I'm actually going through a calc class in high school right now: my teacher ain't even really a math guy, he just got roped into being a math teacher because he originally was going to work on computers, then decided to be a teacher, and since he had taken math for his computer classes, math was what was taught. Anyway, it really plays to his advantage, because he asks the same sort of questions you do about his own material, and so he ends up going over WHY the math works, not just how. Definitely one of my favorite people.
computer science is all about translating pure math into practical forms (rather than more general engineering which is still just plug and chug for the most part)
I was super disappointed in my first calculus course at uni, there was no time in my life other than this situation where I felt like I had a better grasp of the subject than the teacher. I don’t know if he was new, but the guy glossed over trig functions in a single lesson, didn’t appear knowledgeable on how to use them, couldn’t articulate what Cos or Sin meant, seemed really nervous & confused when someone asked a question & really didn’t understand limits, which is insane because limits are based of the initial function’s people learn in college algebra. It was a complete disaster, the entire class complained at the end of the semester & he was removed (probably fired) because I never saw him again. But he really messed it up for those people who really needed to understand calc 1 before they went on to calc 2, I’m blessed that my dad is a metallurgical engineer so he taught us all these concepts at home when my brothers & I were growing up. But seriously, universities need to do a better job of investigating the backgrounds of the people they hired instead of being greedy for money.
I feel like 90% of college lecturers either lack the passion or lack the knowledge to teach. They know how to solve questions, but they don't know how to convey it so that students can benefit from them.
Watching this video made me appreciate my math professor more. He actually spent a significant chunk of time in class going over proofs for limits and derivatives. He was really thorough and had good answers to our questions.
my teacher for this was great. His big reveal towards the end of the derivatives unit in Honors Precalc (which covered calc 1 concepts) was the definition of a limit, and the dreaded "game" (which as he told us regularly was not a very fun game at all) wherein he would give us an equation and an Epsilon and tell us to find the largest delta that would allow us to "win" the "game", as in meet the definition of a limit and trap the function. This was limits at level 3, with level 1 being plugging in close values on a calculator and level 2 being what he called algebraic trickery, where you multiply by 1/the largest power of x in the denominator. Level 4 was in Calc BC, where we extended the concept of the game to win it every single time by actually writing out a proof.
Real analysis goes into almost all of the theory that builds up to calculus. It's also much harder than calculus. So the teaching order, while seemingly bizarre, makes sense in a way. You effectively memorize the building blocks of higher math, and later learn how they themselves are built. I wish they presented it that way.
it "makes sense" in a way that still doesn't make sense in any sort of computational reality of applied mathematics... Leibniz calculus is a paper method designed for paper solving of equations that describe systems. In practice, it is antiquated applied maths that really only serves to teach the jargon so people can read historical works of mathematics. even the integral symbol is what it is because of Leibniz using the ʃ (esh) character, which was available for the german printing press... the more math you learn, the more you'll see that the introduction of calculus, and even real analysis, are just fundamentally flawed. especially in the modern era where computer algebra systems hide a lot of the computational complexity from you. there exists "nonstandard analysis" which instead starts by defining what an infinitesimal is (a number that is not zero, but when squared is zero), then builds algebraically from this number. similar to how the imaginary number (a number that is not -1, but when squared is -1) is used to form the complex numbers. They are called "hyperreal numbers". For whatever reason, this is not covered, despite this being needed under the hood in the implementations of these computer algebra systems that are even hidden in basic calculator apps on phones... it is quite problematic and pervasive on a systemic level. you even defend it claiming it is necessary because you are still blinded by the undergraduate maths program...
@@niikurasu2855 fwiw there's probably institutions in the usa that cover nonstandard analysis, and institutions in europe that skip it... also university here starts after college, not sure what your understanding of these terms is, but know that they are region dependent in case you mean some higher level class on analysis in general, and not the first introduction in what would be a class taken at the same time as a calculus course.
