Mathematical analysis is an important branch of mathematics, which mainly studies the properties of real functions and complex functions, their concepts such as limit, derivative and integral and their applications. The following are some typical problems and methods in mathematical analysis:
Limit: Limit is the basis of mathematical analysis, and its concept and nature are the key to solving mathematical analysis problems. Limit problems usually involve the concepts and properties of infinity and infinitesimal. The solution includes defining limit, judging convergence, calculating limit value and so on.
Derivative: Derivative is an important concept in mathematical analysis, which can describe the rate of change of a function at a certain point. The problem of derivative usually involves derivative, extreme value, monotone interval and so on. The solution includes the use of derivative rules, theorems and derivative tables, and the use of analytical properties such as extremum and monotonicity to solve the maximum value of the function.
Integral: Integral is an important concept in mathematical analysis, which can represent the area or volume of a function in a certain interval. The problem of integral usually involves finding integral, definite integral, multiple integral and so on. The solution includes using the properties and theorems of integrals, such as substitution integral and partial integral, and solving definite integral and multiple integral by numerical method.
Series: Series is an important concept in mathematical analysis, which can represent the sum of a series. The problem of series usually involves judging convergence, summation, finding convergence radius and so on. The solution includes using the properties and theorems of series, such as ratio discrimination and Leibniz discrimination, using series expansion and power series to solve the function and expand it into power series.
Multiple differential: Multiple differential is an important concept in mathematical analysis, which can represent the rate of change of a function at a certain point in multidimensional space. The problem of multiple differentiation usually involves finding partial derivatives, extreme values, curve integrals and so on. The solution includes using the definition and properties of partial derivative, using the concepts and methods of multiple integral and curve integral.