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.
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.
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.
@@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.
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!
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.
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.
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
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 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!
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.
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.
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.
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.
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.
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.
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.
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)
😮*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...📈
My calc teachers taught the mathematics in the notation without the perspective. I wish my Calc 1 teacher put it this way (so simply and intuitively) in my math classes.
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.
I have studied mathematics since the early 90s. I taught myself the fundamental theorem at this time, when I was about 14. Since then I've acquired many degrees, including a PhD. I just want to point out... The fundamental theorem does not require the infinite sum. This discovery came more than 150 years later by B. Riemann. Newton and Leibniz each discovered the theorem independently.
It’s not N cubed in the summation it’s N squared. The. N cancels one N leaving N in the denominator. THEN you can let N go to infinity to end up with the answer.
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.
Indeed, as "A Tour of the Calculus: Berlinski, David" suggests, the fundamental theory of the Calculus is the fundamental theory of the Calculus... (recursive definition being especially significant!) ):-) Simples...
I took only Calc 1 and 2 in undergrad 48 years ago. I struggle with simple algebra now. My brain is shot. I have something more than a vague memory of perhaps 1/100th of 1% of what she is doing. But because of her accent, I'm madly in love with Hanna and now have a strong desire to go back to Algebra 1, then geometry, Trig, Functions, and then the hard stuff. I really feel quite stupid, not because I can't remember this stuff or never quite understood it when I passed the courses, but because I have a crush on a woman and I'm almost old enough to be her grandfather. And if I'm not mistaken she could be pregnant in this video? Brains are truly very sexy.
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.
The first squiggly line she drew as y=f(x) was just an example of some curve. She never defined f(x) for the squiggly and instead used y=x^2 for the real example to work through and find the area under.
9:13 limN(N+1)(2N+1)/((N^3)6) she cancels out the N/N^3 to get 6N^2 in the denominator. Then the next line at 9:25 it became became 1/6 lim(N+1/N)lim(2N+1)/N. Shouldn't the denominators be N^2 not N? When & how did N^2 in the denominator become N?
I don't know if there's a theorem that says the two limits multiplied equals the limit of the two expressions multiplied. But suppose there was: lim(N+1)(2N+1)/N^2 = lim(2N^2 + 3N + 1)/N^2 Which also approaches 2N^2/N^2 = 2
@@divermike8943 Just listen to her words: "...times the limit..." or try it for yourself and calculate: [(N+1)/N] x [(2N+1)/N] = [(N+1)x(2N+1)] / [N x N] = [(N+1)/N] x [(2N+1)/N] / N^2. Right?
It is important to understand that there is NOTHING to understand here. It is just a rendition of agreements on how to write something down, it is all symbols of ideas. This is the one thing people fail to understand. It is just agreements on how to write something down. And a so-called solution to a mathematical problem is just a simpler way of writing something down. Solving a mathematical equation is actually a puzzle. If you understand these two things, maths becomes simple. If not, it is impossibly complex. Most teachers fail to elucidate these two basic tenets of human communication on mathematics.
This is exactly the thing I needed to hear 20 years ago when I was still in school. Realizing this changed everything for me. Thank you for spelling it out so nicely!
3:10 I agree. I really didn't get your first green blocked introduction though. I know what the symbols say but there was no intuition. That was a massive intelectual leap.( ummm sorry, I haven't seen your earlier videos so my comment is moot) Lebesgue, Lebesgue, where is He! Dunno Mate said the builder, off measuring somewhere. Your brain patterns of explanation work different to mine. I am not criticising, just learning. :-)
Thanks for the very thorough video. Slightly off topic question. Could I ask if anyone knows the maker of the fountain pens used? I am a scientist myself and do still enjoy calculating by hand (yes to this day). I have been struggling to find a good fountain pen and this looks like it writes cleanly and Dr. Fry appears to be applying a bit of pressure too. Mine always brake when i do. Thank you
Using a pen requires years of muscle practice. It is not a given, it is learnt by practice. On a good knib the ink flows without any need of pressure. In school I used to bend by knib this way and that way until the ink flowed acording to my writing style. I suppose a good Knib is like your own horse?