@@hetoan2 What he means is that right in 1st semester coming fresh from highschool we start with real analysis and proof based linear algebra (as a math major) without any introduction, but an optional prep course. What has been covered in this video (calc1) is highschool material. This is standard curriculum in germany for analysis: 1st semster: 1-dim analysis in IR 2nd semster: n-dim differentiation and topology 3rd semester: n-dim integration and measure theory In 4th semester i will take functional analysis and ODEs. In all of these courses everything is proven (but the proofs left as an excerise or theorems from other mathematical fields) and while you do some calculation the calculation/ proof ratio is ~ 30/70. Imo this is the correct approach for math majors. I have never studied none-standard analysis, but i am not so sure about your statements eitherway. You learn a lot of technical proof knowhow doing standard analysis and none-standard is more subtle thus probably way less suited for a beginner who has no experience with proof and formal logic. None-standard analysis might be the superior choice for physics students as they argue "none-standard" all the time anyway.
@@IsomerSoma standard analytical methods are the standard because they are easier. However, those proofs will always be less rigorous than constructive proofs that emerge from non-standard analysis. The standard approach relies on infinite sets and "real" numbers, which are ultimately non-constructively defined. In many ways, the non-standard approach is simply more elegant. Many of the non-constructive proofs are just waiting for the constructive version to come about to make them valid. This is literally the case for Leibniz calculus, and I don't see why non-standard approaches shouldn't be introduced early, as they are more solid logically. Also I get that you say physics is non-standard in approach because physicists gloss over the rigor because it aligns with the model and reality. However, nonstandard analysis is more in-line with pure mathematics. So really this comparison is false. For what it's worth, I am simply arguing for the expose of this topic in a general class for 1st years, which would and should also go over topics that would be relevant to other, more applied fields, such as physics, chemistry, and engineering. I'm not surprised to hear german maths education is superior to american... I can read statistics, lol. Calc 1 is also high school math, but it is not uncommon to have to re-teach this to undergrads because of the varying quality of public highschools. Many colleges here will not take high school credit because the foundations are weak. (this was not the case for my school though)
Hey, actual Calc prof here. One way to see why derivatives and integrals are opposite to each other is to remember that derivatives always represent slopes of tangent lines. Now look at a definite integral. If you extend the upper bound a *tiny* bit h, the additional tiny rectangle you get is that tiny amount times f(b+h), so that your area changes by about f(b) per unit of h. But that's the derivative of the integral showing up as f(b) again!
Dr. Peyam has another but longer way of doing it. Basically use the riemann sum definition of an integral and let the integral be of the derivative of a function. let deltax=(b-a)/n, let x subscript i=x+deltax. By the mean value theorem, there exists a f’(c) for c in the interval (x subscript (i-1), x subscript i) such that (f(x subscript i)-f(x subscript (i-1))/deltax=f'(c). Let the height of each rectangle be f'(c) -> integral=sum from k=1 to N as N->inf of [((f(x subscript k)-f(x subscript k-1))/deltax)*deltax]. The deltax’s cancel and we get the limit has N->inf of the sum from k=1 to N of [f(x subscript k)-f(x subscript k-1)]. The sum is also f(x subscript 1)-f(x subscript 0)+f(x subscript 2)-f(x subscript 1)+…+f(x subscript N)-f(x subscript N-1). All terms cancel except -f(x subscript 0) and f(x subscript N). take the limit as n approaches inf and we get f(b)-f(a) is the integral from f’(x) from a to b, thus showing the integral from a to b is the antiderivative of b minus the antiderivative of a. Also, any constants created will be cancelled out.
I was fortunate enough to take Calc 1 with a professor who absolutely refused to skip the rigorous proofs. It was really freakin' difficult to follow those, but it made the concepts of calculus intelligibly and consistently build on one another. The classes also refused to skip the "this case is stupid messy and requires a ton of algebraic manipulation" topics
Here's how I like to think about/justify the fundamental theorem of calculus Derivative = slope = rise over run = x/y Integral = area = x*y I know everyone else is complaining about calc classes but I wanted to throw that out there.
