@@ronniechilds2002That's wonderful! Both you and the person above are truly inspiring! I'm still learning mathematics myself at 35 years old. It's been hard to find as much time and energy as I'd like, but it's definitely worth the effort. I'm currently focusing on Calculus II, because I want to better prepare myself for Differential Equations.
When I started the journey of Calculus 2, This vid impressed me with 2 approaches towards the area. Now I'm back for a replay after completing the first round of learning. In particular, to address the vagueness in the summation approach. The extended summation formula is pretty interesting. But frankly, despite that the explanation is silk smooth, the first approach is a certain headache for many and therefore is a weak point from the planning perspective.
Once upon a time, I wasn't good at math. This fact didn't bode well for my ambition to obtain my degree in mechanical engineering. Ultimately, I found the motivation and put the work in to conquer and subsequently obtained my degree. There was a point when the learning curve abruptly flattened. There is an awesome beauty in it, once I learned how to see. And it was there and always will be. It was like seeing God. While there might be impossibly difficult problems to solve, I would never scared of math. This little lesson made me feel that again. Thank you.
Thanks! We hope to post lessons from our newly launched Precalculus course in the future. In the meantime, check out the course page for more info: www.outlier.org/products/precalculus
From a pedagogical perspective, I found it really important to balance [x -> 0] with the notion that as that happens, the 'n' in the summation also "becomes arbitrarily large". In fact, so much of Calculus is about infinity/division by zero *in tandem* that I'm tempted to point it out at every point in the curve...
If you tell it in language, it sounds easy : the change of accumulation of something, is that something itself. Like distance is an accumulation of speed over time. And the change of distance over time is speed.
This was so well explained but more than that there seemed to be subliminal forces at play to create a cosy safe nostalgic setting...... The soft lighting, the neutral clothing, the analogue watch, the fountain pen and the soft voice. This creates lovely environment for learning - thank you.
I was thinking the same thing. Presentation matters. And Dr. Fry is calming and charming but very persuasive, too. Dr. Loh is different but has the same effect too. I'd buy anything from him or Dr. Fry. Good thing they don't sell used cars.
Where was this video when I was struggling hard to understand this very thing?? I vaguely remember the teacher talking about Darboux's superior and inferior sums and Riemann making an appearance... Oh, and having to memorize a table to primitivations and derivations (my absolute most hated activity in any learning environment).. Excellently explained.
Yeah, big names do without further elaboration and context do leave students mystifies and mystification of something leads to fear as humans have the tendency to fear the unknown.
This video gives a nice easy example of how to USE the Fundamental Theorem of Calculus. But from the title, I thought it was going to explain why the Fundamental Theorem is true. Why is dA/dx equal to y? When I took calculus, I didn't have much trouble using the theorem, but understanding why it's true is something I never fully grasped. Is there another video in this series that proves the theorem?
It always helped me to rearrange and visualize the same equation as dA = y dx. With y being the vertical length of the small dissecting rectangle and dx being the width. And total AREA equals LENGTH times WIDTH. As is with any rectangle or square. Hope that helps a little.
You have identified exactly what handicaps so many people when it comes to calculus. This video, (and most poor teachers) emphasize methods, approaches and rules, while you ask *”but why… how do you know that*” A person can be good at calculus just by ignoring those questions. To be great at calculus you must have someone answer those questions. Alas, most teacher’s don’t even ask themselves those questions, thus handicapping their best and most inquisitive students.
Calculus always gave me fits. I passed the classes, but I never really understood the information. It wasn't until recently that I grasped the concepts. One way I made a connection was taking a known formula, the line equation (y = mx+b) and taking its integral between 0 and 1. That results in A = 1/2x^2. Rewritten a bit, that's the formula for the area of a triangle (A = 1/2bh). I guess my calculus profs didn't want to explain it so easily because they needed to stretch the instruction across three months to justify their tenure.
Brings back some pretty scary memories from my university days long time ago. Today I use a simple casio calculator.... the trick is to use it wisely.. hopefully always with a plus sign in front of the result.
I noticed a couple of non-essential errors in the Riemann sum: one subscript she has as "i", when "1" was intended; and, slightly weightier, the widths of the blocks in her notation is (1 / (N - 1)) rather than (1 / N).