@@alphalunamare We’re Kiwis who lived in France 20 years ago. Handwriting was an important skill in French schools and graduating to use a pen was a big deal.
maybe it's just me, but this seemed to introduce way too much material too quickly, and not explain any details of the notation or concepts. (In the first minute, you write the FTC, then seemingly as an afterthought define integrals as limits of Riemann sums.) Then it looks like you're talking about both parts of the FTC, without mentioning that they're different results. (I think?) Not sure if this was meant to be more of an overview as opposed to a typical lesson in the course, but if I were a student I would have a hard time learning from this. (Also why is the third timestamp called differentiation: both it and the fourth timestamp are integration, just different methods, no?)
This is the problem with modern education: complex concepts are given 3 minutes to be "explained" on the blackboard, 2 minutes for students to write them down on paper, 1 minute to solve an integral several pages long. The original explanations by Newton were more detailed, especially on the concepts of derivatives. Because education then didn't suck - all questions and doubts were answered, with plenty of time to digest.
Thanks, Professor Fry, and I see there is elegance in calculus. But I STILL don't get it. I might check out the pre-calc course, just because I hate being so frustrated by this. In college (many years ago) I struggled through the 2 calculus courses that were required for my B.S. degree. This despite being an A student in High School for algebra, geometry (loved proofs) and trig. As math builds on itself perhaps I missed a concept leading to calculus. Reminds me of when I was in grade school, 10 or 11 years old, where I was out sick for a couple days when the class was taught the signs/operation for "intersection" and "union" between sets. I was clueless for the following weeks until a friend showed me what they meant. On the other hand, maybe trig is all my brain was wired to handle. That's ok too. But for anyone with young kids -- I'd say engage those sponge brains on this stuff.
Sometimes the frustrations of learning math are motivating and sometimes they just shut us down. We're sorry your relationship with math is hitting calculus bumps. Some precalculus knowledge is a prerequisite for this Calculus I course so we hope you do check out our Precalculus course, as it's designed to break down concepts visually and it's taught by world-class math communicators. Here's the course page for more info: www.outlier.org/products/precalculus Happy learning!
@@OutlierOrg No, This video explains a complex matter, much, much too fast, if you want to explain the riemann method for the fundamental therom of calculus you can't just chuck it in a video and assume everyone watching it is familiar with the trapezium rule, summing an infinite series and the b-a/n dx substitution and how to calculate F(Xi). A video like this should have at least links or something to help with explaining the aforementioned, as none of this relates to or would show up in pre-calc anyway.
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.
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/n.gif)
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.
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.
You had me at multiple colors of fountain pens. Great explanation for this theorem.
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.
PID?
@@imacmillexactly, just say PID
@@imacmill meaning "programmed integral derivative"?
@@tmst2199 Proportional-Integral-Derivative.
@@sandworm9528I have no idea what PID means I’m glad they described it the way they did.
Ah! O Level maths, 40 odd years ago! Happy days. I loved this.
Never used it since!
I'm a 82 yo male self-teaching myself calculus and this presentation was very clearly done and I'm finally getting the theorem! thanks, good job,🙂
Keep on learning!
Good luck sir, very inspiring
I'm attempting the same thing. I'm only 73.
@@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.
Im doing the same thing. Im 37.
Math is great for me. Helps me relax.
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!
Very very true
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.
A very ASMR lecture of the fundamental theorem of calculus ✨
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.
that was 2019
Why would you congratulate someone you don't know on her possible pregnancy
@@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.
@@MarcusHCrawford when people you actually know are pregnant. Like your mother, sister , cousin, close friends, maybe coworkers..
