What does the Fundamental Theorem of Calculus connect?

The Fundamental Theorem of Calculus establishes a profound connection between the derivative and the integral, which are two central concepts in calculus. This theorem has two main parts.

The first part states that if a function is continuous over an interval, then the integral of that function over that interval can be defined in terms of its antiderivative (which is a function whose derivative is the original function). Essentially, this shows how integration can be used to recover the original function from its derivative.

The second part provides that if one has an antiderivative of a function, the process of differentiation and the process of integration are essentially inverse operations. This means that applying a derivative to the integral of a function will give back the original function, assuming the function is continuous.

This linkage is crucial as it allows for a deeper understanding of both the notion of finding areas under curves (integration) and the slope of tangent lines to curves (differentiation). Understanding this relationship forms the foundation of further studies in calculus and its applications in mathematics, physics, and engineering. Thus, the connection highlighted in the correct answer is pivotal in understanding how these two fundamental operations interact within the realm of calculus.