Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is a central result in calculus that establishes a connection between differentiation and integration. It consists of two parts that demonstrate the inverse relationship between these two essential operations.

The first part of the FTC states that if a function is continuous on a closed interval [a, b], and its antiderivative (also called the indefinite integral) is F(x), then the definite integral of the function over the interval [a, b] is equal to the difference in the values of the antiderivative at the endpoints of the interval: ∫[a, b] f(x) dx = F(b) - F(a).

The second part of the FTC states that if a function is continuous on an interval [a, b] and has an antiderivative F(x) on that interval, then the derivative of the antiderivative with respect to x is equal to the original function: d/dx [∫[a, x] f(t) dt] = f(x).

The Fundamental Theorem of Calculus has profound implications in mathematics, physics, and engineering. It allows us to find the area under a curve, solve problems involving motion and change, and provides a foundation for many advanced topics in calculus, such as Taylor series, integrals, and the Mean Value Theorem.