Thank you for your explanation of the FTC. In my opinion, Calculus is difficult to comprehend. It is based upon a nebulous, philosophical object, i.e. the limit. For the elementary functions y = x^n, there is a simpler, easier to understand approach, called Algebraic Calculus. It is based upon the invariant in conjunction with box operators.
I taught Industrial Control System. Which is a practical use of calculus as I’m sure you know. I’m 71 years old now but I clicked on this to see your take on this subject. We were trying to control to set point. We would compare a measurement of the process to the desired set point. The difference would be acted on by integral to bring the measurement back to set point. Derivative was use to act on rate of change the measurement was moving away set point. Derivative was used to predict the change.
A simple easy-to-understand concept was made to look 'scary' and 'intimidating' by the fancy marks used to symbolically represent it on a page. Good presentation!
I feel that your explanation is a bit circular. For integration, you say straight away that "we know, don't forget", that y = dA/dx. This is already directly using that integration and differentiation are opposites, so you are somewhat using the fundamental theorem to verify the fundamental theorem. Integration is the "continuous sum" here, that's why we use a stylized S to represent it. That's the part where you should be doing Riemann sums to "integrate" the "parts" forming the area under curve. Differentiation, on the other hand, is all about rates of change... "Instantaneous" rates of change (i.e. slopes of tangent lines). And to the uninitiated, there doesn't seem to be anything obvious linking these two concepts together. This is what makes the fundamental theorem so... "fundamental". It is precisely this theorem that ties these ideas together, and... In the most remarkable way! Not only was that continuous sum (area under the curve) related to the instantaneous rate of change of the function (albeit the ANTI-derivative), but that sum turns out to be completely determined by only evaluating the anti-derivative at the endpoints of the interval. Just those 2 points were all that was needed to completely determine the area under a curve. This still gives me the shivers. Further, this exactly same concept will show up again and again in higher dimensions, with Green's theorem, Stokes theorem, and Gauss' theorems, although with a form of anti-derivative that needs to be made precise... Always, the area or volume being determined by evaluating that anti-derivate on border or boundary of the region or area being summed.
She never explained where the Fundamental theorm came from. As i remember from more than 40 years ago from my calculus, there are two theorems that do the same thing. One is easier to prove than the other. Most of us use these theorems everyday without knowing or caring they ever existed.
Awe, you missed out on so much good stuff especially when you get into vector calculus, analytical geometry within the context of the complex plane such as with Fourier Series and Transforms, Laplace Transforms, and even ODEs (Ordinary Differential Equation Solvers) such as RK2, RK3, RK4 known as Runge Kutta, or even exploring the Hamiltonians such as with Quaternions or Octonions, the Lie Algebras and such. Then again 46 years ago from the posting of your comment was a couple of years before I was even born, lol. You're talking late 70s, I wasn't born until late 80... Calculus itself as in the basic concepts is quite simple, however, the approaches and techniques into using them to solve many various problems are many and can easily become complex. Well, I don't mind the math I've always been decent or really good with it, but that's also in line with my liking towards basic Calculus Based Newtonian Physics, Electroncis or Circuitry both digital and analog. I've always been fascinated with how circuit boards and the various components soldered to them are able to produce various things such as audio or sound, light or images being either pictures or even framed animations, etc. Electromagnetism and wave motion is where it's at! The rest of basic physics is pretty cool too.
Congratulations on your pregnancy, Dr. Fry! I’ve seen your contributions to maths education for years. It’s going to be one smart kid. Thanks for doing this video.
@@xl000 What? Because starting a family is a beautiful thing. That’s what you do when people are pregnant, right? @benthepen6583 Then I guess they’re already a smart kid.
Thank God Dr. Fry isn't selling me stupid things, I'd be broke. I would likely buy anything from her. She's so damn persuasive. I watch a video of her and I instantly want to go do whatever she just taught.
A nice initiative done by teachers but i would like to suggest you that you should try to upload free course on entire calculus and mathematics students will come to know how you teach and a nice exposure for them to clear their concepts it they wanted advance attention so they can refer to paid courses . Actually in INDIA every teacher first available their courses free on youtube then they go for paid .LOVE FROM "INDIA"
Dude, its ok if poor students from third-world developing countries cannot pay for these courses. We are not their target audience anyways.................Remember, people who pay for these also have to earn.
