All the knowledge points of senior one mathematics last semester
Parity of 1. function
(1) If f(x) is an even function, then f (x) = f (-x);
(2) If f(x) is odd function and 0 is in its domain, then f(0)=0 (which can be used to find parameters);
(3) The parity of the judgment function can be defined in equivalent form: f (x) f (-x) = 0 or (f (x) ≠ 0);
(4) If the analytic formula of a given function is complex, it should be simplified first, and then its parity should be judged;
(5) odd function has the same monotonicity in the symmetric monotone interval; Even functions have opposite monotonicity in symmetric monotone interval;
2. Some questions about compound function.
Solution of the domain of (1) composite function: If the domain is known as [a, b], the domain of the composite function f[g(x)] can be solved by the inequality a≤g(x)≤b; If the domain of f[g(x)] is known as [a, b], find the domain of f(x), which is equivalent to x∈[a, b], and find the domain of g(x) (that is, the domain of f(x)); When learning functions, we must pay attention to the principle of domain priority.
(2) The monotonicity of the composite function is determined by "the same increase but different decrease";
3. Function image (or symmetry of equation curve)
(1) Prove the symmetry of the function image, that is, prove that the symmetry point of any point on the image about the symmetry center (symmetry axis) is still on the image;
(2) Prove the symmetry of the image C 1 and C2, that is, prove that the symmetry point of any point on C 1 about the symmetry center (symmetry axis) is still on C2, and vice versa;
(3) curve C 1: f (x, y) = 0, and the equation of symmetry curve C2 about y=x+a(y=-x+a) is f(y-a, x+a)=0 (or f(-y+a,-x+a) =
(4) Curve C 1:f(x, y)=0 The C2 equation of the symmetrical curve about point (a, b) is: f(2a-x, 2b-y) = 0;
(5) If the function y=f(x) is constant to x∈R, and f(a+x)=f(a-x), then the image y=f(x) is symmetrical about the straight line x=a;
(6) The images of functions y=f(x-a) and y=f(b-x) are symmetrical about the straight line x=;
4. The periodicity of the function
(1)y=f(x) for x∈R, f(x+a)=f(x-a) or f (x-2a) = f (x) (a >: 0) is a constant, then y=f(x) is a period of 2a.
(2) If y=f(x) is an even function, and its image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 2 ~ a;
(3) If y=f(x) odd function, whose image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 4 ~ a;
(4) If y=f(x) is symmetric about points (a, 0) and (b, 0), then f(x) is a periodic function with a period of 2;
(5) If the image of y=f(x) is symmetrical (a ≠ b) about straight lines x = a and x = b, then the function y = f (x) is a periodic function with a period of 2;
(6) When y=f(x) equals x∈R, f(x+a)=-f(x) (or f(x+a)=, then y = f (x) is a periodic function with a period of 2;
5. Equation k=f(x) has a solution k∈D(D is the range of f(x));
A≥f(x) keeps a≥[f(x)]max,; A≤f(x) considers a ≤ [f (x)] min;
( 1)(a & gt； 0，a≠ 1，b & gt0，n∈R+)；
(2)logaN =(a & gt； 0，a≠ 1，b & gt0，b≠ 1)；
(3)logab symbols are memorized by the formula of "same positive but different negative";
(4)alogaN = N(a & gt； 0，a≠ 1，N & gt0);
6. When judging whether the corresponding relationship is a mapping, we should grasp two points:
The elements in (1)A must all have images and;
(2) All elements in B may not have original images, and different elements in A may have the same images in B;
7. Prove the monotonicity of the function by using the definition skillfully, find the inverse function and judge the parity of the function.
8. For the inverse function, we should grasp the following conclusions:
The monotone function in (1) field must have an inverse function;
(2) odd function's inverse function is also odd function;
(3) There is no inverse function for even functions whose domain is not a single element set;
(4) The periodic function has no inverse function;
(5) Two mutually inverse functions have the same monotonicity;
(6)y=f(x) and y=f- 1(x) are reciprocal functions. Let the domain of f (x) be a and the domain of f(x) be b, then there is f [f- 1 (x)] = x (x ∈ b).
9. When dealing with quadratic functions, don't forget the combination of numbers and shapes.
Quadratic function must have a maximum in the closed interval, and the problem of finding the maximum is "two views": look at the opening direction; Second, look at the relative position relationship between the symmetry axis and a given interval;
Monotonicity-based 10
By using the sign-preserving property of linear functions on intervals, the problem of finding the range of a class of parameters can be solved.
1 1 solution to the problem;
(1) separation parameter method;
(2) Solving the inequality (group) of distribution table transformed into the root of quadratic equation in one variable;
Exercise questions:
1.(-3,4) The coordinates of the point on the axis of X are _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Symmetrical coordinates about the origin are _ _ _ _ _ _ _.
