Proof of fundamental theorem of calculus | AP Calculus AB

Proof of fundamental theorem of calculus | AP Calculus AB | Khan Academy

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11 жыл бұрын

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The first part of the fundamental theorem of calculus tells us that if we define _(_) to be the definite integral of function Ä from some constant _ to _, then _ is an antiderivative of Ä. In other words, _'(_)=Ä(_). See why this is so. Created by Sal Khan.
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Пікірлер: 165

@celesteacosta3495 4 жыл бұрын

Every-time Sal says "just for fun" Me: " sure, just for fun."

@rajbirvirdi4571 5 жыл бұрын

After watching this 20 times I finally understand!

@fatimakharal4502 4 жыл бұрын

Great 👍

@johnq4841 3 жыл бұрын

i actually watched for 6 days to understand.

@No_BS_policy 3 жыл бұрын

Great. That means we all have different rates of learning. I am by no means a math genius but I understood it the first time I watched it. I actually had to derive the proof myself to see if I really understood it. It was good.

@Akshit.vats. 2 жыл бұрын

@@No_BS_policy you my friend need a lesson on sarcasm!

@ian.ambrose 2 жыл бұрын

@@Akshit.vats. True lol. ''omg I derived the proof myself!!''

@No_BS_policy 3 жыл бұрын

So, Sal casually took the derivative of F(x) just for fun and ultimately produced a proof for the fundamental theorem of calculus? That's genius right there.

@anthonymontanio1012 10 жыл бұрын

this video should be emmy nominated.

@ultimatepirate9589 7 жыл бұрын

if sal had a dollar for every intuition he gave us he'd be bill gates

@Turnamonkey 3 жыл бұрын

nah jeff bezos

@giovannirodriguez3675 3 жыл бұрын

@@Turnamonkey nah Elon Musk

@loneranger4282 3 жыл бұрын

His total views are 1.8 Billion, so sadly not Bill Gates level, but still close

@CyanKash 5 жыл бұрын

Aight I'll just watch it 20 more times

@tornmyhibula 9 жыл бұрын

has this still not won an oscar yet????

@danielgonzalezisaiev9643 11 жыл бұрын

Great vid! Very logical, really breaking the barrier that gives us students the thoughts "How could anyone figure this out? Surely one has to be a genius..." Now I feel like the inventor(s) of the integral proof are actually human and were I one of them at that time I might have figured it out! Big thanks!

@hojiaqian4757 3 жыл бұрын

i really want to cry😂 i finally understand this TvT i have been searching for the proof for 2 days😂 (cuz i really cant accept that formula if i dont understand where it comes from ) thank u

@Akshit.vats. 2 жыл бұрын

same here pal'

@samiabe8686 10 жыл бұрын

Best video on KZfaq.

@AbhinavRawal 5 жыл бұрын

Proofs for the theorems may seem monotonous but they actually give great insights into the concept. That's the beauty of math.

@user-dm1hn9yh8h 5 ай бұрын

I have taken two semesters of calculus and have used this theorem so many times its second nature to me. However I never knew why this theorem worked until just now. I had no idea how the heck an infinite sum could be connected so directly to a derivative, and I didn't imagine it would be so simple. Now I finally understand better where this comes from and I'm so happy about that. Thank you so much Sal!

@andresyesidmorenovilla7888 4 жыл бұрын

I remember watching this video back when I was in my second semester of college. I didn't understanding a thing. Now, being in my sixth semester and watching it again, everything just clicks. It's nice to see some growth for a change. (Btw, beautiful proof and splendid explanation, props!)

@spade8352 3 жыл бұрын

i am an eight grader and yet understood everything thanks to the teaching methods thank you!

@hubenbu 2 жыл бұрын

This is the innermost reasoning of Calculus, it's celestially beautiful!

@liverpooler1997 9 жыл бұрын

you are such a great person. i attend a community college, and out teachers are horrible. my teacher has a huge asian accent and on top of that my registration time for classes was really horrible. i always loved math, but this quarter the only calc B class left open for me was with this asian teacher at 8PM. im really sleepy, hungry, and can't understand a word the teacher is saying. thank you so much for the help khan.

