I. Collection of related concepts
The meaning of 1. set
2. Three characteristics of elements in a set:
(1) element determinism,
(2) Anisotropy of elements,
(3) the disorder of elements,
3. Representation of assembly: {…} For example, {basketball players in our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}
(1) The set is expressed in Latin letters: A={ basketball players in our school}, B={ 1, 2, 3, 4, 5}.
(2) Representation of sets: enumeration and description.
Note: Commonly used digit sets and their symbols:
The set of nonnegative integers (i.e. natural number set) is recorded as n.
Positive integer set N* or N+ integer set z rational number set q real number set r
1) enumeration: {A, b, C...}
R| x-32) Description: A method of describing the public * * * attribute of elements in a set and writing it in braces to represent the set. { x & gt2}，{ x | x-3 & gt； 2}
3) Language description: Example: {A triangle that is not a right triangle}
4) Venn diagram:
4, the classification of the set:
The (1) finite set contains a set of finite elements.
(2) An infinite set contains an infinite set of elements.
(3) An example of an empty set without any elements: {x | x2 =-5}
Second, the basic relationship between sets
1. "Inclusive" relation-subset
Note: There are two possibilities that A is a part of B (1); (2)A and B are the same set.
On the other hand, set A is not included in set B, or set B does not include set A, so it is recorded as A B or B A.
2. "Equality" relationship: A=B (5≥5 and 5≤5, then 5=5)
Example: let a = {x | x2-1= 0} b = {-1,1} "Two sets are equal if their elements are the same".
A means: ① Any set is a subset of itself. A
B Then let's assume that set A is the proper subset of set B, denoted as A B (or B A)B, A2 proper subset: If A.
CC, then AB, b33 if a
B 4 if an a, then at the same time A=B and B
3. A set without any elements is called an empty set and recorded as φ.
It is stipulated that an empty set is a subset of any set and an empty set is a proper subset of any non-empty set.
A set of n elements, including 2n subsets and 2n- 1 proper subset.
Third, the operation of the set.
Complement set of intersection and union of operation types
Defined by all elements belonging to A and B, it is called the intersection of A and B, and marked as A B (pronounced' A crosses B'), that is, AB = {X | X A, and X B}.
A set consisting of all elements belonging to set A or set B is called the union of A and B. Note: A B (pronounced as' A and B'), that is, A B ={x|x A, or X B}).
Let S be a set, A is a subset of S, and the set of all elements in S that do not belong to A is called the complement (or complement) of subset A in S.
Remember, that's
CSA=
leather
elegant
draw
show
nature
Quality A A=A
aφ=φ
A B=B A
one
A B B
A A=A
Aφ= A
A B=B A
one
A B B
(CuA) (cub)
= copper (boron)
(CuA) (cub)
= copper (boron)
A (CuA)=U
a(CuA)=φ。
Example:
1. The following four groups of objects can form a set ().
A all the tall students in a class, B famous artists, C all the big books, the reciprocal of D is equal to its own real number.
2. Set {a, B, c} has a proper subset.
3. If the set m = {y | y = x2-2x+ 1, x r} and n = {x | x ≥ 0}, the relationship between m and n is.
4. let A= and B=. If it is B, the value range of is
5.50 students did two kinds of experiments: physics and chemistry. It is known that 40 students have done physics experiments correctly and 365,438+0 students have done chemistry experiments correctly.
If four people do both experiments wrong, then two people do both experiments right.
6. Represents the set M=. Descriptive method consists of points in the shadow part of the figure (including points on the boundary).
7. Given the set A∩C =φ{ x | x2+2x-8 = 0}, b = {x | x2-5x+6 = 0}, c = {x | x2-MX+m2- 19 = 0}, if b ∩.
Second, the related concepts of function
The concept of 1. function: Let a and b be non-empty number sets. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then F: A → B is called a function from set A to set B, and is denoted as y=f(x). 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. domain: the set of real numbers x that can make the function meaningful is called the domain of the function.
