Summary of Required Knowledge Points and Frequently Asked Questions in Function _ Conceptual Knowledge Points Induction and Frequently Asked Questions Special Exercise of Mathematics Set and Function in Senior High School (with analysis) Conceptual Knowledge Points Induction and Frequently Asked Questions Special Exercise of Mathematics Set and Function in Senior High School (with analysis) Knowledge Points: Chapter I Concept of Set and Function 1. 1 set 1 set's meaning and expression.
2. Three characteristics of elements in a set (1) element certainty; (2) mutual anisotropy of elements; (3) the disorder of elements 2. The concept of "belonging" we usually use uppercase Latin letters A, B, C, ... to represent sets, and lowercase Latin letters A, B, C, ... to represent elements such as: If A is an element of set A, it is said that A belongs to set A, and if A does not belong to set A, it is recorded as A? A 3。 The common number set and its notation non-negative integer set (that is, natural number set) are recorded as: n; Positive integer set is recorded as: N* or n+; The integer set is recorded as: z; The set of rational numbers is written as: q; The real number set is recorded as: R 4 Representation of a set (1) enumeration: list the elements in the set one by one, and then enclose them in braces.
(2) Description: The method of representing a set with the common characteristics of the elements contained in the set is called description.
① Language description: Example: {a triangle that is not a right triangle} ② Mathematical expression description: Example: inequality x-3 > The solution set of 2 is {x ∈ r | x-3 >; 2} or {x | x-3 >；; 2} (3) Graphical method (venn diagram) 1. 1.2 Basic knowledge points of the relationship between sets 1, "inclusion" relationship-subsets Generally speaking, for two sets A and B, if any element of set A is an element of set B, we say that these two sets have an inclusion relationship, which is called set A. B and a? B Then say that set A is the proper subset of set B, and write A? B (or b? A) 4. An empty set without any elements is called an empty set and recorded as φ. What is the basic operation of proper subset 1. 1.3 set that an empty set is a subset of any set and an empty set is any non-empty set? B and b? A knowledge point 1, definition of intersection Generally speaking, the set composed of all elements belonging to A and B is called the intersection of A and B, and it is marked as A∩B (pronounced as "A crosses B"), that is, A∩B={x| x∈A, X ∈ B} Note: A. Generally expressed by u.
(2) Complement Set Let U be a set and A be a subset of U (that is, A? U), the set composed of all elements in U that do not belong to A is called the complement (or complement) of subset A in U Note: CUA, that is, CSA ={x | x? U and x? A} (3) Properties CU(C UA)=A, (Cua) ∩ A = φ, (Cua) ∪ A = U; (C UA)∩(C UB)=C U(A∪B)，(C UA)∩(C UB)= C U(A∪B)。 1.2 function and its representation 1 function. Let any number x in set A have a uniquely determined number f(x) corresponding to it in set B, then F: A → B is called a function from set A to set B. Let it be expressed as: y=f(x), x ∈ a. Where x is called an independent variable, and the range of value a of x is called the domain of the function; The value of y corresponding to the value of x is called the function value, and the set of function values {f(x)| x∈A} is called the range of the function. Note (1) If only the analytical formula y=f(x) is given without specifying its domain, the domain of the function refers to the set of real numbers that can make this formula meaningful; (2) The definition domain and value domain of a function should be written in the form of sets or intervals, and the main basis for finding the definition domain of a function is that the denominator of the (1) fraction is not equal to zero; (2) The number of even roots is not less than zero; (3) The truth value of the logarithmic formula must be greater than zero; (4) The cardinality of exponential and logarithmic expressions must be greater than zero and not equal to 1. (5) If a function is composed of some basic functions through four operations, then its domain is a set of values of x that make all parts meaningful. (6) The exponent is zero, and the base cannot be equal to zero. (7) The definition domain of the function in the actual problem should also ensure that the actual problem is meaningful. (Note: The solution set of inequality group is the domain of function. ) 2. The three elements of a function, namely, the definition domain, the corresponding relationship and the value domain. (1) Note that the three elements of the function are domains. Correspondence and value range. Because the range of values is determined by the domain and the corresponding relationship, if the domain and the corresponding relationship of two functions are exactly the same, the two functions are said to be equal (or the same function).
(2) Two functions are equal if and only if their domains and corresponding relations are completely consistent, regardless of letters representing independent variables and function values.
3. The judgment method of the same function (1) has the same domain; (2) The range of functions with the same expression (two points must be met at the same time) is supplemented by the range (1). No matter what method is used to find the range of a function, its range should be considered first. (2) Familiar with the range of linear function, quadratic function, exponential function, logarithmic function and trigonometric function, which is the basis of solving complex function range.
4. Concept of interval (1) Classification of interval: open interval, closed interval and semi-open and semi-closed interval; (2) Infinite interval; (3) The interval number axis represents the knowledge points of. 1.2.2 function representation, common function representations and their respective advantages (1) Function images can be continuous curves, straight lines, broken lines, discrete points, etc. Pay attention to the basis of judging whether a graph is a function image: make straight lines and curves perpendicular to the X axis at most.
(2) Representation and analysis methods of functions: the definition domain of functions must be specified; Mirror image method: attention should be paid to drawing by tracing point method: determine the definition domain of function; Simplify the analytical formula of the function; Observe the characteristics of the function; List method: the selected independent variables should be representative,
Summary of Required Knowledge Points and Frequently Asked Questions of Function _ Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three, Number Three The relationship between differentiability and connectivity Chapter 2: Monotonicity of functions, properties of continuous functions on the micro-closed interval of function extreme value functions, Rolle's differential theorem, Lagrange's mean value theorem, Cauchy's mean value theorem and Taylor's theorem Chapter 3: Application of definite integral of product of unary functions, existence of limit of differential functions at one point, existence of multiple implicit functions, existence of partial derivatives and total derivatives in continuous chapter 4, existence of partial derivatives, Discussion on the concept, properties and computability of differential calculus of total differential function, the causality between them and the continuity of partial derivative, the calculation and application of causality between them ★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ The relationship between extreme values ★★★★★★★★ Judging the type of continuity and discontinuity of a function ★★★★★★★★★ Finding the limit of a function ★★★★★★★★★ The basic properties of series and the necessity of convergence Chapter 5: Unconditional, comparison and discrimination of positive series, discrimination of convergence and divergence of several series, ratio and root of finite series, Leibniz discrimination of staggered series Chapter 6 is often first-order linear differential equations and homogeneous equations.