1. Induction of five compulsory knowledge points in senior three mathematics.
1. Function idea: Function idea is to express some mutually restrictive variables in a certain change process, study the mutually restrictive relationship between these quantities, and finally solve the problem; 2. It is a key step to solve problems and establish the functional relationship between variables by using the function idea, which can be roughly divided into the following two steps:
(1) Establish the functional relationship between variables according to the meaning of the problem, and transform the problem into a corresponding functional problem;
(2) Construct functions according to needs and solve problems by using relevant knowledge of functions;
(3) Equation idea: In a certain change process, it is often necessary to determine the values of certain variables according to certain requirements. At this time, the equations or (equations) of these variables are often listed and solved by solving the equations (or equations). This is the idea of equation;
3. Function and equation are two closely related mathematical concepts, which permeate each other. Many equation problems need to be solved with the knowledge and methods of functions, and many function problems also need the support of equation methods. The dialectical relationship between function and equation forms the idea of function equation.
2. High school mathematics compulsory five knowledge points induction
1. Problems about parallelism and verticality (straight line, straight line and plane) are repeatedly encountered in the process of solving solid geometry problems, and are indispensable contents in various problems (including argumentation, angle calculation, distance, etc.). Therefore, in the general review of subject geometry, we must first solve the problems about parallelism and verticality. By analyzing and summarizing the problems, we can master the law of solving problems in solid geometry-make full use of the idea of mutual transformation between lines (vertical) and lines (vertical), and improve our logical thinking ability and spatial imagination ability.
2. The method of judging the parallelism of two planes:
(1) According to the definition, it is proved that two planes have no common point;
(2) Judgment Theorem-Prove that two intersecting straight lines in one plane are parallel to another plane;
(3) Prove that two planes are perpendicular to a straight line.
3. The main properties of two parallel planes:
(1) According to the definition, "two parallel planes have nothing in common";
(2) Derived from the definition: "Two planes are parallel, and the straight line in one plane must be parallel to the other plane";
(3) Two-plane parallelism theorem: "If two parallel planes intersect the third plane at the same time, their intersection lines are parallel";
(4) The straight line is perpendicular to one of the two parallel planes and also to the other plane;
(5) The parallel lines sandwiched between two parallel planes are equal;
(6) Only one plane passing through a point outside the plane is parallel to the known plane.
3. Senior three mathematics compulsory five knowledge points induction
(A) the first definition of derivative
Let the function y=f(x) be defined in a domain of point x0. When the independent variable x has increment △ x at x0 (x0+△ x is also in the neighborhood), the corresponding function gets increment △ y = f (x0+△ x)-f (x0); If the ratio of △y to △x has a limit when △x→0, the function y=f(x) can be derived at point x0, and this limit value is called the derivative of function y=f(x) at point x0, which is also called f'(x0), which is the first definition of derivative.
(2) The second definition of derivative
Let the function y=f(x) be defined in a domain of point x0. When the independent variable x changes △ x at x0 (x-x0 is also in the neighborhood), the function changes △y=f(x)-f(x0) accordingly. If the ratio of △y to △x is limited when △x→0, then the function y=f(x) is derivable at point x0. This limit value is called that the derivative of function y=f(x) at point x0 is f'(x0), which is the second definition of derivative.
(3) Derivative function and derivative
If the function y=f(x) is differentiable at every point in the open interval I, it is said that the function f(x) is differentiable in the interval I. At this time, the function y=f(x) corresponds to a certain derivative of each certain value of x in the interval I, and forms a new function, which is called the derivative function of the original function y=f(x), and is denoted as y' and f'. Derivative function is called derivative for short.
Monotonicity and its application
1. General steps to study monotonicity of polynomial functions with derivatives
(1) Find f \u( x)
(2) Make sure that f￠(x) is in (a, b). Symbol (3) If F ￠ (x) >: 0 is a constant on (a, b), then f(x) is a increasing function on (a, b); If the intersection of the solution set of f￠(x)0 and the domain corresponds to an increasing interval; F￠(x)q, it is easy to understand the sufficient condition that p is q.
But why is q a necessary condition for p?
In fact, it is different from "p = >；; The equivalent negative proposition of "q" is "non-q = >；; Not p ". Meaning: If Q is not true, then P must not be true. In other words, q is essential to p, so it is necessary.
(2) Look at "sufficient and necessary conditions"
If there is p =>q and q =>p, then P is both a sufficient condition and a necessary condition, which is called the necessary and sufficient condition of P for short. Write down the concept of "equivalent" that pq learned when he recalled the beginning; If Proposition A holds, it can be inferred that Proposition B holds. On the contrary, if Proposition A holds, A is equivalent to B, which is marked as AB. The meaning of "necessary and sufficient condition" is actually exactly the same as that of "equivalent to". That is to say, if Proposition A is equivalent to Proposition B, then we say that Proposition A holds if and only if Proposition B holds; The necessary and sufficient condition for proposition B to be established simultaneously is that proposition A is established.
(3) Definition and necessary and sufficient conditions
In mathematics, only when A is the necessary and sufficient condition of B can B be defined by A, so each definition contains a necessary and sufficient condition. For example, the definition of "two groups of parallelograms with opposite sides are called parallelograms" means that a quadrilateral is a parallelogram if and only if its two groups of opposite sides are parallel. Obviously, if a theorem has an inverse theorem, then this theorem and the inverse theorem can be expressed by a statement with necessary and sufficient conditions. "Necessary and sufficient conditions" can sometimes be expressed by "if and only if", where "when" means "sufficient". "As long as" means "necessary".
(4) Generally speaking, the conditions in the definition are all necessary and sufficient conditions, the conditions in the judgment theorem are all sufficient conditions, and the "conclusion" in the property theorem can be used as a necessary condition.