I was actually taught the rigorous definition of a limit when I took intro to calculus. however, my teacher didn’t really understand it herself and she abandoned the topic after one confusing lecture lol
Yo dude, I was in the same boat as you because I passed calc 1,2,3 but later realized i didn't know wtf i was really doing, just manipulating symbols. I took one summer to relearn calc 1 and 2 and one winter break to relearn calc 3 all on khan academy. It was well worth the time spent because Sal and Grant Sanderson (the guy from 3blue1brown math youtube channel) explain why things are true in the first place. For example, proving all the derivative rules, the connection between area and slope, why antiderivatives give you area, how double, triple, line and surface integrals really work, and other theorems you learn in calc. Of course, I also had to learn why some concepts in algebra, trig, and linear algebra (like the dot and cross products) were also true before I had to relearn calc. Although the information gained isn't really useful to the average joe since you never really use it unless you work in a specialized field.
This is true. Calc 3 needs a geometric part of the course with scientific computing and graphing by hand. Not sure how I'd figure out how to graph contour maps, etc without checking a real answer.
That's sad bro, but good you relearned, you shouldn't be able to pass any course without understanding basic theory, I guess that's why in my math and physics department they make fun of engenierings
@@43333akjfkgodel not sure why you’re so rude… my teachers taught math as plug and chug without proving anything, pretty sure that doesn’t make it my fault for not understanding it the first time.
@@markdave2456 did you take the whole courses or just what you needed to know? I think i need to do this with algebra because my prof is so plug and chug it’s infuriating
bro, you hit the spot. You're literally pinpointed each problem with the world and calculus and all the people that are teaching it , are missing a lot of things and themselves need to learn it again .
"you probably re-learn it in university anyways." I always taught myself this when I was in high school, and it was a mistake. The lecturers won't bother teaching why things go a certain way because you have learned it in high school anyway. They will go straight to solving problems, skips multiple steps then show you the final answer, leaving you confused.
I'm majoring in Computer Science and Calculus was taught to me in the 2nd semester. The professor pretty much taught it the way you did. He also addressed the same questions that people wonder whenever they take the course. And since I had already learned Linear Algebra in the 1st semester, the behind the scenes stuff with Calculus was kind of a breeze.
3:38 This is very relatable. As a highschool senior, I am currently taking Accelerated Calculus and AP Physics Mechanics. I have a D in Calc and an A in calc-based physics. My calculus class is incredibly dense and boring, and my physics class is fun and exciting. When my calc teacher speaks, my mind drifts (I've literally fallen asleep multiple times in her class) and my physics class is one of the highlights of my day. My physics teacher teaches more calculus in 15 minutes than my calc teacher does in a week, and I actually remember it. I really hate introductory calculus class, and I only put up with it because it is a necessary stepping stone towards the engineering degree I hope to pursue.
funnily enough, I have the opposite situation, though a bit different. I'm more than qualified to take Physics C, but they wouldn't let me, so now I'm in a combined AP Physics 1/Gen physics class, which is shaping up to be just as much of a disaster as it sounds like. My Physics teacher has spent the last two weeks giving exhaustive speeches about how physics is incredibly difficult, and that he doesn't actually expect us to be able to figure it out because of how mind-bogglingly complicated it is. If you were wondering, the thing he was talking about was that even if a velocity line is flat the object is still moving. I suspect we will have covered the AP Physics material around four or five years from now at this rate. I've learned more about physics through explicit connections my calc teacher has explained or by thinking, "huh, that calc thing we learned today seems kind of like this"
4:43 I’ve pondered this question for quite some time. And then i heard someone said derivatives is the operation of subtraction and division which corresponds to slope function and integrals is the opposite operation: addition and multiplication which corresponds to the area under the curve.
