Chapter one, the concepts of set and function.
1. 1. 1, setting
1, the research objects are collectively called elements, and the whole composed of some elements is called set. Set three elements: certainty, mutual difference and disorder.
2. As long as the elements that make up two sets are the same, they are said to be equal.
3. Common set: positive integer set: or, integer set:, rational number set:, real number set:.
4. Representation methods of sets: enumeration and description.
1. 1.2, the basic relationship between sets.
1. Generally speaking, for two sets A and B, if any element in set A is an element in set B, set A is said to be a subset of set B. Write it down.
2. If there are elements in a set, and set A is called the proper subset of set B. Note: A B.
3. Call a set without any elements an empty set. Write as:. And stipulates that an empty set is a subset of any set.
4. If set A contains n elements, set A has a subset.
1. 1.3, the basic operation between sets
1. Generally, a set consisting of all elements belonging to set A or set B is called the union of sets A and B..
Generally speaking, the set consisting of all elements belonging to set A and set B is called the intersection of A and B..
3. Complete works and supplements?
1.2. 1, the concept of function
1, let a and b be non-empty number sets. If any number in set A has a unique number corresponding to it according to a certain correspondence, it is called a function from set A to set B, and it is called:.
2. The constituent elements of a function are: definition domain, correspondence domain and value domain. If two functions have the same domain and the same correspondence, they are said to be equal.
1.2.2, the representation of the function
1, there are three ways to express the function: analytical method, image method and list method.
1.3. 1, monotonicity and maximum (minimum) value
1, pay attention to the general format of function monotonicity proof:
Solution: If sum, then: = …
1.3.2, parity
1. Generally speaking, if there is any one in the domain of a function, then the function is called an even function. Even function images are symmetric about axis.
2. Generally speaking, if there is any one in the domain of a function, then this function is called odd function-odd function's image symmetry about the origin.
Chapter 2, Basic Elementary Functions (Ⅰ)
2. 1. 1, exponential and exponential power operations
1, generally speaking, if, then it is called the second root. One of them is.
2, when odd,;
When it is an even number.
3. We stipulate that:
⑴
⑵ ;
4. Nature of operation:
⑴ ;
⑵ ;
⑶ .
2. 1.2, exponential function and its properties
1, remember the image:
2.2. 1, Logarithm and Logarithm Operation
1、 ;
2、 .
3、 , .
4. When:
⑴ ;
⑵ ;
⑶ .
5. Bottom changing formula:
.
6、
.
2...2.2. Logarithmic function and its properties
1, remember the image:
2.3, power function
1, images of several power functions:
Chapter three, the application of functions.
3. 1. 1, the root of the equation and the zero of the function.
1, the equation has real roots.
The image of the function intersects the axis.
The value of this function is zero.
2. Property: If the image of the function on the interval is a continuous curve and exists, then the function has zero on the interval, that is, it exists, so this is the root of the equation.
3. 1.2, find the approximate solution of the equation by dichotomy.
1, master of dichotomy.
3.2. 1, several functional models of different growth
3.2.2. Application example of function model
1, conventional problem solving method: first draw a scatter plot, then fit it with a suitable function, and finally test it.
Mathematics compulsory 2 knowledge points
1, the structure of space geometry
(1) Common polyhedrons are: prism, pyramid and frustum; Common rotating bodies are: cylinder, cone, frustum of a cone and sphere.
⑵ Prism: Two faces are parallel to each other, the other face is a quadrilateral, and the public sides of every two adjacent quadrilaterals are parallel to each other. A polyhedron surrounded by these faces is called a prism.
⑶ Prism: Use a plane parallel to the bottom of the pyramid to cut the part between the bottom of the pyramid and the cross section. This polyhedron is called a prism.
2. Three views and direct vision of space geometry.
The projection formed by light scattering from a point is called central projection, and the projection lines of the central projection intersect at a point; The projection under a beam of parallel light is called parallel projection, and the projection lines of parallel projection are parallel.
3. Surface area and volume of space geometry
(1) cylindrical side area;
(2) Cone side area:
(3) Side area of frustum of a cone:
④ Volume formula:;
5] surface area and volume of the ball:
.
