The Fundamental Theorem of Calculus is one of the most important concepts in calculus, connecting the two fundamental operations: differentiation and integration. It allows us to solve problems that involve finding the area under curves, calculating the total change of a function, and determining antiderivatives.

Let's begin by understanding the core idea behind the Fundamental Theorem of Calculus. Suppose we have a function f(x) that is continuous on the interval [a, b], and F(x) is the antiderivative of f(x). The first part of the theorem states that if we define a new function by evaluating the definite integral of f(x) from a to x, we get a new function called the accumulation function, denoted as A(x):

A(x) = ∫[a, x] f(t) dt.

This accumulation function tells us the total accumulated change of the original function f(x) as we move from the starting point a to x.

Now, let's dive deeper into the Fundamental Theorem of Calculus and explore its implications in calculus and real-world applications.