It makes sense in a way, but that's not the whole story. The reason why integration is the opposite of differentiation is more of a matter of how antiderivatives relate to the area of a function
I just Understood integration as Summation of Very Tiny Parts of area comprising into a bigger one where the initial and Final ranges may be fixed or not...That's it!😬
Integrals defining the area of a function is an easy concept to grasp in high school, but that was not his point. He was pointing out that somehow the area a function between x=a and x=b is given by just plugging in a and b into an antiderivative of that function and taking their difference, and this is almost never given a proper explanation in high school nor intro calc in college. I personally recall the summer right before I took real analysis when I started wondering why that was so. I had to look up proofs of it online and I was surprised how easy it was to prove so, that even a calc 1 student could understand it
i was so lucky to have such good professors for both my calc 1 and 2 classes. i’m not pursuing any further math than calc 2 because my major doesn’t need it but even tho i don’t like higher math, these two professors made me appreciate it more and understand what the hell these processes mean
Ive been pounding my head against a wall the last 3 days because I feel like i’m just memorizing and regurgitating rules and formulas without it being clearly explained why they work. I can’t find a source that explains the proofs in a way I can comprehend. I’m glad this problem isn’t exclusive to me.
As someone who has taken Calc I and is currently taking Calc II, the biggest thing that can make the difference with actually understanding the subject is definitely the teacher/professor, thankfully I had an amazing professor in Calc I who made sure to explain WHY everything was the way it was instead of just saying "Do this and this, and you get this", him explaining those essential concepts has made my understanding of Calc II so much better. I personally think high schools should give options to kids where they can take classes to better understand Algebra as well, cause if you don't really understand Algebra, then you are going to STRUGGLE in Calculus.
During middle and high school, I always followed formulas, saw a polynomial and thought "Oh I'll just plug it in a formula" and I get the answer a lot of the time. I never understood what those formulas meant. Come my first year of college, I took a remedial math course, and my professor summed up all 4 years of highscool math in a few months, not only that but he explained EVERYTHING perfectly, he explained how formulas were derived, explained why operations worked the way they work, etc. So now when I come accross a polynomial or anything, I don't blindly plug them in a formula anymore, I actually do the long math to make sure I get everything right,
It's been about a year now, and the only concepts I remember are the ones that were explicitly talked about in 3blue1brown's calculus series. I feel like I've learned NOTHING from those classes, and rather were helpful at nailing down the topics I learned in those videos.
I did cherish these five minutes and fifty-one seconds of calculus. It was interesting to hear your perspective on the subject, and I liked that you indicated curiousity. I like calculus, and I also would like to understand the relationship between slope and area under the curve. Calculus is genuinely fun and I think that math is just awe inspiring when you consider its long tradition. It's a gift, passed down from generation to generation and added to, traveling with time along the existence of our species. I will continue to learn more and cherish it. Thank you!
everything you want to know is covered in the first semester of university calc. an integral is an infinite series of rectangles and a derivative is a secant line between two points "infinitely close together" (x +/- delta). If you take the two points defining your secant line and multiply the distance between them by the height of the function somewhere inside that range of points then you get an approximation of the area contained under the curve between those two points. The approximation gets better as the points you choose (again, x +/- d) get closer together. That's why you get the ftc telling you that the derivative is the inverse of calculus. proving it rigorously is for mathematicians and i promise it won't help you apply it to anything unless you're a mathematician or a theoretical physicist.
Good comment. I was thinking the same thing. Rigorously proving this stuff would take forever as a student, it's better to just learn it and then naturally come to an understanding of Calculus as you go!
I took a calculus course in high school for 2 semesters and by the end of it even my teacher was confused. Literally asked me what grade I thought I should get. I got an A.
Thanks bro, my dad constantly reminds me to start form scratch and this vid really shows me why I've been sinking into a chaotic process in learning. But calc is hard to learn by yourself... I'll try and organise my learning a bit.
