Solving math problems is not to regard yourself as a problem-solving machine or a problem-solving slave, but to strive to be a problem-solving master. It is to absorb the methods and ideas of solving problems from solving problems and exercise your own thinking. This is the so-called "math problem should examine the ability of candidates." The following small series brings you the ideas and methods of solving problems in mathematical analysis, hoping to help you.
First, combine numbers with shapes.
The combination of "number" and "shape" is mutual infiltration. Combining the accurate description of algebra with the intuitive description of geometric figures, algebraic problems and geometric problems can be transformed into each other, and abstract thinking and image thinking can be organically combined. Applying the idea of combining numbers and shapes is to fully investigate the internal relationship between the conditions and conclusions of mathematical problems, analyze their algebraic significance and reveal their geometric significance, skillfully combine the quantitative relationship with the spatial form, find ways to solve problems, and solve the problems.
Second, the idea of transformation and transformation.
When we study and solve mathematical problems, we comprehensively use the knowledge and skills we have mastered, and through some means, turn the problems into mathematical methods that can be solved within the existing knowledge.
Generally, complex problems are always transformed into simple problems, difficult problems into easy-to-solve problems, and unsolved problems into solved problems. It can be said that in the process of solving mathematical problems, transformation and transformation ideas are the most common, and all kinds of mathematical problems are solved through continuous transformation. In essence, the combination of numbers and shapes, functional equations and classified discussions commonly used in mathematics can also be understood as manifestations of reduction.
Third, vector thought.
By observing the geometric characteristics of the problem, mining the vector model of algebraic structure, skillfully constructing the vector, and transforming the original problem into the operational function of the vector or the geometric meaning of the vector to solve it, the vector can not only perform rich algebraic operations such as addition, subtraction and number multiplication, but also provide important geometric meaning. Vector builds a bridge between algebra and geometry, so that some difficult algebraic or geometric problems can be easily solved by vector operation. Through vector operation, we can effectively reveal the position and quantity relationship of space (or plane) graphics, and it is the deepening and perfection of the idea of combining numbers with shapes to move from qualitative research to quantitative research.