This Is the Calculus They Won't Teach You

think of a stupidly big number a google a googleplex 10 to the power of a googleplex that number is so massive that it has no meaning in the real world but it's still nothing compared to infinity try to imagine an infinitely small number also called an infinitesimal 0.0000 on and on you need to attach more zeros than whatever massive number you just thought of in order to get a true infinitesimal to put the ridiculousness of those numbers and infinity into perspective the observable universe is about 93 billion light years in diameter if for whatever reason

you wanted to express that monstrosity in terms of the smallest unit we have for distance plank length you would get this number this is the largest possible number you would ever need to use to describe a length and it still doesn't even come close to a google and google doesn't even come close to infinity as anyone living in a finite world with a finite lifespan knows infinity is a horrifying concept i mean who as a child did not lie in bed filled with a slowly mounting terror while sinking into the idea of a universe that

goes on and on forever and ever asks rudy rucker in infinity and the mind we should have absolutely nothing to do with this us trying to play with infinity would be like an ant trying to kill god it's just too powerful of a concept to do anything with we should just erase it from our minds and live out our lives using numbers that actually have meaning to us hey hey wait what are you doing i'm doing calculus is that an infinity i see in there yep nothing too mind-blowing actually why in the world did we

as a species ever agree to do math with infinity oh come on it's not that big of a deal what sick people would ever voluntarily summon this demon of an idea there has to be something wrong with this how did calculus even begin if you've taken a calculus course before you may have been taught in this order this makes absolutely no sense right i mean integrals are the opposite of derivatives and to even define derivatives you need limits right well this ordering is closer to how the idea is developed historically that being said where on

this map do you think the first calculus concepts began here maybe here because it's here and where on this timeline do you think calculus began because it's around here the story begins with the ancient greeks what were they up to let's start with a tour of old math this is math this is kind of bath and this is math right well this would look like nonsense since the hindu arabic numerals and mathematicians weren't used back then and none of this exists yet but this is math in ancient greek philosophy circles geometry is all the rage

with our math knowledge now we can find areas of shapes like this and this and oh god never fear says a certain archimedes if you cut up a circle into pizza slices then rearrange those slices like this it almost looks like a parallelogram if you make those slices smaller and smaller you'll eventually get a perfect rectangle whose side lengths are the radius of a circle and half their circumference so since the circle has the same area as this rectangle the area of a circle is its radius times half its circumference that's ridiculous those tiny pizza

slices won't ever become a rectangle but what if you make them infinitely small you can't do that infinity doesn't exist this was a convincing argument at the time let's travel 150 years before archimedes was born to see what the ancient greeks thought about infinity in order to go from point a to b i need to get halfway there first then i need to go halfway again and so on wait i'll need an infinite amount of halves to get to point b and i can't take an infinite amount of steps therefore i cannot walk to the

other side of my room okay if you really don't believe that the area of a circle is equal to the area of this rectangle then let's assume that this rectangle is smaller than the circle because if the area of the rectangle isn't equal to the area of the circle then it has to be either less than or greater than the circle so let's start with the assumption that the area of the rectangle is less than the area of the circle now let's cut off the curvy parts of the rearranged circle so we get this parallelogram

we know that this parallelogram is smaller than the circle because we literally just cut off these pieces but is the parallelogram bigger or smaller than the rectangle well we define the rectangle to have a height that is equal to the radius of the circle meanwhile since we cut off this extra part to make the parallelogram the parallelogram's height is less than the radius by a small amount so the parallelogram is smaller than the rectangle which we assumed is smaller than the circle now let's cut this circle into tinier pizza slices and let's make a new

parallelogram by cutting off the curvy parts again hang on this parallelogram is slightly taller than the other ones since the curry part we're cutting off also got smaller if we keep cutting the circle into smaller pizza slices then the curvy part that's left over will keep getting smaller and smaller there should be a point where the curvy difference becomes less than whatever difference there is between the circle and the rectangle so eventually the parallelogram should become bigger than the rectangle while still being smaller than the circle but wait how is it even possible for the