@bingodeagle 8 жыл бұрын

+Fled From Nowhere pointing out the fact that he cant understand an Asian accent isn't rasist.

@gaufill 9 жыл бұрын

Thank you so much for what you do. You make a difference in many peoples' lives, and I appreciate it.

@smerdis6274 4 жыл бұрын

the best one so far. every video I've watched before had left me with lots of questions. but this video gave me Intuitive understanding and mathematical understanding at the same time. thanks a lot and big ups

@joelgerard7869 5 ай бұрын

Choice of words: "RESORT to Squeeze Theorem". That's sort of how I feel about using the Squeeze Theorem as well.

@stefan_dobre 11 жыл бұрын

its amazing how your new videos are always synced with what im currently doing in class...

@jadhavnamdev1 5 жыл бұрын

Really enjoyed watching like a movie. Every step is quite interesting. Thank you sir.

@shalev1234 8 жыл бұрын

amazing explanation, I tried understanding it from my teacher and FAILED, but here its so flowing.. thanks!!

@Mrnoob2uu 9 жыл бұрын

I've watched a lot of your videos, and I have to say, this is your masterpiece. Good job and thank you Mr. Khan

@MrBrendanpdx 3 жыл бұрын

Thank you so much! My math text is so hard to follow and this really helped me understand how these are connected!

@anjumanara550 4 жыл бұрын

awesome video ,u just cleared all my doubts thank you so much

@arthurthegreat216 11 жыл бұрын

Beautiful proof. Thank you Sal.

@vko89 11 жыл бұрын

Well both t and x are placeholders for numbers that lie on the interval [a,b]. What the theorem says is that F' and f always have the same value when you evaluate them at the same number. The main reason that t is used instead of x in the integral is because there x is used as a fixed point denoting the upper bound of the integral and we must integrate with respect to a variable. Just as easily we could have written F(t) = integral from a to t of f(x)dx.

@dktchr3332 4 жыл бұрын

Nicely integrates (no pun/integral intended) the MVT into the explanation. Well done proof.

@funfair-bs7wf 2 жыл бұрын

Great ! Thank you for you work !

@GreenDayxRock1 11 жыл бұрын

For a while in my first calculus course it's been bugging me A LOT why I was anti-differentiating when what I was writing was talking about an infinite sum.. Seriously, thank you so much for tying everything together

@MrTanorus 9 жыл бұрын

Thanks. it helped me a lot.

@shauryaverma2705 3 жыл бұрын

Thanks a lot sir 👍👍👍👍👍

@unknownvariablex7 7 жыл бұрын

love the way he teaches

@charlotteshi 3 жыл бұрын

Best video on KZfaq:] u made my day

@chsuresh009 4 жыл бұрын

this is so nice, thanks

@whitecrackerhardcore 11 жыл бұрын

Good video. Helped me out. Thanks.

@darkinferno4687 7 жыл бұрын

the real mvp!!! thank you sir!

@tanujam.4152 2 жыл бұрын

Very clearly explained. Thankyou.

@ugurylmaz7138 7 жыл бұрын

We use the mean value theorem for definite integrals while prooving the fundamental theorem of calculus. However when prooving m.v.t for definite integrals we also use the fundamental theorem of calculus. What exactly is going on in here?

@yassershubbar3876 7 жыл бұрын

No fucking idea.

@gustavo_m32 6 жыл бұрын

This is bugging me out

@carlo2074 6 жыл бұрын

You can use the second fundamental theorem of calculus to prove the M.V.T and then use the M.V.T to prove the first fundamental theorem of calculus

@cameronspalding9792 5 жыл бұрын

MVT applies to any function that is differentiable

@Ltellin669957 5 жыл бұрын

you can prove mvt without tfc

@orz6 11 жыл бұрын

For all intents and purposes, x in the theorem represents any t value provided it's between some continuous region in f(t). F'(t) = f(t) would be a way you would express that if you knew the whole function beyond 'a' (in both directions) is continuous. It would be more confusing getting to that result expressing the integrals in the proof this way however. The statement is true if the whole function is continuous as it says we get f(t) from the derivative of the antiderivative (now) for all t

@idreamcsgobhop7021 2 жыл бұрын

Really good video thanks for it :)

@cezarywystup1505 5 жыл бұрын

Sal is a legend!

