First, understand the related concepts in the set.

Characteristics of elements in (1) set: certainty, mutual difference and disorder.

(2) The relationship between sets and elements is represented by the symbol =.

(3) Symbolic representation of common number sets: natural number set; Positive integer set; Integer set; Rational number set, real number set.

(4) Representation of sets: enumeration method, description method and Wayne diagram.

(5) An empty set refers to a set without any elements.

An empty set is a subset of any set and a proper subset of any non-empty set.

Second, function

I. Mapping and function:

The concept of (1) mapping: (2) one-to-one mapping: (3) the concept of function:

Second, the three elements of function:

The judgment method of the same function: ① correspondence rule; (2) Domain (two points must exist at the same time)

Solution of resolution function (1):

① definition method (patchwork method): ② substitution method: ③ undetermined coefficient method: ④ assignment method:

(2) The solution of functional domain:

(1) The universe with parameters should be discussed by classification;

(2) For practical problems, after finding the resolution function; We must find its domain, and the domain at this time should be determined according to the actual meaning.

(3) The solution of function value domain:

① Matching method: transform it into a quadratic function and evaluate it by using the characteristics of the quadratic function; Often converted into:;

(2) Inverse solution: the value range obtained by inverse solution is expressed by and then obtained by solving inequality; Commonly used to solve, such as:

(4) Substitution method: transforming variables into functions of assignable fields and returning to ideas;

⑤ Triangular Bounded Method: Transform it into a function containing only sine and cosine, and use the boundedness of trigonometric function to find the domain;

⑥ Basic inequality methods: transformation and modeling, such as: using the average inequality formula to find the domain;

⑦ Monotonicity method: The function is monotonous, and the domain can be evaluated according to the monotonicity of the function.

⑧ Number-shape combination: According to the geometric figure of the function, the domain is found by the method of number-shape combination.

Third, the nature of the function:

Monotonicity, parity and periodicity of functions

Monotonicity: Definition: Note that the definition is relative to a specific interval.

The judgment methods are: definition method (difference comparison method and quotient comparison method)

Derivative method (for polynomial function)

Composite function method and mirror image method.

Application: compare sizes, prove inequalities and solve inequalities.

Parity: Definition: Pay attention to whether the interval is symmetrical about the origin, and compare the relationship between f(x) and f(-x). F (x)-f (-x) = 0f (x) = f (-x) f (x) is an even function;

F (x)+f (-x) = 0f (x) =-f (-x) f (x) is odd function.

Discrimination methods: definition method, image method and compound function method.

Application: function value transformation solution.

Periodicity: Definition: If the function f(x) satisfies: f(x+T)=f(x) for any x in the definition domain, then t is the period of the function f(x).

Others: If the function f(x) satisfies any x in the domain: f(x+a) = f (x-a), then 2a is the period of the function f (x).

Application: Find the function value and resolution function in a certain interval.

Fourth, graphic transformation: function image transformation: (key) It is required to master the images of common basic functions and master the general rules of function image transformation.

Regularity of common image changes: (Note that translation changes can be explained by vector language, which is related to vector translation)

Translation transformation y = f (x) → y = f (x+a), y = f (x)+b.

Note: (1) If there is a coefficient, first extract the coefficient. For example, the image of the function y = f (2x+4) is obtained by translating the function y = f (2x+4).

(2) Combining with the translation of vector, understand the meaning of translation according to vector (m, n).

Symmetric transformation y = f (x) → y = f (-x), which is symmetric about y.

Y = f (x) → y =-f (x), which is symmetrical about x.

Y=f(x)→y=f|x|, keep the image above the X axis, and the image below the X axis is symmetrical about X.

Y=f(x)→y=|f(x)| Keep the image on the right side of the Y axis, and then make the right part of the Y axis symmetrical about the Y axis. (Note: it is an even function)

Telescopic transformation: y=f(x)→y=f(ωx),

Image transformation of Y=f(x)→y=Af(ωx+φ) reference trigonometric function.

An important conclusion: if f (a-x) = f (a+x), the image of function y=f(x) is symmetrical about the straight line x=a;

Five, the inverse function:

(1) Definition:

(2) Conditions for the existence of the inverse function:

(3) The relationship between the domain and the value domain of reciprocal function:

(4) Steps to find the inverse function: ① Take it as an equation about and solve it. If there are two schemes, pay attention to the choice of the scheme; (2) will also be exchanged; ③ Write the domain of the inverse function (that is, the value domain of).

