Function and equation thought

Function thought refers to analyzing, reforming and solving problems with the concept and nature of function. The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solve the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations are mutually transformed and interrelated, thus solving problems. Descartes' equation thought is: practical problem → mathematical problem → algebraic problem → equation problem. The universe is full of equality and inequality. We know that where there are equations, there are equations; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations ... and so on; The inequality problem is also closely related to the fact that the equation is a close relative. Column equation, solving equation and studying the characteristics of equation are all important considerations when applying the idea of equation. Function describes the relationship between quantities in nature, and the function idea establishes the mathematical model of function relationship by putting forward the mathematical characteristics of the problem, so as to carry out research. It embodies the dialectical materialism view of "connection and change". Generally speaking, the idea of function is to use the properties of function to construct functions to solve problems, such as monotonicity, parity, periodicity, maximum and minimum, image transformation and so on. We are required to master the specific characteristics of linear function, quadratic function, power function, exponential function, logarithmic function and trigonometric function. In solving problems, it is the key to use the thought of function, be good at excavating the implicit conditions in the problem, and construct the properties of distinguishing function and ingenious function. Only by in-depth, full and comprehensive observation, analysis and judgment of a given problem can we have a trade-off relationship and build a functional prototype. In addition, equation problems, inequality problems, set problems, sequence problems and some algebraic problems can also be transformed into related functional problems, that is, solving non-functional problems with functional ideas. Function knowledge involves many knowledge points and a wide range, and has certain requirements in concept, application and understanding, so it is the focus of college entrance examination. The common types of questions we use function thought are: when encountering variables, construct function relations to solve problems; Analyze inequality, equation, minimum value, maximum value and other issues from the perspective of function; In multivariable mathematical problems, select appropriate main variables and reveal their functional relationships; Practical application of problems, translation into mathematical language, establishment of mathematical models and functional relationships, and application of knowledge such as functional properties or inequalities to solve them; Arithmetic, geometric series, general term formula and sum formula of the first n terms can all be regarded as functions of n, and the problem of sequence can also be solved by function method.

A combination of numbers and shapes

"Numbers are invisible, not intuitive, and numerous shapes make it difficult to be nuanced", and the application of "combination of numbers and shapes" can make the problem to be studied difficult and simple. Combining algebra with geometry, such as solving geometric problems by algebraic method and solving algebraic problems by geometric method, is the most commonly used method in analytic geometry. For example, find the root number ((A- 1)2+(B- 1)2)+ root number (A 2+(B- 1)2)+ root number ((A- 1) 2+B).

Classified discussion thinking

When a problem may lead to different results because of different situations of a certain quantity or number, it is necessary to discuss the various situations of this quantity or number in categories. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A in different categories.

Equal thinking

When a problem may be related to an equation, we can solve it by constructing the equation and studying its properties. For example, when proving Cauchy inequality, Cauchy inequality can be transformed into a discriminant of quadratic equation.

Holistic thinking

Starting from the overall nature of the problem, we emphasize the analysis and transformation of the overall structure of the problem, find out the overall structural characteristics of the problem, and be good at treating some formulas or figures as a whole with the "overall" vision, grasping the relationship between them, and carrying out purposeful and conscious overall treatment. The holistic thinking method is widely used in simplification and evaluation of algebraic expressions, solving equations (groups), geometric proof and so on. Integral substitution, superposition multiplication, integral operation, integral demonstration, integral processing and geometric complement are all concrete applications of integral thinking method in solving mathematical problems.

Change idea

It is through deduction and induction that unknown, unfamiliar and complex problems are transformed into known, familiar and simple problems. Mathematical theories such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus, and even rulers and rulers of ancient mathematics are permeated with the idea of transformation. Common transformation methods include: general special transformation, equivalent transformation, complex and simple transformation, number-shape transformation, structural transformation, association transformation, analogy transformation and so on.

Implicit conditional thinking

Conditions that are not explicitly stated but can be inferred from existing explicit expressions, or conditions that are not explicitly stated but are routines or truths.

Edit the analogy thought of this paragraph.

Comparing two (or two) different mathematical objects, if they are found to have similarities or similarities in some aspects, it is inferred that they may also have similarities or similarities in other aspects.

Modeling thinking

In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a language that is generally regarded as strict to describe various phenomena, which is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects and carry out corresponding experiments. The experiment itself is also a theoretical substitute for the actual operation.

Change ideas

The idea of transformation is to transform the problem A to be solved or difficult to solve into the problem B with a fixed solution mode or easy to solve by some transformation means, and solve the problem A by solving the problem B. The principles of transformation include turning the unknown into the known, making it complicated, making it easy, reducing dimension and order, and standardizing.

Inductive reasoning thinking

The reasoning that some objects of a certain kind of things have certain characteristics, and all objects of this kind of things have these characteristics, or the reasoning that generalizes general conclusions from individual facts is called inductive reasoning (induction for short). In short, inductive reasoning is reasoning from the local to the whole and from the individual to the general. In addition, there are mathematical ideas such as probability and statistics. For example, probability statistics refers to solving some practical problems through probability statistics, such as the winning rate in lottery tickets and the comprehensive analysis of an exam. In addition, some area problems can be solved by probability method. Let me give you an example. There is an angular bisector in the picture, which can be perpendicular to both sides. You can also look at the picture in half, and there will be a relationship after symmetry. Angle bisector parallel lines, isosceles triangles add up. Angle bisector plus vertical line, try three lines. Perpendicular bisector is a line segment that usually connects the two ends of a straight line. It needs to be proved that the line segment is double-half, and extension and shortening can be tested. The two midpoints of a triangle are connected to form a midline. A triangle has a midline and the midline extends. A parallelogram appears and the center of symmetry bisects the point. Make a high line in the trapezoid and try to translate a waist. It is common to move diagonal lines in parallel and form triangles. The card is almost the same, parallel to the line segment, adding lines, which is a habit. In the proportional conversion of equal product formula, it is very important to find the line segment. Direct proof is more difficult, and equivalent substitution is less troublesome. Make a high line above the hypotenuse, which is larger than the middle term. Calculation of radius and chord length, the distance from the chord center to the intermediate station. If there are all lines on the circle, the radius of the center of the tangent point is connected. Pythagorean theorem is the most convenient for the calculation of tangent length. To prove that it is tangent, carefully distinguish the radius perpendicular. Is the diameter, in a semicircle, to connect the chords at right angles. An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely. There are two chords on the corner of the circle, and the diameters of the two ends of the chords are connected. Find tangent chord, same arc diagonal, etc. If you want to draw a circumscribed circle, draw a vertical line in the middle on both sides. Also make a dream circle with inscribed circle and bisector of inner angle. If you meet an intersecting circle, don't forget to make it into a string. Two circles tangent inside and outside pass through the common tangent of the tangent point. If you add a connector, the tangent point must be on the connector. Adding a circle to the equilateral angle makes it not so difficult to prove the problem. The auxiliary line is a dotted line, so be careful not to change it when drawing. If the graph is dispersed, rotate symmetrically to carry out the experiment. Basic drawing is very important and should be mastered skillfully. You should pay more attention to solving problems and often sum up the methods clearly. Don't blindly add lines, the method should be flexible. No matter how difficult it is to choose the analysis and synthesis methods, it will be reduced. Study hard and practice hard with an open mind, and your grades will soar. Limit thought Limit thought is the basic idea of calculus, and a series of important concepts in mathematical analysis such as function continuity, derivative and definite integral are defined by means of limit. If you want to ask, "What is the theme of mathematical analysis?" Then it can be summed up as follows: "Mathematical analysis is a subject that studies functions with extreme ideas".