@MaryashrafBaly 3 жыл бұрын

Amazing!

@marcoponzio1644 Ай бұрын

Wonderful 🤩

@abidaliseikh8351 3 жыл бұрын

At last got a proper video 🧡💛💚💙💜🤎

@juancuneo8346 5 жыл бұрын

Amazing video

@loocoo8877 7 жыл бұрын

is it weird that i got asmr tingles from this?

@ultimatepirate9589 7 жыл бұрын

couchpotatos same here

@rhoadess 11 жыл бұрын

I always thought of the point at which the line is tangent to a function as a kind of tinny little top to a trapezoid, and if we added up every little area for each trapezoid we would get the area within that interval. I guess what this is saying is that if we have an area as a function and we take the derivative, the y value f(x) is the slope of the top of our little trapezoid at x. I know it is saying more, but I am trying to picture this out loud any thoughts would be helpful.

@1213yaya 10 жыл бұрын

your explanation is amazing!! thank you very much!!!

@ap-pv7ug 4 жыл бұрын

I still don't intuitively understand why that integral always equals the anti derivative regardless of what the arbitrary lower bound is? Shouldn't it depend on what a equals?

@2funky4u88 4 жыл бұрын

the integral equals the anti-derivative evaluated at the endpoints e.g. F(b)-F(a) so yeah it does depend, but only on the actual evaluation of the area. If you are just looking for the anti derivative of a function it would be F(x).

@mikaylaliang9323 2 жыл бұрын

god bless this man

@joyneelrocks 8 ай бұрын

That is a really great video, however I did find the mean value theorem a little abrupt and thought that it would’ve been better to use the Riemann Sums, which does get you to the same result, but is more intuitive for others to understand as I’m pretty sure Riemann Sums is the bare minimum that is taught to people with respect to the various approximation methods that have been invented. But anyways, great video 👍

@sofiarivero0808 Жыл бұрын

Beautiful👌

@tomashernandez8711 2 жыл бұрын

what a wonderful video, my god, I UNDERSTAND

@ayoubdiri4553 7 жыл бұрын

ohh it was priceless video

@JavierBonillaC 8 ай бұрын

Finally, after watching this video 10 times I think I know what the source of all (my) confusion was. In my humble opinion it is bad nomenclature or lack of explanation of the nomenclature. f(t) is a function. f(c) and f(x) are representations for the f(t) function when t=c and when t=x respectively. They are not new functions in themselves. It is pretty strange to see t as something that has undetermined (variable) values and then look at f(x) and f(c) as specific values. Am I right or am I missing the forest for the tree or viceversa?

@Dharmarajan-ct5ld 3 жыл бұрын

Could we keep it simple!! As ∆x tends to 0, you may assume f is monotone, region approximates to trapezium (lower classes) ... This finishes it due to continuity.O ne may avoid mean value theorem etc. Kindly consider

@apmcx 11 жыл бұрын

He proved them in this video

@tincho15neem 7 жыл бұрын

The theorem also says that F(x) is a continuous function even if f(t) isn't. You need to proof that also.

@Silky0925 3 жыл бұрын

Why is there a need for C? The area can be written as f(x)dx so the limit is just f(x) as dx approaches zero.

@78anurag 3 жыл бұрын

This is insanely beautiful Period

@user-uj7tw1vv4n 8 ай бұрын

Not our calculus prof giving this to proof in final exam

@This_comeback_is_personal 2 жыл бұрын

We proved that this F function gives you the area below the graph from point a to point b. How do we know that point a is 0?

@logeshtu2485 4 жыл бұрын

it should be dealt T because it is x axis named as time 't'

@jamest3592 4 жыл бұрын

but I want to ask a qustion if every continus funcction has an antaiderivtive and that e^(-x^(2)) is a continus why their is no antidrivtive for it

@mashedpotatoez99 4 жыл бұрын

there is an antiderivative. It's just not "elementary" in the sense that it cannot be written using polynomials, trigonometric functions, logarithms, exponentials, inverse trig functions, hyperbolic trig etc. Notice that there is a very big difference between asking "does f(x) have an antiderivative" vs "does f(x) have an elemetary/simple anti derivative". The fundamental theorem of calculus proves to you that EVERY continuous function $f$ has an antiderivative, but it says nothing about whether the result can be then expressed using such familiar functions.

