High school mathematics knowledge points
A preliminary study on solid geometry
Structural characteristics of 1, column, cone, platform and ball
(1) prism:
Definition: Geometry surrounded by two parallel faces, the other faces are quadrangles, and the common edges of every two adjacent quadrangles are parallel to each other.
Classification: According to the number of sides of the bottom polygon, it can be divided into three prisms, four prisms and five prisms.
Representation: Use the letter of each vertex, such as a five-pointed star, or use the letter at the opposite end, such as a five-pointed star.
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
Definition: One face is a polygon, the other faces are triangles with a common vertex, and the geometric figure enclosed by these faces.
Classification: According to the number of sides of the bottom polygon, it can be divided into three pyramids, four pyramids and five pyramids.
Representation: Use the letters of each vertex, such as a pentagonal 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:
Definition: Cut the part between the pyramid, the section and the bottom with a plane parallel to the bottom of the pyramid.
Classification: According to the number of sides of the bottom polygon, it can be divided into three prisms, four prisms and five prisms.
Representation: Use the letters of each vertex, such as a pentagonal pyramid.
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: Geometry surrounded by surfaces that rotate on one side of a rectangle and on the other three sides.
Geometric features:
① The bottom surface 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: Geometry surrounded by the surface formed by the circle rotating with the right-angled side of the right-angled triangle as the rotation axis.
Geometric features:
① The bottom surface 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: Cut the part between the cone, the section and the bottom with a plane parallel to the bottom of the cone.
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 where the diameter of the semicircle is located as the rotation axis and the semicircle surface rotates once.
Geometric features:
① The cross section of the ball is circular;
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) and top view (from top to bottom)
Note: the front view reflects the position relationship of the object, that is, it reflects the height and length of the object;
The top view reflects the position relationship between the left and right of the object, that is, the length and width of the object;
The side view reflects the up-and-down and front-and-back positional relationship of the object, that is, it reflects the height and width of the object.
3. Intuition of space geometry-oblique two-dimensional drawing method.
Characteristics of oblique mapping;
(1) the original line segment parallel to the X axis is still parallel to X, with the same length;
② The line segment originally parallel to the Y axis is still parallel to Y, and its length is half of the original.
Mathematics knowledge point 2
Linear sum equation
(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.
Mathematics knowledge point 3
power function
Definition:
A function in the form of y = x a (a is a constant), that is, a function with the base as the independent variable and the exponent as the dependent variable is called a power function.
Domain and Value Domain:
When a is a different numerical value, the different situations of the domain of the power function are as follows: if a is any real number, the domain of the function is all real numbers greater than 0; If a is a negative number, then X must not be 0, but the definition domain of the function must also be determined according to the parity of Q, that is, if Q is even at the same time, then X cannot be less than 0, then the definition domain of the function is all real numbers greater than 0; If q is an odd number at the same time, the domain of the function is all real numbers that are not equal to 0. When x is different, the range of power function is different as follows: when x is greater than 0, the range of function is always a real number greater than 0. When x is less than 0, only when q is odd and the range of the function is non-zero real number. Only when a is a positive number will 0 enter the value range of the function.
Nature:
For the value of a nonzero rational number, it is necessary to discuss their respective characteristics in several cases:
First of all, we know that if a=p/q, q and p are integers, then x (p/q) = the root of q (p power of x), if q is odd, the domain of the function is r, if q is even, the domain of the function is [0, +∞). When the exponent n is a negative integer, let a =-k, then x = 1/(x k), obviously x≠0, and the domain of the function is (-∞, 0)∩(0, +∞). So we can see that the limitation of X comes from two points. One is that it may be used as the denominator but not 0, and the other is that it may be under the even root but not negative, so we can know:
Rule out two possibilities: 0 and negative number, that is, for x>0, then A can be any real number;
The possibility of 0 is ruled out, that is, for X.
The possibility of being negative is ruled out, that is, for all real numbers with x greater than or equal to 0, a cannot be negative.
Mathematics knowledge point 4
exponential function
The domain of (1) exponential function is the set of all real numbers, where a is greater than 0. If a is not greater than 0, there will be no continuous interval in the definition domain of the function, so we will not consider it.
(2) The range of exponential function is a set of real numbers greater than 0.
