What exactly is calculus What exactly is calculus

What is Calculus? Meaning of Calculus:

Calculus is the branch of mathematics that studies the differentiation and integration of functions, as well as concepts and applications. It is a fundamental subject of mathematics. The content mainly includes limits, differential calculus, integral science and their applications.

Differential calculus consists of the operation of finding derivatives and is a set of theories about the rate of change.

It makes it possible to discuss functions, velocities, accelerations, and slopes of curves in a common set of notations. Integralism, including the operation of finding integrals, provides a general set of methods for defining and calculating area, volume, etc.

1. The "micro" of differentiation means small, segmented, and very finely divided; The "product" of an integral means accumulation, totaling, and summing.

2. The simple term of differentiation is to calculate the relevant rate of change and implication rate of change, which can be summarized in terms of thinking and method: segmentation, comparison, and limit; The geometric meaning is to transition from finding the slope of the secant to the slope of the tangent.

3. The basic idea of integration can be summarized as: division, summing, and limiting. Geometry is the sum of the areas of the microelements.

4. The application of calculus is ubiquitous, including physics, chemistry, biology, geology, meteorology, oceanography, hydrology, astronomy, electronics, computers, electrical machinery, machinery, chemical engineering, and smelting. In the economy, finance, accounting, management. It also has an extremely wide range of applications.

It can be said that without calculus, there would be no modern technology; If you don't know calculus, you don't know the most basic mathematical logic. **)

5. To add that the derivative is the quotient of the differential, and the derivative of y = dy dx. And dy, dx are differentials, respectively, so the derivative is the differentiation divided by the differentiation – we call it the microquotient.

Calculus includes differentiation and integration, and the operations of differentiation and integration are opposite, and the two are inverse operations of each other.

Integrals include definite and indefinite integrals.

A definite integral means that there is a fixed integral interval and its integral value is determined.

An indefinite integral does not have a fixed integration interval, and its integral value is indefinite.

Applications of Calculus:

1) The problem of finding each other between velocity and distance in motion.

2) Find the tangent of the curve.

3) Find the length, area, volume, and center of gravity problems.

4) Finding the maximum and minimum values of the problem (quadratic functions, belonging to a class of calculus) Application of definite integrals:

1. Solve the problem of finding the area of curved edge figures.

Example: Find the area s. of a plane figure d surrounded by a parabola and a straight line

2. Find the distance of linear motion at variable speed.

The distance s traveled by an object in linear motion with variable speed is equal to the definite integral of its velocity function v=v(t) (v(t) 0) over the time interval [a,b].

3. Change force to do work.

Integration is the inverse of differentiation, that is, knowing the derivative of the function, the original function is reversed. In terms of application, the integral effect is not only that, it is widely used in summation, in layman's terms, to find the area of the curved triangle, this ingenious solution method is determined by the special properties of the integral. It is mainly divided into definite integrals, indefinite integrals, and other integrals.

The properties of integration mainly include linearity, number preservation, maximum minimum, absolute continuity, absolute value integration, etc.

224 commonly used integration formulas for higher numbers.

3 Common derivatives and integral formulas.

Calculus is a branch of mathematics that studies the rate of change and limit concepts and uses these concepts to study the properties of functions and curves. It is widely used in science, engineering, economics, computer science, physics, and other fields, and is an important mathematical tool for modern science and engineering.

Calculus is divided into two parts, differentiation and integration. Differentiation is the study of the derivative of a function, describing the rate of change of a function at a certain point. Integral is the study of the indefinite integral and definite integral of the function, which can find the area, volume and curve length of the function over a period of time.

Calculus has a wide range of applications. For example, in physics, calculus is used to study the motion, mechanics, and fluid mechanics of objects. In economics, calculus is used to analyze the supply and demand and changes in the market.

In engineering, calculus is used to design and analyze a variety of mechanical, electrical, chemical, and materials engineering systems, among other things.

In conclusion, calculus is a fundamental discipline in modern science and engineering, and it is a powerful mathematical tool that can be used to solve complex problems and understand various phenomena and real-world laws.

Learning calculus requires a certain foundation in mathematics, including algebra, trigonometry, exponential, and logarithmic functions. If you already have these foundations, there are a few ways to start learning calculus:

Derivatives and Differentiation: Derivatives are the most fundamental concepts in calculus and describe the rate of change of a function at a certain point. Once you learn the derivative, you can learn differentiation, which is an operation on the derivative that can be used to find the local rate of change of a function.

Integral and indefinite integrals: Integration is another important concept in calculus, which can be used to find the area of a function over a certain interval or the length of a curve, etc. An indefinite integral is a primitive function of a function that can be used to calculate definite integrals.

Definite integral: Definite integrals can be used to calculate the area under a curve or the volume of an object, etc. You need to learn integral and differential before you can learn definite integral.

Applications of Calculus: After learning calculus, you can learn the applications of calculus, such as motion and mechanics in physics, optimization and marginal analysis in economics, dynamics and control in engineering, etc.

When learning calculus, it is necessary to master certain mathematical thinking and methods, such as understanding and applying limit concepts, building functional models, and using calculus tools to solve problems. In addition, it takes a lot of practice and practice to deepen the understanding and mastery of calculus in order to truly master the basic knowledge and skills of calculus.