2. The distance from point B (-5, -2) to the X axis is _ _ _ _, the distance to the Y axis is _ _ _ _, and the distance to the origin is _ _ _ _.
3. With the point (3,0) as the center, the coordinates of the intersection of the circle with radius of 5 and the X axis are _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
The coordinates intersecting the Y axis are _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
4. If point P (a-3,5-a) is in the first quadrant, the range of value of A is _ _ _ _ _ _ _ _.
Xiaohua bought a commodity with a unit price of 3 yuan with 500 yuan, and the remaining money was Y (yuan) and the number of pieces of this commodity was X (pieces).
The functional relationship between them is _ _ _ _ _ _ _ _ _, and the value range of x is _ _ _ _ _ _ _ _.
6. The value range of the independent variable X of the function y= is _ _ _ _ _ _ _ _
7. When a=____, the function y=x is a proportional function.
8. The image of function y=-2x+4 passes through _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
The circumference is _ _ _ _ _
9. The image of linear function y=kx+b passes through the point (1, 5) and intersects the Y axis at three places, then k = _ _ _ _, and b = _ _ _ _.
10. If the point (m, m+3) is on the image of the function y=-x+2, then m=____
1 1.y is proportional to 3x. When x=8 and y=- 12, then the resolution function of y and x is _ _ _ _ _ _ _.
12. The image with function y=-x is a straight line passing through the sum of the origin and (2, _ _ _), and the straight line passes through the _ _ _ quadrant.
When x increases, y follows _ _ _ _ _
13. the function y=2x-4, when x _ _ _ _ _ _ _ _ y0, B0 and b > 0; c、k
All the knowledge points of senior one mathematics last semester
Definition of 1. sequence
A series of numbers arranged in a certain order is called a series, and each number in the series is called an item of the series.
(1) As can be seen from the definition of series, the numbers of series are arranged in a certain order. If the numbers that make up a series are the same but in different order, then they are not the same series. For example, the series 1, 2,3,4,5 is different from the series 5,4,3,2, 1.
(2) The definition of series does not stipulate that the numbers in the series must be different. Therefore, multiple identical numbers can appear in the same series, such as:-1, 1, 2, 3, 4, … to form a series:-1,-1.
(4) The term of a series is different from its number. The term of a series refers to a certain number in this series, which is a function value, equivalent to f(n), while the term of a number refers to the position serial number of this number in the series, which is the value of an independent variable, equivalent to n in f(n).
(5) Order is very important for a series. There are several identical figures. Because of their different arrangement order, the series is also different. Obviously, there is an essential difference between a series and a group of numbers. For example, if the five numbers 2, 3, 4, 5 and 6 are arranged in different order, different series will be obtained, while {2, 3, 4, 5 and 6}.
2. Series classification
(1) According to the number of items in the series, the series can be divided into finite series and infinite series. When writing a series, the last item should be written for a finite series, for example, the series 1, 3, 5, 7, 9, …, 2n- 1 indicates a finite series. If the sequence is written as 1,
(2) According to the relationship between items or the increase or decrease of series, it can be divided into the following categories: increasing series, decreasing series, swinging series and constant series.
3. General term formula of sequence
A sequence is a series of numbers arranged in a certain order, and its essential attribute is to determine the law of this number, which is usually expressed by formula f(n).
Although these two general formulas are different in form, they represent the same series, just as not every functional relationship can be expressed by analytical formula, and not every series can write its general formula. Although some series have general formulas, they may not be formally established. They only know the finite term in front of a series, and there is no other explanation. The series cannot be determined, and the general formula does not exist. For example, the series 1, 2, 3, 4, ...
The items written after the formula are different. Therefore, the induction of general term formula depends not only on its first few terms, but also on the synthesis law of series, and more observation and analysis are needed to truly find the internal law of series. There is no general way to write its general term formula from the first few terms of a series.
In order to understand the general formula of series, emphasize the following points again:
The general term formula of (1) series is actually that the domain is positive integer set n or its finite set {1, 2, ..., n}.
(2) If we know the general term formula of the series, then we can use 1, 2, 3, … instead of N in the formula in turn to find out the terms of this series; At the same time, we can also use the general term formula of series to judge whether a number is an item in the series, and if so, what item it is.
(3) Just as all functional relationships do not necessarily have analytical formulas, not all series have general formulas.
If the approximation is less than 2, the sequence shall be accurate to 1, 0. 1, 0.0 1, 0.00 1, 0.000 1, ... 1, 1.
(4) Some general formulas of series are not necessarily formal, such as:
(5) Some series only give the first few terms, but do not give their composition rules, so the general term formula of series derived from the first few terms is not.