(5) The relationship between reciprocal images:

(6) The original function and the inverse function have the same monotonicity;

(7) If the original function is odd function, its inverse function is still odd function; The original function is even, so there must be no inverse function.

Seven, commonly used elementary functions:

(1) unary linear function:

(2) One-variable quadratic function:

general formula

Two-point type

Vertex type

The problem of finding the maximum value of quadratic function: firstly, the collocation method should be adopted and transformed into a general formula.

There are three types of questions:

(1) Vertex is fixed and interval is fixed. For example:

(2) Vertex contains parameters (i.e. vertex changes) and the interval is fixed. At this time, it is necessary to discuss when the abscissa of the vertex is in the interval and when it is outside the interval.

(3) The vertex is fixed and the interval is variable, so we should discuss the parameters in the interval.

There are two equivalent propositions in the interval, two in the interval and one in or above the interval.

Note: If the equation has a real number solution in the closed interval, we can first use the real root distribution in the open interval to get the result and check the endpoints.

(3) Inverse proportional function:

(4) Exponential function:

Exponential function: y =(a>；; O, a≠ 1), the constant intersection point of the image (0, 1), and the monotonicity is related to the value of A. In solving problems, A is often graded as A >；; 1 and 0

(5) Logarithmic function:

Logarithmic function: y =(a>；; O, a≠ 1) images have a constant intersection (1, 0), and the monotonicity is related to the value of A. In solving problems, A is often graded as A >；; 1 and 0

note:

(1) The basic method to compare two exponents or logarithms is to construct corresponding exponents or logarithms. If the base is different, it will be converted into an exponent or logarithm with the same base, and it should also be compared with 1 or 0.

VIII. Derivative

1. Derivation rule:

(c)/=0, where c is a constant. That is, the derivative value of the constant is 0.

(xn)/= nxn- 1 Especially: (x)/=1(x-1)/= ()/=-x-2 (f (x) g (x))/= f/(x) g. f/(x)

2. Geometric and physical meanings of derivatives:

K = f/(x0) represents the slope of the tangent of point P(x0, f(x0)) on curve y=f(x).

V = s/(t) represents the instantaneous speed. A=v/(t) stands for acceleration.

3. The application of derivative:

① Find the slope of the tangent.

② The relationship between the derivative of function and monotonicity.

The domain of (1) analysis is known; (2) finding the derivative; (3) solving inequalities; The part of the solution set in the definition domain is an increasing interval; (4) solving inequalities; The solution set in the definition domain is a decreasing interval.

When we use derivatives to judge the monotonicity of a function, we must make clear the following three relationships in order to accurately judge the monotonicity of the function. Take increasing function as an example to make a simple analysis. The precondition is that the function is differentiable in a certain interval.

③ Find the extreme value and the maximum value.

Note: Extreme value ≠ maximum value. The maximum value of the function f(x) in the interval [a, b] is the maximum value and the maximum value in f(a) and f(b). The minimum value is the smallest sum of f(a) and f(b).

You can't get F/(x0) = 0. When x=x0, the function has an extreme value.

However, when x=x0, the function has an extreme value f/(x0) = 0.

Judging extreme value needs to explain the monotonicity of function.

4. The standard of derivative products:

(1) representation function (more accurate and subtle than the elementary method);

(2) the connection with tangent in geometry (the tangent of plane curve can be studied by derivative method);

(3) Application problems (elementary methods often require high skills, while derivative methods are relatively simple) and other derivative problems about sub-polynomials are difficult types.

2. There are many problems about the maximum value of function characteristics, so it is necessary to discuss them specially. The derivative method is faster and simpler than the elementary method.

3. The mixed problem of derivative and analytic geometry or function image is an important type, and it is also a direction of comprehensive ability in college entrance examination, which should be paid attention to.

IX. Inequality

First, the basic properties of inequality:

Note: (1) special value method is a method to judge whether inequality propositions are true, especially for those that are not true.

(2) Pay attention to several attributes of textbooks, and pay special attention to:

(1) If ab>, then 0. That is, when the signs on both sides of the inequality are the same, the two sides of the inequality take the reciprocal, and the direction of the inequality will change.

