Mathematical analysis originated from calculus in17th century. Mathematicians at that time developed the theory and method of calculus in order to solve practical problems, such as calculating the length, area and volume of curves. With the passage of time, mathematical analysis has gradually developed into a huge system, including real number theory, function theory, differential equation, series theory and many other branches.
In mathematical analysis, function is a basic concept. Function describes the correspondence between input and output, and it maps each element in one set (domain) to an element in another set (range). Functions can be linear, such as linear functions and quadratic functions, or nonlinear, such as exponential functions and logarithmic functions. The continuity, differentiability and integrability of functions are important contents of mathematical analysis and research.
Limit is another basic concept in mathematical analysis. Limit describes the process that a variable approaches a certain value infinitely in a certain direction. The concept of limit plays a key role in mathematical analysis, which enables us to study the local properties of a function at a certain point, such as derivative and differential.
Derivative is one of the core concepts in mathematical analysis. Derivative describes the local rate of change of a function at a certain point. The calculation and analysis of derivatives can help us to study the properties of functions, such as monotonicity, concavity and convexity, extremum and so on. Derivative can also be applied to practical problems, such as velocity and acceleration, tangent and normal, maximum problem and so on.
Integral is another important concept in mathematical analysis. Integral describes the overall change of a function in a certain interval. The calculation and analysis of integral can help us to study the properties of function such as area and volume. Integral can also be applied to practical problems, such as the area, volume, center of mass and moment of inertia of geometric figures.
Mathematical analysis is not only widely used in mathematics, but also has important application value in physics, engineering, economy, biology and other fields. By studying mathematical analysis, we can cultivate students' logical thinking, abstract thinking and ability to solve practical problems, and lay a solid foundation for future academic research and career development.