Explain or Discuss the significance of the Fundamental Theorem of Calculus in the field of Mathematics.

Answer 1

This is quite the dense question! The Fundamental Theorem of Calculus (both part 1 and part 2) is one of the most important theorems in calculus - hence the term "fundamental." At its core, part one of the Theorem (the one that says that the antiderivative of f(x) is equal to the integral from a to x of f(t) dt) represents the relationship between differentiation and integration and implies the inverse nature of the two. It also ensures that all integrable functions have antiderivatives. Furthermore, since all continuous functions are integrable, it implies that all continuous functions have antiderivatives.

The second part of the theorem is what allows us to evaluate definite integrals analytically. Deeper still, it can represent linear transformations on the space of continuous, real-valued functions. There are so many applications to the theorem, that it feels inadequate to leave it there, but that should give you a taste.