The two kinds of calculus we've been doing - integral differential, seem unrelated at first glance, but, as Newton and his crack team of mathematicians discovered, they're closely related. The two processes are actually the inverse of each other, and that will enable us to not do these long processes from the last two chapters except on tests from now on. At least, that's what I'm hoping.
We start this off by defining a function as g(x) = ⌠(a to x) f(t)dt, where f is a continuous function on [a,b] and x can be either a or b. The value of g is only dependent on x, which is the top number of our interval. If x is fixed, then the integral is a constant, but if x is a variable, then the integral varies and is a function of g(x).
If f(t) is positive, then g(x) is the area under the graph of f from a to x, and x can vary. g(x) is the "area so far," as the book puts it, which I like.
This is the first entry where I'm doing the writing before having the class on the topic, so I don't fully understand what's going on, but I'll write my best understanding of it and probably talk about my clarifications after class.
Anyway, we're shown a curvy graph for y=f(t) and, given that the integral from 0 to x of that function is a function of x, evaluate that function [g(x)] for a series of numbers. We can see from looking at it that g(0) will be 0, and the rest of them can be interpreted as areas of fairly easy geometric shapes. Except a couple of them, which they eyeball to say are about 1.3, because I guess that's okay here.
We plot these points on a graph of g(x). Since the areas switched to negative areas at one point, the graph begins decreasing there and we know we have a maximum. Using an exercise my teacher didn't assign so I had to look up, we know that the integral of a variable function is half of the difference between b^2 and a^2, which, if we choose a as 0 and b as x, we get (x^2/2) for the integral, and thus g(x).
Now, the book says that we take f(t) to equal t, which I don't fully understand the reasoning behind and will ask about in class tomorrow. But, just taking their word from, if we take the derivative of g(x), we get x. So, we can say g'=f, meaning that g is an antiderivative of f.
Anyway, if we estimate the derivative of g(x) and sketch our graph out, it should look pretty close to the graph we started with, which confirms our suspicion.
To make it more mathematically sound than "Hey, those graphs look like each other," we think about any graph where f(x)≥0, so that g(x) (being the integral from a to x of f(t)) is the area under the graph from a to x. If we're going to compute g'(x) using the limit-taking method none of us remember how to do because we haven't used it since section 2, using g(x) and g(x+h) actually results in us subtracting areas. Since we're subtracting the area from the area of h further than our first area, what we really get is the area of the rectangle whose width is h, and whose height is f(x) for that point, or hf(x).
So g'(x) is equal to the limit as h→0, which is also equal to f(x). I don't fully understand this step, but I think it's because I don't really remember this method for taking derivatives. Again, I'll ask about it in class and go look that process up again later.
So we get the first part of the Fundamental Theorem of Calculus. If f is continuous on [a, b], then g(x) (which is the integral from a to x of f(t) as long as x is between a and b) is continuous and differentiable on [a, b] and, more importantly, g'(x)=f(x).
Reworded, assuming the normal constraints of continuity and whatnot, if we take the integral of a function from a to x, and define it as a function of x, then the derivative of that new function is the antiderivative of the original integrated function. As the book puts it, if we integrate f, then differentiate the result, we get back where we started.
This realization allows us to look at principles discovered with integral calculus (Like the Fresnel function, provided by the book as a function important to optics) and let us play with it in all the ways we can do with differential calculus. It's like realizing you can stick other games into Sonic and Knuckles other than just a Sonic game and play new bonus stages.
The next part of the FTC provides a much easier method of integration than chaining together all of the Riemann sums that took forever and made me stay up until all hours doing my homework.
FTC2: The integral of a function from A to B is the antiderivative of B minus the antiderivative of A. Or, if you know an antiderivative of f, then we can integrate the function by subtracting the the antiderivatives at the endpoints.
I must say...that's irritatingly simple.
So the Fundamental Theorem of Calculus is basically 2 parts. If a function is integrated then differentianted, you end up back where you started, and the integral of a function equals any of its antiderivatives at its two given endpoints. Or, integration and differentiation are inverse processes.
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