The Fundamental Theorem of Calculus establishes a powerful connection between antiderivatives and derivatives. Let's explore this connection in more detail.

Consider a function f(x) and let F(x) be its antiderivative. According to the first part of the Fundamental Theorem of Calculus, if f(x) is continuous on an interval [a, b], then the definite integral of f(x) from a to b can be evaluated by finding the difference between the antiderivative F(x) evaluated at the endpoints: F(b) - F(a). It is important to note that a and b are real numbers within the interval.

Here's an example to illustrate this concept:

Suppose we have the function f(x) = 2x. To find the definite integral of f(x) from 1 to 3, we need to find the antiderivative of f(x) first. Integrating f(x) with respect to x gives us F(x) = x^2 + C, where C is the constant of integration. Then, evaluating F(x) at the endpoints, we have F(3) - F(1) = (3^2 + C) - (1^2 + C) = 9 - 1 = 8.

This demonstrates how the Fundamental Theorem of Calculus enables us to calculate definite integrals by finding the difference between antiderivatives.

Moving on to derivatives, the second part of the Fundamental Theorem of Calculus states that if F(x) is continuous on an interval [a, b] and F'(x) is the derivative of F(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a). In other words, the integral of the derivative of a function over an interval is equal to the value of the function evaluated at the endpoints of that interval.

Let's consider the same function f(x) = 2x. The derivative of F(x) = x^2 + C with respect to x is f(x) = 2x. Now, finding the definite integral of f(x) from 1 to 3 is as simple as evaluating F(x) at the endpoints: F(3) - F(1) = (3^2 + C) - (1^2 + C) = 9 - 1 = 8, just as before.

By connecting antiderivatives with derivatives, the Fundamental Theorem of Calculus provides a powerful tool to evaluate definite integrals and find the slope of a curve using antiderivatives.

With this understanding, you are ready to apply the Fundamental Theorem of Calculus in various problem-solving scenarios. Keep practicing and exploring the connections between calculus concepts, and you'll master this fundamental theorem in no time!