@@xl000
I didn’t know there was a rule that you couldn’t congratulate people on growing their family.
Make more of this kind of lucid explanative lecture. These are very easy to understand
Remarkable initiative .....Guys, Could you upload some free series here on precalculus.
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
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.
Welcome back to UA-cam. This was very entertaining, reminding me how to do things right (I aced AP calculus 41 years ago so I sorta knew it already).
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!
Anglophilia irritates me.
You're an excellent lecturer.
Hot for teacher.
Nice fountain pens! Doing calculus in ink = confidence squared!
How did we ever come up with this? I have no idea what she’s talking about….but I can’t look away!
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.
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.
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.
A smart woman doing maths with a fountain pen: nothing’s missing.
Just us I presume
I like her format
Please keep enlightening us with interesting tips like the unification of dx into one symbolic script.
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.
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.
Didn’t learn a thing. Probably because I fell fast asleep after a couple minutes. Very soothing voice.
Well done. Very.
I don't understand anything she's saying I just love that there's a polished English woman on my screen talking to me. God bless Tim Bernes Lee!
I am in love with the accent. British accents are the best.
One of my fav subject. Always love the bow shape of integral symbol, which also exist on violin body.
I got to Green's theorem and realized I was finished with calculus, I aced the class but it was a struggle, that was 46 years ago
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.
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.
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)
Impressive. Wish I could understand any of it.
The amazing Hannah Fry 🙏🏻
😮*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...📈
Wow. Learned this in the 1980's whilst a mathematics undergraduate and largely forgotten almost everything 😉
My calc teachers taught the mathematics in the notation without the perspective. I wish my Calc 1 teacher put it this way (so simply and intuitively) in my math classes.
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.
Outstanding!!
I have studied mathematics since the early 90s. I taught myself the fundamental theorem at this time, when I was about 14. Since then I've acquired many degrees, including a PhD. I just want to point out... The fundamental theorem does not require the infinite sum. This discovery came more than 150 years later by B. Riemann. Newton and Leibniz each discovered the theorem independently.
It’s not N cubed in the summation it’s N squared. The. N cancels one N leaving N in the denominator. THEN you can let N go to infinity to end up with the answer.
I wish my Calculus teacher speaks English like her. Mine is a Russian lady who no doubt is brilliant in math but has an incomprehensible thick accent.
Anyone else fall asleep to this video her voice is so soothing lol.
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.
Indeed, as "A Tour of the Calculus: Berlinski, David" suggests, the fundamental theory of the Calculus is the fundamental theory of the Calculus... (recursive definition being especially significant!) ):-) Simples...
you do know an averaging the xn of known heights of the section multplyed by the known length approximates the required area
I took only Calc 1 and 2 in undergrad 48 years ago. I struggle with simple algebra now. My brain is shot. I have something more than a vague memory of perhaps 1/100th of 1% of what she is doing. But because of her accent, I'm madly in love with Hanna and now have a strong desire to go back to Algebra 1, then geometry, Trig, Functions, and then the hard stuff. I really feel quite stupid, not because I can't remember this stuff or never quite understood it when I passed the courses, but because I have a crush on a woman and I'm almost old enough to be her grandfather. And if I'm not mistaken she could be pregnant in this video? Brains are truly very sexy.
Easy to look at ! 😊
Fundamental theorem basically says the derivative is equal to the integral of the second derivative, basically
I would think mixing the indefinite integral at this level would confuse students
Brilliant! and LOVE the fountain pens and the lovely handwriting!! Looks like a Cross.....is it??
I'm 71, and learnt calculus around 50 years ago, and got lost in the first 30 seconds of this!
Was a bit confused when the answer provided (10:13 mins) had no units and was not squared... (may to be assumed?)
Thank you so much
I don’t like the way she writes her integral symbol, but clear explanation.
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.
So I still don’t know the area under the squiggly line. What is it?
The first squiggly line she drew as y=f(x) was just an example of some curve.