4. Series of images
For series 4, 5, 6, 7, 8, 9, 10, the corresponding relationship between the serial number of each item and this item is as follows:
Serial number: 1234567
Item code: 456789 10
In other words, the above can be regarded as a mapping from one set of serial numbers to another. Therefore, from the perspective of mapping and function, a sequence can be regarded as a positive integer set n (or its finite subset {1, 2,3, ..., n}). When the independent variable takes the value from small to large, the corresponding function value list. The function here is
Because the term of the series is a function value and the serial number is an independent variable, the general term formula of the series is the corresponding function and analytical formula.
Sequence is a special function, which can be expressed intuitively by images.
The sequence is represented by an image. With the serial number as the abscissa and the corresponding item as the ordinate, you can draw a picture to represent an order. When drawing, for convenience, the unit length taken on the two coordinate axes of the plane rectangular coordinate system can be different. From the image representation of the sequence, we can directly see the change of the sequence, but it is not accurate.
Compared with function, sequence is a special function, which is a group of positive integers or a group of finite continuous positive integers headed by 1, and its image is infinite or finite isolated points.
5. Recursive series
A pile of steel pipes is stacked in seven layers, and the number of steel pipes in each layer forms a sequence from top to bottom: 4, 5, 6, 7, 8, 9, 10.
The order ① can also be given by the following method: the number of steel pipes in the first floor from top to bottom is 4, and the number of steel pipes in each floor below is more than that in the previous floor 1 root.
Exercise questions:
1. If the sum of the first n terms of arithmetic progression {an} is Sn, and S33-S22= 1, the tolerance of sequence {an} is ().
A. 12B
Analysis: from Sn=na 1+n(n- 1)2d, we get S3=3a 1+3d, S2=2a 1+d, and substitute S33-S22= 1 to get d=2, so we choose.
Answer: c
2. given the sequence a 1= 1, a2=5, an+2=an+ 1-an(n∈N), then a20 1 1 is equal to ().
A. 1B。 -4C.4D.5
Analysis: From the known, a 1= 1, a2=5, a3=4, a4=- 1, a5=-5, a6=-4, a7= 1, a8=5, …
So {an} is a series with a period of 6.
∴a20 1 1=a6×335+ 1=a 1= 1.
A: A.
3. Let {an} be arithmetic progression, Sn be the sum of its first n terms, S5S8, then the following conclusion is wrong ().
Ad & lt0B.a7=0
C.S9 & gtS5D。 S6 and S7 are both values of Sn.
Analysis: ∵S50. S6=S7，∴a7=0.
S7 & gts8,∴a8<； 0.
Assuming S9 & gtS5, then A6+A7+A8+A9 >; 0, that is 2(a7+a8)>0.
a7 = 0，a8 & lt0,∴a7+a8<； 0. The assumption is not true, so S9
Answer: c
All the knowledge points of senior one mathematics last semester
The Meaning and Expression of 1. Set
1, the meaning of set: a set is the sum of some different things, and people can realize these things and judge whether a given thing belongs to the whole.
The research objects are collectively called elements, and the whole composed of some elements is called set, which is called set for short.
2. Three characteristics of elements in a set:
(1) Determinism of elements: If the set is certain, then whether an element belongs to this set is certain: yes or no.
(2) Mutual dissimilarity of elements: the elements in a given set are affirmative and unrepeatable.
(3) The disorder of elements: the position of elements in the set can be changed, and changing the position does not affect the set.
3. Representation of set: {…}
(1) indicates the set in capital letters: A={ basketball players in our school}, B={ 1, 2,3,4,5}.
(2) Representation of sets: enumeration and description.
Enumeration: Enumerate the elements in the set one by one {a, B, c...}
B, description method:
① Interval method: describe the common attributes of the elements in the set, and write them in braces to represent the set.
{x？ r | x-3 & gt； 2}，{ x | x-3 & gt； 2}
② Language description: Example: {A triangle that is not a right triangle}
③ venn diagram: Draw a closed curve, which represents the set.
4, the classification of the set:
(1) finite set: a set with finite elements.
(2) Infinite set: a set containing infinite elements.
(3) Empty set: a set without any elements.
5, the relationship between elements and sets:
(1) If an element is in a set, it belongs to the set, that is, a? A
(2) If the element is not in the set, it does not belong to the set, that is, a ￠ a
Note: Commonly used digit sets and their symbols:
The set of nonnegative integers (i.e. natural number set) is denoted as n.
Positive integer set n or N+
Integer set z
Rational number set q
Real number set r
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★ Summary of knowledge points in the first volume of senior one mathematics.
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★ Comprehensive summary of mathematics knowledge points in senior one.
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