I'm not sure if anyone has pointed this out yet, but integration IS NOT the inverse of derivatives! Derivatives look at the rate of change of something (i.e., this is why the derivative of a linear equation is a constant = the rate at which the line changes is constant). The second order derivatives are usually called concavity, but they are simply measuring the rate of change of the rate of change, and so on for higher order derivatives. If you look at the limit definition, it is simply taking a slope! lim as h->0, (f(x+h) - f(x))/h, right? But note that h is defined as some distance from x. So, x2 is simply x+h, and x is, well x. Then delta x is (x+h) - x = h, just like in the denominator. The numerator is the function evaluated at x+h (which is x2), making this your y2. Similarly, the function evaluated at x is y1. So, you are simply doing the standard slope formula of (y2-y1)/(x2-x1), as the distance between the x2 and x1 values becomes VERY small. The extra condition of making the distance "very small" is justified because the tangent to a curve gives its rate of change at that point, hence, the limit definition of the derivative is simply finding the slope of the tangent line at the point x! Integration is the area under the graph, which is correct. However, that graph can be shifted up or down by any constant value C (it doesn't have to be at 0). So, if my curve is a sinewave centered at y = 1000, my area covered by the curve would be no different than the area covered by a sinewave centered at 0. If you understand this concept, you will understand that integration gives you a RANGE of possible functions (essentially, it gives you a function that can vary infinitely by a constant). So, if I take the derivative of sin(x) + 1000, I'll get cos(x), just as I would if I had done sin(x) - 100, or sin(x) + 35, etc. When I integrate cos(x), my answer is sin(x) + C, where the C signifies an infinite variation of the sin(x) functions - notice how this is not an inverse?! An inverse is, for example, multiplying 4 by 2, then dividing by 2, returns back 4. So, division is an inverse of multiplication (same with addition and subtraction). Finally, consider integrating a second order derivative twice: suppose our second order derivative gave us sin(x). Then, integrating it once will give us -cos(x) + C. Integrating this again will give us -sin(x) + Cx + C1! That means that not only are we getting an infinite constant shift, but ALSO an infinite linear function shift! Hopefully, this helps you understand derivation and integration better, and why integration IS NOT an inverse of derivation :) Best regards, Your friendly neighbourhood Math teacher ^_^
I'm planning to attend grad school for math next year, and I'll likely be teaching a calc 1 course in the near future. What do you wish you could've learned differently or more of that would've made your experience better? Stuff like rigorous proofs of the limit and Fundamental theorem of calculus are relegated to an introductory real analysis course that builds the fundamentals of calculus from the ground up in a completely rigorous manner. However, only math and maybe CS majors usually take this course.
I think the best way to teach is that there should be a clear reason and buildup to every new topic you introduce to newcomers since it is such a foreign concept. Rigourous proofs are boring yes but if someone manages to explain the reasoning behind it in an understandable maybe relatable way, i think it would entice students to want to learn more about the subject.
Use the hyperreal numbers to define an infinitesimal, then use that definition to define what a limit really is, instead of using the epsilon-delta definition. This comes from nonstandard analysis. Math should be one done over an infinitesimal system. It is infinitely better. If you need evidence, just read that statement: "one over an infinitesimal" is "infinity". That's a joke, but it truly is way easier to learn than you'd think, and the 2x2 matrix definition allows you to also introduce early the more accurate formulation of the complex numbers as well. It is also helpful in that it's just literally how it's programmed... so it aligns with our computational tools which come from linear algebra.
As a student and as a future teacher, i think the most important thing to start explaining something is the context. Why was the need to "invent" calculus? How was it created? Who did it? Why is it useful? And after that: RIGOROUSNESS
1) Remember - or teach - that Function is a model of causal relationship between two phenomena: f(x) - the "output" or "effect" - represents the evolution of one of them caused by (or used as reference for) the evolution of the other (x, or the "input" or "cause"). 2) When this relationship is continuous (like a "flow") we need tools to "break it" in measurable parts. So we start measuring the instantaneous rate of change of the "effect" phenomenon using Derivatives. In other words: this phenomenon is changing rapidly or subtle in real-time? 3) Integrals shows the cumulative "interference" of x over f(x) along the evolution until now - the net change 4) To understand the continuous behaviour of this relation, we need study Limits
I'm happy I took AP calculus courses at my high school. Having a full year for what is typically a semester course really let us go through more material and actually build on everything from precalc.
Wow this is so true. For me it was now over 20 years ago, I had to take calc I two times before actually understanding what a derivative was in an intuitive sense. Literally after getting through twice, while mid way into calc 2 I finally got it. And it was because I talked to my uncle who was an engineer and he gave me real world usage of what integrals and derivatives are used for, and it suddenly made intuitive sense. And now I realize I wasn't bad at math, I did poorly because of how I was taught.