parallelogram to become bigger than the rectangle the rectangle's height is the radius of the circle that's how we defined it but the parallelogram's height will always be the radius of the circle minus some really really small number we have two conflicting conclusions from following two logical ways of thinking so maybe that means the rectangle isn't smaller than the circle after all what if the rectangle is bigger than the circle this time let's make a parallelogram by extending the sides of the slices until they become triangles the parallelogram is bigger than the rectangle because the height

is a little bigger than the radius the parallelogram is bigger than the rectangle which is bigger than the circle if we cut the circle into small slices again and repeat the parallelogram making process we start getting smaller parallelograms the more we cut the circle eventually the parallelogram should be smaller than the rectangle but that conflicts with the fact that the rectangle's height is the radius of the circle while the parallelogram's height is always the radius plus some small number so this rectangle can't possibly be bigger than the circle well if this special rectangle is neither

smaller than nor bigger than the circle then it must be equal to the circle and therefore the area of the circle is its radius times its circumference over two alright fine i guess you win this time if you got lost there it's okay i did the first 20 times i looked at it as well but the key parts that you should appreciate about this argument are that one it cuts something into tinier and tinier pieces than adds them back up again that's what integration is and two even though it uses the idea of infinity it

never really uses infinity it's almost like archimedes is very carefully trying to tame a beast there is a really nice lesson about math that you can take away from this proof notice how archimedes started with a hazy argument about how you can make a rectangle out of pizza slices of a circle and then created a rigorous proof for it after that the big part of math is about finding what people call a moral solution to a problem which is the solution that should be right and then coming up with a formal proof to justify it

archimedes wrote about this in a text called the method and this method is still recognized today hey here's even a quote from one of grant sanderson's podcast talking about this idea the mathematical equivalent of you playing with your ribbon for the undergrad result where you first figure out what's true and then when you need to make it rigorous that's that's the second step but just into like what do you want to be true that should be treated as a a step worth highlighting and holding up as just as valuable as the rigorous follow exactly exactly

and you know who did this archimedes if you so i give an example it's the 800s and you are a scholar in the baghdad house of wisdom you're sporting a cool beard and turban and you just finished reading the works of archimedes and other greek philosophers what a wonderful time to contribute to the development of algebra you think to yourself algebra i thought this video was about calculus you're right but the development of algebra was crucial to the development of calculus and it just so happens to have happened in the middle of our story so

let's look at how this algebra thing is going you know these now they're starting to get popular especially because muhammad imnemusa al-kawarismi is using them in his book al-jabret but these aren't popular yet instead equations are written as sentences and numbers are thought of as geometric shapes a square is a square a cube is a cube and ah yeah they didn't really do these they also didn't do these since you can't have negative area or volume 800 years later these ideas caught on to a new group of mathematicians who not only spread hindu arabic numerals

but also added their own notation francois viet introduced the use of letters as symbols for unknowns and constants and rene descartes standardized the use of these letters for variables and these four constants hey math is now starting to look a little bit more like math but the best is soon to come after he said i think therefore i am the guy that looked like a cartoon villain had the idea to combine geometry with algebra to make a coordinate system this was revolutionary look at what happens when you graph a quadratic it's a parabola to understand

just how crazy this is the ancient greeks have been cutting up cones to study different shapes for centuries archimedes even wrote the whole book specifically about the parabola they had been studying these shapes without a standard way to easily manipulate them but with descartes analytic geometry you have the power to analyze any curve as a family of equations another thing is up until this point the idea that you could compare two variables in this way was ridiculous this number represents money and this number represents cabbages those are two very different numbers you cannot put them