A very poor explanation. Uses terms not previously defined. Derivative, limit etc...Talks about δx as a strip but never makes it clear. As an introduction it is useless. Just read any of a thousand introductions to calculus in maths textbooks and you will be better off. Clearly not a maths teacher.
Nicely presented, but this kind of A-level style explanation is more or less the reason why I failed my maths A-level many years ago, ironically at Harlow college 🙂. The use of y instead of writing the function explicitly always confused me. Then there's the dy/dx notation (thanks Leibniz), which looks like a fraction but isn't and is often treated like one through an unstated application of the chain rule. It's really confusing too. Also, terms like "tends to" presented in a hand wavy kind of way don't help either. Personally, I would have done better if A-level study had been extended by a year to introduce FOL and epsilon-delta, plus a much more thorough account of the reals and their construction.
Happy to be able to balance a check book add subtract. Hatsl off to those who aspire to higher learning. Ihave always considered mathematics the language of the Gods.
😮*Clarification* Xi is actually equal to i/N So that makes X0 = 0 X1 = 1/N and XN = 1 when doc wrote X1 = 0, she was really saying these above statements in a shorter way... 5:04 this video explains the main idea behind the subject of Calculus as a whole... She's right, they don't call it the fundamental theorem for nothing... 15:43 without this idea, there wouldn't be a Calculus How did Hannah Fry give me math with such a straight face ...and still made it so fun?...She's the only Fry i enjoy...📈
The necessity to incorporate lots of jargon and symbols in relating this impedes students' understanding. The student must first grasp that the rate of change ("derivative") of the area under a curve ("integral") of a function at any point (x) equals the value of the function at x. The derivative of the integral of function is the function.
Thought I was maximally infatuated with Professor Fry and then I saw her handwriting. I really hope they hired some artist for the closeups because if that's really her it's just too many talents. Ist this just ASMR for überdorks?
This is the first time I saw the connection to area - maybe I missed something earlier in my life, but you have opened the door for me to Maxwells and Faradays equations (I love electronics)
I flippen loved Calculus at school. Especially integrals that took like 2 A4 pages to solve...given all the tips and tricks learned. Differentiation was just a doddle. However, h o w e v e r, what we were actually doing, by solving it, was lost on me...and it is only slightly touched on at the start of this good lesson. Tbh, i am still not getting it, except that integrating is to calculate the area under a curve. For what practical purpose? who knows. I still don't know. hahahaha
The first method through the use of summations can be considered or linked to the Riemann Sums. The second method is based on the Definite Integral. There is another form of Integration where there are no bounds which is typically considered either the Indefinite Integral or simply the Antiderivative. Typically, the Definite Integral will result in either an Area or a Volume depending on the types of integrations where the indefinite integral will return a family of functions in which you must also append or include the constant of integration. This is because the derivative of any constant value converges to 0. So, when we evaluate an indefinite integral, we must also incorporate the constant C. Now outside of the scope of limits themselves which are a fundamental part of Calculus we can treat the process of taking a derivative and the process of integration by considering them to be what addition is to subtraction, integration to derivative, or multiplication to division, or exponentiation to finding the radicals. They are in a sense; inverses of each other except that they are usually almost always associated with a given variable, an unknown that we are trying to solve for. Sometimes it might require a derivative to solve for a given unknown, sometimes it might require integration, sometimes it might require the 2nd derivative, sometimes it might require combinations of them, and sometimes we just might not be able to solve it at all. Understanding that which was mentioned above, there is a very useful shortcut or technique in evaluating the integral of a given algebraic polynomial. It is simply taking a given polynomial of the form f(x) = ax^n as the given function or differentiable smooth curve and its Antiderivative or Indefinite Integral will have the form: F(x) = (a/(n+1)x^(n+1)) + C. The above formula is simply stating that we take its current exponent of n and add 1 to it, then we divided ax by the new exponent where ax is both the variable of integration dx with respect to x and its coefficient. Simple examples: f(x) = x^2, F(x) = (1/3)x^3 + C f(x) = 9x^4, F(x) = (9/5)x^5 + C f(x) = 12x^5, F(x) = (12/6)x^6 + C = 2x^6 + C Now as for other forms of integration such as with other algebraic forms, trigonometric forms, logarithms, etc... there are many other techniques. This is just a simple explanation for any who are willing to read this that may need some help into understanding the connections relationships between derivatives and integrations. The reason why we have the constant C within the context of indefinitely integrals is simply due to the fact that when we take the derivatives of the following set of functions: f(x) = x^2 + 7, x^2 - 8, x^2 + 9, etc... They will all evaluate to the same derivative of 2x. This is why the indefinite integrals is known to produce a family of functions as opposed the definite integral between two bounds that yields some value either it being an area, volume, hyper volume, etc... Just some fun facts!!! Hope this helps those who may need it.