As a 1 off you're hella good at this, just finished 4U functions functions from an incomprehensible "this is how yo do it, can't tell you why" teacher, headfirst into ONLINE calculus class, this has helped me get a much better grasp of the overall paths calculus takes and what I need to really work on
My fist calculus class in highschool (grade 11, for 16-17 year old students) actually started with epsilon-delta, and we had quite a few advanced integrals at the end of it. Even at uni level I could use most of that.
Calculus to me is more about applying concepts that are seemingly left out or forgotten when the class is being taught. Those concepts are the connections between algebra and the next level that you speak of. Like defining an integral to say it is a function that sweeps from left(a) to right(b) within the bounds of the function and the x-axis depending on if the function is above or below the x-axis.
the 3blue1brown course is so good it gives a vague idea of the connection between slopes and areas in literally the first video, before using the limit definition of a derivative and still make complete intuitive sense better than any math class i ever had
The issue with bring theoretical math all the way to first year is that it'll throw a bunch of students off, particularly for those who just want to use calculus. What's best is for students to be able to choose whether they want the rigorous math treatment right in first year, and know what they'll learn or miss out on. Of course, there are unis that do this already
Currently at midterms. Going to take my first calculus class next semester. Shits looks scary af. My highschool was as shitty as it could be in terms of math.
Okay this may sound slightly different, but I have to point out just knowing how to do the computations is a big win for when you actually learn the concepts rigorously. For example, I would always try my best to learn and answer questions about limits/derivatives/integrals, why they come together, but no matter what my high school courses only cared that i knew HOW to do them(obv i tried researching beyond the scope of my course) But in uni when you finally learn the rigorous stuff(like epsilon-delta proof) you see how amazingly everything fits together, and at that point you don't have to worry about learning HOW to do stuff, because you already know it, but instead WHY stuff happens. It's the same thing with other stuff in math too, you need to just sometimes grind through it before the puzzle starts to come together. And if you weren't fluent in computation you probably wont be able to give the theory behind it time. Grinding on problems to just get good at computation is underrated Like you got fluent in basic arithmetics (addition/substraction...) before you actually started applying them in word problems right? Same analogy.
I get your point and I felt the same way. The entirely of Calc 1 made me feel like a complete idiot because of all of my classmates could jump through hoops like monkeys on demand while I still questioned why sin became cos. Almost flunked the class but after college I feel like I’m part of the 1% that still uses calc from my class while my former classmates couldn’t use calc to save their life. But I have to ask, what is the alternative? Is there really much utility in going through the algebraic tricks needed to produce some of the more obscure derivatives? At the end of the day, calculus is a tool to achieve a goal much like algebra. Should we have spent a few more months on the foundations of calculus?
I took calc 1 during the summer and my teacher spent ample time going through the rigorous proofs for several concepts (including the limit proof you showed) instead of assigning us exercises to practice computations. Needless to say, I ended up dropping calc 2 on the fall and taking it the following spring ‘cause I had no clue what I was doing. Got an A the second time around at least lol
Here's my understanding of 4:45 . When you integrate a function, say f(x), from a to b, you get the area under it, which is also F(b)-F(a). f(x) is the derivative of F(x), so it is the instantaneous slope m of F(x) at every value of x. Therefore, when you integrate f(x), you are actually adding together all the mx for every infinitesimal change in x. Since y=mx, what you get is therefore the change in y from a to b on F(x), which is the same as F(b)-F(a).
The ED proof was the first thing covered in my high school calculus class. The teacher was nice and simplified it down into normal graph talk, explaining in terms of functions.
I really feel bad for y’all who got introduced to calculus this way. I didn’t take it until college after taking years away from math courses and they went through a lot and had us construct a lot of proofs which helped establish the theoretical foundations of the work we were doing (epsilon delta proofs of limits, riemann sums, fundamental theorem, etc). This was even in courses mainly for science and engineering students and not just the math majors. It was actually interesting and it really felt like I was taking a peak at just how vast and deep mathematics can really be by being taught some fundamentals tools of analysis. I used to hate math but college (along with good youtube math communicators and me getting farther in my science education) really helped me change that view
i'm glad i did, i would have had 15 seizures at once if my freshman year calculus courses went any deeper into proofs than they did. although it also was a large classroom and i didn't know i was near-sighted until halfway through integrals.
