The Fundamental Theorem of Calculus

The fundamental theorem of calculus will allow us to formally link together differential and integral calculus. The fundamental theorem of calculus is typically presented in two parts, where the first part implies that there exists an antiderivative for a continuous function, and the second part states how we can computer an integral using an antiderivative. With this understanding, we will be able to compute the area under any continuous curve that we can get the antiderivative of.

The Fundamental Theorem of Calculus Part 1: If f is a function that is continuous on [a,b], then the function: g(x) = \int_{a}^{x} f(t) dt is continous on [a,b], differentiable on [a,b], and g'(x) = f(x).

As we discussed above, this theorem is essentially asserting that if we have a function f(x) that is continuous on an interval, then there exists a g(x) that is continuous and differentiable on the same variable. In addition to that, g(x) has a derivative that is equal to f(x). This theorem gives us a solid link between differentiable and integral calculus, making what we discussed about definite integrals more concrete. The second part of the fundamental theorem goes one step further and explains how the integrals can be used to compute the area under a curve.

The Fundamental Theorem of Calculus Part 2: If f is continuous on [a,b], then \int_{a}^{b} f(x) dx = F(b) - F(a), where F is any antiderivative of f.

Of the two parts of the theorem, the second part is the one you will use most actively. The first part is mostly to formalize the idea of antiderivatives being related to integrals. You’ll find that once we find the antiderivative, the actual act of integrating the function is rather easy. You simply evaluate the antiderivative at the points x = a and x = b, and subtract them as described in the theorem. The harder part, and the part that we will focus on, is how we can find the antiderivative of a given function f(x).

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