The main basis for finding the domain of function is:
The denominator of (1) score 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 bases of exponential and logarithmic expressions must be greater than zero and not equal to 1.
(5) If a function is a combination of some basic functions through four operations, then its domain is a set of values of x that make all parts meaningful.
(6) The index is zero, and the bottom 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.
The judgment method of the same function: ① the expression is the same (regardless of letters representing independent variables and function values); (2) Domain consistency (two points must be met at the same time)
(See related example 2 on page 2 1 of the textbook)
2. Range: consider its definition range first.
(1) observation method
(2) Matching method
(3) substitution method
3. Function image knowledge induction
(1) Definition: The set c of points P(x, y) with functions y=f(x) and (x ∈ a) as abscissa and function value y as ordinate in a plane rectangular coordinate system is called the image of functions y=f(x) and (x ∈ a).
(2) Painting
First, the tracking method:
B, image transformation method
There are three common conversion methods.
1) translation transformation
2) Telescopic transformation
3) Symmetric transformation
4. The concept of interval
Classification of (1) interval: open interval, closed interval and semi-open and semi-closed interval.
(2) Infinite interval
(3) The number axis representation of the interval.
map
Generally speaking, let A and B be two nonempty sets. If any element X in set A has a unique element Y corresponding to it according to a corresponding rule F, then the corresponding F: A B is the mapping from set A to set B ... Write F: A → B.
6. Piecewise function
(1) Functions with different analytic expressions in different parts of the domain.
(2) The values of the independent variables of each part.
(3) The domain of piecewise function is the intersection of the domain of each segment, and the range is the union of the range of each segment.
Supplement: composite function
If y = f (u) (u ∈ m) and u = g (x) (x ∈ a), then y=f[g(x)]=F(x)(x∈A) is called the composite function of f and g.
Two. Properties of functions
Monotonicity of 1. Function (Local Property)
(1) incremental function
Let the domain of function y=f(x) be I, if for any two independent variables x 1, x2 is in an interval d within the domain I, when X 1
If the values of any two independent variables in the interval d are both x 1, x2, when X 1 f (x2), then f (x) is said to be a decreasing function in this interval. The interval d is called monotonically decreasing interval y=f(x).
Note: the monotonicity of a function is a local property of the function;
(2) the characteristics of image
If the function y=f(x) is increasing function or subtraction function in a certain interval, it is said that the function y=f(x) has (strict) monotonicity in this interval, and the image of increasing function rises from left to right, and the image of subtraction function falls from left to right.
(3) The method of judging monotone interval and monotonicity of function.
(1) Definition method:
○ 1 Let x 1, x2∈D, x 1
2 difference f (x1)-f (x2);
3 deformation (generally factorization and formula);
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○967
○5 Draw a conclusion (point out the monotonicity of the function f(x) in the given interval d).
(b) Image method (looking up and down from the image)
(c) Monotonicity of composite functions
The monotonicity of the compound function f[g(x)] is closely related to the monotonicity of its constituent functions u=g(x) and y=f(u), and its law is "same increase but different decrease".
Note: The monotone interval of a function can only be a subinterval of its domain, and the intervals with the same monotonicity cannot be summed together to write its union.
8. Parity of function (global property)
(1) even function
Generally speaking, f (-x) = f(x) exists for any x in the domain of function f(x), so f (x) is called even function.
(2) odd function
Generally speaking, F (-x) =-f(x) exists for any X in the definition domain of function f(x), so F (x) is called odd function.
(3) Features of images with parity function
The image of even function is symmetrical about y axis; Odd function's image is symmetrical about the origin.
Judging the parity of a function according to its definition;
○ 1 First, determine the definition domain of the function and judge whether it is symmetrical about the origin;
2 determine the relationship between f (-x) and f(x);
○3 The corresponding conclusion is drawn: if f (-x) = f(x) or f (-x)-f (x) = 0, then f(x) is an even function; If f (-x) =-f(x) or f (-x)+f (x) = 0, then f(x) is odd function.