together but descartes did it and somehow it worked now we can visually represent relationships between variables and it's all thanks to descartes and his method speaking of relationships hey descartes look there's this vermont guy who wants to talk to you he says he developed analytic geometry a decade before you did he says he did what and so began a fierce lifelong rivalry between two huge names in math and physics armed with analytic geometry this battle is where some of the origins of the modern derivative lay and where we can start talking about calculus again now

the idea of an instantaneous rate of change goes back to at least zeno who other than the argument about not being able to get out of his room also famously riddled that an arrow moving through the air is really an infinite collection of moments where the arrow is unmoving in each some practical uses of derivatives can be found in the 500s where indian mathematician aryabhatta discovered that the rate of change of the sine function is the same as the cosine function and used this to approximate specific values of sine but the derivative in the graphical

sense starts here descartes discovered a way to find the tangential slope by finding the radius of a circle centered at where this line hits the x-axis verma found the tangential slope by doing something similar to solving this as h gets smaller and smaller which you may be a bit more familiar with but more than just finding the derivative verma also tackled some other problems that influenced later calculus development like using derivatives in physics finding the maxima and minima of curves and finding a general integral power rule the cart and for moz work will pave the

way for others to make discoveries that will forever change the world hey let's check in on the development of integral calculus what have they been up to while algebra was being developed in the 800s some scholars in baghdad were working on finding the volume of certain shapes quick side note if you've ever taken the calculus course before have you ever wondered why in the world people would ever need to find the volume of solids that revolve around an axis and why the ap test keep checking that we know this formula oh oh i see in

1600's italy bonaventura cavalieri performed integration with a controversial method where he cut a 2d shape into infinitesimally small slices then rearranged them his student evangelista toricelli later used his invisibles to find the shape with an infinite surface area and a finite volume but notice that unlike archimedes cavalieri and toricelli play directly with infinity and mathematicians really didn't like that now hold on to that thought for later because calculus is about to change fundamentally ideas behind integral and differential calculus have so far been developing separately over thousands of years but we're about to see the birth

of a new calculus and it starts with two stunning men i'm talking about galileo and kepler of course these guys actually came before descartes and firma but i bring them up now because i think their work transitions better with newton and leibniz than the others and the way that they reflect the beginnings of mathematicians discovering the relationship between differentiation and integration the key reason why these two mark the beginning of this relationship is because get this isolated behavior in the real world a lot of the examples and mathematicians i brought up so far have been

doing what feels a lot like pure math or math for the sake of doing math these two will do what feels a lot more like applied math where they use math to model physics galileo galilei isn't called the father of modern science for nothing when he asked the question he made sure that he could answer it by focusing on that one question only one of the questions he asked at the time was how objects fell and he went about answering it by creating the most ideal system for measuring the effect of gravity on an object

a ramp where he could time how long it takes for a ball to roll down to different lengths using a water clock when the ball starts moving he opens a valve and lets water flow into this bucket when it reaches a checkpoint or stops he closes the valve and sees how much water has accumulated i wonder if there's any pattern to how this ball falls wait a minute in one second the ball moves one unit then the second second the ball moves three units then in the third second the ball moves five units that's odd

gravity follows some rule to do with odd numbers if you know a bit of physics you might know that this is because gravity on earth's surface can be roughly modeled by a parabola which follows this pattern one unit traveled is one squared units one unit plus three units is two squared units one unit plus three plus five is three squared units and so on you may also start to notice how this is the first hint toward discovering the connection between the integral and the derivative the total distance that the ball travels which feels a lot

like integration because we're adding up the distances it travels in each second can be calculated by a pattern in the rate at which the ball covers distance which feels like a derivative not only that even the way that galileo measures any of this feels a lot like integration and differentiation he first tracks how much water accumulates in a bucket then subtracts differences in how much water there is to find the rate of change of the ball while galileo was looking at gravity on earth johannes kepler studied gravity in the solar system he discovered three laws

