I've returned to calculus later in life to refresh for an exam I'm taking. It's absolutely astonishing how I perceive this to be an overcomplication (not on Khans part, but mathematics overall). It almost seems like the fundamental theorem of calculus is constructed in such a way as to deter people from attempting to wrap their heads around it, even though it is an absolutely simple concept in itself. If you have an area under a curve on an interval [a, b] you can find the area under the curve from [a, x] so long as x < b. Like I said, absolutely mind boggling that such a thing is made out to be what is is. It's literally common sense.
The fundamental theorem of calculus relates differentiation and integration: taking the derivative of an integral with a variable upper bound gives you the original function. What you said is, "A small area can be inside of a big area."
I finished my calculus 1, 2 and 3 last semester. and yet I still enjoy watching these videos. Also thank a lot sal, I got an A in all my calculus courses and in my ODE course. and thanks a great deal to your clear intuitive explaination !
A nice video, the benefit of which was greatly augmented by my reading of the same topic in Wikipedia. Grateful thanks to all these philanthropic types!
The intuitive explanation is rather simple: the rate of change of the area under the curve, from x=a to x, evaluated at x, per small change of x, is in fact the height of the rectangle f(x) delX, i.e value of f(x) itself, at x.
This part simply tells us that the derivative with respect to x of the definite integral from a to x of the function f(t) is f(x). This is the part which informs us that the instantaneous rate of change with respect to x of F(x) is actually just f(x). The intuition as to why this is the case remains obscure to me despite the rigorous proof that dF/dx=f(x).
I wished I had this available to me when I was learning calculus, college students are so fortunate to have resources like this. There is no excuse for failing calculus these days, and BTW I passed all my calc classes but I had to work really hard at it.
FToC literally reads "the rate of change of the definite integral of f(t) for t=a to t=x depends on the value of the function f at x". If you consider integration as a summation of slices of f(t), FToC looks like a pretty obvious statement: the rate of change of the area depends on the value of the next slice i.e., f(x).
Oh yeah, and he's also using Camtasia Studio for the screen casting. This software will run you a couple hundred dollars, but there is a free alternative called "ScreenCastOMatic" that I really like, but you only have a 15 minute recording ceiling.
when you replace t s with x variables its it because basically, you are finding the definite integral of that function where its F(upper bound which is x in this case) - F(lower bound). And then taking the derivative of the function you get the original function in terms of x and since the integral of the lower bound gives you a number, the derivative of any number is just zero. so the net effect is like replacing the t with x??
This explains it well. I think mostly because of the graph. I feel bad for my prof, he's really passionate. However, I can't understand any of them. I guess he expects me to be a genius which I
Probably hasn't synchronised his module with the others, so he's assuming knowledge you are yet to obtain. It's a common thing everywhere, in the sciences and in life.
OMG. Thank you so much! I spent the last two days trying to make sense of this because the book I'm using for class sucks!!! It's so horrible with explanations.
So (correct me if I'm wrong please) does that mean for that last example in pink we need to show that f(t) is a continuous function on that interval for the theorem to apply? i.e. to skip ahead and say what f(x) is.
Is there seriously no other book better than Rudins? I have it, with along 2 or 3 others and they all suck. My teacher pretty much recited Rudin verbatim in class, so he didn't help much, and reading it on your own doesn't help that much, neither. Now, if Khan could do to analysis what he has done with his other math vids, i'd be REALLY REALLY REALLY thoroughly impressed.
I believe he is using SmoothDraw (which is free), accompanied by a Wacom Tablet. These tools used together enable him to create these stylish diagrams.
They should change how colleges teach since I'm learning more by videos such as these than having to listen to a lecture without having the ability to rewind and grasp the concept.
I’ve always struggled to fully comprehend the FTC. After this vid, I still struggle. 😂 it’s something like the derivative of the anti derivative equals the derivative of the anti derivative which equal the original function. And I’m like, so what? In my head it sounds like a+b = b+a = 1a + 1b. AMAZING!! 😂
I think it means that the integral from "a" to "b" of f(x)·dx es equal to the antiderivative of f(x) at point "a" minus the antiderivative of f(x) at point "b". We are so used to associating integrals to antiderivatives that we give it for granted that they are the same thing, but they are not. The integral is an infinite sum of infinitesimal areas and if you discovered calculus for the first time, you wouldn't find it so obvious that it's the same thing than an antiderivative.
