Fundamental Theorem of Calculus
Fundamental theorem of calculus
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- Let's say I have some function f,
- that is continuous on an interval between a and b,
- and I have these brackets here, so it also includes a and b and the interval.
- So let me graph this, just so we get a sense of what I'm talking about
- So that's my vertical axis, this is my horizontal axis. I'm gonna label my horizontal axis t,
- so that we can save x for later, and I can still make this y right over there
- and let me graph, this right over here is the graph of: y is equal to f(t)
- Now, our lower endpoint is a, so that's a, right over there
- Our upper boundary is b,
- ...to make that clear, and actually just to show that we're including that endpoint,
- let me make them bold lines, filled-in lines.
- So lower boundary a, upper boundary b, we're just saying, and I've drawn it this way,
- that f is continuous on that.
- Now, let's define some new function that's the area under the curve
- between a and some point that's in our interval.
- Let me pick this right over here, x.
- So let's define some new function, to capture the area under the curve,
- between a and x.
- Well, how do we denote the area under the curve between two endpoints?
- Well, we just use our definite intergral, that's our Riemann integral, that's really..
- that right now, before we come up with the conclusion of this video,
- it really just represents the area under the curve between two endpoints.
- So this right over here, we can say is,
- the definite integral from a to x, of f(t), dt.
- Now, this right over here is going to be a function of x, let me make it clear,
- where x is in the interval between a and b. This thing right over here
- is going to be another function of x. This value is going to depend
- on what x we actually choose.
- So let's define this, as a function of x, so I'm gonna say that this is equal to
- uppercase F(x).
- So all fair and good, uppercase F(x) is a function, if you give me an x value
- that's between a and b, it'll tell you the area under lowercase f(t)
- between a and x.
- Now, the cool part, the fundamental theorem of calculus,
- the fundamental theorem of calculus tells us, let me write this down,
- it's a big deal. Fundamental theorem of calculus tells us,
- that if we were to take the derivative of our capital F,
- so if I were to take the derivative of capital F with respect to x,
- which is the same thing as taking the derivative of this, with respect to x,
- which is equal to the derivative of all of this business, let me copy this
- So, copy, and then paste. Which is the same thing,
- I've defined capital F as this stuff, so if I'm taking the derivative of the left hand side,
- it's the same thing as taking the derivative of the right hand side.
- The fundamental theorem of calculus tells us,
- that this is going to be equal to,
- it's going to be equal to f, lowercase f(x)
- Now, why is this a big deal? Why does it get such an important title,
- as the fundamental theorem of calculus?
- Well, it tells us, that for any continuous function f,
- If I define a function, that is the area under the curve between a and x right over here,
- that the derivative of that function is going to be f.
- So let me make it clear,
- every continuous f has an antiderivative F(x).
- That by itself is a cool thing, but the other really cool thing,
- or I guess these are somewhat related, is, remember, coming into this,
- all we did is, we just viewed the definite integral as symbolizing the area under the curve
- between two points. That's where that Riemann definition
- of integration comes from. But now we see a connection,
- between that and derivatives. When you're taking the definite integral,
- one way of thinking, especially if you're taking the definite integral between a lower boundary and an x,
- one way of thinking about it is that you're essentially taking an antiderivative.
- So we now see a connection,
- this is why it is the fundamental theorem of calculus,
- it connects differential calculus and integral calculus. Connection between derivatives,
- and (maybe I should say antiderivatives), derivatives and integration,
- which before this video, we just viewed integration as area under the curve.
- Now we see it has a connection to derivatives.
- So how would you actually use the fundamental theorem of calculus, maybe in the context of a calculus class,
- and we'll do the intuition for why this happens,
- or why this is true, and maybe a proof in later videos,
- but how would you actually apply this right over here?
- Well, let's say someone told you, that they want to find the derivative,
- Let's say someone wanted to find the derivative with respect to x
- of the integral from, I don't know, I'll pick some random number here,
- Pi to x, of, I'll put something crazy here,
- cosine squared of t, over the natural log of t minus the square root of t, dt.
- So they want you to take the derivative with respect to x of this crazy thing,
- Remember, this thing in the parentheses, is a function of x.
- Its value, it's going to have a value that is dependent on x, you give it a different x,
- and it's going to have a different value. So what's the derivative of this with respect to x?
- Well, the fundamental theorem of calculus tells us it can be very simple.
- We essentially, and you can even pattern match up here, we'll get more intuition of why this is true in future videos,
- but essentially, everywhere where you see this right over here is an f(t)
- everywhere you see a t, replace it with an x, and it becomes an f(x).
- So this is going to be equal to
- cosine squared of x, over the natural log of x minus the square root of x,
- you take the derivative of the indefinite integral, where the upper boundary is x right over here,
- just becomes, whatever you were taking the integral of, that as a function of, instead of t,
- that is now a function of x.
- So it can really simplify sometimes, taking a derivative,
- and sometimes, you'll see on exams,
- these trick problems where you have this really hairy thing that you need to take a definite integral of
- and then take the derivative,
- you just have to remember the fundamental theorem of calculus,
- the thing that ties it all together, it connects derivatives and integration,
- you can just simplify it, by realizing that this is just going to be,
- instead of a function lowercase f(t), it's going to lowercase f(x),
- let me make it clear, in this example right over here,
- this right over here was lowercase f(t),
- and now it became lowercase f(x).
- This right over here was our a,
- and notice, it doesn't matter what the lower boundary of a actually is.
- You don't have something on the right hand side that is in some way dependent on a.
- Anyway, hope you enjoyed that and in the next few videos, we'll think about the intuition,
- and do more examples making use of the fundamental theorem of calculus.
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