She never defined f(x) for the squiggly and instead used y=x^2 for the real example to work through and find the area under.
defining delta x would be nice ...
fascinating way of summarizing that concept.
Am I wrong? I think in the example the xi = 0 is a typo, and it should be x1 = 0.
But do you have the division rule for 7 memorized?
Why am I watching this when I haven’t got a clue what is being talked about?
No one speaks more beautifully of things I don't get, than my lovely Dr. Hannah..
BTW: Sex is like math. I don't get it.
1/n^3 how?
Wow, never thought I'd learn FTC from the author of Hello World
Well I can balance my checkbook and that's pretty much all I need math for.
How happy am I that I learned this subject from a standard classical book on Calculus ...
no offense, but the level here is unbelievably low ...😢😢😢
Having such a Nerd-Crush on this lady right now 😍
A fountain pen, nice
9:13 limN(N+1)(2N+1)/((N^3)6) she cancels out the N/N^3 to get 6N^2 in the denominator. Then the next line at 9:25 it became became 1/6 lim(N+1/N)lim(2N+1)/N. Shouldn't the denominators be N^2 not N? When & how did N^2 in the denominator become N?
I don't know if there's a theorem that says the two limits multiplied equals the limit of the two expressions multiplied.
But suppose there was:
lim(N+1)(2N+1)/N^2 = lim(2N^2 + 3N + 1)/N^2
Which also approaches 2N^2/N^2 = 2
No, that is correct. Multiplication takes place between the brackets in the numerator, not addition.
Otherwise you would be right.
@stephandanzeisen No. Look again. At 9:13 she cancels N / N^ 3 x 6 to get N^2 x 6. The next line at 9:25 the denominator becomes 6 N. not 6 N^2 .
@@divermike8943(N+1)(2N+1)/N^2
= ((N+1)/N)×((2N+1)/N)
@@divermike8943 Just listen to her words: "...times the limit..." or try it for yourself and calculate: [(N+1)/N] x [(2N+1)/N] = [(N+1)x(2N+1)] / [N x N] = [(N+1)/N] x [(2N+1)/N] / N^2. Right?
You could tell me how a toilet works and I’d still be transfixed.
It is important to understand that there is NOTHING to understand here. It is just a rendition of agreements on how to write something down, it is all symbols of ideas. This is the one thing people fail to understand. It is just agreements on how to write something down. And a so-called solution to a mathematical problem is just a simpler way of writing something down. Solving a mathematical equation is actually a puzzle. If you understand these two things, maths becomes simple. If not, it is impossibly complex. Most teachers fail to elucidate these two basic tenets of human communication on mathematics.
Mind blown. I always thought a solution was finding some secrete sentence amidst an endless amount of words.
Underrated comment
This is exactly the thing I needed to hear 20 years ago when I was still in school. Realizing this changed everything for me. Thank you for spelling it out so nicely!
You didn't explain a thing.
@@reasonsreasonably I sure did. It is the way I relieve people from their fear of maths.
Nice video. I enjoyed it a lot.
But I was expecting a formal proof of the fundamental theorem of calculus.
Orthogonality and non orthogonality.
3:10 I agree. I really didn't get your first green blocked introduction though. I know what the symbols say but there was no intuition. That was a massive intelectual leap.( ummm sorry, I haven't seen your earlier videos so my comment is moot) Lebesgue, Lebesgue, where is He! Dunno Mate said the builder, off measuring somewhere. Your brain patterns of explanation work different to mine. I am not criticising, just learning. :-)
We did this at age 15
Thanks for the very thorough video. Slightly off topic question. Could I ask if anyone knows the maker of the fountain pens used? I am a scientist myself and do still enjoy calculating by hand (yes to this day). I have been struggling to find a good fountain pen and this looks like it writes cleanly and Dr. Fry appears to be applying a bit of pressure too. Mine always brake when i do. Thank you
Another fountain pen fan! I use either my father’s Schaeffer (from before I was born) or a cheap Lamy. I’m a retired aeronautical engineer.
