Why is calculus so ... EASY ? 

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

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.

@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.

Really ?🤔

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.

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 🤔.

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

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.

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.

I wish he'd also written "Algebra Made Easy".

And every normal person does not

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.

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.

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.

it's already in the video timeline

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...

still dont get it xD

Probably Gauss or Leibniz

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!

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.

That's great. Thanks for sharing :)

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.

:)

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

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.

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 ?

No please don't study calculus, it will make me have competition in future

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.

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.

Great :)

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

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.

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 . 🤷‍♂️

The best of all possible worlds!

@@gordonglenn2089 Shut it, Pangloss. :)

:)

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?

Love the kick ass music at the end

Great :)

Actually the calculus of variations is one of my all times favourite topics in maths :)

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.

Anyone who thinks Analysis isn't hard, give me an epsilon that proves the series 1/(2n+1)(2n-1) converges.

@@High_Priest_Jonko Maybe first place the parentheses correctly :) Right now it is ambiguous how the terms are multiplied/divided. Also, I'm missing the summation in your series (including limits).

@@vicktorioalhakim3666 I meant 1/[(2n+1)(2n-1)] or [(2n+1)(2n-1)]^-1 if you will. Simply summed from 1 to infinity.

@@High_Priest_Jonko Nice, thanks. Split [(2n+1)(2n-1)]^-1 = 1/2(1/(2n - 1) - 1/(2n + 1)), and notice that the sum is telescoping, so 1/2*\sum_{n=1}^k 1/(2n - 1) - 1/(2n + 1) = 1/2*(1 - 1/(2k +1)) ... Now take k -> inf, or use your favorite delta-epsilon proof of the function 1/x.

The difficult part is remembering all of that stuff.