As a seasoned expert in the field of mathematics, I often find myself delving into intricate questions that challenge the very core of mathematical concepts. Today, let's embark on a journey through the labyrinth of calculus, specifically focusing on a master level question that delves into the depths of the Fundamental Theorem of Calculus. Join me as we unravel the mysteries of this theorem and shed light on its profound implications.

The Fundamental Theorem of Calculus stands as a cornerstone in the realm of calculus, serving as a bridge between the concepts of differentiation and integration. It beautifully connects these seemingly disparate operations, offering profound insights into the behavior of functions and their accumulated effects over intervals. At Math Assignment Helper, we understand the significance of mastering this theorem, as it lays the groundwork for advanced mathematical analysis and applications across various fields.

Question:

Consider a continuous function f(x) defined on the closed interval [a, b]. If F(x) is an antiderivative of f(x) on [a, b], prove that ∫(a to b) f(x) dx = F(b) - F(a).

Answer:

The Fundamental Theorem of Calculus elegantly showcases the relationship between differentiation and integration, providing a powerful tool for evaluating definite integrals. To prove this theorem, let's begin by considering an antiderivative F(x) of the function f(x) on the interval [a, b]. By the definition of an antiderivative, we have:

F'(x) = f(x)

for all x in [a, b]. Integrating both sides of this equation with respect to x over the interval [a, b], we obtain:

∫(a to b) F'(x) dx = ∫(a to b) f(x) dx

By the Second Fundamental Theorem of Calculus, the integral of a derivative over an interval yields the difference of the function evaluated at the endpoints of the interval. Therefore, we have:

F(b) - F(a) = ∫(a to b) F'(x) dx

Substituting F'(x) = f(x), we conclude:

F(b) - F(a) = ∫(a to b) f(x) dx

Thus, we have proven that ∫(a to b) f(x) dx = F(b) - F(a), as desired.

Conclusion: In conclusion, the Fundamental Theorem of Calculus serves as a testament to the elegance and power of mathematical theory. Through rigorous analysis and application of fundamental principles, we have unveiled the connection between antiderivatives and definite integrals, paving the way for deeper understanding and exploration of calculus. At Math Assignment Helper, we thrive on unraveling such complex mathematical concepts, providing students with the expertise and guidance needed to excel in their studies