② If both sides of the inequality are multiplied by an algebraic expression at the same time, pay attention to its sign. If the constellation is uncertain, pay attention to the classification discussion.

③ Image method: directly compare the sizes by using the images of correlation functions (exponential function, logarithmic function, quadratic function and trigonometric function).

④ Median method: first compare the algebraic expressions to be compared with "0" and "1", and then compare their sizes.

Second, mean inequality: the arithmetic mean of two numbers is not less than their geometric mean.

Basic applications: ① Scaling and deformation;

② Find the maximum value of the function: Note: ① One positive, two definite, three-phase and so on; ② The sum of definite products is minimum, and the sum of definite products is maximum.

The common methods are: splitting, gathering and square;

Third, absolute inequality:

Note: the conditions for the above equal sign "=" to be established;

Four, commonly used basic inequalities:

Five, the common methods to prove inequality:

(1) comparison method: compare the differences:

Steps of difference comparison:

⑴ Difference: distinguish two numbers (or formulas) with different sizes.

⑵ Deformation: decompose or formulate the difference factor into the complete sum of squares of several numbers (or formulas).

⑶ Symbol with poor judgment: Symbol with poor judgment combined with deformation results and problems.

Note: If it is difficult to distinguish two positive numbers, you can use their square difference to compare the sizes.

(2) comprehensive method: cause and effect.

(3) Analysis method: the reason of fruit holding. Basic steps: get a certificate ... a certificate of fairness ... a certificate of fairness. ...

(4) reduction to absurdity: if it is difficult, it will be reversed.

(5) Scaling method: the inequality side is appropriately enlarged or reduced to prove the problem.

Scaling methods include:

(1) Add or omit some items,

(2) Enlarge (or shrink) the numerator or denominator.

⑶ Using basic inequalities,

(6) method of substitution: method of substitution's aim is to reduce the variables in inequality, thus making the problem difficult and simplifying the complex. The commonly used substitutions are trigonometric substitution and algebraic substitution.

(7) Construction method: prove inequality by constructing function, equation, sequence, vector or inequality;

Ten, the solution of inequality:

(1) One-dimensional quadratic inequality: If the quadratic coefficient of one-dimensional quadratic inequality is less than zero, the same solution is transformed into a quadratic coefficient greater than zero; Note: To discuss:

(2) Absolute inequality: if, then; ;

note:

(1) To solve the problem of absolute value, you can consider removing absolute value. The method of removing absolute value is as follows:

(1) Discuss that the absolute value is greater than, equal to and less than zero, and then remove the absolute value;

(2). Divide the absolute value by squares on both sides; It should be noted that both sides of the inequality sign are non-negative.

(3) Inequalities with multiple absolute value symbols can be solved by the method of "discussing by zero points".

(4) Solving the fractional inequality: transforming the general solution into algebraic expression inequality;

(5) Solution of inequality group: Find the solution set of each inequality in the inequality group, and then find its intersection, which is the solution set of this inequality group. In the intersection, the solution set of each inequality is usually drawn on the same number axis, and their common parts are taken.

(6) Solving inequalities with parameters:

When solving inequalities with parameters, we should first pay attention to whether it is necessary to discuss them in categories. If you encounter the following situations, you generally need to discuss them:

① When two ends of inequality multiply and divide a formula with parameters, we need to discuss the positive, negative and zero properties of this formula.

② When monotonicity of exponential function and logarithmic function is needed in solving, their bases need to be discussed.

(3) When solving a quadratic inequality with letters, we need to consider the opening direction of the corresponding quadratic function, the conditions of the roots of the quadratic equation with one variable (sometimes we need to analyze △), compare the sizes of two roots, and let the roots be (or more) but contain parameters, all of which should be discussed.

XI。 sequence

This chapter is one of the main contents of the proposition of college entrance examination. We should review it comprehensively and deeply, and focus on solving the following problems on this basis: (1) The proof of arithmetic and geometric series must be proved by definition. It is worth noting that if the sum of the first few items of a series is given, the general items can be written if it is satisfied. (2) The calculation of series is the central content of this chapter. Using the general formula, antecedents, formulas and their properties of arithmetic progression and geometric progression to make clever calculations is the key content of the college entrance examination proposition. (3) When solving the problem of sequence, we often use various mathematical ideas. It is our goal to be good at using various mathematical ideas to solve the problem of sequence. (1) Function Thought: The summation formula of the general term formula of arithmetic geometric progression can be regarded as a function, so some problems of arithmetic geometric progression can be solved as function problems.

