Summary of senior high school mathematics knowledge 1
Related concepts of functions
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.
U The judgment method of the same function: ① The expressions are the same (regardless of the letters indicating the independent variables and function values); (2) Domain consistency (two points must be met at the same time)
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
Summary of mathematics knowledge points in senior one and two
gather
(1) The number of subsets of the set containing n elements is 2 n, and the number of proper subset is 2n-1; The number of non-empty proper subset is 2n-2;
(2) Note: Don't forget the situation when discussing.
(3) Part II Functions and Derivatives
1. Mapping: Note ① Elements in the first group must have images; ② One to one or many to one.
2. Solution of function value domain: ① analysis method; ② Matching method; ③ discrimination method; ④ Using monotonicity of functions; ⑤ substitution method; ⑥ Using mean inequality; ⑦ Use the combination of numbers and shapes or geometric meaning (meaning of slope, distance, absolute value, etc. ); ⑧ Use the boundedness of function; Pet-name ruby derivative method.
3. Some questions about compound function.
Solve the domain of (1) composite function;
① If the domain of f(x) is [a, b], the domain of compound function f[g(x)] is solved by inequality a≤g(x)≤b ② If the domain of f[g(x)] is [a, b], then the domain of f(x) is found, which is equivalent to X. 。
(2) Determination of monotonicity of composite function:
Firstly, the original function is decomposed into basic functions: internal function and external function;
Secondly, the monotonicity of inner and outer functions in their respective domains is studied respectively.
③ According to "the same sex increases and the opposite sex decreases", we can judge the monotonicity of the original function in its domain.
Note: The domain of the outer function is the domain of the inner function.
4. piecewise function: range (maximum), monotonicity, image and other issues, first solve them in segments, and then draw a conclusion.
5. Parity of functions
The symmetry of the domain of (1) function about the origin is a necessary condition for the function to have parity;
(2) odd function has the same monotonicity and even functions have the opposite monotonicity in the monotonic interval symmetrical about the origin;
(3) If the analytic formula of a given function is complicated, equivalent deformation should be carried out before judging its parity;
Summary of Mathematics Knowledge Points of Grade One 3 in Senior High School
1. Definition of arithmetic series
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, and this constant is called arithmetic progression's tolerance, which is usually represented by the letter D.
2. arithmetic progression's general formula
If the first term of arithmetic progression {an} is a 1 and the tolerance is d, its general term formula is an = a1+(n-1) d.
3. Arithmetic average term
If A=(a+b)/2, then a is called the arithmetic average of a and b.
4. The common nature of arithmetic progression
The generalization of (1) general formula: an=am+(n-m)d(n, m∈N_).
(2) If {an} is arithmetic progression, m+n=p+q,
Then am+an=ap+aq(m, n, p, q∈N_).
(3) If {an} is a arithmetic progression with a tolerance of d, then ak, ak+m, ak+2m, ... (k, m∈N_) are arithmetic progression with an error of MD. 。
(4) The series Sm, S2m-Sm, S3m-S2m, … are also arithmetic progression.
(5)S2n- 1=(2n- 1)an .
(6) If n is even, then parity = nd/2;
If n is odd, then s odd -S even = mean term.
note:
inference
Derive the first n terms and formulas of arithmetic progression by anti-addition;
Sn=a 1+a2+a3+…+an，①
Sn=an+an- 1+…+a 1，②
①+②: Sn=n(a 1+an)/2
Two technologies
Knowing that three or four numbers constitute a kind of arithmetic progression, we should be good at setting elements.
(1) If the odd number is arithmetic progression and the sum is constant, it can be set to …, a-2d, a-d, a, a+d, a+2d, ….
(2) If the even number is arithmetic progression and the sum is a constant value, it can be set as …, a-3d, a-d, a+d, a+3d, …, and the rest items are set symmetrically according to arithmetic progression's definition.
Four methods
Arithmetic progression's Judgment Method
(1) Definition: For any natural number with n≥2, verify that an-an- 1 is the same constant;
(2) Arithmetic average method: verify that 2an- 1=an+an-2(n≥3, n∈N_) is true;
(3) General formula: verify an = pn+q;
(4) The first n terms and formula method: verify Sn=An2+Bn.
Note: The latter two methods can only be used to judge whether it is arithmetic progression, but not to prove arithmetic progression.
Summary of Mathematics Knowledge Points in Senior One 4
Definition of equality of two complex numbers:
If the real and imaginary parts of two complex numbers are equal, then we say that these two complex numbers are equal, that is, if A, B, C and d∈R, then a+bi=c+di.
A=c, B = D. Especially, when A, b∈R, a+bi=0.
a=0，b=0。
The necessary and sufficient conditions for the equality of complex numbers provide a way to turn complex problems into practical problems.
Special reminder of plural equality:
Generally speaking, two complex numbers can only be said to be equal or unequal, but their sizes cannot be compared. If both complex numbers are real numbers, the sizes can be compared, and only if both complex numbers are real numbers can the sizes be compared.
The method steps to solve the complex equation problem:
(1) Converts a given complex number into the standard form of a complex number;
(2) Solving complex numbers according to necessary and sufficient conditions.
High school mathematics knowledge summary scientific induction method 5
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 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;
Rule out the possibility of 0, 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.
Expanding reading: the skills of mathematics examination in college entrance examination
1, repeated regularly.
Even the reviewed content should be consolidated regularly, but the number of reviews should be gradually reduced with the increase of time, and the interval can be gradually lengthened. You can consolidate new knowledge on the same day, make weekly summary every week, make periodic summary every month, and conduct comprehensive and systematic semester review at the middle and end of the semester. From the content, we should immediately review the knowledge of each lesson, sort out the knowledge of each unit, summarize the knowledge of each chapter, connect the relevant knowledge together, form a knowledge network, and grasp the knowledge and methods as a whole.
2, scientific and reasonable arrangement
Review can generally be divided into centralized review and decentralized review. Experiments show that the effect of decentralized review is better than that of centralized review except in special circumstances. Decentralized review can properly classify the materials that need to be memorized and alternate with other studies or entertainment or rest, so as not to use a certain way of thinking monotonously and form fatigue. Decentralized review should also combine your own cognitive level and the characteristics of memory materials, and grasp the number of repetitions and the interval. The longer the interval, the better, but it should be suitable for your own review rules.
3. Carefully examine questions, patiently answer questions, standardize and be accurate, and reduce mistakes.
Computational ability and logical reasoning ability are two training abilities clearly stipulated in the examination syllabus. It can be said that they are the two most basic abilities to learn mathematics well, and the examination in mathematics papers is everywhere. Moreover, in the annual marking, the loss caused by these two abilities accounts for a considerable proportion. Therefore, when reviewing mathematics, we should not only do a good job in teaching knowledge, problems and methods, but also improve and standardize students' computing ability and logical reasoning ability through various channels and opportunities.