Great video! My school does not teach this, since of course I am in grade 7 only. I must say, I found this lesson quite unique... You have made this video so easy to understand, and I am 12.. Thank you very much!
This maybe the fundamental theorem of calculus... the fundamental point of calculus and my mind is that it is a tool belt... calculus is a way of doing things that arent really possible to do another way. It uses extraordinarily powerful concepts to help us optimize and engineer things. It gets us the most for the least. The point of calculus is engineering.
This guy is awesome and I love Khan academy but I find his repeating of words when writing very distracting. Edit: after reading the comments the suggestion to increase playback speed to 1.25 really helped alot
"dt" means delta t. It represents infinitely small interval of t to calculate the exact area under f(t) between points a and b, provided f(t) is continuous between a and b.
Just curious . Is he using the mouse to write and draw? Or is it some tablet with a pen? He is a legend if he were using a pen, and superhuman if it is a mOuSe.
what if instead of t , there is an x inside of integrant with t : ) lot more complicated for example d/dx integral from 0 - x third root of x+ sin t dt . how about that ? I was trying different methods but couldn't get anything reasonable
6:16 my question is.. does this imply that " x " has to be greater than π, since in the set up " x " is greater than " a " (i.e. the lower bound of our integral) ???
The fundamental theorem of calculus (FTC) relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. The historical relevance of the (FTC) is not the ability to calculate these operations, but the realization that the two seemingly distinct operations are actually closely related.-Wiki
🔊 THE FUNDAMENTAL THEOREM OF ⚡CALCULUS, Calculus, calculus,...⛈🌪🌧🌧 🎵🎶 "Riders on the storm..."🎵🎶 (How it sounds in my head when you say it: echoes with thunder & lightening, then slowly fades out with "the Doors.")😆
I'm curious why this is said to be a theorem and not an axiom, it looks like this defines the notation of an integral and its right inverse making me think of an axiom. Otherwise, is there a proof from axioms?
Related video had that exact title, "Proof of Fundamental Theorem of Calculus" /watch?v=pWtt0AvU0KA I haven't watched it yet, but it's 14 minutes long so probably is what you are looking for.
This is a pretty good example of a bad example. You could have summed saying that a derivative is basically the opposite of an integral and everybody would have got it. You could have also said that a remanns sum is like having lots and lots of really small rectangles under the curve and the area of all of those really small rectangles added together approximates the area under the curve and everybody would have got it. There are much easier ways to explain these things.
f(x) is the same as f(t) only with X's. t is basically just a placeholder value inside the integral because if x was used, since the integral is inside of a function of x (F(x)) then the function would change when x should just represent the mark to which we want to find the area (starting from a). So basically f(x) isn't on that graph because there's no x axis on that graph, there's a t axis. But if you were to plot f(x) it would require the x axis instead of the t axis, and it would be the exact same as f(t)
f(x) is the derivative of F(x). It tells us the instantaneous rate of change of F(x) with respect to x. Sal didn't have to come up with the graph of f(x) because there is no point doing that.
My debt to Khan Academy is one without end.
Heartofadracky donate to em
Facts
+1
Anyone else just looks at the subject (AP in particular but any other works) and just don’t know what to learn next?
Even mine....😁
I like how he says, THE FUNDAMENTAL THEOREM OF CALCULUS
i HEARD your comment somehow
I read that comment as he was saying it LMAO
@@DaveAgbayani literally couldn't have said that better XD
Lol , the way he says that makes me laugh a lot 😂
I was looking for this comment.
Your computer hand-writing is amazing.
+ender2999 its not a computer... its like those projecter thingies with pens
Oh
@Jordan Schiltz its a pen and and attachment to the computer
Khan have iPad
Khan Academy uses holograms
I've returned to calculus later in life to refresh for an exam I'm taking. It's absolutely astonishing how I perceive this to be an overcomplication (not on Khans part, but mathematics overall). It almost seems like the fundamental theorem of calculus is constructed in such a way as to deter people from attempting to wrap their heads around it, even though it is an absolutely simple concept in itself. If you have an area under a curve on an interval [a, b] you can find the area under the curve from [a, x] so long as x < b. Like I said, absolutely mind boggling that such a thing is made out to be what is is. It's literally common sense.