(3) The function graph is concave.
(4) If a is greater than 1, the exponential function increases monotonically; If a is less than 1 and greater than 0, it is monotonically decreasing.
(5) We can see an obvious law, that is, when a tends to infinity from 0 (of course, it can't be equal to 0), the curves of the functions tend to approach the positions of monotonic decreasing functions of the positive semi-axis of Y axis and the negative semi-axis of X axis respectively. The horizontal straight line y= 1 is the transition position from decreasing to increasing.
(6) Functions always infinitely tend to a certain direction on the X axis and never intersect.
(7) The function always passes (0, 1).
Obviously the exponential function is unbounded.
odevity
definition
Generally, for the function f(x)
(1) If any x in the function definition domain has f (-x) =-f(x), then the function f(x) is called odd function.
(2) If any x in the function definition domain has f (-x) = f(x), the function f(x) is called an even function.
(3) If f (-x) =-f(x) and f (-x) = f (x) are true for any x in the function definition domain, then the function f(x) is both a odd function and an even function, which is called an even-even function.
(4) If neither F (-x) =-f(x) nor F (-x) = f (x) can be established for any x in the function definition domain, then the function F (x) is neither a odd function nor an even function, which is called a nonsingular non-even function.
Summary and formula of high school mathematics knowledge points
1.
1) Set: set some specified objects together to form a set. Each object is called an element.
Note: ① Set and its elements are two different concepts, which are given by description in textbooks, similar to the concepts of points and lines in plane geometry.
② The elements in the set are deterministic (A? A and a? A, the two must be one), different from each other (if a? A, b? A, then a≠b) and disorder ({a, b} and {b, a} represent the same set).
③ A set has two meanings: all eligible objects are its elements; As long as it is an element, you must sign the condition.
2) Representation methods of sets: enumeration method, description method and graphic method are commonly used.
3) Classification of sets: finite set, infinite set and empty set.
4) Common number set: n, z, q, r, N*
2. Concepts such as subset, intersection, union, complement, empty set and complete set.
1) subset: if there is x∈B for x∈A, then A B (or ab);
2) proper subset: A B has x0∈B but x0 A；; Marked as b (or, and)
3) Intersection: A∩B={x| x∈A and x∈B}
4) and: A∪B={x| x∈A or x∈B}
5) Complement set: cua = {x | x but x ∈ u}
3. Understand the relationship between sets and elements, sets and sets, and master relevant terms and symbols.
4. Several equivalence relations about subsets
①A∩B = A A B； ②A∪B = B A B； ③A B C uA C uB；
④A∩CuB = empty set cuab; ⑤CuA∪B=I A B .
5. The essence of intersection and union operations
①A∩A=A，A∩B = B∩A； ②A∪A=A，A∪B = B∪A；
③Cu(A∪B)= CuA∪CuB，Cu(A∪B)= CuA∪CuB；
6. Number of finite subsets: If the number of elements in set A is n, then A has 2n subsets, 2n- 1 nonempty subset and 2n-2 nonempty proper subset.
Expanding reading: the learning method of senior high school mathematics
1. First of all, you should be familiar with the basic steps and methods of solving problems. Usually practice and exam are the same. We should pay attention to every step. The process of solving problems is a process of thinking. Pay attention to a high degree of concentration and don't let your thinking go astray, but we usually follow our own ideas and follow familiar steps to find the answer easily.
It is very important to carefully examine the questions when you get them, which directly determines the correct rate and speed of your answer. Knowledge will make many detours, waste a lot of time, make mistakes, and lose more than you gain. Therefore, it is very important to read every known condition, analyze the relationship between the question and the condition, and think about the calculation before starting to answer the question.
3. Do a good job of induction and summary at ordinary times, so that it will be easy to classify the questions during the exam. Often the same type of questions will have the same points or even give you the same ideas, which will enable you to sum up the problem-solving methods well, and then draw inferences, so that when you see the same type of questions, you can greatly shorten the time for answering them.
It is also important to learn to draw. People's brains remember pictures better than documents, so learning to use known conditions to assume scenes and draw corresponding pictures is very beneficial to solving problems and the correct rate is relatively high. General problems come from solving practical problems in life, which also helps you to connect textbook knowledge with reality.