(2) judging by f (-x) f (x) = 0 or f (x)/f (-x) = 1;
(3) Using the image judgment of theorems or functions.
9. Analytic expression of function
(1). The analytic formula of the function is a representation of the function. When the functional relationship between two variables is needed, the corresponding law between them and the definition domain of the function are needed.
(2) The main methods for finding analytic expressions of functions are:
1) matching method
2) undetermined coefficient method
3) Alternative methods
4) Parameter elimination method
10. Maximum (minimum) value of the function (see textbook p36 for definition).
○ 1 Use the property of quadratic function (collocation method) to find the maximum (minimum) value of the function.
○2 Use images to find the maximum (minimum) value of the function.
○3 Use monotonicity of function to judge the maximum (minimum) value of function;
If the function y=f(x) monotonically increases in the interval [a, b] and monotonically decreases in the interval [b, c], then the function y=f(x) has the maximum value f (b) at x=b;
If the function y=f(x) monotonically decreases in the interval [a, b] and monotonically increases in the interval [b, c], then the function y=f(x) has a minimum value f (b) at x=b;
Example:
1. Find the domain of the following function:
⑴ ⑵
2. Let the domain of the function be _ _
3. If the domain of the function is, the domain of the function is.
4. Function, if, then =
6. Know the function and find the analytical formula of the function.
7. If the known function is satisfied, then =.
8. Let it be odd function on R, if, then if =
The analytical formula of r is
9. Find the monotone interval of the following functions:
⑴ (2)
10. Judge the monotonicity of the function and prove your conclusion.
1 1. Set a function to judge its parity and prove:
High school mathematics compulsory 2 knowledge points
I. Lines and equations
(1) inclination angle of straight line
Definition: The angle between the positive direction of the X axis and the upward direction of the straight line is called the inclination angle of the straight line. In particular, when a straight line is parallel or coincident with the X axis, we specify that its inclination angle is 0 degrees. Therefore, the range of inclination angle is 0 ≤α < 180.
(2) the slope of the straight line
① Definition: A straight line whose inclination is not 90, and the tangent of its inclination is called the slope of this straight line. The slope of a straight line is usually represented by k, that is. Slope reflects the inclination of straight line and axis.
At that time,; At that time,; It didn't exist then.
② Slope formula of straight line passing through two points:
Pay attention to the following four points: (1) At that time, the right side of the formula was meaningless, the slope of the straight line did not exist, and the inclination angle was 90;
(2)k has nothing to do with the order of P 1 and P2; (3) The slope can be obtained directly from the coordinates of two points on a straight line without inclination angle;
(4) To find the inclination angle of a straight line, we can find the slope from the coordinates of two points on the straight line.
(3) Linear equation
① Point-oblique type: the slope of the straight line is k, passing through the point.
Note: When the slope of the straight line is 0, k=0, and the equation of the straight line is y=y 1.
When the slope of the straight line is 90, the slope of the straight line does not exist, and its equation can not be expressed by point inclination. But because the abscissa of each point on L is equal to x 1, its equation is x=x 1.
② Oblique section: the slope of the straight line is k, and the intercept of the straight line on the Y axis is b..
③ Two-point formula: () Two points on a straight line,
(4) Cutting torque type:
Where the straight line intersects the axis at the point and intersects the axis at the point, that is, the intercepts with the axis and the axis are respectively.
⑤ General formula: (A and B are not all 0)
Note: Various equations with special application scope, such as:
A straight line parallel to the X axis: (b is a constant); A straight line parallel to the Y axis: (A is a constant);
(5) Linear system equation: that is, a straight line with some * * * property.