Thank you for stating this! I really wish my text book would’ve had a side note with this comment. Text books always over complicated the concepts!
The fundamental theorem of calculus relates differentiation and integration: taking the derivative of an integral with a variable upper bound gives you the original function. What you said is, "A small area can be inside of a big area."
I'm in calc 2 and did really well with the fundamental theorem of calculus in calc 1, but I still didn't even understand what you just said
I see I left a reply with an unfavorable tone. I am sorry for that.
EVERYONE! Watch this video at 1.25x regular speed! It's much faster and just as understandable! Good luck!
Nathan Alspaugh more like 2.5x
Nathan Alspaugh Your'e Awesome mate
Nathan Alspaugh nope. I got lost quickly
@Nathan Alspaugh 2x is much better
Nilay Gupta at times2 he sounds as if he's out of breath
I learned more in 8 minutes and 2 seconds than a 50 minute lecture.
Aye.
Methodology wins everytime
I cannot count the amount of times Khan here was doing the teaching and not the person who's job is to you know, teach.
idek why people dislike this video! Khan Academy is really help (for me at least)!
I finished my calculus 1, 2 and 3 last semester. and yet I still enjoy watching these videos.
Also thank a lot sal, I got an A in all my calculus courses and in my ODE course.
and thanks a great deal to your clear intuitive explaination !
do you still watch his videos lol
Thank you, you you guys are making this world a better world
you are gay
@@dsdsspp7130 your dad is gay
@@abcdxx1059 no u
A nice video, the benefit of which was greatly augmented by my reading of the same topic in Wikipedia. Grateful thanks to all these philanthropic types!
Thank You Khan Academy, this video was a HUGE help!
Thank you for making math so much more interesting! :)
wonderful...I hadn't understand a single word of my instructor, but now it is clear to me what the fundamental theorem is...THANKS KHAN ACADEMY....
The intuitive explanation is rather simple: the rate of change of the area under the curve, from x=a to x, evaluated at x, per small change of x, is in fact the height of the rectangle f(x) delX, i.e value of f(x) itself, at x.
Saying i appreciated you would be not enough. You are the best.
Wow. It finally clicked. Wow. Im on cal 2 and it finally clicked. TY so much! Now off to make this intuitive! :\
I'm still learning algebra 2 and I understand Calculus far better than the way I learned algebra at first, and bearly a Junior in high school
+DarkCrimson barely*...
+qbwkp thanks
+DarkCrimson No problem. I'm always happy to correct and educate.
You are a lifesaver!
deadass would give my first born to this man if he asked. You js saved me from having my 8th mental breakdown this weeek thank yoy!
Fundamental theorem is important for AP calculus AB/BC .
Thanks a lot .
this is a really weird question, but what microphone did you use? I like its tone
Alex schwalb he has a kinky voice
I have my AP calc test tomorrow, and the second multiple choice section on this test is looking brutal
This part simply tells us that the derivative with respect to x of the definite integral from a to x of the function f(t) is f(x). This is the part which informs us that the instantaneous rate of change with respect to x of F(x) is actually just f(x). The intuition as to why this is the case remains obscure to me despite the rigorous proof that
dF/dx=f(x).
Hey Sal! You should do a video on Pascal's Triangle (Algebra)
I wished I had this available to me when I was learning calculus, college students are so fortunate to have resources like this. There is no excuse for failing calculus these days, and BTW I passed all my calc classes but I had to work really hard at it.
FToC literally reads "the rate of change of the definite integral of f(t) for t=a to t=x depends on the value of the function f at x". If you consider integration as a summation of slices of f(t), FToC looks like a pretty obvious statement: the rate of change of the area depends on the value of the next slice i.e., f(x).
LOVE CONJECTURE PROVEN.
Really good description of the fundamental theorom of calculus. Helped a lot! Really grateful for this.
Oh yeah, and he's also using Camtasia Studio for the screen casting. This software will run you a couple hundred dollars, but there is a free alternative called "ScreenCastOMatic" that I really like, but you only have a 15 minute recording ceiling.
thank you
Thanks...so much!
