@@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.
This lesson was the exact moment in high school that I fell in love with Calculus. I remember having a brief moment where I thought to myself "this is actual genuine magic using numbers"
Agreed, most people are not fond of maths. I marvelled at how it could predict how a system could behave like a trolley rolling down a slope in my physics class.
Holy hell, Navy Nuke Pwr School has a 48% failure rate. All candidates already do calculus to even have a chance. Sumbitch, 'at's the Navy SEALS of math. The elite selected wear civilian clothes on active duty. That is badassed.
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.
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.
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.
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.
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.
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.
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 book “Calculus” by Spivak has a more careful look at this sort of thing. But ultimately the answer is fairly intuitive: if we already know the area under some section of the curve, the change (aka derivative) in the area if we were to add a new “strip” is obviously going to be proportional to the level of the function. If the function is zero at that point, the area will not change at all as we add the new strip. If the function level is very high, we will add a lot of area with our new strip.
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!
She explains beautifully Teachers like her students will learn and show interest in math and learning. You can only explain like her when you have command of the subject.
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
Here is the short answer if anyone is confused: 1. Take the derivative of x squared and get 2x 2. You will find that the area under the curve y=2x is equal to x squared 3. now notice x squared is the reverse derivative of 2x 4. so the area under the curve of any function is called the integral, and it equals the reverse derivative of that function
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!
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.
I've come to appreciate the work that's been put in to defining such fundamental theorems of calculus, such that we can now rely on it to calculate things like distance traveled over 0
Beautiful theorem, in beautiful English ... by a beautiful teacher. My old professor insisted that teachers should be attractive: female teachers - beautiful and male teachers - handsome. Dr. Hannah Fry, you have it all.
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.
Thank you for this clear explanation. Like several of you in the comments, I do not consider myself good at math either, but I discovered that what is more important is your drive to understand.
This is such an elegant presentation. Thank you! But now I need fancy pens to work my math problems. Also an elegant lamp and desk. An euphonious English accent would be nice too.
Good to see a young person using a proper pen. I did differential calculus for my Maths O level in 1970. I was a whizz at it whilst not having the foggiest idea what I was doing or why.
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 don’t think i could have started this video without pictures in my head from 3blue1brown. But this presentation definitely gave me more understanding than that by itself.
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...
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.
Really good video, I always enjoy your content. In your example you should have used x^2 + c, the c's would have cancelled, but I believe they should have been there in your example even if we don't usually do it. Also when you did the limits you didn't really deal with the +1 in each limit. i think it would have been better to have divided each term of the numerator by the N and then just evaluate 1/N as N tends to infinity.
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).
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.
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.
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.
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.
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.
It's easy to state that integration is simply the reverse of differentiation. The way it feels to me is differentiation is like squeezing the toothpaste out of the tube and integration is "simply" putting the toothpaste back in.
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.
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.
😮*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...📈
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.
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.
Would have loved to had you for my Calc II and III classes. Calc I : English speaking fun class was not difficult Calc II : Indian woman, omg soft spoken and medium heavy accent ... struggled had to move around to find where I could at least hear her and then understand. Calc III : Oriental, and new teacher, very heavy accent and I think I only heard/understood 30% of what he said omg struggled. I wouldn't mind having a teacher with a thick accent for teaching basic skills classes. However, as the subj material progresses fighting to simply understand what word is spoken let alone the meaning of the entire dicta makes learning stuff difficult, IMO.
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.
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 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'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.
This lesson was the exact moment in high school that I fell in love with Calculus. I remember having a brief moment where I thought to myself "this is actual genuine magic using numbers"
Agreed, most people are not fond of maths. I marvelled at how it could predict how a system could behave like a trolley rolling down a slope in my physics class.
Holy hell, Navy Nuke Pwr School has a 48% failure rate. All candidates already do calculus to even have a chance. Sumbitch, 'at's the Navy SEALS of math. The elite selected wear civilian clothes on active duty. That is badassed.
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.
"wasn't good at math" you just were further down on the learning cureve.
@@DonMayfield true statement
You had me at multiple colors of fountain pens. Great explanation for this theorem.
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.
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.
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.
Thank you for your comment
i wish when i was going though high school and uni that resources like this were available, would have made life so much easier
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.
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.
Nice!
thank you. this makes much more sense than anything in the video
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.
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 book “Calculus” by Spivak has a more careful look at this sort of thing.
But ultimately the answer is fairly intuitive: if we already know the area under some section of the curve, the change (aka derivative) in the area if we were to add a new “strip” is obviously going to be proportional to the level of the function.
If the function is zero at that point, the area will not change at all as we add the new strip. If the function level is very high, we will add a lot of area with our new strip.
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
She explains beautifully Teachers like her students will learn and show interest in math and learning. You can only explain like her when you have command of the subject.
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
Welcome back to TH-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).
Here is the short answer if anyone is confused:
1. Take the derivative of x squared and get 2x
2. You will find that the area under the curve y=2x is equal to x squared
3. now notice x squared is the reverse derivative of 2x
4. so the area under the curve of any function is called the integral, and it equals the reverse derivative of that function
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.
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.
I've come to appreciate the work that's been put in to defining such fundamental theorems of calculus, such that we can now rely on it to calculate things like distance traveled over 0
Beautiful theorem, in beautiful English ... by a beautiful teacher.