(2) The idea of classified discussion: the summation formula of equal proportion series should be divided and summed; When the time is known, it should also be classified;

③ Holistic thinking: When solving the sequence problem, we should pay attention to getting rid of the rigid thinking mode solved by formulas and use integers.

Body and mind solutions.

(4) When solving the related application problems of series, we should carefully analyze and abstract the actual problems into mathematical problems, and then use the knowledge and methods of series to solve them. Solving this kind of application problem is a comprehensive application of mathematical ability, and it is by no means a simple imitation and application. Pay special attention to the items of geometric series related to years.

First, the basic concept:

1, definition and representation of sequence:

2. Items and number of items in the series:

3, finite sequence and infinite sequence:

4, increasing (decreasing), swing, cycle order:

5. The general formula of sequence {an} an:

6. The first n terms of the sequence and the formula Sn:

7. Structure of arithmetic progression, Tolerance D and arithmetic progression:

8. The structure of geometric series, Bi Gong Q and geometric series;

Second, the basic formula:

9. the relationship between the general term an and the first n terms and Sn of a general sequence: an=

10, the general formula of arithmetic progression: an = a 1+(n-1) Dan = AK+(n-k) d (where a1is the first term and AK is the known k term), when d≠0.

1 1, the first n terms of arithmetic progression and its formula: Sn= Sn= Sn=

When d≠0, Sn is a quadratic form about n, and the constant term is 0; When d=0 (a 1≠0), Sn=na 1 is a proportional formula about n.

12, the general formula of geometric series: an = a1qn-1an = akqn-k.

(where a 1 is the first term, ak is the known k term, and an≠0).

13, the first n terms of geometric series and their formulas: when q= 1, Sn=n a 1 (this is a direct ratio formula about n);

When q≠ 1, Sn= Sn=

Third, the conclusion about arithmetic and geometric series.

Arithmetic progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 14 is still arithmetic progression.

15, arithmetic progression {an}, if m+n=p+q, then

16, geometric series {an}, if m+n=p+q, then

Geometric progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 17 is still geometric progression.

18, the sum and difference of two arithmetic progression {an} and {bn} series {an+bn} is still arithmetic progression.

19, a sequence consisting of the product, quotient and reciprocal of two geometric series {an} and {bn}

{an bn},,, or geometric series.

20. arithmetic progression {an} Any equidistant series is still arithmetic progression.

2 1, the series of any equidistant term of geometric progression {an} is still geometric progression.

22. How to make three numbers equal: A-D, A, A+D; How to make four numbers equal: A-3D, A-D, A+D, A+3D?

23. How to make three numbers equal: A/Q, A, AQ;

Wrong methods of four numbers being equal: a/q3, a/q, aq, aq3.

24.{an} is arithmetic progression, then (c>0) is a geometric series.

25 、{ bn }(bn & gt； 0) is a geometric series, then {logcbn} (c >; 0 and c 1) are arithmetic progression.

Four, the common methods of sequence summation: formula method, split item elimination method, dislocation subtraction, reverse addition, etc. The key is to find the general item structure of the sequence.

26. Find the sum of series by grouping method: for example, an=2n+3n.

27. Sum by offset subtraction: for example, an=(2n- 1)2n.

28. Sum by split term method: for example, an= 1/n(n+ 1).

29, reverse order addition sum:

30. The method of finding the maximum and minimum term of series {an}:

① an+ 1-an = ... For example, an= -2n2+29n-3.

② an=f(n) Study the increase and decrease of function f(n)

3 1. In arithmetic progression, the problem of the maximum value of Sn is usually solved by the adjacent term sign change method;

(1) When >: 0, d < When 0, the number of items m meets the maximum value.

(2) When

We should pay attention to the application of the transformation idea when solving the maximum problem of the sequence with absolute value.

Twelve, the plane vector

1. Basic concepts:

Definition of vector, modulus of vector, zero vector, unit vector, opposite vector, * * * line vector, equal vector.

2. Algebraic operations of addition and subtraction:

(1) if A = (x 1, y 1), B = (x2, y2), AB = (x 1+x2, y 1+y2).