Chapter two: the positional relationship among points, lines and surfaces.
1, axiom 1: If two points on a straight line are on the same plane, then the straight line is on this plane.
2. Axiom 2: When three points that are not on a straight line intersect, there is only one plane.
3. Axiom 3: If two non-coincident planes have a common point, then they only have a common straight line passing through the point.
Axiom 4: Two lines parallel to the same line are parallel.
Theorem: If two sides of two angles in space are parallel to each other, then the two angles are equal or complementary.
6. The positional relationship between lines: parallel, intersecting and out of plane.
7. Line-plane positional relationship: the straight line is in the plane, the straight line is parallel to the plane, and the straight line intersects the plane.
8. Face-to-face positional relationship: parallelism and intersection.
9, a straight line parallel to the plane:
⑴ Judgment: If a straight line out of the plane is parallel to a straight line in the plane, the straight line is parallel to the plane.
⑵ Property: If a straight line is parallel to a plane, the intersection line between any plane passing through this straight line and this plane is parallel to this straight line.
10, parallel face to face:
⑴ Judgment: If two intersecting straight lines in one plane are parallel to another plane, the two planes are parallel.
⑵ Nature: If two parallel planes intersect with the third plane at the same time, their intersection lines are parallel.
1 1, the straight line is perpendicular to the plane:
⑴ Definition: If a straight line is perpendicular to any straight line in a plane, it is said that the straight line is perpendicular to the plane.
⑵ Judgment: If a straight line is perpendicular to two intersecting straight lines in the plane, the straight line is perpendicular to the plane.
⑶ Property: Two straight lines perpendicular to the same plane are parallel.
12, face-to-face vertical:
⑴ Definition: Two planes intersect. If the dihedral angle they form is a straight dihedral angle, the two planes are said to be perpendicular to each other.
⑵ Judgment: If one plane passes through the vertical line of the other plane, the two planes are vertical.
⑶ Property: If two planes are perpendicular to each other, the straight line perpendicular to the intersection in one plane is perpendicular to the other plane.
Chapter 3: Lines and Equations
1, dip angle and slope:
2, linear equation:
(1) point inclination:
2 inclined type:
(3) Two-point type:
(4) General formula:
3. For straight lines:
There are:
⑴ ;
(2) Intersection;
(3) and coincidence;
⑷ .
4. For straight lines:
There are:
⑴ ;
(2) Intersection;
(3) and coincidence;
⑷ .
5, the distance between two points formula:
6, point to straight line distance formula:
Chapter 4: Circle and Equation
1, equation of circle:
(1) standard equation:
⑵ General equation:.
2. The relationship between two circles:
(1) Depart:
(2) circumscribe:
(3) Intersection:
4 cut:;
(5) Including:
3, the distance between two points in space formula:
Mathematics compulsory 3 knowledge points
Chapter 1: Algorithm
1, algorithms in three languages:
Natural language, flow chart, programming language;
2. Three basic structures of the algorithm:
Sequence structure, selective structure, cyclic structure.
3. The framework in the flowchart:
Normative representation methods such as start-stop box, input-output box, processing box, judgment box and assembly line;
4. Two common structures in circular structure:
When cycle structure, until cycle structure
5. Basic algorithm statement:
① Assignment statement: "=" (sometimes "↓")
② Input and output statements: "Input" and "Print"
③ Conditional statement:
If ... then
…
Otherwise ...
If ... it will be over.
④ loop statement: "Do" statement
do
…
Until ...
end
"While" statement
When ..
…
line
[6] Algorithm Case: Coincidence Thought Division
Chapter II: Statistics
1, sampling method:
① Simple random sampling (less in total)
② Systematic sampling (large number of people)
③ Stratified sampling (significant overall difference)
Note: N individuals are selected from the population of N individuals, and the chance (probability) of each individual being selected is 0.
2. Overall distribution estimation:
(1) Table 1 and Figure 2:
① Frequency Distribution Table-Detailed Data
② Histogram of frequency distribution-intuitive distribution.
