1, Limit Pinch Theorem in Queue Waiting

In the pinch theorem of sequence limit, draw three straight lines perpendicular to the axis, representing three planes perpendicular to the plane respectively, and label them as Yn, a and Zn from left to right. Suppose A is in a fixed form, and both Yn and Zn are infinitely close to A, but at this time, put the plane Xn between Yn and Zn arbitrarily, and these values are infinitely close to A, which is the image description of the pinch theorem.

2. The application of "infinitesimal method" in calculating solid volume in vegetable cutting.

When learning definite integral to calculate the volume of three-dimensional space with known parallel cross-sectional area, it is assumed that a solid plane in the space is surrounded by a curved surface and two planes perpendicular to the X axis. If any point is perpendicular to the plane of the X axis, the cross-sectional area obtained is also a known continuous function, and the three-dimensional volume can be expressed by definite integral. And draw a conclusion through "differential element method".

The application of this method in life can be considered as follows: when cutting cucumber rings, put the washed cucumber on a horizontally placed chopping board, and cut off both ends of the cucumber with a kitchen knife in the direction perpendicular to the chopping board, that is, the required volume of three-dimensional space. That is to say, a cucumber slice is cut perpendicular to the cutting board at a small distance and regarded as a pillar. This volume is equal to the cross-sectional area times the thickness.

By analogy, if the cucumber is cut into several pieces, the approximate volume of the cucumber can be obtained by calculating the area of each piece. If the cucumber piece is thin, the volume value is probably accurate. That is to subdivide it infinitely, and then get infinite sum, which is the best application of definite integral.

The formative period of calculus

1,1the first half of the 7th century:

During this period, almost all the masters of science devoted themselves to solving problems such as speed, extremum, tangent and area, especially the infinitesimal algorithm to describe motion and change, and made great progress in a short time.

Astronomer Kepler discovered the three laws of planetary motion, and used the idea of summation of infinitesimal to find the area of curved polygon and the volume of rotator. Cavalieri, an Italian mathematician, also discovered the cavalieri principle, and defined the integral formula of power function by the undivided method.

In addition, cavalieri also proved Gilding's theorem (the volume of a three-dimensional figure obtained by rotating a plane figure around an axis is equal to the product of the circumference of a circle formed by the center of gravity of the plane figure and the area of the plane figure). ) has a profound influence on the formation of the rudiment of calculus.

In addition, the algebraic method of Descartes, the founder of analytic geometry, also greatly promoted the development of calculus. Fermat, a great French mathematician, has made great contributions to finding the tangent of curve and the extreme value of function. Among them, there is Fermat's theorem about mathematical analysis: let the function f(x) be defined on a certain interval χ, and take the maximum (minimum) value at the inner point c of this interval. If there is a finite derivative f'(c) at this point, there must be f'(c)=0.

2,1the second half of the 7th century:

British scientist Newton began to study calculus. Inspired by Wallis' arithmetica infinitorum, he extended algebra to analysis for the first time. 1665, Newton invented downstream counting (difference), and the following year he invented countercurrent counting. After that, we summarized the flow number technology and wrote a brief introduction to flow number, which marked the birth of calculus.

Then, Newton studied the variable flow generation method, and thought that variables were generated by the continuous movement of points, lines or surfaces. Therefore, he called variable flow and variable rate flow.

In the later period of Newton's creation of calculus, he denied that variables were a set of static infinitesimal elements, and no longer emphasized that mathematical quantities were composed of inseparable smallest units, but were generated by the constant movement of geometric elements. He no longer thinks that the flow number is the ratio of two real infinitesimals, but the initial ratio of the original quantity or the final ratio of the vanishing quantity, which leads to the infinite division process of the quantity from the original real infinitesimal point of view, that is, the potential infinity point of view.

At the same time, German mathematician Leibniz also independently founded calculus. He published his first differential paper in 1684, defined the concept of differential, and adopted differential symbols dx and dy. 1686 published an integral paper, which discussed differential and integral, and used the integral symbol ∫. The invention of symbols makes the expression of calculus easier. In addition, he also discovered the Leibniz formula for finding higher derivatives and Newton Leibniz formula for connecting differential and integral operations, which contributed as much to calculus as Newton.