(1) parallel linear system
A linear system parallel to a known straight line (a constant that is not all zero): (c is a constant)
(2) Vertical linear system
A linear system perpendicular to a known straight line (a constant that is not all zero): (c is a constant)
(3) A linear system passing through a fixed point
(i) Linear system with slope k: a straight line passes through a fixed point;
(ii) The equation of the line system at the intersection of two lines is
(is a parameter), where the straight line is not in the straight line system.
(6) Two straight lines are parallel and vertical.
When, when,
;
Note: When judging the parallelism and verticality of a straight line by using the slope, we should pay attention to the existence of the slope.
(7) The intersection of two straight lines
stride
The coordinates of the intersection point are a set of solutions of the equation.
These equations have no solution; The equation has many solutions and coincidences.
(8) Distance formula between two points: Let it be two points in a plane rectangular coordinate system,
rule
(9) Distance formula from point to straight line: distance from point to straight line.
(10) Distance formula of two parallel straight lines
Take any point on any straight line, and then convert it into the distance from the point to the straight line to solve it.
Second, the equation of circle
1. Definition of a circle: The set of points whose distance to a point on a plane is equal to a fixed length is called a circle, the fixed point is the center of the circle, and the fixed length is the radius of the circle.
2. Equation of circle
(1) standard equation, center and radius r;
(2) General equation
At that time, the equation represented a circle. At this point, the center is and the radius is.
At that time, I said a point; At that time, the equation did not represent any graph.
(3) Method of solving cyclic equation:
Generally, the undetermined coefficient method is adopted: first set, then seek. Determining a circle requires three independent conditions. If the standard equation of a circle is used,
Demand a, b, r; If you use general equations, you need to find d, e, f;
In addition, we should pay more attention to the geometric properties of the circle: for example, the vertical line of a chord must pass through the origin, so as to determine the position of the center of the circle.
3, the position relationship between straight line and circle:
The positional relationship between a straight line and a circle includes three situations: separation, tangency and intersection:
(1) Set a straight line and a circle, and the distance from the center of the circle to L is, then there is; ;
(2) Tangent to a point outside the circle: ①k does not exist, so verify the existence of ②k, establish an oblique equation, and solve k with the distance from the center of the circle to the straight line = radius, and get two solutions of the equation.
(3) The tangent equation of a point passing through a circle: circle (x-a)2+(y-b)2=r2, and a point on the circle is (x0, y0), then the tangent equation passing through that point is (x0-a) (x-a)+(y0-b) (y-b) =
4. The positional relationship between circles: it is determined by comparing the sum (difference) of the radii of two circles with the distance (d) between the center of the circle.
Set a circle,
The positional relationship between two circles is usually determined by comparing the sum (difference) of the radii of the two circles with the distance (d) between the center of the circle.
At that time, the two circles were separated, and there were four common tangents at this time;
At that time, the two circles were circumscribed, and the connection line crossed the tangent point, with two outer tangents and one inner common tangent;
At that time, the two circles intersect, and the connecting line bisects the common chord vertically, and there are two external tangents;
At that time, two circles were inscribed, and the connecting line passed through the tangent point, and there was only one common tangent;
At that time, two circles included; It was concentric circles.
Note: when two points on the circle are known, the center of the circle must be on the vertical line in the middle; It is known that two circles are tangent and two centers are tangent to the tangent point.
The auxiliary line of a circle generally connects the center of the circle with the tangent or the midpoint of the chord of the center of the circle.
Third, preliminary solid geometry
Structural characteristics of 1, column, cone, platform and ball
(1) prism:
Geometric features: the two bottom surfaces are congruent polygons with parallel corresponding sides; The lateral surface and diagonal surface are parallelograms; The sides are parallel and equal; The section parallel to the bottom surface is a polygon that is congruent with the bottom surface.
② Pyramid
Geometric features: the side and diagonal faces are triangles; The section parallel to the bottom surface is similar to the bottom surface, and its similarity ratio is equal to the square of the ratio of the distance from the vertex to the section to the height.