I could not be more greatful to have a good AP calc teacher. I'm comming up on the exam now and evryone in the class actually understands everything bc he taught it well enough
mickpuffi believe you could be one of the best math teachers on youtube if u continued this gen z way of teaching. so please do make more. if you struggle to find where to start, you can do the NSW Stage 6 Math Adv or math ext syllabus
This video resumes how I felt my 1st year of physics at uni, just learning how to make calculations that my phone can do faster and better, without truly understanding what I was doing. Glad I switched to math.
I haven’t built the proper lebesgue integral yet, but we did do Riemann. Basically, the integral from a to b is defined as the area under the curve, and by building it from piece wise functions and then using uniform approximations you can have a generalised integral. Then from that construction you can prove the well known properties such as fundamental theorem of calculus, linearity, substitution, ibp. The thing is it relies on having a mesured space so vector spaces with norms work but more general topological spaces are problematic hence the lebesgue integral which my professor has hinted at many times
I’m in math analysis rn (10 grader) and my teacher has actually done a really good job explaining the logic behind limits and derivatives so far. Luv you ms Rosenberger you a real one 🫶
all this video did was make me appreciate how my own cal a class was structured, starting with limits and then going from limits into the idea of a tangent line to a curve representing the “slope” of the whole curve at that point, which then led into derivatives. Then, anti derivatives were introduced before integrals, shortly followed by the indefinite integral, which was explained as representing the set of all antiderivatives of the integrand function. From there, we briefly covered finite sums and sigma notation and then Riemann sums, which served as the basis for definite integrals. (Most of the other methods for solving integrals were also gone over before the Fundamental Theorem was introduced). So, yeah. Thanks Dr. Bossaller
I could not have understood what you are saying a few weeks back, but as my 12th grade board exams are approaching, I had to learn Differentiation and Indefinite integration and I can totally agree with everything you said! Though it's easy, the questions related to Trigonometric values or other algebraic values inserted in it makes it difficult and you really need to memorize a lot of stuff to manipulate those values and get correct answer!
Integrals are like asking for a running total. So you're adding things. Differentiation is like asking, "How is this thing getting bigger or smaller?" so it's like subtraction. That's kind of why they undo each other. Remember that dy is kind of like a small interval between values of why. y2 - y1 => Δy => dy. If you let the difference between y2 and y1 approach the infinitesimal, you get dy (this is lazy talk for intuitive purposes). Thus the derivative dy/dx is kind of like the difference in y over the difference in x. Or rather, how y changes with respect to x. So derivatives are essentially fancy subtraction. As for integrals, you are summing up all the infinitesimal dy's or dx's. . One other way to think of it, which you will see later in calculus, is that differentiation gives you a lower dimension while integration gives you a higher dimension. For example, take (4/3)pi*r^3. What is that? That is the volume of a sphere. What happens if you take the derivative? You get 4*pi*r^2. Guess what that is? A _constant_ times the area of a circle. You turned a 3D circle into a 2D one, except you have that multiplied constant. Now, take 4*pi*r^2 and take another derivative. What do you get? 8*pi*r. What's that? Well that's 4 times 2*pi*r. And what is 2*pi*r? That's the circumference of a circle. So what's important here? (4/3)pi*r^3 is a _cubic_ function. 4*pi*r^2 is a _squared_ function (a parabola). And 2*pi*r is a LINE. That's right, it's a linear function (just connected together into a circle. Think y = mx + b, except m = pi, x = r, and b = 0. So what happens if you go in reverse? Let's integrate 2*pi*r. What do we get? Using the power rule, you get 2*pi*r^2 * (1/2) + C = pi*r^2 + C. So now we have almost the same thing as when we went backwards. The difference is that instead of some constant C, we had a coefficient of 4. But the point is, you're getting back the same type of function, with the only difference between the coefficients and constants. But you can change a constant into a coefficient anyway. For example, 5 + 26 = 31, but also 5 *(31/5) = 31.
