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Course: AP®︎/College Calculus BC > Unit 6
Lesson 4: The fundamental theorem of calculus and accumulation functions- The fundamental theorem of calculus and accumulation functions
- Functions defined by definite integrals (accumulation functions)
- Functions defined by definite integrals (accumulation functions)
- Finding derivative with fundamental theorem of calculus
- Finding derivative with fundamental theorem of calculus
- Finding derivative with fundamental theorem of calculus: chain rule
- Finding derivative with fundamental theorem of calculus: chain rule
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Finding derivative with fundamental theorem of calculus: chain rule
The Fundamental Theorem of Calculus tells us how to find the derivative of the integral from 𝘢 to 𝘹 of a certain function. But what if instead of 𝘹 we have a function of 𝘹, for example sin(𝘹)? Then we need to also use the chain rule.
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I hope this helps someone understand what we are doing here. It's quite simple to understand actually. Think back to when we learnt the chain rule. We learnt to think about composite functions as outside functions and inside functions. For example (2n)^2. The outside function is x^2 and the inside function is 2n. It's the same concept here. sin(x) is the inside function and the result of that is then inputted into the outside function which is the integral h(x). So to find the derivative we simply apply the chain rule here.
First, find the derivative of the outside function and then replace x with the inside function. So the derivative of the integral h(x) is 2x-1 and we replace the x with the inside function sin(x) giving us 2(sin(x)).
Now we multiply 2(sin(x)) by the derivative of the inside function sin(x) which is cos(x). Now we have 2(sin(x)) * cos(x).
It's initially very confusing if you don't remember about outside and inside functions. But I hoped this helped :)(12 votes)
what is the point of setting g(x) = sin(x)?(2 votes)
To magnify that sin(x) is a function of x, so the chain rule applies(9 votes)
Sorry, I'm not getting this. Why can we write F(x)=h(g(x)) ? FTIC2 surely only leads us to F'(x) = h(g(x)) ?(3 votes)
∫(2𝑡 − 1)𝑑𝑡 = 𝑡² − 𝑡 + 𝐶
FToC2 gives us
𝐹(𝑥) = sin²𝑥 − sin 𝑥 + 𝐶 − (1² − 1 + 𝐶) = sin²𝑥 − sin 𝑥
Thereby,
𝐹 '(𝑥) = 2 sin 𝑥 cos 𝑥 − cos 𝑥 = (2 sin 𝑥 − 1)cos 𝑥(4 votes)
If F(x) is a function of x and you can find its derrivative, then does that mean that you can graph F, where F(x) is the area that different x and x is the different upper bounds?(3 votes)
Yep, if you evaluate the integral in terms of t, and then substitute the limits, you get a function of x which can of course be graphed(1 vote)
- So, what does the fundamental theorem actually mean, if it's not just plugging in the top boundary into the integrand?(2 votes)
- the fundamental theorem of calculus is the fundamental connection between the operations of differentiation and integration as inverses of each other. When that concept is crystal clear in your head, the rest of it makes sense. I suggest you check out 3 blue 1 brown's essence of calculus series. he explains calculus very intuitively.(1 vote)
How would you solve it if sin(x) was the a of the integral notation (the value below the integral symbol). Would the methods be the same?(2 votes)- remember integral (a->b) fxdx = -integral (b->a) fxdx(1 vote)
- why doesn't F'(x)=h(sinx) or 2sinx-1?(2 votes)
- because it’s F(x) which is equal to h(sin x) so you need to derivate it to get the h prime. and you can’t say that (F(x)=2sin(x)-1) because F(x) is a function of( sin (x)) not in x so that's why you need to do it in that way.(1 vote)
- At1:34, Sal says that F(x)=h(g(x)). Why isn't that what F'(x) is equal to? The integral sign seems to be missing in h(x).(1 vote)
- It's h(g(x)) because the integral (on the upper bound) approaches sin(x) and not x, and this makes it a composite function because h(x) = the integral but with x as the upper bound rather than sin(x) and g(x) = sin(x) which makes F(x) = h(g(x)) and F'(x) = h'(g(x)) * g'(x) by the chain rule. This might seem weird but if the bound is not x, and is something else like a function of x then we need to represent it as a composite function like this problem and solve from there. I hope this helped, if this seems confusing or unclear, please let me know.(2 votes)
I'm not exactly sure what is the intent behind changing the upper bound from x to sin(x) or any other complex type of function. Also, why does chain rule take upper bounds into effect when computing area?(1 vote)
I would say so long you know what the chain rule is and what the fundamental theorem of calculus is you are good to go.
This is from the video description.
The Fundamental Theorem of Calculus tells us how to find the derivative of the integral from 𝘢 to 𝘹 of a certain function. But what if instead of 𝘹 we have a function of 𝘹, for example sin(𝘹)? Then we need to also use the chain rule.(2 votes)
Is there a "u-substitution" sort of method for solving these, or do I have to use the chain rule?(1 vote)- u-substitution can apply depending on the integral. But, you'll need the chain rule when differentiating if your bounds aren't just x or numbers.(2 votes)
1:34Video transcript
- [Instructor] Let's say
that we have the function capital F of x, which we're going to define
as the definite integral from one to sine of x, so that's an interesting
upper bound right over there, of two t minus one, and of course, dt, and what we are curious about is trying to figure out
what is F prime of x going to be equal to? So pause this video and see
if you can figure that out. All right. So some of you might have
been a little bit challenged by this notion of hey, instead of an x on this upper bound, I now have a sine of x. If it was just an x, I could have used the
fundamental theorem of calculus. Just to review that, if I had a function,
let me call it h of x, if I have h of x that was
defined as the definite integral from one to x of two t minus one dt, we know from the fundamental
theorem of calculus that h prime of x would be simply this inner function with the t replaced by the x. It would just be two x minus
one, pretty straightforward. But this one isn't quite
as straightforward. Instead of having an x up here, our upper bound is a sine of x. So one way to think about it
is if we were to define g of x as being equal to sine of x, equal to sine of x, our capital F of x can be
expressed as capital F of x is the same thing as h of, h of, instead of an x, everywhere we see an x, we're replacing it with a sine of x, so it's h of g of x, g of x. You can see the g of x right over there. So you replace x with g of x for where, in this expression, you get h of g of x and that is capital F of x. Now why am I doing all of that? Well, this might start making you think about the chain rule. Because if this is true, then that means that capital F prime of x is going to be equal to h prime of g of x, h prime of g of x times g prime of x. And so what would that be? Well, we already know
what h prime of x is, so I'll need to do this in another color. This part right over
here is going to be equal to everywhere we see an x here, we'll replace with a g of x, so it's going to be two, two times sine of x. Two sine of x, and then minus one, minus one. This is this right over here, and then what's g prime of x? G prime of x, well g prime of x is just, of course, the derivative of sine
of x is cosine of x, is cosine of x. So this part right over here is going to be cosine of x. And we could keep going. We could try to, we could try to simplify this a little bit or rewrite it in different ways, but there you have it.