1. Important attributes:
(1) symmetry a >: B (2) transitive a >; b，b & gtc
(3) Empathy Law a>b
(4) Addition rules a>b, c>d
(5) multiplication rules a>b, c & gt0a & gt；; b & gt0
a & gtb，c & lt0c & gt； d & gt0
(6) Reciprocity rules a>b, ab>0
(7) Power Law a>b & gt0(n∈N, n> 1)
(8) Root rule a>b & gt0(n∈N, n> 1)
2. Important inequalities
(1) a, b∈R, a2+b2≥2ab (equal sign is true if and only if a=b)
(2) a, b∈R+, a+b≥2 (the equal sign is true if and only if a=b)
(3)ab & gt； 0, (if and only if the equal sign holds)
(4)ab & lt； 0, (if and only if the equal sign holds)
3. Inequalities with absolute symbols
||||| A |-| B||||≤| AB|≤| A+B | (called trigonometric inequality)
Two. Proof of inequality
1. comparison method (1) difference comparison method: (2) quotient comparison method:
2. Comprehensive method:
3. Analysis method:
4. Other methods: reduction to absurdity, mathematical induction, scaling and trigonometric substitution.
Three. Solutions to inequality
1 unary linear inequality ax-b > 0
2 the relationship between quadratic inequality, quadratic function and quadratic equation (A > 0)
Discriminant? △=b2-4ac △>0 △=0 △ 0 solution
The solution of inequality AX2+BX+C < 0
Note: A < 0 can be converted into A > 0.
3 solution set of simple absolute inequality | x-a | > b(b > 0)_ a-b
Solution set of | x-a | < b(b > 0)_ a-b _
Solutions of four higher inequalities (x-x 1) (x-x2)...(x-xn) > 0 (graph)
The solution of (x-x 1) (x-x2) ... (x-xn) < 0 (graph)
Four. Application of inequality
When using the mean value theorem to find the maximum value of a function, the following requirements should be met: (1)
(2)
(3)
Ordered sequence
I. General concept of sequence
1. Definition:
The essence of a sequence is a function defined on natural number set or its finite subset. General formula: a n=f(n)
2. Recursive formula: a n+ 1= f(a n) (n∈N) is a method to give a series, and the first n items of the series can be written when a 1 is known.
Two. Arithmetic series and geometric series
Name geometric series, arithmetic series
definition
General term formula
The first n terms and formulas
Zhongfu
nature
Sum of Series: Let {{a n}} be arithmetic progression and {{b n}} be geometric progression.
1. Find the sum of the first n items of the sequence {{a n+b n}}.
2. Find the sum of the first n items of the sequence {{a n×b n}}.
3. Find the sum of the first n items of the sequence {N2+BN+C}-sum by the formula method.
4. Find the angular relationship in the triangle: (1) the sum of the internal angles of the triangle and the first n terms of the theorem series {}-the sum of the split terms.
：
(2)sin(A+B)= (3)
= =
= =
(4) Sine theorem:
(5) Cosine theorem: variant form:
(6) area formula of triangle:
(7) Types of declination triangle:
(8) Judging the shape of a triangle:
Related knowledge:
1. Symbol of trigonometric function value: 2 Relationship of trigonometric functions with the same angle:
3 inductive formula
α
sine
cosine
tangent
cotangent
Analytic geometry-straight line
I. Basic concepts and formulas
1 inclination angle of straight line:
2 Slope of straight line: (slope definition and slope formula)
3 Direction vector of straight line:
4 formula for the distance between two points:
5 midpoint coordinate formula:
Second, the linear equation
Several forms of 1. linear equation
Description of conditional equation with known name
Point-oblique type
slope intercept form
Two-point type
Intercept form
general formula
Parameter formula
2. Special linear equation:
Straight line perpendicular to x: x = straight line perpendicular to y: y=b
The straight line passing through the origin (0,0) _ _ _ _ _ _ _ _ The straight line (two straight lines) passing through the point P(x0, y0) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Lines with equal intercepts _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
A straight line parallel to the straight line L 1: a1x+b1y+c1= 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
A straight line perpendicular to the straight line L 1: a1x+b1y+c1= 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
The positional relationship between three points m and straight line l
(1)M is on a straight line, and Ax0+By0+C=0.
(2)M is outside the straight line, and the distance from m to l is
4. The positional relationship of the sum of two straight lines
Let l1:a1x+b1y+c1= 0, and L2: A2X+B2Y+C2 = 0.
If the slope exists, then l1:y = k1x+b1,L2: y = K2X+B2.
1 Parallel:
L 1‖L2
If the slope exists, L 1‖L2
Distance formula between parallel lines: d=
2. Vertical:
L 1⊥L2
If the slope exists, L 1⊥L2
Five about symmetry
Let P(x0, y0) be: the coordinate of point p with respect to the axisymmetrical point of X is ().
The coordinate of point P about the Y-axis symmetry point is ()
The coordinate of point P about the symmetrical point of origin is ()
The coordinate of the point p about the line y=a is ()
The coordinate of the point P with respect to the line y=-b is ()
The coordinate of point P about the symmetrical point of M(a, b) is ().
The coordinate of the symmetrical point of the point P about the straight line y=x is ()
The coordinate of the point p about the line y=-x is ()
The coordinate of point P with respect to the line L symmetric point Q(x, y): ax+by+c = 0 is satisfied.
Five linear programming problems:
Six-dimensional rectangular coordinate system, divination limit and distance formula between two points in space;
Analytic Geometry —— Circle
Definition of 1 circle:
2 standard equation of circle:
General equation of three circles:
4. The positional relationship between circle and straight line:
(Method 1) Set a straight line L: Y = KX+B.