My old professor insisted that teachers should be attractive: female teachers - beautiful and male teachers - handsome.
Dr. Hannah Fry, you have it all.
Make more of this kind of lucid explanative lecture. These are very easy to understand
A very ASMR lecture of the fundamental theorem of calculus ✨
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.
Thank you for this clear explanation. Like several of you in the comments, I do not consider myself good at math either, but I discovered that what is more important is your drive to understand.
Even if we are not on the same linguistic spectrum, it is always better to see how another person perceives and compares two methods
This is such an elegant presentation. Thank you!
But now I need fancy pens to work my math problems. Also an elegant lamp and desk. An euphonious English accent would be nice too.
Please keep enlightening us with interesting tips like the unification of dx into one symbolic script.
the idea that Newton just figured this out, out of thin air is incredible to me
Good to see a young person using a proper pen. I did differential calculus for my Maths O level in 1970. I was a whizz at it whilst not having the foggiest idea what I was doing or why.
Thank you Mr Newton for giving us such a mighty tool.
Didn't understand a word but would happily listen to the Doctor read the phone book.
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!
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 don’t think i could have started this video without pictures in my head from 3blue1brown. But this presentation definitely gave me more understanding than that by itself.
Nice fountain pens! Doing calculus in ink = confidence squared!
Rigorous derivation! Thank you! A function for Area! Deep stuff! Love mathematics!
in my own experience: success in a calculus class is a function of not only the student but also the teacher
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...
Really nice video. Happy to get a recap of my old studies. Maybe i now remember it for ever :)!
Hannah is the best teacher.
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.
You're an excellent lecturer.
Hot for teacher.
Ah! O Level maths, 40 odd years ago! Happy days. I loved this.
Never used it since!
Guess you never went on to study any engineering or physics then?
? I hope you can still...calculate. What a f.cked
@dumb generation (Luke 11:29)?
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 starts out great
Really good video, I always enjoy your content. In your example you should have used x^2 + c, the c's would have cancelled, but I believe they should have been there in your example even if we don't usually do it. Also when you did the limits you didn't really deal with the +1 in each limit. i think it would have been better to have divided each term of the numerator by the N and then just evaluate 1/N as N tends to infinity.
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).
One of my fav subject. Always love the bow shape of integral symbol, which also exist on violin body.
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.
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.
Thanks Hannah, and Thanks Outlier
You can do the calculus of finite differences with a spreadsheet. Very practical, and no need for limits.
Love you. I started learning calculus
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.
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.
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.
It's easy to state that integration is simply the reverse of differentiation. The way it feels to me is differentiation is like squeezing the toothpaste out of the tube and integration is "simply" putting the toothpaste back in.
Wow. Learned this in the 1980's whilst a mathematics undergraduate and largely forgotten almost everything 😉
This is taking me back to my college days. Weren't those "hops" called Riemann squares or something like that?
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.
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.
😮*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...📈
this is a superb explanation
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.
Can you please recall the theorem and explain the same here for us.
I am in love with the accent. British accents are the best.
So you were able to get 100k + views, kudos to you
It’s odd how every math education video on TH-cam is for an audience who already knows the math.
This is so well done
Didn’t understand a word. But now madly stricken ❤
Didn’t learn a thing. Probably because I fell fast asleep after a couple minutes. Very soothing voice.
What textbooks would you recommend?
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.
Differential calculus from first principles is worth learning...ie y+dy = (x+dx)^2...etc
Thank you teacher, very helpful video.
How did we ever come up with this? I have no idea what she’s talking about….but I can’t look away!
love how she draws the x
Paul A. Foerster, who taught me Calculus, is also a Brit, now a Texan.
Would have loved to had you for my Calc II and III classes.
Calc I : English speaking fun class was not difficult
Calc II : Indian woman, omg soft spoken and medium heavy accent ... struggled had to move around to find where I could at least hear her and then understand.
Calc III : Oriental, and new teacher, very heavy accent and I think I only heard/understood 30% of what he said omg struggled.
I wouldn't mind having a teacher with a thick accent for teaching basic skills classes. However, as the subj material progresses fighting to simply understand what word is spoken let alone the meaning of the entire dicta makes learning stuff difficult, IMO.
Impressive. Wish I could understand any of it.
Outstanding!!
what fountain pen is she using?
As far as I can tell, it is a cross centrury classic. by the line weight, I would say M nib.
I was a little confused until I realized lower case delta was being used instead of capital delta.
I have brain envy. As a left hander, I also have fountain pen envy.
Brilliant! and LOVE the fountain pens and the lovely handwriting!! Looks like a Cross.....is it??
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.
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.
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.
The amazing Hannah Fry 🙏🏻
Lost at 1:00 🤷🏼♂️
And that’s how maths goes for me. One concept I can’t get and it just carries on without me until the exams come round.
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.
you do know an averaging the xn of known heights of the section multplyed by the known length approximates the required area
Was a bit confused when the answer provided (10:13 mins) had no units and was not squared... (may to be assumed?)
What kind of paper was that?
Well done. Very.
Best ASMR ever.
What is the model of the fountain pen?
Hannah Fry is a badass. This was a great explanation of the FTC.
Anyone else fall asleep to this video her voice is so soothing lol.
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...