after so much trial and error but we're only interested in the second one that the planets orbit in a way that for two time intervals the area of these pizza slices are the same ah i wonder why that happens ah what that's because [angular momentum] is constant while acceleration is radial if an object keeps moving at the same speed it sweeps out the same area in the same time interval if the angle at which an object moves changes, but the object keeps moving at the same speed [with the same angular momentum] it'll still sweep out

the same area in the same time interval who the hell are you isaac newton born on christmas had a father who died before he was born and a mother who left him for a rich second husband went to college poorer even though his mother definitely could have supported him and was understandably antisocial would end up on this list be simultaneously hated and loved by billions of physics and math students over the centuries and make everything about modern life possible it's around 1666 and there's an epidemic in england hey what did you do over quarantine i

learned to play the guitar i got good at painting i started putting together the ideas that would lead to my calculus in my world changing book the principia show off after doing some light reading newton went on an intellectual rampage in the midst of discovering laws of gravity and inventing telescopes one day he must have gotten pretty bored because he started tackling some casual puzzles i wonder if there's anything special about a curve that has an equation that can express its area oh huh the height of the curve is the same as the rate at

which the curve's area increases i also wonder if there's anything special about the equation that gives the area of the curve oh the area given between two bounds is just the equation at the first bound minus the equation at the second bound now what did newton do with most of his findings he kept them hidden but they got leaked and one day he started getting annoying fan mail from a certain gutfried leibniz was a german lawyer who worked as a secretary to a politician so he traveled a lot on diplomatic missions he got to meet

a lot of people and one person he happened to meet was christian huygens who was the mathematician of the time hey you seem like a cool guy wanna learn advanced math from me so you can publish the first articles on calculus in 15 years and one day have a way longer wikipedia page than your own boss that sounds awesome thanks man hey you should go talk to this newton guy i heard he's really smart yo isaac i heard about your contributions to math would you be so kind as to show me some of your work

that would make my day no come on man what if i show you this cool infinite series i found that's amazing you're a mathematical mastermind it's incredible to see that someone else is able to do something i already know how to do in three different ways this son of a magnet discovered the fundamental theorem of calculus after newton but he was the first to publish his calculus and he packaged it all in neat understandable notation that we still use today there are a few key differences between newton and leibniz's calculus though for leibniz this the

part in his dydx notation represents a differential or an infinite testament you might be familiar with this explanation of imagining the d as a tiny tiny nudge in the direction of a variable and d y d x as a ratio of tiny nudges that's what it is newton also played with infinitesimals but he presented them in a different way he used what he called flexions which treat each variable as changing with time rather than directly comparing how one variable changes with another this is kind of like treating every variable like a parametric equation so basically

leibniz looked at infinite testimony lodges in variables while newton looked at infinitesimal nudges in time notice still how both of them explicitly used infinitesimals in their calculus this was definitely frowned upon and they both knew it newton trivialized his work since it was only good for finding solutions not proving them levnitz said that philosophically speaking i know more believe in infinitely small quantities than in infinitely great ones that is in infinitesimals rather than infinite tuples i consider them both as fictions of the mind for succinct ways of speaking appropriate to the calculus to newton and

leibniz their early calculus was just a thinking tool it wasn't rigorous it was kind of hand wavy the question of what to do with infinite decimals and infinity won't be answered until a bit later but before we get into that leibniz did another thing to calculus that newton could never ever have done remember how he was a secretary who went on diplomatic missions that made him really good at dealing with people and he used those people skills to popularize calculus one genius can only do so much but a lot of geniuses working in the same

field can do a lot more and this was honestly one of leibniz's biggest contributions to the early study of calculus that he got the ideas to the people who are ready to work with them you might be able to notice that a key theme in how i presented the development of calculus is that it involved many pairs both in people and also in ideas with the pairs of algebra and geometry integral and differential calculus and math and physics this fairness of calculus to me is telling of how powerful relationships between ideas and people can be

