fundamental theorem of calculus part 1 practice problems

Write this down because this is a big deal. Now the cool part, theįundamental theorem of calculus. That's between a and b, it'll tell you theĪrea under lowercase f of t between a and x. Over here is going to be another function of x. Going to be a function of x- and let me make Here, we can say is the definite integralįrom a to x of f of t dt. Now before we come up with the conclusionīetween two endpoints. The area under the curve between two endpoints? Well, we just use ourĭefinite integral. Some new function to capture the area under New function that's the area under the curveīetween a and some point that's in our interval. We're including that endpoint, let me make them bold My horizontal axis t so we can save x for later. So it also includes a and b in the interval. So, while we can create a mathematical model and imagine cutting it into infinitely many pieces just so we can minimize our approximation errors, that doesn't mean that something in the real world can be cut up into infinitely many slices. And, if current scientific hypotheses are correct, we will never get close to measuring time that small. But, the dispute amongst scientists is not over the measurement, but over whether there even exists a time period shorter than 1 Planck time.Īt present, we don't have the technology to measure time anywhere near as brief as a Planck time (though we have other reasons for knowing that is the limit of measurement). So, you could only ever measure whole number increments of a Planck time. What is fairly well accepted by scientists is that it is impossible by any means to measure time shorter than 1 Planck time, whether or not there is such a thing as part of a Planck time. This unit is the Planck time, which is equal to about 5.39×10⁻⁴⁴ sec. While this is not a well-confirmed idea, and many scientists disagree, there are some scientists who think that there is a smallest unit of time, something like the time equivalent of an atom - you just can't get any briefer a period of time. These are mathematical models of reality, not reality itself. When the curve is at y = 3, for example, the area under the curve is expanding at a rate of 3. Ultimately we're trying to prove something that is, in a way, quite obvious: that the rate of change in the area under a curve is given by the value of the curve. The reason is that we're trying to get at the rate of change in F(x), and the rate at which F(x) is changing doesn't depend on how much area there is to the left of x. The other thing that many people find confusing is that we get the same result no matter what point we choose for a, the starting point of the region we're studying. We use t in place of x for the curve, and sometimes this is called a dummy variable. We set it up initially as f(t) because we're using x for a different purpose (area under the curve) and it would create a logical contradiction for x to be both the independent variable that produces the curve and the independent variable that produces the area under the curve.

It can operate on t or x or any other variable or constant within its domain. The key to understanding this is to realize that f, by itself, is a function.

Some people have a hard time with this duality of f(t) and f(x).

We're trying to show that if we have a function F(x) that provides the area under the curve f(t), the derivative of F(x) is f(x). If you get that much firmly in mind, the rest should be easier, but there are a couple of other points of confusion. We're given a function f(t) and asked to think about another function, not displayed, that is the area under the curve, not the value displayed on the curve. The graph doesn't show F(x) at all in fact, it doesn't have an x-axis. As x moves to the right, this area increases, even if f(t) is decreasing. Instead, F(x) is the area under this graph between point a and point x. So the first thing I would offer in trying to understand this better is to get a clear picture that this graph does not depict F(x). In this case, we're studying a function F(x) but looking at a graph of a different function, f(t). Part of the problem is that in almost all our other work we're looking at a graph of the function we're studying.

Fundamental theorem of calculus part 1 practice problems