Unearthing the Magic of Calculus!
The Fundamental Theorem of Calculus connects differentiation and integration, two core ideas in calculus. It serves as a cornerstone of analysis and offers a unique and vital connection between these two mathematical operations.
The theorem is divided into two parts:
- First Part: This part guarantees that every continuous function has an antiderivative.
- Second Part: It states that the integral of a function over an interval can be computed using one of its antiderivatives.
Mathematically, this can be expressed as:
Let \( f \) be a continuous real-valued function defined on a closed interval \([a, b]\). Then, if \( F \) is defined by \( F(x) = \int_a^x f(t) \, dt \), the first part guarantees \( F \) is continuous on \([a, b]\), differentiable on the open interval \((a, b)\), and \( F'(x) = f(x) \) for all \( x \) in \((a, b)\).
In the second part, for any antiderivative \( F \) of \( f \) on \([a, b]\), we have:
\(\int_a^b f(x) \, dx = F(b) - F(a)\)