Sorting out the knowledge points of compulsory mathematics in the first volume of senior one.
Properties of functions
Monotonicity of Functions (Local Properties)
(1) incremental function
Let the domain of the function y=f(x) be I, if for any two independent variables x 1 and x2 in the interval d within the domain I, when x 1,
If the values of any two independent variables in the interval d are x 1, x2, and when x 1f(x2), then f(x) is said to be a decreasing function in this interval. The interval d is called monotonically decreasing interval y=f(x).
Note: the monotonicity of a function is a local property of the function;
(2) the characteristics of image
If the function y=f(x) is increasing function or subtraction function in a certain interval, it is said that the function y=f(x) has (strict) monotonicity in this interval, and the image of increasing function rises from left to right, and the image of subtraction function falls from left to right.
(3) The method of judging monotone interval and monotonicity of function.
(1) Definition method:
(1) take x 1, x2∈D, x 1.
(2) difference f (x1)-f (x2); Or do business.
(3) Deformation (usually factorization and formulation);
(4) Number (that is, judging the positive and negative difference f (x1)-f (x2));
(5) Draw a conclusion (point out the monotonicity of the function f(x) in the given interval d).
(b) Image method (looking up and down from the image)
(c) Monotonicity of composite functions
The monotonicity of the compound function f[g(x)] is closely related to the monotonicity of its constituent functions u=g(x) and y=f(u), and its law is "same increase but different decrease".
Note: The monotone interval of a function can only be a subinterval of its domain, and the intervals with the same monotonicity cannot be summed together to write its union.
Parity of Function (Global Property)
(1) even function: Generally speaking, for any x in the definition domain of function f(x), f(-x)=f(x), then f(x) is called even function.
(2) odd function: Generally speaking, for any x in the definition domain of the function f(x), there is f (-x) =-f(x), so f(x) is called odd function.
(3) Characteristics of the image of the parity function: the image of the even function is symmetrical about the Y axis; Odd function's image is symmetrical about the origin.
9. Use the definition to judge the parity of the function:
1 First determine the definition domain of the function and judge whether it is symmetrical about the origin;
2 determine the relationship between f(-x) and f(x);
3. It is concluded that if f(-x)=f(x) or f(-x)-f(x)=0, then f(x) is an even function; If f(-x)=-f(x) or f(-x)+f(x)=0, then f(x) is odd function.
Summary of five knowledge points of compulsory mathematics in senior one.
Arithmetic progression with a tolerance of (1) is still arithmetic progression, and its tolerance is still D. 。
⑵ For arithmetic progression whose tolerance is d, the sequence obtained by multiplying each term by the constant k is still arithmetic progression, and its tolerance is kd.
(3) If {a} and {b} are arithmetic progression, {a b} and {ka+b}(k and b are nonzero constants) are also arithmetic progression.
(4) For any m and n, arithmetic progression {a} has: a=a+(n-m)d, especially when m= 1, arithmetic progression's general formula is obtained, which is more general than arithmetic progression's general formula.
5. Generally speaking, if L, K, P, …, M, N, R, … are all natural numbers, l+k+p+…=m+n+r+… (the number of natural numbers on both sides is equal), then when {a} is arithmetic progression, there is: A+A+.
[6] arithmetic progression with a tolerance of d, from which equidistant terms are extracted, forms a new series, which is still arithmetic progression, and its tolerance is kd(k is the difference between the number of extracted terms).
(7) If {a} is a arithmetic progression with a tolerance of d, then A, A, …, A and A are also arithmetic progression with a tolerance of -d; In arithmetic progression {a}, a-a=a-a=md (where m, k,).
In arithmetic progression, from the first term, every term (except the last term of a finite series) is the arithmetic average of the two terms before and after it.
Levies when the tolerance d >. 0, the number in arithmetic progression increases with the increase of the number of terms; When d < 0, the number in arithmetic progression decreases with the decrease of the number of terms; When d=0, the number in arithmetic progression is equal to a constant.
⑽ Let A, A and A be three terms in arithmetic progression, and the ratio of the distance difference between A and A, A and a=(≦- 1), then A =.
(1) The necessary and sufficient condition for the sequence {a} to be arithmetic progression is that the sum of the first n terms of the sequence {a} can be written in the form of S=an+bn (where a and b are constants).
(2) In arithmetic progression {a}, when the number of terms is 2n(nN), S-S=nd, =; When the number of terms is (2n- 1)(n), S-S=a, =.
(3) If the sequence {a} is arithmetic progression, then S, S-S, S-S, ... are still arithmetic progression with an error of.
(4) If the sum of the first n terms of two arithmetic progression {a} and {b} is s and t respectively (n is odd), then =.
5] In arithmetic progression {a}, S=a, s = b (n >; M), then S=(a-b).
[6] In arithmetic progression {a}, it is a linear function of n, and all points (n,) are on the straight line y=x+(a-).
(7) Remember that the sum of the first n items of arithmetic progression {a} is S.① If a >；; 0, tolerance d
Reference of mathematics learning methods in senior one.
Foundation is the key, and textbooks are the first choice.
First of all, senior one students should make it clear that senior one mathematics is the key foundation of senior high school mathematics. Just entering senior one, some students still don't adapt. If you learn the skills of the college entrance examination directly, it seems that you want to run if you can't learn to walk well. Any skill is based on solid basic knowledge, so it is recommended that senior one students pay more attention to the basics and read more textbooks.
In exam-oriented education, only by memorizing formulas, mastering problem-solving skills, being familiar with various types of questions and turning yourself into a problem-solving machine can we achieve good results in the exam. It is not enough to do the college entrance examination questions. It is necessary to add "proficiency" on the basis of the meeting, and the small questions are generally controlled at about two minutes at a time.
Senior one has a lot of mathematics knowledge, and the test questions account for about 70% of the college entrance examination results. You have to study five books in a school year. As long as you firmly grasp the mathematics of senior one, senior two and senior three are only the review and supplement of senior one. Therefore, after entering high school, we should adapt to the new environment as soon as possible, listen carefully in class and take more notes, and we will certainly learn math well.
Therefore, freshmen should do more exercises on the basis of memorizing concepts in order to learn math well.
First, math preview
Preview is a necessary prerequisite for learning mathematics well, which can be described as the "east wind" needed to "burn Chibi". Generally speaking, preview can be divided into the following two steps.
1. Preview the textbook knowledge of the chapters to be learned. In the process of previewing the textbook, we should memorize the definitions and theorems in the textbook for flexible use. We should do well the examples in the textbook and the exercises after class. These basic things are often the most important.
2. Consciously complete the self-study draft. Self-learning draft is a popular learning method since the new curriculum reform! First of all, you should finish the preview test part of the self-study draft, and then look at the following questions. At first, some people may not be able to do it. Remember not to study hard, it will often get twice the result with half the effort!
Second, math listening
Listening is an important part of learning mathematics well. It can be said that if you don't listen, you won't get good grades.
1. Listen carefully in class and speak actively. Mark the questions you don't understand and ask the teacher in time after class!
2. Recording is often a tiny link. Pay attention to the teacher's repeated statements and a lot of words written on the blackboard (math teachers usually don't write very much), and record them in a small book in time, and over time, a knowledge booklet is formed.
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