③ Line chart of frequency distribution-convenient for observing the overall distribution trend.
Note: The area enclosed by the density curve of the overall distribution and the horizontal axis is 1.
2 stem leaf diagram:
① The stem-leaf diagram is suitable for the case of less data, from which it is easy to see the distribution of data, as well as the median and mode.
(2) Single digits are leaves, and ten digits are stems. The data on the right is from small to uppercase, and the same medicine is written repeatedly.
3. Estimation of total feature number:
(1) average:
If the frequency of = is, the average value is;
Note: The frequency distribution table should take the median value of this group to calculate the average value.
⑵ Variance and standard deviation: a set of sample data.
Variance:;
Standard deviation:
Note: The smaller the variance and standard deviation, the more stable the sample data.
The average value reflects the overall level of data; Variance and standard deviation reflect the stability of data.
⑶ Linear regression equation
① Two relationships between variables: functional relationship and correlation relationship;
(2) Make a scatter plot and judge the linear correlation.
③ Linear regression equation: (least square method)
Note: Linear regression straight line passes through fixed point.
Chapter 3: Probability
1, random events and their probabilities:
(1) event: every possible result of the test is represented by capital letters;
(2) Characteristics of inevitable events, impossible events and random events;
(3) the probability of random event A:
2. Classical probability:
(1) Basic events: every basic result that may appear in the test;
(2) The characteristics of classical probability:
① The number of basic events is limited;
(2) the possibility of every basic event is equal.
⑶ Classical probability calculation formula: There are n equal possible basic events * * * in a test, and event A contains m basic events, then the probability of event A.
3. Geometric probability:
Characteristics of (1) geometric probability:
① All basic events are infinite;
(2) the possibility of every basic event is equal.
⑵ Geometric probability calculation formula:;
Among them, the measurement is determined according to the topic, which is generally line segment, angle, area, volume and so on.
4. mutually exclusive events:
(1) Two events that cannot occur at the same time are called mutually exclusive events;
(2) If any two events are mutually exclusive events, they are said to be mutually exclusive.
(3) If events A and B are mutually exclusive, then the probability of event A+B is equal to the sum of the probabilities of events A and B,
Namely:
(4) If the events are mutually exclusive, there are:
5] Opposing events: If one of the two mutually exclusive events is bound to happen, these two events are called opposing events.
(1) contrary to the events recorded as follows
(2) The opposing event must be mutually exclusive events, and mutually exclusive events is not necessarily an opposing event.
Mathematics compulsory 4 knowledge points
Chapter one, trigonometric function.
1. 1. 1, any angle.
1, the concepts of positive angle, negative angle, zero angle and quadrant angle.
2. A set of angles that are the same as the ending edge of the angle:
.
1. 1.2, arc system
1. An angle that the central angle of an arc with a length equal to the radius is 1 radian.
2、 .
3. Arc length formula:.
4. Sector area formula:.
1.2. 1, trigonometric function of any angle.
1, let it be an arbitrary angle, and its terminal edge intersects the unit circle at this point, then:
.
2. If the set point is any point on the edge of the angle terminal, then: (Set)
, , .
3. Four-quadrant symbol and trigonometric function line drawing.
4. Inductive formula 1:
(Among them:)
5. Special angles of 0, 30, 45, 60,
The trigonometric function value is 90,180,270.
1.2.2, the basic relationship of trigonometric functions with the same angle
1, square relation:.
2. business relations:
1.3, inductive formula of trigonometric function
1, inductive formula 2:
2. Inductive formula 3:
3. Inductive formula 4:
4. Inductive formula 5:
5. Inductive Formula 6:
1.4. 1 image, sine and cosine functions
1, remember sine and cosine function images:
2. The definition range, value range, maximum and minimum value, symmetry axis, symmetry center, parity, monotonicity, periodicity and other related properties of sine and cosine functions can be described by comparing with images.
3, you can use the five-point method to draw.
1.4.2, Properties of Sine and Cosine Functions
1, definition of periodic function: For a function, if there is a non-zero constant t that makes every value in the defined domain available, then this function is called a periodic function, and the non-zero constant t is called the period of this function.