Using a pen requires years of muscle practice. It is not a given, it is learnt by practice. On a good knib the ink flows without any need of pressure. In school I used to bend by knib this way and that way until the ink flowed acording to my writing style. I suppose a good Knib is like your own horse?
@@alphalunamare We’re Kiwis who lived in France 20 years ago. Handwriting was an important skill in French schools and graduating to use a pen was a big deal.
Now take the ∫ (education costs) dx for how many years someone spends in college vs learning math on your own. 💰
Black widow teaches maths. You have my full attention.
Which pens are those?
I never got past basic math…. can you dumb this down a bit?
I love how it makes sense to say "naught"
I like Turtles.
the baby will know calculus early!
why do you care
@@xl000 I care as much as you cared to reply to the post
Real pens!
"You have some function, y"? I have no clue, either.
The unknow / variable function name Y is pronounced why for some reason.
My PhD officemate is doing very well.
maybe it's just me, but this seemed to introduce way too much material too quickly, and not explain any details of the notation or concepts. (In the first minute, you write the FTC, then seemingly as an afterthought define integrals as limits of Riemann sums.) Then it looks like you're talking about both parts of the FTC, without mentioning that they're different results. (I think?) Not sure if this was meant to be more of an overview as opposed to a typical lesson in the course, but if I were a student I would have a hard time learning from this.
(Also why is the third timestamp called differentiation: both it and the fourth timestamp are integration, just different methods, no?)
No you're not wrong this video is ridiculous for anyone who doesn't already understand it and the timestamps are clearly incorrect
If that’s the fundamental theorem I’d hate to see the complicated version.
+ c
❤❤
Yeah no clue.
This is the problem with modern education: complex concepts are given 3 minutes to be "explained" on the blackboard, 2 minutes for students to write them down on paper, 1 minute to solve an integral several pages long. The original explanations by Newton were more detailed, especially on the concepts of derivatives. Because education then didn't suck - all questions and doubts were answered, with plenty of time to digest.
Breaking News: We no longer need integrals (Whaat!)
Fountain pen stylish lady
🇮🇳
This is not helping me get back to sleep
Thanks, Professor Fry, and I see there is elegance in calculus. But I STILL don't get it. I might check out the pre-calc course, just because I hate being so frustrated by this. In college (many years ago) I struggled through the 2 calculus courses that were required for my B.S. degree. This despite being an A student in High School for algebra, geometry (loved proofs) and trig. As math builds on itself perhaps I missed a concept leading to calculus. Reminds me of when I was in grade school, 10 or 11 years old, where I was out sick for a couple days when the class was taught the signs/operation for "intersection" and "union" between sets. I was clueless for the following weeks until a friend showed me what they meant.
On the other hand, maybe trig is all my brain was wired to handle. That's ok too. But for anyone with young kids -- I'd say engage those sponge brains on this stuff.
Sometimes the frustrations of learning math are motivating and sometimes they just shut us down. We're sorry your relationship with math is hitting calculus bumps. Some precalculus knowledge is a prerequisite for this Calculus I course so we hope you do check out our Precalculus course, as it's designed to break down concepts visually and it's taught by world-class math communicators. Here's the course page for more info: www.outlier.org/products/precalculus Happy learning!
@@OutlierOrg No, This video explains a complex matter, much, much too fast, if you want to explain the riemann method for the fundamental therom of calculus you can't just chuck it in a video and assume everyone watching it is familiar with the trapezium rule, summing an infinite series and the b-a/n dx substitution and how to calculate F(Xi). A video like this should have at least links or something to help with explaining the aforementioned, as none of this relates to or would show up in pre-calc anyway.
You are my dream.
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.
It does assume quite a lot of foreknowledge (which I don’t have) so it was lost on me.😢