Geometric representation of vector addition and subtraction: parallelogram rule and triangle rule.

Vector addition has the following laws:+=+(commutative law); +( +c)=(+)+c (law of association);

3. Product of real number and vector: The product of real number and vector is a vector.

( 1)| |=| | | |;

(2) when a > 0, it is in the same direction as a; When a < 0, it is opposite to a direction; When a=0, a = 0.

Necessary and sufficient conditions for two vector lines;

The necessary and sufficient condition for the straight line between (1) vector b and non-zero vector * * * is that there is only one real number, so b =.

(2) If = () and b = (), then ‖ b 。

The basic theorem of plane vector;

If e 1 and e2 are two nonlinear vectors on the same plane, there is only one pair of real numbers for any vector on this plane, so = e 1+ e2. ..

4. The ratio of P-divided directed line segments:

Let P 1 and P2 be two points on a straight line, and point P is any point in the world different from P 1 and P2, then there is a real number that makes =, which is called the ratio of point P to directed line segment.

When point p is on the line segment, > 0; When point P is on the extension line of line segment or, < 0;

Formula of vernal equinox coordinates: if =；; The coordinates of are (), () and () respectively; Then (≦- 1), the midpoint coordinate formula:.

5. Quantity product of vectors:

(1). Vector angle:

Given that two nonzero vectors and b make =, =b, then ∠AOB= () is called the included angle between the vector and b.

(2). Quantity product of two vectors:

If two nonzero vectors and b are known and their included angle is, then b = |||| b | cos.

Where | b | cos is called the projection of vector b in the direction.

(3) Properties of the product of vector numbers:

If = () and b = (), then e = e = || cos (e is the unit vector);

⊥ b b = 0 (,b is a non-zero vector); | |= ;

cos = =。

(4) Vector product algorithm:

b = b()b =(b)=(b)； (+b) c= c+b c。

6. Main ideas and methods:

This chapter mainly sets up the viewpoint of number-shape transformation and combination, handles geometric problems with algebraic operation, especially the relative position relationship of vectors, correctly uses the basic theorems of * * * line vector and plane vector to calculate the modulus of vectors, the distance between two points and the included angle of vectors, and judges whether the two vectors are vertical or not. Because vectors are new tools, they are often combined with trigonometric functions, sequences, inequalities and solutions. And it is the intersection of knowledge.

Thirteen, solid geometry

The basic properties of 1. plane: If you master three axioms and inferences, you will explain the problems of * * * points, * * lines and * * * planes.

Able to draw by tilt measurement.

2. The positional relationship between two straight lines in space: the concepts of parallelism, intersection and nonplanarity;

Will find the angle formed by straight lines in different planes and the distance between straight lines in different planes; Generally, two straight lines are proved to be non-planar straight lines by reduction to absurdity.

3. Lines and planes

① positional relationship: parallel, straight line in the plane, and straight line intersects with the plane.

(2) The method and nature of judging the parallelism between a straight line and a plane, and the judgment theorem is the basis of proving the parallelism problem.

(3) What are the methods to prove that a straight line is perpendicular to a plane?

④ The angle formed by a straight line and a plane: the key is to find its projection on the plane, and the range is {00.900}.

⑤ Three Verticality Theorem and Its Inverse Theorem: This theorem should be examined in college entrance examination questions every year. The theorem of three perpendicular lines and its inverse theorem are mainly used to prove the vertical relationship and the measurement of spatial graphics, such as proving that straight lines in different planes are vertical, determining the plane angle of dihedral angle, and determining the perpendicular line from point to straight line.

4. Airplanes and Airplanes

(1) positional relationship: parallel, intersecting, (vertical is a special case of intersection)

(2) Master the proof method and nature of plane parallel to plane.

(3) Master the proof method and property theorem of the plane perpendicular to the plane. Especially, it is known that two planes are perpendicular, which can be proved by property theorem.

(4) Distance between two planes → Distance from point to surface →

(5) dihedral angle. Method and solution of dihedral angle plane intersection;

(1) definition method, generally using the symmetry of graphics; Generally, the oblique triangle should be solved in the calculation;

(2) The vertical line, diagonal line and projection method generally require that the vertical line of the plane is easy to find, and a right triangle should be solved in the calculation.