Summary of Mathematics Knowledge Points in Senior High School (1)
1. Some basic concepts:
(1) Vector: It has both magnitude and direction.
(2) Quantity: only size, no directional quantity.
(3) Three elements of a directed line segment: starting point, direction and length.
(4) Zero vector: a vector with a length of 0.
(5) Unit vector: a vector with a length equal to 1 unit.
(6) Parallel vector (* * * line vector): non-zero vector with the same or opposite direction.
A zero vector is parallel to any vector. ※ 。
(7) Equal vectors: vectors with equal length and the same direction.
2. Vector addition operation:
⑵ The characteristics of triangle rule: end to end.
⑵ Characteristics of parallelogram rule: * * starting point
Summary of Mathematics Knowledge Points in Senior High School (2)
Roots of equations and zeros of functions
1, the concept of function zero: for a function, the real number that makes it true is called the zero of the function.
2. The meaning of the zero point of the function: the zero point of the function is the real root of the equation, that is, the abscissa of the intersection of the image of the function and the axis. Namely:
The equation has a real root function, the image has an intersection with the axis, and the function has a zero point.
3, the role of zero solution:
Find the zero point of a function:
1 (algebraic method) to find the real root of the equation;
2 (Geometric method) For the equation that can't be solved by the root formula, we can relate it with the image of the function and find the zero point by using the properties of the function.
4. Zero point of quadratic function:
Quadratic function.
1 、△& gt； 0, the equation has two unequal real roots, the image of the quadratic function has two intersections with the axis, and the quadratic function has two zeros.
2.△=0, the equation has two equal real roots (multiple roots), the image of the quadratic function intersects with the axis, and the quadratic function has a double zero or a second-order zero.
3 、△& lt； 0, the equation has no real root, the image of the quadratic function has no intersection with the axis, and the quadratic function has no zero.
Summary of Mathematics Knowledge Points in Senior High School (3)
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, which is marked as AB or BA.
2. "Equality" relationship (5≥5, and 5≤5, then 5=5)
Example: let a = {x | x2-1= 0} b = {-1,1} "The elements are the same".
Conclusion: For two sets A and B, if any element of set A is an element of set B and any element of set B is an element of set A, we say that set A is equal to set B, that is, A = B.
(1) Any set is a subset of itself. Aiya
② proper subset: If AíB and A 1B, then set A is the proper subset of set B, and it is recorded as AB (or BA).
③ If aí b and bí c, then aí c.
④ If AíB and BíA exist at the same time, then a = 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.
Summary of Mathematics Knowledge Points in Senior High School (4)
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 it can be seen that the limitation of X comes from two points. First, it can be used as a denominator, but it cannot be used as a denominator.
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.
To sum up, when a is different, the different situations of the domain of 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 greater than 0, the range of the 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.
Since x is greater than 0, it is meaningful to any value of a, so the following gives the respective situations of power function in the first quadrant.
You can see:
(1) All graphs pass (1, 1).
(2) When a is greater than 0, the power function monotonically increases, while when a is less than 0, the power function monotonically decreases.
(3) When a is greater than 1, the power function graph is concave; When a is less than 1 and greater than 0, the power function graph is convex.
(4) When a is less than 0, the smaller A is, the greater the inclination of the graph is.
(5)a is greater than 0, and the function passes (0,0); A is less than 0, and the function has only (0,0) points.
(6) Obviously the power function is unbounded.
Summary of Mathematics Knowledge Points in Senior High School (5)
Frequently tested knowledge points
Sets are usually represented by uppercase Latin letters, such as: a, b, c… while elements in sets are represented by lowercase Latin letters, such as: a, b, C… Latin letters are just equivalent to the names of sets and have no practical significance.
The method of assigning Latin letters to a set is represented by an equation, for example, in the form of A={…}. The left side of the equal sign is capitalized Latin letters, and the right side is enclosed in curly braces. In brackets are some mathematical elements with the same nature.
Commonly used are enumeration method and description method.