(3) Prism:
Geometric features: ① The upper and lower bottom surfaces are similar parallel polygons; ② The side is trapezoidal; ③ The sides intersect with the vertices of the original pyramid.
(4) Cylinder: Definition: It is formed by taking a straight line on one side of a rectangle as the axis and rotating the other three sides.
Geometric features: ① The bottom is an congruent circle; ② The bus is parallel to the shaft; ③ The axis is perpendicular to the radius of the bottom circle; ④ The side development diagram is a rectangle.
(5) Cone: Definition: A Zhou Suocheng is rotated with a right-angled side of a right-angled triangle as the rotation axis.
Geometric features: ① the bottom is round; (2) The generatrix intersects with the apex of the cone; ③ The side spread diagram is a fan.
(6) frustum of a cone: Definition: Take the vertical line of the right-angled trapezoid and the waist of the bottom as the rotation axis, and use Zhou Suocheng to rotate.
Geometric features: ① The upper and lower bottom surfaces are two circles; (2) The side generatrix intersects with the vertex of the original cone; (3) The side development diagram is an arch.
(7) Sphere: Definition: Geometry formed by taking the straight line with the diameter of the semicircle as the rotation axis and the semicircle surface rotating once.
Geometric features: ① the cross section of the ball is round; ② The distance from any point on the sphere to the center of the sphere is equal to the radius.
2. Three views of space geometry
Define three views: front view (light is projected from the front of the geometry to the back); Side view (from left to right),
Top view (from top to bottom)
Note: the front view reflects the height and length of the object; The top view reflects the length and width of the object; The side view reflects the height and width of the object.
3. Intuition of space geometry-oblique two-dimensional drawing method.
The characteristics of oblique bisection method are as follows: ① The line segment originally parallel to the X axis is still parallel to X, and its length remains unchanged;
② The line segment originally parallel to the Y axis is still parallel to Y, and its length is half of the original.
4. Surface area and volume of cylinders, cones and platforms.
The surface area of a (1) geometry is the sum of all the surfaces of the geometry.
(2) The surface area formula of special geometry (C is the perimeter of the bottom, H is the height, and L is the generatrix)
(3) Volume formulas of cylinders, cones and platforms.
(4) Formula of surface area and volume of sphere: v =；; S=
4, spatial point, straight line, plane position relationship.
Axiom 1: If two points of a straight line are on a plane, then all points of the straight line are on this plane.
Application: judging whether a straight line is in a plane.
Express axiom1in symbolic language;
Axiom 2: If two non-coincident planes have a common point, then they have one and only one common straight line passing through the point.
Symbol: Plane α and β intersect, the intersection line is A, and it is denoted as α ∩ β = A..
Symbolic language:
The role of axiom 2:
(1) It is a method to determine the intersection of two planes.
② Explain the relationship between the intersection of two planes and the common point of two planes: the intersection must pass through the common point.
(3) It can be judged that a point is on a straight line, which is an important basis for proving several points.
Axiom 3: After passing through three points that are not on the same straight line, there is one and only one plane.
Inference: a straight line and a point outside the straight line determine a plane; Two intersecting straight lines define a plane; Two parallel straight lines define a plane.
Axiom 3 and its corollary: ① It is the basis for determining planes in space ② It is the basis for proving plane coincidence.
Axiom 4: Two lines parallel to the same line are parallel to each other.
The positional relationship between spatial straight lines and straight lines
① Definition of non-planar straight lines: two straight lines that are different from each other on any plane.
② Properties of straight lines in different planes: neither parallel nor intersecting.
③ Determination of out-of-plane straight line: A straight line passing through a point out of plane and a point in plane is an out-of-plane straight line.
(4) Angle formed by straight lines on different planes: parallel, so that two lines intersect to get an acute angle or a right angle, that is, the formed angle. The angle range formed by two straight lines in different planes is (0,90). If the angle formed by straight lines of two different planes is a right angle, we say that the straight lines of two different planes are perpendicular to each other.