Fundamental theorem of calculus (Part 1) | AP Calculus AB | Khan Academy
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The fundamental theorem of calculus shows how, in some sense, integration is the opposite of differentiation. Created by Sal Khan.
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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.
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.
Your computer hand-writing is amazing.
+ender2999 its not a computer... its like those projecter thingies with pens
Oh
@@vivianraxe i thought that too but the hand follows, which makes me think it is computer hand-writing, because the pen would have to be hovering pretty close every time for the hand to know where he's holding the pen.
@@Schiltzy_ISU its a pen and and attachment to the computer
Khan have iPad
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.
idek why people dislike this video! Khan Academy is really help (for me at least)!
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 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.
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!
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.
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.
Saying i appreciated you would be not enough. You are the best.
Sal "Let me write this down, this is a big deal." XD
Fundamental theorem is important for AP calculus AB/BC .
Thanks a lot .
You are a lifesaver!
I have my AP calc test tomorrow, and the second multiple choice section on this test is looking brutal
Lol, it's true, Sal is actually really good at writing with that mouse, something most of us absolutely cannot do
Really good description of the fundamental theorom of calculus. Helped a lot! Really grateful for this.
this is a really weird question, but what microphone did you use? I like its tone
Alex schwalb he has a kinky voice
Hey Sal! You should do a video on Pascal's Triangle (Algebra)
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 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....
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.
learning from Bangladesh
liked it!
I love to watch his videos after class and 10 minutes before 4:30....
Thanks...so much!
thank you
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.
Very well explained thanks Sir...
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??
Dear mr.Sal,
How can I give u hug?
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.
LOVE CONJECTURE PROVEN.
i think you can have a numerable number of discontinous points inside the interval and it still holds.
Really appreciated the help!!
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.
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.
breh, ur amazing
Beautiful. A work of art
amazing explanation. bravo
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.
So dF/dx = f(x), not f(t)?
Personal misconception officially gone. Thank you very much.
I am still confused, Someone Help!
Very helpful! Thank you!!
Very interesting. Thank you.
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....
th-cam.com/video/pWtt0AvU0KA/w-d-xo.html
How are wave functions relevant to investing?
idk google it
Why can the lower bound be disregarded?
Mashallah Sal
Why does the lower bound not change anything? Be it pi or a or whatever?
Merciii
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.
can u suggest any video for leibnitz theorem of derivative????
it would really help me out......
Continuity in a closed interval?
Yes! Please prove this!
Finished
thanks sal
you're a genius
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
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.
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.
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!
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?
What program is used to make this video? Is it a special software to write with the mouse and then convert it to TH-cam?
smoothdraw 4, and he's not using a mouse but a digital drawing board. Check out wacom bamboo for example:)
Hallie Richie
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
Bravo
you son are a master at drawing on a computer
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.
lol just learned this today!!!
Beautiful.
hi, what is the name of this program? and do I need a light pen ?
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
Light work
Does anyone know the notation that is used in the video for the width of the rectangle?
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 TH-cam just search "Proof of fundamental theorem of calculus Khan academy"
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.
But you would need to be sure that your f(t) is continous on the intregration-interval though right?
Yes.
pretty cool
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.
So basically let me get this straight: when you differentiate and integral you just swap all the variables for X (or whatever your value is )? If I’m wrong do tell me
Yes, but only when the bounds of your integral go from a to x. If your bounds go from, say, a to x^2, you would plug into x^2 in for t, which means you need to you the chain rule. Khan Academy may have videos that go into examples of this, but I know The Organic Chemistry Tutor has an excellent video on the topic.
Isn't this kind of like saying "If you add 1 to 1 you arrive at 2. But, wait a second. If you subtract 1 from 2, you get, gasp, ONE." Obviously the derivative of an integral is the integral before it's solved. Am I missing something here?
I must be missing something. If you solve a function for it's integral OBVIOUSLY the original function is it's derivative?!?!??!?!!??
I'm so freakin confused.
Example: Integral of X^2 = x^3/3 and the derivative of x^3/3 = x^2
ya its that obvious. like there isn't anything too it really except that you can then define a fucking F(x)=integralf(x)dx or whatever
+Marc Rover What you are missing is the fact that the indefinite integral (i.e., the antiderivative) and the definite integral (area under a function) are defined as two completely distinct operations. Nothing in the world connects them, yet.
The fundamental theorem of calculus links these two operators and proves that they are, in fact, the same thing. Without this theorem, nothing would connect them. Most students miss this point, and it's partially the result of denote the indefinite and definite integral operations with the same symbol, thus foreshadowing that they are the same thing.
It's an absolutely non-obvious result. If you become confused for a second and think it's obvious, try focusing on proving why the area under a function and the antiderivative of that function are equal.
Unfortunately, most teachers don't clarify this at all.
The gasp part isn't the part where anti derivatives are the backwards processes of derivatives, as you compared with adding and subtracting. That's what humans have decided. The gasp part is that the anti derivative is the area of a function! That's the surprise. Just think about, the slope of the tangents of some unknown Area function is equal to the original function????!! That sounds insane and is seemingly is so intuitive. Everything else is humanly constructed definition.
In reality, before this was proved, the anti derivative had no real use. Only after it is proved that the derivative of the area function is the original function ( dA(x)/dx= f(x) ), then did we desire an anti derivative "operator", so that we could "undo" the derivative on the area function in the equation above, and solve for the Area function, and thus, the symbol ∫ for anti derivatives was born, and now we could solve for the area function:
dA(x)/dx= f(x)
dA(x)= f(x) dx
∫dA(x)=∫f(x) dx
A(x)=∫f(x) dx
Now the question becomes how to actually evaluate this ∫, aka the anti derivative "operator". And that's what Sal explains further on.
Marc Rover
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.
i dont get it
ya me neither homie
dF = f(x)dx 😊
🔊 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.")😆
all i thought about was how the blue, orange ..or peach and yellow look good together.
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.
whats a power of mathematics.
Ok , but what happened to the let bound a
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.
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.
The FUNDAmental THEOrem of CalCULus
“Like the old man said, together.”
-Not me