@yuzhe6054 4 жыл бұрын

This is a work of art.

@cuber64 4 жыл бұрын

What software is this video using to writing the formulas?

@nafrost2787 4 жыл бұрын

I have a question. In your proof, you used the mean value theorem for integrals and then proved that the value of t with the mean height approaches x as delta x-> 0, but I noticed that as delta x-> 0, the size of the interval [x, x + delta x] also approaches 0. So if the size of the interval approaches 0, can't we say that the area under the curve on the interval [x, x +delta x] approaches to the area of a rectangle whose base is delta x, and height is f(x)? That path would reach the same conclusions, and would also prove the fundamental theorem of calculus, but it is faster.

@chappie3642 3 жыл бұрын

I suppose that isn't rigorous enough

@001stLove 11 жыл бұрын

Or maybe his class is using Khan Academy's videos as a course guide :P

@FlareGunDebate 3 жыл бұрын

I now associate the color magenta with Sal Khan.

@hypotherima1 11 жыл бұрын

Yet another awesome math class from Salman

@Yusa1804 2 жыл бұрын

at 7:10 can you guys help me answer why f(c).dx = area under the curve I mean why f(c).dx I think it should added a limit when dx-->0

@renzovallejos6129 7 жыл бұрын

try taking real analysis guys. It is basically restarting calculus but with proofs. 10x harder but much more enjoyable

@etherealstars5766 4 жыл бұрын

@@amberheard2869 HAHA yeah, 2 years and 5 months later, you ask, and now i like your comment 7 more months after that. Where has life taken you guys, if a reply may come??

@etherealstars5766 4 жыл бұрын

@@amberheard2869 Interesting! I am in an AP Calculus class in high school. These videos are really useful, lol. Its fun to learn.

@Coolimre 4 жыл бұрын

William John We used Adam & Essex - Calculus: A Complete Course for our first and second semester of real analysis. Would recommend if you’re doing real analysis.

@mt_xing 9 жыл бұрын

MIND = BLOWN

@Amir4v 11 ай бұрын

who is the teacher? does anyone have his social media accounts? or website or whatever? his awesome

@eidlebanon5245 7 жыл бұрын

People used the fundemental theorem of calculus to prove the mean value theorem for integrals not the other way around.

@carloscerritoslira328 7 жыл бұрын

did i waste my time?

@lukapopovic5802 6 жыл бұрын

hobo doc Can you post the text of that pages in this comment section ? I would apreciate it.

@siddharthkapoor1056 7 жыл бұрын

What software is he using?

@iamlymoa 7 жыл бұрын

HOLY I've never been so enlightened

@sanjitrao2761 3 жыл бұрын

I like resorting to the Sandwich Theorem. Very delicious.

@ZoboZodiac 9 жыл бұрын

I'm a little confused, if we can prove F'(x)=f(x) for the function f(t) does that mean when we take a definite integral normally we should change the variable, technically?

@paulwang7229 7 жыл бұрын

The thing is like fist of all you have a function f(t) on a closed interval [a,b]. You define a NEW function F(x)="integral from a to x "f(t)dt (imagine i've got the integral sign right there). This F(x) is related to f(t) by the fact that it represents the area under f(t) and a horizontal "t-axis" between a and x on f(t). Now this F(x) is itself a function with respect to x: for each x we choose in the interval [a,b], we get a distinct "area under curve" value out of F(x). Bear in mind that F(x) is itself a function w.r.t. x and has its own graph. We now try to find the derivative of F(x). The fundamental theorem of Cal tells us that this F'(x) equals f(x). So what is f(x)? I remember seeing a f(t), but where does this f(x) come from?Now try to recall what we first learned when we studied functions: the letters we use to represent variables does not matter. If I have a function g(x)=cosx, then this means the same thing as "g(t)=cost". It's not the x or t that makes you recognize the function. Rather, it's the "g" in front of it. You see that g, and you know it stands for cos in this case. If you put g(k), then it's cosk; if you put g(party hats), then it's "cos (party hats)"(as long as "party hats" represent a variable). Returning to the problem at hand. We know F'(x)=f(x). Also, we have an expression of f(t). Let's say f(t)=ln(arcsin t). Then what's f(x)? We know the letters do not matter. Everywhere we see t, we replace it with x. So we have f(x)=ln(arcsin x). Therefore, F'(x)=ln(arcsin x)------a nice and pretty derivative expression that we should be family with. So to directly answer your question, no, we don't change variables. The "identity" of a function is its expression. What letters we use is irrelevant. They might look different, but x, t, k, and party hats in fact stand for the same thing.

