Basic theorem of Calculus

Looking back, in 1908, calculus (also known as "micro-accumulation") was introduced into China. At the beginning, only a few people in China knew about calculus. After liberation, especially after the proposal of "entering science" in 1956, China began to learn about the Soviet Union in an all-round way. At that time, the Soviet scholar fekhincz wrote the "calculus tutorial" (three volumes of 9, ye yanqian translated) popular throughout the country, cultivate a large number of China's new generation of mathematical workers. I am also the "product" of that age ".

As an example of the "11th Five-Year Plan for General Colleges and Universities": "Advanced Mathematics" (Tongji University) and "Mathematical Analysis" (Fudan University ), they all inherit from Fei's "calculus tutorial" (or the theoretical system. The choice of the core content of calculus is somewhat biased. For example, the basic theorem of calculus (Theorem) originally created by Newton leveniz was intentionally diluted, it is called only the "basic calculus formula" or the "Newton-laveniz formula" (fomula ). The Theorem and formula are of course different in importance.

In 1960, German mathematician. robinson created "non-standard analysis", and his theory rigorously restored the Historical Appearance of calculus. From then on, the title (or argument) of the basic theorem of calculus emerged historically. This theorem embodies and is highly concentrated on the essence (or core) of calculus, improving people's understanding of calculus.

In 1976, J. Keisler's basic calculus textbook reflected this historical change. In section 4.2 (called the basic theorem of calculus) of the pocket ebook in this textbook, I will meet you today. This is a historical moment that deserves our nostalgia. (Note: Search for the keyword "section 4.2 basic theorem of calculus .)

The basic theorem of calculus is as follows:

Fundamentaltheorem of Calculus

Suppose f is continuous on its domain, which is an open interval I.

(I)For each point A in I, the definite integral
Of F from A to X considered as a function of X is an antiderivative(Inverse derivative)Of
F. That is

2. If F is any antiderivative of F, then for any two points (a, B) in I the definite integral of F from A to B is equal to the difference
   F (B)-f (),

What is the basic theorem of calculus? J. Keisler pointed out:"
Fundamental theorem of calculus is important for two reasons. first, it shows the relation between the two main notions of calculus: The derivative, which corresponds to velocity, and the integral, which corresponds to area. it shows that differentiation and
Integrationare "inverse" processes. second, it gives a simple method for computing extends definite integrals. "means that the basic theorem shows that the differential method and the integral method are two" reciprocal "processes, and a simple calculation method for the definite integral is given.