The investigation of the nature of basic elementary function, the function problem with derivative knowledge as the background; Function problems based on vector knowledge: from the examination of specific functions to the examination of abstract functions; From focusing on results to focusing on processes; From the examination of familiar scenes to the examination of novel scenes.
2, vector knowledge:
Vector has the duality of number and shape, and the proposition trend of vector test questions in college entrance examination is: to examine the basic concept and operation law of plane vector; Examine the coordinate operation of plane vector; Examine the synthesis of plane vectors and geometry, trigonometry, algebra and other disciplines.
3, inequality knowledge:
It is a new orientation of inequality proposition to highlight instrumentality, downplay independence and highlight solutions. The proposition trend of inequality test questions in college entrance examination: taking the basic linear programming problem as the compulsory content, combining exponential function, logarithmic function, trigonometric function and intersection function to investigate the nature, maximum value and monotonicity of inequality; The questions to prove inequality are mostly based on functions, sequences, analytic geometry and other knowledge, and are put forward at the intersection of knowledge networks, which is comprehensive and requires high ability. The test questions for solving inequalities are often associated with the discussion of formulas, roots and parameters. Examining students' ability of equivalent transformation and classified discussion; Application questions based on the current economy, social production and life and inequality synthesis will still be the hot spot in the college entrance examination, mainly to examine students' reading comprehension ability and problem-solving ability.
4, solid geometry knowledge:
The year of 20xx has become simple, but the difficulty of 20xx is still not great, and it is not difficult to examine the basic three views. The combination of ball and geometry involves the problems of cutting and connecting, the investigation of the vertical and parallel position relationship between line and plane, the included angle between line and plane, the included angle between plane, the volume calculation of geometry and so on. , are the key exam content.
5, analytic geometry knowledge:
The minor questions mainly involve conic equation, the positional relationship between straight line and circle, the investigation of geometric properties of conic curve, the knowledge of analytic geometry in polar coordinates, the knowledge of straight line and circle, the knowledge of straight line and conic curve, the simultaneous, fixed point, fixed value and range of straight line and conic equation, which reduces the difficulty of examination.
6. Derived knowledge:
The examination of derivatives is still given in the form of science 19 questions and liberal arts 20 questions. Starting with the commonly used functions, the functions (tangent and monotonicity) of derivative tools are comprehensively investigated, which requires high ability; It is often associated with the discussion of formulas, derivatives and parameters to examine the ability of transformation and reduction, but this year the overall difficulty is low.
7. Open innovation issues:
The answer is no, or logical reasoning questions, as well as the examination of open questions in the answer, are the key points, science 13, liberal arts 14 questions.
Summary and sharing of compulsory mathematics knowledge points of 2 inverse proportional function in senior one.
A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
The range of the independent variable x is all real numbers that are not equal to 0.
Inverse proportional function image properties:
The image of the inverse proportional function is a hyperbola.
Since the inverse proportional function belongs to odd function, there is f (-x) =-f (x), and the image is symmetrical about the origin.
In addition, from the analytical formula of inverse proportional function, it can be concluded that any point on the image of inverse proportional function is perpendicular to two coordinate axes, and the rectangular area surrounded by this point, two vertical feet and the origin is a constant, which is ∣k∣.
The function images when k is positive and negative (2 and -2) are given above.
When K>0, the inverse proportional function image passes through one or three quadrants, it is a decreasing function.
When k < 0, the inverse proportional function image passes through two or four quadrants, which is increasing function.
The inverse proportional function image can only move towards the coordinate axis infinitely, and cannot intersect with the coordinate axis.
Knowledge points:
1. Any point on the inverse proportional function image is a vertical line segment of two coordinate axes, and the area of the rectangle surrounded by these two vertical line segments and coordinate axes is |k|.
2. For hyperbola y=k/x, if you add or subtract any real number on the denominator (that is, y = k/(x m) m is a constant), it is equivalent to translating the hyperbola image to the left or right by one unit. (When adding a number, move to the left, and when subtracting a number, move to the right)
Summary and sharing of compulsory mathematics knowledge points in senior three 1, parity of 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));
6.a≥f(x) is considered as A ≥ [f (x)] max; A≤f(x) considers a ≤ [f (x)] min;
7 、( 1)(a & gt； 0，a≠ 1，b & gt0，n∈R+)； (2)log a N =(a & gt； 0，a≠ 1，b & gt0，b≠ 1)；
(3) 3) The symbol of LOGA B is memorized by the formula of "same positive but different negative"; (4) logarithm a n = n (a >; 0，a≠ 1，N & gt0);
8. When judging whether the correspondence is a mapping, grasp two points: (1) All elements in A must 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;
9. Skillfully use definitions to prove monotonicity of functions, find inverse functions and judge parity of functions.
10. For the inverse function, we should grasp the following conclusions: (1) The monotone function on the domain must have the 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; (5)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).
1 1, don't forget the combination of numbers and shapes when dealing with quadratic functions; 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;
12. According to monotonicity, the range of a class of parameters can be solved by using the sign-preserving property of linear functions on intervals.
13, work out the solution to the invariant problem:
(1) separation parameter method;
(2) Solving the inequality (group) of distribution table transformed into the root of quadratic equation in one variable;
Summary and sharing of compulsory knowledge points in senior one mathematics The general form of 4 logarithm function is that it is actually the inverse function of exponential function. Therefore, the stipulation of a in exponential function is also applicable to logarithmic function.
