Final answer:
The Fundamental Theorem of Calculus links differentiation with integration. The first part tells how to calculate the integral of a function using its antiderivative. The second part, or Evaluation Theorem, describes how the integral can represent the accumulation of a function's values.
Step-by-step explanation:
The Fundamental Theorem of Calculus connects differentiation and integration, two main concepts in calculus, establishing a relationship between the derivative and the integral. It is divided into two parts. The first part of the theorem tells us that if we have a continuous function f(x) on an interval [a,b], and if F(x) is an antiderivative of f(x), then the integral of f(x) from a to b is equal to F(b) - F(a).
For example, consider the function f(x) = x^2. One antiderivative of this function is F(x) = (1/3)x^3. According to the Fundamental Theorem, the integral from 0 to 1 of x^2 would be F(1) - F(0) = (1/3)(1)^3 - (1/3)(0)^3 = 1/3.
The second part, also known as the Evaluation Theorem, allows us to evaluate the integral as the accumulation of a function's values. It states that if f(x) is continuous on [a,b], then the function F(x) defined by F(x) = ∫_a^x f(t) dt for x in [a,b] is continuous on [a,b] and differentiable on (a,b), and F'(x) = f(x).
An example of this is if we take f(x) = cos(x) and we want to find the integral from 0 to π/2, we have F(x) = ∫_0^x cos(t) dt. We can say F'(π/2) = cos(π/2) = 0, meaning the rate of change of the accumulation of cos(x) from 0 to π/2 is zero at π/2.