Lol, it's true, Sal is actually really good at writing with that mouse, something most of us absolutely cannot do
liked it!
Very well explained thanks Sir...
learning from Bangladesh
Sal "Let me write this down, this is a big deal." XD
thanks sal
when you replace t s with x variables its it because basically, you are finding the definite integral of that function where its F(upper bound which is x in this case) - F(lower bound). And then taking the derivative of the function you get the original function in terms of x and since the integral of the lower bound gives you a number, the derivative of any number is just zero. so the net effect is like replacing the t with x??
I love to watch his videos after class and 10 minutes before 4:30....
Really appreciated the help!!
i think you can have a numerable number of discontinous points inside the interval and it still holds.
This explains it well. I think mostly because of the graph. I feel bad for my prof, he's really passionate. However, I can't understand any of them. I guess he expects me to be a genius which I
am
Probably hasn't synchronised his module with the others, so he's assuming knowledge you are yet to obtain. It's a common thing everywhere, in the sciences and in life.
Yes! Please prove this!
OMG. Thank you so much! I spent the last two days trying to make sense of this because the book I'm using for class sucks!!! It's so horrible with explanations.
So (correct me if I'm wrong please) does that mean for that last example in pink we need to show that f(t) is a continuous function on that interval for the theorem to apply? i.e. to skip ahead and say what f(x) is.
Is there seriously no other book better than Rudins? I have it, with along 2 or 3 others and they all suck. My teacher pretty much recited Rudin verbatim in class, so he didn't help much, and reading it on your own doesn't help that much, neither. Now, if Khan could do to analysis what he has done with his other math vids, i'd be REALLY REALLY REALLY thoroughly impressed.
amazing explanation. bravo
Merciii
Very helpful! Thank you!!
I believe he is using SmoothDraw (which is free), accompanied by a Wacom Tablet. These tools used together enable him to create these stylish diagrams.
Very interesting. Thank you.
Beautiful. A work of art
I watched this instead of going to my lecture
i'm a bad student
honestly i learned more from this video than from going to the lectures lol
They should change how colleges teach since I'm learning more by videos such as these than having to listen to a lecture without having the ability to rewind and grasp the concept.
It means that you made a Good decision..
No , you're a better one then
you see, right there is why I keep balking at going back to school. So much good stuff online now....
breh, ur amazing
can u suggest any video for leibnitz theorem of derivative????
it would really help me out......
Finished
I’ve always struggled to fully comprehend the FTC. After this vid, I still struggle. 😂 it’s something like the derivative of the anti derivative equals the derivative of the anti derivative which equal the original function. And I’m like, so what? In my head it sounds like a+b = b+a = 1a + 1b. AMAZING!! 😂
I think it means that the integral from "a" to "b" of f(x)·dx es equal to the antiderivative of f(x) at point "a" minus the antiderivative of f(x) at point "b".
We are so used to associating integrals to antiderivatives that we give it for granted that they are the same thing, but they are not. The integral is an infinite sum of infinitesimal areas and if you discovered calculus for the first time, you wouldn't find it so obvious that it's the same thing than an antiderivative.
Dear mr.Sal,
How can I give u hug?
you're a genius
How are wave functions relevant to investing?
idk google it
Great video! My school does not teach this, since of course I am in grade 7 only. I must say, I found this lesson quite unique... You have made this video so easy to understand, and I am 12.. Thank you very much!
just replying to remind you of this comment lol
Why can the lower bound be disregarded?
Continuity in a closed interval?
Why does the lower bound not change anything? Be it pi or a or whatever?
What program is used to make this video? Is it a special software to write with the mouse and then convert it to UA-cam?
smoothdraw 4, and he's not using a mouse but a digital drawing board. Check out wacom bamboo for example:)
Hallie Richie
Light work
you son are a master at drawing on a computer
Beautiful.
This maybe the fundamental theorem of calculus... the fundamental point of calculus and my mind is that it is a tool belt... calculus is a way of doing things that arent really possible to do another way. It uses extraordinarily powerful concepts to help us optimize and engineer things. It gets us the most for the least. The point of calculus is engineering.
This guy is awesome and I love Khan academy but I find his repeating of words when writing very distracting.
Edit: after reading the comments the suggestion to increase playback speed to 1.25 really helped alot
Me too!
