Mathematics Beijing Normal University Edition Compulsory II Outline
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.
(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.
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, 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, verify ②k exists, establish an oblique equation, and solve k with the distance from the center of the circle to the straight line = radius to get 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 x***.
(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 reasoning function:
(1) It is the basis for determining the spatial plane.
(2) It is the basis of 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.
To find the angles formed by straight lines on different planes:
A, using the defined structural angle, one can be fixed, the other can be translated, or both can be translated to a special position at the same time, and the vertex can be selected at a special position.
B, prove that the angle is the angle.
C, use triangle to find the angle
(7) Equiangular theorem: If two sides of one angle and two sides of another angle are parallel respectively, then the two angles are equal or complementary.
(8) The positional relationship between spatial straight line and plane.
A straight line is in a plane-there are countless things in common.
Symbolic representation of three positional relationships: aαa∪α= Aa‖α.
(9) positional relationship between planes: parallel-no common point; α‖β
Intersection-there is a straight line. α∪β= b
5. Parallel problems in space
Determination and properties of (1) parallel lines and planes
Theorem for judging the parallelism between a straight line and a plane: A straight line out of the plane is parallel to a straight line in the plane, then the straight line is parallel to the plane.
Lines, lines, parallel lines, parallel planes.
Theorem of parallelism between straight lines and planes: If a straight line is parallel to a plane and the plane passing through it intersects with this plane, then the straight line is parallel to the intersection line. Line, plane, parallel line, parallel line
(2) The judgment and nature of parallelism between planes.
Theorem for judging the parallelism of two planes
(1) If two intersecting lines in one plane are parallel to the other plane, the two planes are parallel.
(parallel lines and planes → parallel planes),
(2) If two sets of intersecting straight lines are parallel in two planes, the two planes are parallel.
(parallel lines → parallel planes),
(3) Two planes perpendicular to the same straight line are parallel,
Property Theorem of Parallel Two Planes
(1) If two planes are parallel, a straight line in one plane is parallel to the other plane. (Face-to-face parallelism → Line-to-Line parallelism)
(2) If two parallel planes intersect with the third plane, their intersection lines are parallel. (Face-to-face parallelism → Line-to-Line parallelism)
7. Vertical problem in space
(1) Definition of line, surface and line-surface verticality
(1) Perpendicularity of two straight lines with different planes: If the angle formed by two straight lines with different planes is a right angle, the two straight lines with different planes are said to be perpendicular to each other.
② Line-plane verticality: 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.
③ Plane is perpendicular to the plane: if two planes intersect, the dihedral angle (the figure formed by two half planes starting from a straight line) is a straight dihedral angle (the plane angle is a right angle), which means that the two planes are perpendicular.
(2) Determination of vertical relation and property theorem.
(1) The judging theorem and property theorem of the perpendicularity between a straight line and a plane.
Decision theorem: If a straight line is perpendicular to two intersecting straight lines on a plane, then the straight line is perpendicular to the plane.
Property theorem: If two straight lines are perpendicular to a plane, then the two straight lines are parallel.
(2) The judgment theorem and property theorem of vertical plane.
Decision theorem: If one plane passes through the vertical line of the other plane, then the two planes are perpendicular to each other.
Theorem of nature: If two planes are perpendicular to each other, then the straight line perpendicular to their intersection on one plane is perpendicular to the other plane.
9. Spatial angle problem
(1) Angle between straight lines
① Angle formed by two parallel straight lines: specified as.
(2) The angle formed by the intersection of two straight lines: the angle formed by the intersection of two straight lines is not greater than the right angle, which is called the angle formed by these two straight lines.
(3) Angle formed by two straight lines with different planes: when passing through any point o in space, make the straight line parallel to the two straight lines with different planes A and B to form two intersecting straight lines, and the angle formed by these two intersecting straight lines is called the angle formed by two straight lines with different planes.
(2) The angle formed by a straight line and a plane
① The angle formed by the parallel lines between the plane and the plane: specified as.
② The angle between the plane and the perpendicular to the plane: specified as.
(3) The angle formed by the oblique line of the plane and the plane: the acute angle formed by an oblique line of the plane and its projection in the plane is called the angle formed by this straight line and this plane.
The idea of finding the angle between diagonal and plane is similar to finding the angle formed by straight lines on different planes: "one work, two certificates and three calculations"
When making an angle, project according to the definition key. From the definition of projection, the key lies in the point on the diagonal to the perpendicular to the surface.