even when they might not seem to work well together at first galileo conducted experiments by creating an ideal environment while kepler only had data to try to use to discover his laws descartes was pretty famous in his lifetime verma was a lot more reserved newton kept to himself leibniz was a people person but despite the differences they all contributed to the development of pivotal ideas and altogether they were able to make calculus what it is not only does this show how big ideas developed from the contributions of many people rather than from a single prodigy

it also shows how much more far-reaching things can become when different ways of thinking find a common ground and can have free reign to influence each other so far i've mostly been talking about infinity as if it was a value but there's also another type of infinity that i'll quickly mention where you treat it like a process that never ends enter the second half of calculus 2. remember zeno's paradox about movement an infinite series is when you add up a sequence of numbers in this case one-half plus one-fourth plus one-eighth and so on to get

a finite value they are too much to get into now and i know i'm skipping a really big part of math but for the purposes of this narrative of calculus these are like tools that kind of developed alongside the fundamental theorem and helped newton leibniz and the people after them make a lot of discoveries notably leonard euler's use of infinite series moved calculus away from the language of geometry and more toward the language of functions and he even derived his famous number from them this time period just after newton and leibniz is also to blame

for the convergence test that you may have had to memorize and all the funny names that you may be familiar with attention everybody there is not a single infinite series whose sum has been rigorously determined in other words the most important parts of mathematics stand without foundation yeah but what about the xenoseries i'm pretty sure that it converges to one yeah it converges to one but what does that even mean to add an infinite amount of terms in the 1800s there was another revolution in calculus and all of mathematics as a whole after mathematicians decided

to revisit the foundations of math now is the time that we finally get to talking about limits i've been implicitly referencing them throughout this video and mathematicians since before newton and leibniz had been kind of using them this whole time what leibniz said about thinking of infinitesimals as fictions of the mind is kind of an example of what a limit tries to do it's like a tool of thinking that gets you to express infinite numbers in a way that doesn't actually summon infinite numbers a limit is usually described as a value that some function gets

close to so if we scroll to the right of this function forever it approaches zero so how is this limit idea better than just saying imagine that infinitesimals are real like like this did first off let's look at what a function is a function is defined by pairs of x and f of x let's say we have a function and we think that the value it approaches at a is l if you aren't convinced that a pairs with l then i dare you to give me any range around l that you think only contains values

that don't pair with a i guarantee that no matter what range you pick i can give you a range around a that only has values that pair with numbers inside your range meaning that my a has to pair with one number in your range you can keep picking the smallest number you can think of but i guarantee that any range you come up with will always be matched by a range around a that proves that your number isn't small enough for my value it's clear that since you can't give me a range that excludes my

a the limit of f at a has to be l it doesn't really matter if you understood that or not the key to all of this is that now when we do calculus we aren't all agreeing to pretend like we're playing with infinity writing l-i-m means that we agree that if we wanted to we could keep on going forever and ever picking smaller and smaller or bigger and bigger numbers that slowly push the function closer to the limit we now have a language where the only thing you have to know logically to prove anything is

if one number is bigger or smaller than another number it's very similar in spirit to the initial problem i gave at the beginning of the video where archimedes proved the area of a circle and it's beautiful the attitude of the epsilon delta proof perfectly complements the attitude of the mathematician who asks how can i know this for certain by answering if you're really up for it i guarantee that you can ask that question over and over again until you are satisfied the epsilon delta limit is almost like an atom for calculus and in that way

it captures the whole idea of calculus in itself by defining calculus with the language of limits we get the tame infinity archimedes would be proud but of course there's more to the story than that there's still the question of infinity itself the limit was able to redefine calculus without explicitly using infinity but it didn't get rid of the idea of infinity there are still debates today about infinity's place in math in the real world ranging from views that argue that calculus can work fine with infinitesimals to views that infinity doesn't exist at all and that