2:02 what does the "dt'' after 'f(t) symbolise,signify... or represent?
"dt" means delta t. It represents infinitely small interval of t to calculate the exact area under f(t) between points a and b, provided f(t) is continuous between a and b.
hi, what is the name of this program? and do I need a light pen ?
what if the upper bound of the integral is in terms of two different integrals- i.e. x and t. would this change the derivative?
also you at 6:33 "we'll get more intuition of why this is true in future videos".. anyone know where i can find said videos
Download his mobile app or go to his website where all the topics are in order and in coordination making it easier for us to learn any topic....
ua-cam.com/video/pWtt0AvU0KA/v-deo.html
So dF/dx = f(x), not f(t)?
Personal misconception officially gone. Thank you very much.
I am still confused, Someone Help!
pretty cool
Mashallah Sal
The FUNDAmental THEOrem of CalCULus
Just curious . Is he using the mouse to write and draw? Or is it some tablet with a pen? He is a legend if he were using a pen, and superhuman if it is a mOuSe.
what if instead of t , there is an x inside of integrant with t : ) lot more complicated for example d/dx integral from 0 - x third root of x+ sin t dt . how about that ? I was trying different methods but couldn't get anything reasonable
Bravo
lol just learned this today!!!
whats a power of mathematics.
But you would need to be sure that your f(t) is continous on the intregration-interval though right?
Yes.
dF = f(x)dx 😊
Does anyone know the notation that is used in the video for the width of the rectangle?
6:16 my question is.. does this imply that " x " has to be greater than π,
since in the set up " x " is greater than " a " (i.e. the lower bound of our integral) ???
yes u are right
The fundamental theorem of calculus (FTC) relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. The historical relevance of the (FTC) is not the ability to calculate these operations, but the realization that the two seemingly distinct operations are actually closely related.-Wiki
thankyou, that was a nice piece of information :)
🔊 THE FUNDAMENTAL THEOREM OF
⚡CALCULUS, Calculus, calculus,...⛈🌪🌧🌧
🎵🎶 "Riders on the storm..."🎵🎶
(How it sounds in my head when you say it: echoes with thunder & lightening, then slowly fades out with "the Doors.")😆
Absolutely love the contents and the way you cover these materials...but allow me to say that you've got a very distracting voice there.
Read the book...you'll learn more.
What are u using to graph it? What program
Andrew Oneal if you find out can you tell me? it's really cool
Ben Spooner I will, but he hasn't said anything yet.
all i thought about was how the blue, orange ..or peach and yellow look good together.
what program is he using?
I'm curious why this is said to be a theorem and not an axiom, it looks like this defines the notation of an integral and its right inverse making me think of an axiom. Otherwise, is there a proof from axioms?
Yes there is a proof of this theorem by Khan him self you can check it on UA-cam just search "Proof of fundamental theorem of calculus Khan academy"
Related video had that exact title, "Proof of Fundamental Theorem of Calculus"
/watch?v=pWtt0AvU0KA
I haven't watched it yet, but it's 14 minutes long so probably is what you are looking for.
This is a pretty good example of a bad example. You could have summed saying that a derivative is basically the opposite of an integral and everybody would have got it. You could have also said that a remanns sum is like having lots and lots of really small rectangles under the curve and the area of all of those really small rectangles added together approximates the area under the curve and everybody would have got it. There are much easier ways to explain these things.
where does f(x) appear on the graph though?
f(x) is the same as f(t) only with X's.
t is basically just a placeholder value inside the integral because if x was used, since the integral is inside of a function of x (F(x)) then the function would change when x should just represent the mark to which we want to find the area (starting from a).
So basically f(x) isn't on that graph because there's no x axis on that graph, there's a t axis.
But if you were to plot f(x) it would require the x axis instead of the t axis, and it would be the exact same as f(t)
f(x) is the derivative of F(x). It tells us the instantaneous rate of change of F(x) with respect to x. Sal didn't have to come up with the graph of f(x) because there is no point doing that.
it would have been more interesting if you made a proof of the theorem
Why is it that numbers are not used for like an example? It is so much more difficult to follow when there is not a reference number.
Once upon a time I could understand this. I feel like I had just watched a foreign film with foreign subtitles that I can't understand.