The figure on the right shows the function diagram of different size A:
You can see that the graphs of logarithmic functions are only symmetric graphs of exponential functions about the straight line y=x, because they are reciprocal functions.
The domain of (1) logarithmic function is a set of real numbers greater than 0.
(2) The range of logarithmic function is the set of all real numbers.
(3) The function always passes (1, 0).
(4) When a is greater than 1, it is monotone increasing function and convex; When a is less than 1 and greater than 0, the function is monotonically decreasing and concave.
(5) obvious logarithmic function.
Summary and sharing of compulsory knowledge points in senior one mathematics 5 1, "inclusive" relation-subset
Note: There are two possibilities.
(1)A is a part of B;
(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: A=B(5≥5 and 5≤5, then 5=5)
Example: let a = {x | x2-1= 0} b = {-1,1} "Two sets are equal if their elements are the same".
Namely: ① Any set is a subset of itself. Answer? A
② proper subset: If a? B and a? B then says that set A is the proper subset of set B, and it is denoted as AB (or BA).
3 if a? B，B？ C, then a? C
4 if a? 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.
A set of n elements, including 2n subsets and 2n- 1 proper subset.
4. Sets and elements
Whether a thing is a set or an element is not absolute, and it is often relative. A set is a set of elements, and elements are elements that make up a set. For example, your class is a collection of dozens of students of the same age, and you are an element of this class collection; The whole school is a collection of many classes, and your class is just one of them, an element. Class is relative to your collection, and it is relative to the elements of the school. Different references lead to different conclusions. It can be seen that neither the set nor the elements are absolute.
Knowledge point 2, the key to solving the set problem
The key to solve the problem of set is to find out which elements make up the set, that is, to concretize and visualize the abstract problem, to represent the set represented by feature description by enumeration, or to represent the abstract set by Wayne diagram, or to represent the set by graph, such as number axis, or the elements of the set are ordered real numbers, and to represent the related set by graph in plane rectangular coordinate system.
Summary and sharing of compulsory mathematics knowledge points of 6 basic elementary functions in senior one.
I exponential function
(A) the operation of exponent and exponent power
1, the concept of radical: generally speaking, if, then it is called n-th root, where >; 1, and∈
In odd numbers, the power root of a positive number is a positive number and the power root of a negative number is a negative number. At this point, the second root of is represented by a symbol. The formula is called radical, here it is called radical component, and it is called radical.
When it is an even number, a positive number has two power roots, and the two numbers are opposite. At this time, the positive power roots of positive numbers are represented by symbols, and the negative power roots are represented by symbols-. Positive and negative power roots can be combined into +(>: 0). It can be concluded that negative numbers have no even roots; Any power root of 0 is 0, which is recorded as.
Note: In odd numbers, even numbers,
2. Power of fractional exponent
The meaning of the power of the positive fractional index stipulates:
A positive fractional exponent power of 0 is equal to 0, and a negative fractional exponent power of 0 is meaningless.
It is pointed out that after defining the meaning of fractional exponent power, the concept of exponent is extended from integer exponent to rational exponent, and the operational nature of integer exponent power can also be extended to rational exponent power.
3. Operational Properties of Exponential Power of Real Numbers
(B) Exponential function and its properties
1, the concept of exponential function: Generally speaking, a function is called an exponential function, where x is the independent variable and the domain of the function is R.
Note: The base range of exponential function cannot be negative, zero 1.
2. Images and properties of exponential function
Summary and sharing of compulsory and compulsory mathematics knowledge points in senior one 7 knowledge points summary
The knowledge in this section includes monotonicity, parity, periodicity, maximum, symmetry and images of functions. Monotonicity, parity, periodicity, maximum and symmetry of functions are the basis of learning function images, and function images are their synthesis. So understand the previous knowledge points, and the image of the function will be solved.
First of all, the monotonicity of the function
1, the definition of monotonicity of function
2. Judgment and proof of monotonicity of function;
(1) definition method
(2) Complex variable function analysis method
(3) Derivative proof method
(4) Image method
Second, the parity and periodicity of the function
1, the definition of parity and periodicity of function
2. Methods to judge and prove the parity of functions.
3. The method of judging the periodicity of the function
Third, the function of image.
1, the method of function image
(1) stippling
(2) Image transformation method
2. Image transformation includes images: translation transformation, expansion transformation, symmetry transformation and folding transformation.
Common inspection methods
This part is an indispensable part of Duan and the college entrance examination, and it is the focus and difficulty of Duan and the college entrance examination. There are multiple-choice questions, fill-in-the-blank questions and solutions, and the questions are more difficult. In solving problems, we can combine each chapter of high school mathematics, mostly advanced questions. More attention should be paid to the monotonicity, maximum and image of the function.
Misunderstanding reminder
1. To find the monotone interval of a function, the domain of the function is required first, that is, the principle that the domain of the function problem takes precedence is followed.
2. Monotone interval must be expressed by interval, not by set or inequality. Monotone interval is generally written as an open interval, regardless of the endpoint problem.
3. Generally, the image as a function is to simplify the analytical formula first, and then determine the image as a function by chasing points or image transformation.
4. To judge the parity of a function, we should first consider the domain of the function. If the domain of a function is not symmetric about the origin, then the function must be a odd function or even function.