our current foundation for math set theory is flawed because it directly plays with infinity but that is an incredible surprisingly heated debate to save for another time i have long embraced the belief that every course should be built around a story a quest to answer certain burning questions in writing this book i sought to unearth the questions that drove the historical development of calculus says dr david persue about his book calculus reordered which is one of the books i used to research for this video for me the question that took me down this quest had

to do with infinity is infinity really this simple of an idea that we could just say the derivative is like finding an infinitely small ratio between two changing variables it's easy to get desensitized to the idea of infinity when first learning calculus each time a new subject is introduced there's almost like a mental countdown to when the teacher says something like now imagine that we take this process on to infinity what happened to that whole other side of infinity that leads to paradoxes and is a massive headache to think about and come to think of

it why do we even have these limits again i was surprised and felt almost validated when i learned that the use of infinity in math was a problem even to the mathematicians who developed ideas that led to calculus and after the initial shock from hearing that integration came before differentiation everything started to make more sense infinity only became this simple after we developed the limited trap it but the idea of cutting something into infinitely many pieces was a thing before that and for most of the history of calculus mathematicians struggled with trying to limit infinity

to just that one interpretation the whole story of calculus is a story of how mathematicians over millennia develop different ways of jousting with infinity to get simple results leading up to the twist that integration and differentiation are connected and coming to a resolution with the development of formal limits and the following shift from calculus to analysis as dr brasau says the progression we now use for teaching is appropriate for the student who wants to verify that calculus is logically sound however that describes very few students in first year calculus by emphasizing the historical progression of

calculus students have a context for understanding how these big ideas developed now this begs the question what if we always teach introductory calculus like this i've talked with with many many engineers and no engineer that i have ever met says that in their job they actually need to find derivatives or evaluate integrals i taught at penn state for many years and uh interviewed students and most of them when i asked them what what you want to get from this course they said i wanted to understand calculus they've been told that this is important mathematics that

this laid the groundwork for for much of our modern era they want to know why it's important to try to understand what's important about it a lot of students they're pressed for time at college or university and so and again there have been some great studies on this that show that that once students get to a point where they're having difficulty maintaining this this mastery of the ideas behind calculus they fall back on mastering the technique so that they can pass the next test and unfortunately that's what happens to most students uh during their calculus

course and i think part of the problem there is that uh that so much of the way that we teach the calculus facilitates this we we have a great many ideas but we don't really spend the time to allow students to to to absorb these ideas we spend a lot of our time just just explaining well here's how you apply this particular technique and i interviewed penn state students this was toward the end of the semester and i asked them to describe what goes on in a typical class at this point one of the most

common responses i got well at the beginning of the class the professor tries to explain what's going on in the mathematics he's going to present and i know i can ignore that and then he starts doing examples and that's when i know i have to start paying attention so what i think the history is very important because it can help illuminate uh the ideas that are behind the the concepts of calculus and and it motivates the subject i find students are much more interested in the calculus if they know where the ideas are coming from

and also know how people had to struggle to come to these ideas i know that you mentioned that infinitesimals and these tiny pieces um yeah that's something that philosophers wrestled with for hundreds of years i think the history of calculus is a great way of informing instructors you know here's a place where the field as a whole stumbled and had difficulty for a long time expect your students to stumble and have difficulty for a long time calculus is notorious for being a difficult subject riddled with hazy ideas and limitless rules to hastily memorize but it

doesn't have to be this way some mathematical pedagogists have considered mixing calculus with its history to contextualize the concepts and leave a lasting impression on the main ideas of calculus there have been efforts to reform the way calculus is taught for decades notably there was a reform movement in the early 1990s but support for it dwindled in the years following i bring this up because i want to emphasize that the current way that calculus is taught may give many people a flawed impression on what calculus and math in general is i know that personally i

walked out of calculus only having a vague idea of how this supposedly revolutionary math i just learned was important i hope that if you were like me this video was able to go beyond what you might learn in the classroom and show the humanity of math and i hope now you have a different perspective on what calculus is thanks for watching you