From the Renaissance in Europe at the beginning of15th century, the large-scale development of industry, agriculture, navigation and merchant trade has formed a new economic era. The religious reform and suspicion of the imprisonment of the church thought, the introduction of advanced science and technology from the East through Arabia, and the influx of Greek documents into Europe after the collapse of the Byzantine Empire all gave intellectuals at that time a brand-new look. /kloc-in the 0/6th century, Europe was in the embryonic stage of capitalism, and its productivity was greatly developed. The development of production practice puts forward new topics for natural science, which urgently needs the development of basic disciplines such as mechanics and astronomy, and these disciplines are deeply dependent on mathematics, thus promoting the development of mathematics. The requirements of science for mathematics are finally summarized as several core issues:
Mutual Solution of Velocity and Distance in (1) Motion
That is to say, knowing that the moving distance S of an object is expressed as a function of time, we can use the formula S=S(t) to find out the speed and acceleration of the object at any time. Conversely, it is known that the accelerometer of an object is a function of time, and the speed and distance are obtained. This kind of problem appears directly when studying sports. The difficulty is that the speed and acceleration studied are constantly changing. For example, to calculate the instantaneous speed of an object at a certain moment, we can't divide the moving distance by the moving time like calculating the average speed, because at a given moment, the moving distance and the time used by the object are both 0,0/0, which is meaningless. However, according to physics, there is no doubt that every moving object must have a speed at every moment of its movement. The problem of finding the moving distance with the known speed formula also encounters the same difficulty. Because the speed changes all the time, we can't get the moving distance of an object by multiplying the moving time by the speed at any moment.
(2) Find the tangent of the curve
The problem itself is pure geometry, which is of great significance to scientific application. Because of the need of studying astronomy, optics is an important scientific research in the17th century. In order to study the light passing through the lens, the lens designer must know the angle at which the light enters the lens in order to apply the reflection law. What matters here is the angle between the light and the normal of the curve, which is perpendicular to the tangent, so it is always to find the normal or tangent. Another scientific problem involving the tangent of a curve appears in the study of motion, that is, to find the direction of motion of a moving object at any point on its trajectory, that is, the tangent direction of the trajectory. (3) Find the length, area, volume and center of gravity.
These problems include finding the length of a curve (such as the distance a planet moves in a known period), the area enclosed by a curve, the volume enclosed by a curved surface, the center of gravity of an object, and the attraction of a larger object (such as a planet) to another object. In fact, the problem of calculating the length of an ellipse puzzles mathematicians, so that for a period of time, mathematicians failed in their further work on this issue, and new results were not obtained until the next century. Another example is the problem of finding the area. As early as in ancient Greece, people used the exhaustive method to find some areas and volumes. For example, they use the exhaustive method to find the area S enclosed by parabola and X axis and straight line x= 1 on the interval [0, 1]. When n is smaller and smaller, the result at the right end is closer and closer to the exact value of the required area. However, the application of exhaustive method must add many skills, and it lacks universality and often has no numerical solution. When Archimedes' work became famous in Europe, his interest in finding length, area, volume and center of gravity revived. The exhaustive method was first revised gradually, and then it was fundamentally revised because of the creation of calculus. (4) The problem of finding the maximum and minimum values.
The horizontal distance, that is, the range, of the projectile from the barrel depends on the inclination angle of the barrel to the ground, that is, the shooting angle. A "practical" problem is to find the launching angle that can get the maximum range. /kloc-At the beginning of the 0/7th century, Galileo concluded that the maximum range (in vacuum) was reached when the launch angle was 45; He also obtained different maximum heights of the shells after they were fired from different angles. Studying the motion of planets also involves the problems of maximum and minimum, such as finding the distance between planets and the sun.