1. enumeration: usually used to represent a finite set. All the elements in the collection are listed one by one and enclosed in braces. This method of representing a set is called enumeration. { 1,2,3,……}
2. Description: It is often used to represent an infinite set. The public * * * attribute of the elements in the collection is described by words, symbols or expressions and enclosed in braces. This method of representing a set is called description. {x|P}(x is the general form of the elements of this set, and p is the * * * same property of the elements of this set) For example, a set composed of positive real numbers less than π is represented as {x|0.
3. Graphical method (venn diagram): In order to visually represent a set, we often draw a closed curve (or circle) and use its interior to represent a set. gather
Symbols of commonly used number sets in natural languages;
(1) The set of all non-negative integers is usually called the set of non-negative integers (or the set of natural numbers), and is recorded as n; A set of natural numbers excluding 0, denoted as N+
(2) A set excluding 0 from a non-negative integer set, also called a positive integer set, is denoted as z+; The negative integer set also excludes the set of 0, which is called the negative integer set and recorded as Z-
(3) The set of all integers is usually called the set of integers, and is denoted as z..
(4) The set of all rational numbers is usually referred to as rational number set for short, and is denoted as Q..Q={p/q|p∈Z, q∈N, p and q coprime} (the set of positive and negative rational numbers is denoted as Q+Q- respectively).
(5) The set of all real numbers is usually referred to as real number set for short, and recorded as r (positive real number set is recorded as r+; Negative real numbers are recorded as R-)
(6) The operation of counting a complex set as a C set: set exchange law A ∩ B = B ∩ A ∪ B = B ∪ A set associative law (A∪B)∩C = A ∪( B∪C)(A∪)
Cu(A∩B)= CuA∪CuBCu(A∪B)= CuA∪CuB The principle of inclusion and exclusion will encounter problems about the number of elements in the set, so we will write the number of elements in the limited set A as card (a).
Law of set absorption A ∩ (A ∩ B) = AA ∩ (A ∪ B) = Law of set complement A∪Cua = UA∪Cua =φ Let A be a set, and let the set composed of all subsets of A be called A-(BUC) = (A-).
Mathematics learning methods in senior one.
1, moderate exercise to maintain vitality
Many students have this feeling. They have to take an exam without doing math problems for a few days, and they are slow and hesitant in examining the questions, which makes them get off to a bad start, run poorly and make mistakes easily. Therefore, we must insist on doing appropriate exercises every day, especially key and hot topics, to prevent ideological degradation and laziness, and to keep our thinking flexible and smooth. When doing the problem, especially when doing the comprehensive paper, you should finish it within a limited time, otherwise it is easy to form a procrastination style, lack of passion for thinking on the spot, leading to time out of control and failure to play its due level.
2, induction, sublimation into a classic.
The teacher of Yanboyuan Education advised the students to master mathematical methods skillfully, which can be unchanged or changed. Mastering mathematical thinking methods can be started from two aspects. The first is to summarize important mathematical thinking methods. For example, an algebraic problem can be communicated with geometric problems through association, and the method of combining numbers and shapes can be used. Such as correlation slope, intercept, function image, equation curve, etc. The second is to summarize the problem-solving methods of important questions. Such as formula method, dislocation subtraction method, split item elimination method, iterative method, inductive proof method, undetermined coefficient method and so on. Also pay attention to the scope of application and conditions of use of typical methods to prevent errors caused by formal application.
3. Check for leaks and fill gaps, and strive for perfection.
A considerable number of senior one students have low test scores, and many of them make mistakes in the questions they can do, especially the basic questions. The reasons are both knowledge and methods. Therefore, it is necessary to strengthen the study of previous wrong questions, find out the causes of errors, list the knowledge points that are prone to errors, and summarize the methods that are prone to misuse. For example, when crossing a point to make a straight line, the slope does not exist, the discussion of q= 1 is ignored in the summation of geometric series, the discriminant is ignored in Vieta theorem, and the range is ignored in substitution or elimination. Students can ask each other questions together and make corrections in arguments and discussions, which is more effective. If you find out the cause of the mistake, you can prescribe the right medicine, so that the mistakes you have made will not happen again, and the problems you can do will not be wrong again.