This video is the best of the 4 part series on Fundamental Theorem of Calculus, starting from the origin of the function f(x), then to its derivative f(x) [!! I am always stunned by the confusing notation], to definite integral and Riemann Sum, then go backwards to F(x), with brief mentioning of F'(x) too. This way is more efficient to teach the FTC. Thank you for the work!
The derivative of f applied to x is most commonly denoted by f'(x) or d/dx f(x). If you use the letter f to represent a function and the derivative of that function at the same time, that would imply that the derivative of the function is actually itself; in other words, f'(x) = f(x). This is a differential equation, and it implies that f is defined by f(x) = ke^x, for some constant multiple k.
@@zack_120 Then what exactly were you referring to? Also, what do you mean by "unique"? Do you mean it's unique in being its own derivative? As I pointed out, it's not; you can put any constant multiple there, and the function you get will also be its own derivative.
I like the way you included the reiman sum definition of the integral , most proofs don’t show it from the foundations of the reiman sum , and I was looking for how the reiman sum is included in the FTC
In my math book, the second fundamental is the derivative of the definite integral from a constant to x is the function in terms of x. What you explained is the 1st fundamental theorem in my book
I agree. I have a textbook that tried to explain it but it doesn't make a lot of sense. It uses first principles differentiation to prove it. Thumbs up so Sal sees this comment !!!
The area is not the antiderivative, the area under the line of the curve of the antiderivative graph is the "space" between the two given points. The space between the two given points is also S(a) - S(b), so, the definite integral from a to b of the antiderivative of S (the area under the antiderivative curve) is equal to S(a) - S(b). In common terms, the definite integral from a to b of f(x) is equal to the antiderivative of f(a) minus the antiderivative of f(b).
i hope i really understood this. right now i think i did and would be a shame if im just thinking i understand but i dont because the examination is closing in soon... nobody gonna see this message anyway it just feels good to vent
Luan Cristian Thums Yes, antiderivative is not the same as area. But the rest is not correct at all. The area under the "antiderivative graph" is not the "space" between the two points. If you take the area of the antiderivative graph you would be integrating the antiderivative, and that is not what it is being done in the video. There is zero concern about the area under the graph of the antiderivative. In the video, all that matters from that graph is that it gives you the final and initial values of the antiderivative of velocity, which can be used to find the area under the graph of velocity, since velocity is a derivative of space with respect to time.
I don't understand why only that Mich views for such a quality video ...people who make 1000° knife videos r getting more views than this what the heck man !!
Because most people hate calculus, i only watch because i have to. We have had snow days from school in the PNW for the past two weeks, and we had winter break before so I don't remember anything and I have a test tomorrow, the first day we get back. ugh
This video is the best of the 4 part series on Fundamental Theorem of Calculus, starting from the origin of the function f(x), then to its derivative f(x) [!! I am always stunned by the confusing notation], to definite integral and Riemann Sum, then go backwards to F(x), with brief mentioning of F'(x) too. This way is more efficient to teach the FTC. Thank you for the work!
The derivative of f applied to x is most commonly denoted by f'(x) or d/dx f(x). If you use the letter f to represent a function and the derivative of that function at the same time, that would imply that the derivative of the function is actually itself; in other words, f'(x) = f(x). This is a differential equation, and it implies that f is defined by f(x) = ke^x, for some constant multiple k.
@@isavenewspapers8890The case of d(e^x)/dx = e^x is unique and not what referred to here.
@@zack_120 Then what exactly were you referring to? Also, what do you mean by "unique"? Do you mean it's unique in being its own derivative? As I pointed out, it's not; you can put any constant multiple there, and the function you get will also be its own derivative.
I like the way you included the reiman sum definition of the integral , most proofs don’t show it from the foundations of the reiman sum , and I was looking for how the reiman sum is included in the FTC
I looked at this a couple days ago and justified this using the exact same reasoning. Letsss goo
The video is correct. This is the second part of the fundamental theorem.
In my math book, the second fundamental is the derivative of the definite integral from a constant to x is the function in terms of x.
What you explained is the 1st fundamental theorem in my book
Which is first and which is second varies among the different treatments of different authors.
I agree.
I have a textbook that tried to explain it but it doesn't make a lot of sense. It uses first principles differentiation to prove it. Thumbs up so Sal sees this comment !!!
Amazing Explanation
Probably the best proof found in youtube
Thank you!
Thanks Sal. Great. But will you do the video where you give a rigorous proof of Fundamental theorem of Calculus and why the area is antiderivative?
You know something disastrous is about to happen if the instructor says "Nothing EarTh Shattering So Faar"
Loads of thanks
The area is not the antiderivative, the area under the line of the curve of the antiderivative graph is the "space" between the two given points. The space between the two given points is also S(a) - S(b), so, the definite integral from a to b of the antiderivative of S (the area under the antiderivative curve) is equal to S(a) - S(b). In common terms, the definite integral from a to b of f(x) is equal to the antiderivative of f(a) minus the antiderivative of f(b).
that's what he said in the video........
Oh boy🤦♂️
thank you sal!
i hope i really understood this. right now i think i did and would be a shame if im just thinking i understand but i dont because the examination is closing in soon... nobody gonna see this message anyway it just feels good to vent
Appreciate it dude
Luan Cristian Thums Yes, antiderivative is not the same as area. But the rest is not correct at all. The area under the "antiderivative graph" is not the "space" between the two points. If you take the area of the antiderivative graph you would be integrating the antiderivative, and that is not what it is being done in the video. There is zero concern about the area under the graph of the antiderivative. In the video, all that matters from that graph is that it gives you the final and initial values of the antiderivative of velocity, which can be used to find the area under the graph of velocity, since velocity is a derivative of space with respect to time.
There are a lot of videos on intuition. It would be nice if Sal devoted at least one video to the actual rigorous proof
I believe they call this the Fundamental Theorem of Calculus.
+Hannah Pham
We all know that It's written in the title of the video
I don't understand why only that Mich views for such a quality video ...people who make 1000° knife videos r getting more views than this what the heck man !!
It's a whole lot easier to watch videos about 1000 degree knives than it is to watch videos about calculus
Because most people hate calculus, i only watch because i have to. We have had snow days from school in the PNW for the past two weeks, and we had winter break before so I don't remember anything and I have a test tomorrow, the first day we get back. ugh
I thought this was the First Fundamental Theorem of Calculus.
@@MoodiFLEX I thought you were my grandson
I thought you were my great-grandson
I thoguht you were my great-great-great grandson
@@milee105 i thought you were my great-great-great-great grandson
I thought you were my adopted son
willwen645 Yes, you are right. These videos have their first and second fundamental theorems confused.
S of b ... hmmmm
TerriblesHorse Best comment on this video
Do more proofs it is easier to remember formulas that way They only take a few seconds,at most one minute
Does anybody know what program he uses to draw?
Smoothdraw3
@@johnperez3454 thank you!
Did you just call me an SOB?!!
No dear
Why quality is too low ,cant see anything properly 👁️
I lost you after a-b=sb
6:47 shouldn't we multiply the the term (t sub n-1) in the velocity function by the step size which in this case is delta t?
Shouldn't we care about what happened to the c, constant of integration?
When it is a definite integral, it doesn't matter. Just let c=0, or account for c, and then see that it subtracts itself, and isn't needed.
dude i want you to be my dad
time wasted