The fundamental theorem of calculus is a powerful concept in calculus that connects the two major operations: differentiation and integration. It states that if a function is continuous over a closed interval, then the definite integral of its derivative over that interval is equal to the difference in the values of the original function evaluated at the endpoints of the interval.

Let's consider an example to illustrate this theorem. Suppose we have a function f(x) = 3x^2, and we want to find the integral of its derivative over the interval [1, 4]. First, we find the derivative of f(x) which is f'(x) = 6x. Then, we integrate f'(x) over the interval [1, 4] to get the definite integral. Evaluating the integral, we obtain the value of the definite integral as the difference in the values of the original function evaluated at the endpoints, f(4) - f(1). This concept is foundational in calculus and plays a crucial role in various mathematical applications.

Understanding the fundamental theorem of calculus is essential as it enables us to find the area under curves, evaluate definite integrals, and solve many real-world problems involving rates of change and accumulation. In the upcoming posts in this series, we will explore the intricacies of this theorem and delve into its applications in more detail.