@paulwang7229 7 жыл бұрын

"should be familiar with" on the second to last paragraph. Autocorrect must have changed that.

@deepakbellur9676 2 жыл бұрын

@@paulwang7229 Autocorrect looked at the context and found "nice and pretty' and went to work!

@hypotherima1 11 жыл бұрын

Even though it seemed obvious because the derivative of an intergral of f(x) is just f(x) ,you still managed to amaze me by doing this in a mathematical way that was still helpful :D

@carloscerritoslira328 7 жыл бұрын

are you still alive?

@mrnosy1 11 жыл бұрын

SALAM!

@somniad 6 жыл бұрын

I'm still somewhat confused... what is f(x)? How is that function defined? I see there's an f(t) but what is f(x)? I'm clearly missing something.

@carlo2074 6 жыл бұрын

The graph is using t on the horizontal axis. F(x) = area under the curve f(t) from t=a to t=x on the horizontal axis. So instead of a definite integral from some constant 'a' to another constant 'b', we have a fixed 'a' and a variable end 'x'. f(x) = F'(x) -the derivative of F(x)

@carvantes 11 жыл бұрын

What was that scary sound at 6:05?

@NaderM 9 жыл бұрын

dang this video rules

@johnbroflovski1252 7 жыл бұрын

you used the mean value theorem for integrals to prove the FTC . problem with this is that the MVT for integrals relies on the FTC. You get caught up in a loop.

@johnbroflovski1252 7 жыл бұрын

hobo doc hi hobo, for MVT you need FTC2.

@carloscerritoslira328 7 жыл бұрын

@sapwho 25 күн бұрын

I feel insane since the comments are all expressing support of this video. This is not a proof of the fundamental theorem of calculus. This is surely the first step, but this is absolutely incomplete. Edit: Oh it’s because he calls this the first fundamental theorem and the full one the second fundamental theorem of calculus. That’s historically inaccurate, but my fault regardless 😂

@maxprezas92 3 жыл бұрын

How would integrals be solved without knowing the fundamental theorem? They always teach integrals as the inverse process of derivatives.

@alberto3071 3 жыл бұрын

With infinite series, a true nightmare.

@isabellapark5101 5 жыл бұрын

Wow I was the 1.2Kth like!

@MrBoo303 11 жыл бұрын

hello viewers

@user-uj7tw1vv4n 8 ай бұрын

I lost it at 10:00🥲🥲

@EpiCuber7 5 жыл бұрын

1:48 wait why

@chappie3642 4 жыл бұрын

Bruh it's literally the definiton of an intrgral

@EpiCuber7 4 жыл бұрын

@@chappie3642 Hahaha fair at this point I was just trying to learn about integration from scratch (literally without even learning much differential calc), thankfully it's all g now

@chappie3642 4 жыл бұрын

@@EpiCuber7 understandable xD, I'm glad you realized your mistake

@someone229 6 жыл бұрын

Still a little bit complicated...

@benquinneyiii7941 11 ай бұрын

Matilda Panzer III

@benlyman7880 7 жыл бұрын

This business

@mittmasai1678 11 жыл бұрын

T HANKYOU ORZ6

@zDoser 11 жыл бұрын

The reason the derivative of an integral of f(x) is f(x) is because of the fundamental theorem of calculus. So the claim you made is actually not obvious without proof. Math doesn't work of the basis of this is equal to that just because i say so. To really understand you need proof.

@merlinthegreat100 7 жыл бұрын

My motto for the last couple of years

@carloscerritoslira328 7 жыл бұрын

:( so, i wasted my time?

@chappie3642 3 жыл бұрын

What do you mean? This is literally the proof of that