep them coming !
Great soundtrack! Really captures the mood continuously and smoothly throughout the whole video! 🤘
Luke 🙃
I’m glad there’s smart people out there that understand calc so that the rest of us don’t have to 🙏
Me after failing my calculus final : yes very easy
Weeelllll, integral calculus can also be considered in terms of the first derivative. If we wish to compute f(x) from f'(x), while area is indeed correct for considering what an integral is, it is also correct to state that it is the surface defined by a set of infinitesimally small vectors such that if continuous change in position were possible, traveling by a distance of dx along the vector defined by f'(x) would trace the image of f(x). So, put simply: the derivative concerns continuous change in direction with regard to some f(x); the integral concerns continuous change in position with regard to some set of vectors f(x). Of course, if we apply this directly, what we would get is the parametric representation of the integral of f'(x) as (x(t), y(t)) where t represents time or arc length, viz. the time travelled along the curve defined by f(x), so to get f(x), just solve for the inverse of x(t) to get f(t) = y(x^(-1)(t)). In terms of how we might get x(t) and y(t), it might be easiest to evaluate the limit in such a way that x(t) = cos(a(t), and y(t) = sin(a(t)), by finding an identity f'(x) = -1/tan(a(x)) first which is then trivially split apart into a parametric form which can be converted to its cartesian equivalent. This as a general method of computing the antiderivative of f'(x) is probably a lot easier than memorizing or just looking up an identity, or fiddling with f'(x) to make it possible to find its antiderivative using the existing rules, but if the effort of finding a(x) is worth it to you, then why not? You can manipulate circular/hyperbolic functions with relative ease, so the utility is quite obvious and should make the effort of computing an inverse all the more easy by converting everything into identities of the exponential function in principle.
This has made math seem so straightforward in a way that had never even occurred to me.
Fun fact: you can use the velocimeter to measure the velocity of your car, but you can't use the odometer to measure the odor of your car.
The tachometer?
@@robertcampomizzi7988tachometer is rpm of the engine
@jepsmcsmackin2507 Yeah but a tacometer(rotation) would have a magnitude as speedometer doesn't. They mentioned velocity. So I was going off of that... I think? It was a while ago. So if I had to guess that's why I typed what I typed.... or I was dazing off for some reason. Edit: "velocity OF CAR" ok.. I see now . 🤷♂️
What I disliked about calculus was seemingly endless questions asking me to integrate or differential nonsensical mashups of sins and cosines like wtf would be the actual utility of integrating sin^2(5x)cos^3(x) aside from exam questions so annoying
One practical application of complicated trig integrals (a rather simple example, but the idea is the important part), is the theory behind the RMS values of electrical power. When you talk about 240V AC electricity, it really is not 240V in amplitude, but rather 340V instead. The ratio between 340V and 240V is sqrt(2). But why? The reason is that we are ultimately interested in finding the average value of the function V(t)^2, over the time domain of 1 cycle. You will end up integrating (340*sin(t))^2 from 0 to 2*pi, and then dividing by (2*pi), and then taking the square root of the entire result to find an equivalent DC voltage. This is called root mean square, or RMS. It is the equivalent DC voltage that provides the same power to a simple resistive load. Low and behold, you get the nominal 240V that we know and love.
Another example of complicated trig mashups in real life, is the theory behind FM radio, and other frequency modulation signals. Suppose you listen to radio station 100.1 FM. The 100.1 value is a frequency in Megahertz, and is the carrier frequency of the wave. The frequency will shift around from its center of 100.1 MHz, and carry the audio signal you are superimposing on it. If you want the station 100.1 FM to carry a tone of middle A at 440 Hz, then you need its frequency to also be a sinusoidal function of time. The ultimate wave W(t) will take the following form. Notice the nested trig functions: W(t) = A*sin(2*pi*(B*sin(2*pi*f*t) + F)*t) where: A is the amplitude of the waveform, related to the signal strength B is the amplitude of the frequency band, dedicated to carrying the signal. F is the carrier frequency, 100.1*10^6 Hz f is the audio signal frequency, 440 Hz t is the time in seconds Of course, the waveform it's carrying isn't going to be a simple single tone, otherwise that wouldn't be very appealing to people. But this is the general idea.
I loved the end of the video, beautiful visualization to understand how these properties appear, amazing Journey, even with a Tool vibe at the end hahaha, greetings!!
Thank you sir for this wonderful representation ,I believe I've learnt a lot from this one video