My final exam for calc 1 was 3 hours, no notes, no calculator. Prof had basically skimmed over trig functions in the course, literally every single question on the final bar one or two were complex trig equations.
In Israel (I'm personally in the most accelerated "normal" course (technoscientific reserve (amat))), the teacher shortly explained limits and then we did the power rule and most of the other rules, except the chain rule for some reason? It's currently been half a year and the most advanced derivative we've done is fractions with square roots, so I don't know how that compares to somewhere like the US.
I took calc 1-3, diff eq and linear algebra before Covid then dropped out, came back to finish engineering degree this year and remember nothing. I want to go back and relearn the proper way (I never did proofs besides in Lin alg) but don’t have time between work and classes. Great video
I took the advanced first-year calculus course at my uni this semester and it spent the first 5 or so weeks defining sets, the notation, then going into sequences and limits (rigorously proving everything along the way), and then finally went into derivatives, integrals (using the riemann definition with partitions and so forth), and finally we are now onto series. It definitely isnt introductory since it assumes you did well in the advanced maths at high school which include a lot of calculus
I’m in my junior year of Highschool rn. Taking both AP Calc and AP Stat made me realize the “trust me bro” meme so much better for the reasons stated in this video
Watched a video on RU-vid that explains why differentiation was invented and how it was found, helped clarify the concept for me and also helped clarify how limits and differentials were connected. I greatly recommend doing that
Im taking real anaysis now as a senior in university. Love the epsilon delta proof of a limit. Makes a lot of sense. But, I'm not sure if its necessary to learn as an average student that will not study some sort of stem major. Sometimes getting familiar with the process using real numbers (instead of variables like in rigorous proofs) is easier. However, understanding that the derivative is the slope of tangent like helped me tremendously when I learned Calculus. I had a great professor that explained the "why it works" aspect and it would help when I was able to follow it along.
I saw some other comments explaining the fundamental theorem of calculus, but the way I intuit it is like this. If you're driving in a car and want to know approximately how far you're driving, every second or so you can look down at your speed, say 25 meters per second, and assume that your car will be maintaining that speed until next you record your speed. In that second, then, you know that you just travelled 25 m/s * 1s = 25 meters. If you put down where you are when you start the car, then keep up this method of recording how far you've gone each second, you'll get an approximation of how far you've driven. When we put this on a graph, it turns out to be exactly the Riemann approximation of the area, where the base of each rectangle is the amount of time between recordings and the height is the current speed. If you squint your eyes a bit, you can say from here that for smaller and smaller times between recordings, the approximation will become more and more accurate, and you get the integral.
That f(x+h) - f(x) all over h part is sooo true, learned it in physics this year before we even began learning about derivatives in math and felt my brain actively dissolve
@@hashtags_YT that's because that's the intended idea of a derivative. The definition of a derivative comes from the rise over run equation (y2 - y1)/(x2 - x1). Just that by adding h that approaches 0, f(x + h) - f(x) wouldn't be 0 since h is approaching 0 and never at 0, then dividing it by h will give you the slope of an exact point at a curve. Then the power rules and other derivative rules are generalised versions of the definition of a derivative to make things easier to compute.
@@sek5372 Yeah, I get it now, but I had absolutely no clue when we began. I had no idea what a limit was, or that you could even calculate the slope of a non linear function. And that was on top of my physics teacher not being good at teaching and more or less just writing down the formula and an example on the board before calling it a day. It wasn't until the very next day that I was properly taught that stuff in my math class that it actually began to make sense.
Integrals and derivatives are opposites cause one is variation and other accumulation... so the accumulation of variations would be total variation F(b) - F(a) but F(x) was previously defined as the area under such f
think if they tried getting the students to understand calc in a more rigorous fashion it'd take up the entire school curriculum and the souls of both the teachers and students LOL
Is that so? In my first calculus class we had all the "basic" derivatives and "basic" integrals (exponentials, logarithms, trig, square roots, and combinations of those). We had proofs for the fundamental theorem of calculus and had a mention of the formal limit definition