When solving problems, pay attention to mining two main information in the problem setting:
(1) The vertical line from a point on the diagonal to the surface;
(2) The diagonal or a point on the plane of the diagonal is perpendicular to the known surface, and the vertical line can be easily obtained from the vertical nature of the surface.
(3) The dihedral angle of dihedral angle and plane angle
① Definition of dihedral angle: The figure formed by two half planes starting from a straight line is called dihedral angle, this straight line is called the edge of dihedral angle, and these two half planes are called the faces of dihedral angle.
② Plane angle of dihedral angle: Take any point on the edge of dihedral angle as the vertex and make two rays perpendicular to the edge in two planes. The angle formed by these two rays is called the plane angle of dihedral angle.
③ Straight dihedral angle: A dihedral angle whose plane angle is a right angle is called a straight dihedral angle.
If the dihedral angle formed by two intersecting planes is a straight dihedral angle, then the two planes are vertical; On the contrary, if two planes are perpendicular, the dihedral angle formed is a straight dihedral angle.
(4) Calculation method of dihedral angle
Definition method: select the relevant point on the edge, and make a ray perpendicular to the edge in two planes through this point to get the plane angle.
Vertical plane method: when the vertical lines from one point to two surfaces in dihedral angle are known, the angle formed by the intersection of two vertical lines as the intersection of plane and two surfaces is the plane angle of dihedral angle.
Methods to improve mathematics performance in a short time
1, find out what efforts can be made in knowledge. The key is to be prepared for knowledge. Before the exam, you should check if there are any loopholes in the math knowledge you have learned in junior high school, and if there are any forgotten or confused places. Secondly, be prepared to solve common mistakes, and then look at your own error notes. If there is no wrong book, you can find out the papers you have done before. Look at the revised part, that is, the wrong place, and try not to make or make fewer mistakes in the past when answering questions. In other words, don't make mistakes.
You must have confidence in yourself and the future. Mentality is the most important reason that affects the exam. When you enter the examination room, you must have a domineering attitude. Be confident, believe that you have studied 1000 days in junior high school, reviewed for more than 300 days, worked out 3,000 to 4,000 math problems in junior high school, and trained 1000 days, which took a while. Now is the harvest time, and you will get good grades.
3. After reading the book, put away the textbooks and do exercises. By doing exercises, you can check what you did badly again. If you can't, you can read the textbook again. In this way, I believe you will be deeply impressed.
Math answering skills
1, direct debit method
Starting directly from the conditions given by the proposition, using concepts, formulas, theorems, etc. Reasoning or operating, drawing a conclusion and choosing the correct answer are traditional methods to solve problems.
2. Verification method
Find out the appropriate verification conditions from the questions, and then find out the correct answer through verification. You can also substitute alternative answers into conditions to verify and find the correct answer. This method is called verification method (also called substitution method). This method is often used when encountering quantitative propositions.
3. Special element method
Substitute appropriate special elements (such as figures or numbers) into the conditions or conclusions of the topic to get the answer. This method is called the special element method.
4. Exclusion method
For multiple-choice questions with only one correct answer, according to mathematical knowledge or reasoning and calculus, the incorrect conclusion is excluded, and the remaining conclusions are screened, so that the solution to the correct conclusion is called exclusion screening.
5. Graphical method
According to the nature and characteristics of the graphics or images that meet the requirements of the topic, it is called graphical method to make the right choice. Graphic method is one of the common methods to solve multiple-choice questions.
Mathematics Beijing Normal University Edition compulsory two outline related articles:
★ Compulsory Mathematics for Grade One of Beijing Normal University 2 Chapter 2 Analysis of Geometry Knowledge Points
★ Mathematics knowledge points of Grade 2 of Beijing Normal University Edition
★ Prepare materials
★ Mathematics syllabus for primary schools of Beijing Normal University
★ Two compulsory questions of senior three mathematics in Beijing Normal University
★ Beijing Normal University Edition Junior High School Mathematics Knowledge Point Outline
★ Summary of Knowledge Points in Volume II of Grade Two Mathematics of Beijing Normal University Edition
★ Beijing Normal University Edition Grade Two Mathematics Textbook
★ Summary of knowledge points in the first volume of primary school mathematics published by Beijing Normal University
★ A set of knowledge points is required for senior one mathematics in Beijing Normal University.