Differential and integral are reciprocal operations, and this concept is refined from the basic theorem of calculus. This means that we can discuss calculus from either of them, but in teaching, differential calculus is usually introduced first.
Calculus has three main branches: limit, differential calculus and integral calculus. The basic theory of calculus shows that differential and integral are reciprocal operations. Newton and Leibniz discovered this theorem, which caused other scholars to study calculus enthusiastically. This discovery enables us to switch between differential and integral. This basic theory also provides a method to calculate many integral problems by algebra. This method does not really perform limit operation, but looks for indefinite integral. This theory can also solve some differential equations.
The basic concepts of calculus include function, infinite sequence, infinite series and continuity, and the operation methods mainly include symbolic operation skills, which are closely related to elementary algebra and mathematical induction.
Calculus has been extended to differential equations, vector analysis, variational methods, complex analysis, time domain differentiation and differential topology. Modern version of calculus is real analysis.
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limit
The most important concept in calculus is "limit". Wechat business (derivative) is a limit. Definite integral is also a limit.
Mathematicians have struggled for more than 200 years from Newton's practical use of it to the formulation of a careful definition. The definition used now was given by Westeros in the middle of19th century.
The limit of a sequence means that when an ordered sequence extends forward, if there is a finite number, the sequence can be infinitely close to this number, and this number is the limit of this sequence.
The representation method of sequence limit is:
Where x is the limit value. For example, its limit at that time was x = 0. That is to say, the greater n (the farther it extends), the closer this value is to 0.
derivant
We know that in kinematics, the average speed is equal to the distance traveled divided by the time spent. Similarly, in a short time interval, except for the short distance traveled, it is equal to the speed in this short time. When the time of this short interval tends to zero, the speed at this time is instantaneous and cannot be calculated according to the usual division. At this time, speed is the derivative of time. It must be calculated by derivative method. That is to say, when the independent variable of the function approaches a certain limit, the limit of the quotient of the increment of the dependent variable and the increment of the independent variable is the derivative. On the issue of speed, distance is a dependent variable of time, which changes with time. When time approaches a certain limit, the limit of distance increment divided by time increment is the derivative of distance to time.
The geometric meaning of derivative is the tangent slope of function curve at this point.