This is all going back 30+ yrs for me. From what I recall of the methods used here, The First is Newton's method and the second Method, which I use, is Leibniz's Method. IMHO, Leibniz method is much faster to use and less prone to errors.
Lovely ginger-haired woman with a cool, smooth British accent, explaining the precepts of calculus to me. Delicately sketching the equations with various gold-nibbed fountain pens. It was like being secured to an overstuffed winged back chair as this beauty ran her fingers through my hair and waved a scalpel before me describing which parts of my anatomy she was going to slice off. I tried to follow along. My head ached. I just stopped caring and listened to her voice.
This is a little too advanced for me. I understand the geometry of calculus, but can't understand how she's getting 1/N^3, unless it'd be 1/N^4 if you just expanded it one more term, but that just wouldn't fit the formula, as the formula is meant to work through the first three terms. As I know a lot of calculus is formulas that men far smarter than me make. And I'm also seeing this is like quadratic equations, where the math represents a shape, which that 1/3 would ratio to any area in a number. Like I'm noting wherever you graph the two points on the parabola, for x^2 it will always equal 1/3 underneath it. Like a quadratic equation, it just fits into a shape, that happens to be 1/3 of the area of the rectangle you make out of it.
The other day I was reading about the indian mathematicians and I found the work of Bhramagupta and Madhava. I played with the Madhava series and I realized that if I multiplied each result for a high I was getting the area of a curve.
I could never jive the idea that the area under a curve of ((fx)) applies in Electricity and Magnetism or other naturally occurring phenomenon. E.G. Voltage and Current/amperage and other forces. Or even in business models.
I want this type of video for whole of calculus. I cant link any thing in calculus with each other. Understanding differential equation would be great. I failed 5 times in structural dynamics which involve differential equations to solve the equations of oscillations or simple harmonic motion. please make something that glues all parts of calculus with a simple example that is easy to understand and wont test my attention span.
I always struggled with calculus, it’s one of those topics that I know if I could just master it I could really do some serious stuff. I’m ashamed to say I am an electronics engineer and although I understand in practice about rate of change and the gradient of the slope etc as well as taking an RF AM Modulated carrier and that the area of the envelope for 2 given limits of time is described by integration … I have managed to get by but when K was a kid my maths teacher called me stupid because I dared to ask him to explain what he said a few minutes ago one more time… are you stupid son? he said…. Well it took me until I was 52 and when I was diagnosed with ADHD that I now understand the problem. This is no disrespect mathematicians for me have to assume a lot of things when they are explaining a concept, which is ok otherwise we would be here all day going back to first principles …. But when you encounter someone with an ADHD brain, we want to stop you at every point and say…. Whoa whoa whoa… just rewind and go back, can you just explain that bit a bit deeper there … what’s happening is that for someone like me to take it in, I would need to like put you on pause whilst I go away and fully understand and prove with examples the assumption that you as the teacher has to just assume the class knew if that makes sense. So and this is for anyone else out there who is reading this with ADHD, remember if you don’t understand something it’s not because you are stupid it’s just that your brain keeps interrupting and says …explain explain…. Explain….so if you want to learn something listen to the instructor like this wonderful,y clever young lady here and each time you do t get it,,,pause the vid and go off and research the bit and when you understand it come back and rewind until you get it and then watch some more… we Ai t stupid we just have to work at some things a bit more ….😂
Not sure what the intended audience for this is. Sounds like on the level of plausible derivation but is not really a rigorous introduction to definite integrals.
This is brilliant and very concisely explained. The teacher’s accent is also very relaxing to listen to, like ASMR, so it makes it much easier to focus. Thanks for the video!