1.4.3, Properties of Image and Tangent Function
1, remember the image of tangent function:
2. We can tell the related properties of tangent function: domain, range, symmetry center, parity, monotonicity and periodicity.
1.5, image of function
1, can tell the translation and expansion transformation relationship between function image and function image.
2. For the function:
There are: amplitude a, period, initial phase, phase and frequency.
1.6, Simple application of trigonometric function model
1, familiar with textbook examples is required.
Chapter 2, Plane Vector
2. 1. 1, the physical background and concept of vector
1. Understand four common vectors: force, displacement, velocity and acceleration.
2. A quantity with both magnitude and direction is called a vector.
2. 1.2, geometric representation of vector
1. A directed line segment is called a directed line segment, which contains three elements: starting point, direction and length.
2, the size of the vector, that is, the length of the vector (or modulus), recorded as; A vector with zero length is called a zero vector; A vector with a length of 1 unit is called a unit vector.
3. Non-zero vectors with the same or opposite directions are called parallel vectors (or * * * line vectors). Specifies that the zero vector is parallel to any vector.
2. 1.3, equation vector and * * * line vector
1, vectors with equal length and the same direction are called isometric vectors.
2.2. 1, vector addition and its geometric significance
1, triangle rule and parallelogram rule.
2、 ≤ .
2.2.2, vector subtraction operation and its geometric significance
1, vectors with the same length and opposite directions are called inverse quantities.
2.2.3 Multiplication of Vector Numbers and Its Geometric Significance
1. stipulates that the product of a real number and a vector is a vector, and this operation is called vector multiplication. It is written as:, and its length and direction are specified as follows:
⑴ ,
(2) When appropriate, the direction of is the same as that of; When, the direction is opposite to that of.
2. Plane vector * * * straight line theorem: vector and * * * straight line, if and only if there is only one real number, make.
2.3. 1, fundamental theorem of plane vector
1. Basic theorem of plane vectors: If there are two non-linear vectors on the same plane, there are only one pair of real numbers for any vector on this plane.
2.3.2. Orthogonal decomposition and coordinate representation of plane vectors
1、 .
2.3.3, coordinate operation of plane vector
1, and then:
⑴ ,
⑵ ,
⑶ ,
⑷ .
2. Set, and then:
.
2.3.4, coordinate representation of plane vector * * * line.
1, and then
The midpoint coordinates of (1) line segment AB are,
⑵ The barycenter coordinate of △ ABC is.
2.4. 1, the physical background and significance of plane vector product.
1、 .
2. The projection in the direction is:
3、 .
4、 .
5、 .
2.4.2. Coordinate representation, module and included angle of plane vector product.
1, and then:
⑴
⑵
⑶
2. Set, and then:
.
2.5. 1, vector method in plane geometry
2.5.2. Examples of vector application in physics
Chapter three, trigonometric identity transformation.
3. 1. 1, cosine formula of the difference between two angles
1、
2. Remember the trigonometric function value of 15?
3. 1.2, sum and difference formula of sine, cosine and tangent.
1、
2、
3、
4、 .
5、 .
3. 1.3, Sine, Cosine and Tangent Double Angle Formulas
1、 ,
Deformation:
2、
Deformation 1:,
Deformation 2:.
3、 .
3.2. Simple trigonometric identity transformation
1. Pay attention to chord cutting and square reduction.
Five knowledge points required in mathematics
Chapter 1: Solving Triangle
1, sine theorem:
.
2, cosine theorem:
3, triangle area formula:
Chapter 2: Sequence
1, the relationship between and in the sequence:
2. Arithmetic series:
⑴ Definition: If a series starts from the second term and the difference between each term and its previous term is equal to the same constant, then this series is called arithmetic progression.
(2) General formula:
(3) Sum formula:
3. Equal ratio series
⑴ Definition: If a series starts from the second term and the ratio of each term to its previous term is equal to the same constant, then the series is called geometric series.
(2) General formula:
(3) Sum formula:
Chapter III: Inequality.
1、
2、
3. Deformation: