A, set, simple logic, function
1. When studying sets, we must pay attention to the characteristics of set elements, that is, three properties (certainty, mutual difference and disorder); Known set A={x, xy, lgxy}, set
B = {0, | x |, y}, A=B, then x+y=
2. To study a set, we must first understand the representative elements in order to understand the meaning of the set. Given the set m = {y | y = x2, x ∈ r}, n = {y | y = x2+ 1, x ∈ r}, find m ∩ n; Find the difference between m = {(x, y) | y = x2, x ∈ r}, n = {(x, y) | y = x2+ 1, x ∈ r}.
3. When assembling A and B, did you notice "extreme" situations: or; When you find a subset of a set, do you forget it? For example, when finding the constant planting range of A, have you ever discussed the situation that A = 2?
4. For a finite set m with n elements, the number of subsets, proper subset, nonempty subset and nonempty proper subset is the number of the set M*** that meets the conditions in turn.
5. The basic tool to solve the set problem is Wayne diagram; There are * * members of a literary and art group 10, each of whom can sing and dance at least, of which 7 can sing and dance well and 5 can sing and dance well. Now choose a person who can sing and dance and perform a song and dance program. How many different ways are there?
6. The relationship between two sets.
7.(CUA)∩(CU B)= CU(A∪B)(CUA)∩(CUB)= CU(A∪B)； ;
8. A statement that can be judged true or false is called a proposition.
Logical conjunctions include or, and right and wrong.
The truth table of compound proposition in p and q forms;
P q P and q P or q
Really, really, really.
True or false.
False, true, false, true
Fake, fake, fake.
9. Four forms of propositions and their relationships.
If the original proposition is p, then q
If q is the inverse proposition of p
There is no proposition if ﹃p and then ﹃ ask.
If the negative proposition is ﹃q ﹃p
Inverted easily
Interaction
Interaction
No, no, no, no.
No, no.
No, no.
There is no reciprocity.
The original proposition and the negative proposition are the same as true and false; The truth and falsehood of an inverse proposition are the same.
10, do you understand the concept of mapping? Mapping F: In A → B, what kinds of corresponding relationships can be mapped between the arbitrariness of elements in A and the uniqueness of corresponding elements in B?
Several important properties of the function 1 1;
① If the function has everything or f(2a-x)=f(x), the image of the function is symmetrical about a straight line.
② Function and function image are symmetrical about a straight line;
Functions and images of functions are symmetrical about a straight line;
The function and its image are symmetrical about the origin of coordinates.
(3) If odd function is the increasing function on the interval, then it is also the increasing function on the interval.
④ If an even function is a increasing function in an interval, it is a decreasing function in an interval.
⑤ Shift the image of the function by one unit to the left along the X axis to get the image of the function; The image of the function (obtained by translating the image of the function to the right along the X axis;
The image of the function +a is obtained by translating the auxiliary image of the function by one unit along the Y axis. The image of the function +a is obtained by translating the auxiliary image of the function by units along the Y axis.
12. When finding the analytic expression of the function, is the domain of the function marked?
13. Do you remember the common types of function domains? The domain of function y= is;
Is the domain of compound function clear? The domain of the function is [0, 1], and the domain of the function is []. Find the domain of the function.
14. Remember to discuss the range and maximum value of quadratic function with parameters. If the minimum value of the function y=asin2x+2cosx-a-2(a∈R) is m, find the expression of m.
17. When judging the parity of a function, have you noticed the necessary and sufficient conditions for whether the domain of the function is symmetric about the origin? In the public domain: the product of two odd function is an even function; The product of two even functions is an even function; The product of odd function and even function is odd function;
18. What is the canonical format when the monotonicity of a function is proved by definition? Don't forget that derivative is also an important method to judge the monotonicity of a function.
19, do you know the monotone interval of the function? (This function monotonically increases in sum; This is a widely used function!
20. When solving the logarithmic function problem, did you notice the limitation of real number and base? (The true number is greater than zero, and the radix is greater than zero and not equal to 1) The letter radix needs to be discussed.
Have you mastered the formula of 2 1 and logarithm and its deformation? ( )
22. Do you remember the logarithmic identity? ( )
23. "Quadratic equation with real coefficients has real solution" is converted into "". Have you noticed the necessity? When a=0, "the equation has a solution" cannot be transformed into. If the original question does not indicate that it is a "quadratic" equation, function or inequality, do you consider the situation that the coefficient of the quadratic term may be zero?
Second, triangle, inequality
24. Do you remember the triangle formula? The sum and difference formula of two angles _ _ _ _ _ _ _ _ _ _ _ _ _; Double angle formula: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ The basic skills are: changing the angle skillfully, using the formula to deform, cutting into strings, and using the double angle formula to reduce the higher order.
25. When solving the trigonometric problem, did you notice the domain of tangent function and cotangent function? Is the tangent function monotonic in the whole region? Notice the boundedness of sine and cosine functions?
26. Do you know how much 1 equals in a triangle? (
These substitutions of the constant "1" have a wide range of applications. (There are also formulas for the same angle relationship: quotient relationship, reciprocal relationship and square relationship; Inductive public examination: parity remains unchanged, symbols look at quadrants)
27. In the constant deformation of a triangle, special attention should be paid to various transformations of angles (such as. )
28. Do you remember what the requirements of triangulation are? The formula with the least number of items, the least function types, the denominator without trigonometric function, and the value can be found must calculate the value)
29. Do you remember the general method of triangle simplification? (chord cutting, power reduction formula, transformation with triangle formula, special angle appears. Different angles are the same, different names are the same, high order and low order); Do you remember the formula of decreasing power? cos2x =( 1+cos2x)/2； sin2x=( 1-cos2x)/2
30. Remember some trigonometric functions with special angles?
( )
3 1, do you still remember the arc length formula and the sector area formula under the arc system? ( )
32. Auxiliary angle formula: (in which the quadrant of the angle is determined by the symbols of A and B, and the value of the angle is determined by) plays an important role in finding the maximum value and simplification.
33. Can the sketch of trigonometric functions (sine, cosine and tangent) be drawn quickly? Can you write their monotone region, symmetry axis and the set of x values when taking the maximum value? (don't forget k Z)
Remember the properties of trigonometric functions. Images and properties of function y= k;
Amplitude |A|, period T=, if x=x0 is the symmetry axis of this function, x0 is the point that makes y get the maximum value, on the contrary, the set of x that makes y get the maximum value is ————————————————————————.
Five-point drawing method: find X and Y in sequence and draw point by point.
34, trigonometric function image transformation, remember?
Translation formula (1) If the point P(x, y) is translated to p ′ (x ′, y ′) by vector, then
(2) The equation after the curve f(x, y)=0 is translated along the vector is f(x-h, y-k)=0.
35. Some conclusions about oblique triangles: (1) Sine theorem: (2) Cosine theorem: (3) Area formula.
36. When using the inverse trigonometric function to express the inclination angle of a straight line and the angle formed by two straight lines in different planes, have you noticed their respective range and significance?
① The angles formed by straight lines on different planes, angles formed by straight lines and planes, and included angles of vectors are in the range of.
② The range of inclination angle, included angle and included angle of a straight line is.
(3) The values of the sine, cosine and tangent functions are as follows.
37. Can the same inequality be subtracted and divided?
38. What is the standard writing format of inequality solution set? (generally written as an expression of a set)
39. What is the general idea of solving fractional inequality? (When the term is moved to general division, the numerator and denominator decompose the factor, and the coefficient of X becomes positive, which is odd and even)
40. What problems should we pay attention to when solving inequalities? Monotonicity of exponential function and logarithmic function, and the real number of logarithm is greater than zero.
4 1. How to find the absolute value of an inequality with two absolute values? (Generally, discussions are classified by definition. )
42. When finding the maximum value of a function with important inequalities and variants, have you noticed the conditions that A, B (or A, B are non-negative) and "equal sign holds", and one of the products ab or a+b should be a constant value? (One positive, two fixed, three phases, etc. )
43. (If and only if, take the equal sign); A, b, c R, (if and only if, take the equal sign);
44. How to discuss when solving inequalities with parameters? (especially the bottom sum of exponent and logarithm) After discussion, write: To sum up, the solution set of the original inequality is ...
45. The general method to solve inequality with parameters is "domain as the premise, function increase and decrease as the basis, and classification discussion as the key."
46. What are the commonly used methods to deal with the problem of constant inequality? (becomes a maximum problem)
Third, the order
47. The important nature of arithmetic progression: (1) If, then; (2) ;
(3) If three numbers are arithmetic progression, they can be set as a-d, A, A+D; If it is four numbers, it can be set to a-, a-, a+, a+;
(4) In arithmetic progression, the idea of finding the maximum (minimum) value of Sn is to find an item, so that both this item and the item before it take a positive (negative) value or 0, and the following items take a negative (positive) value, then the sum of the items from the first item to this item is the maximum (minimum). That is, when A 1 >: 0, d<0, the value of n when Sn reaches the maximum can be obtained by solving the inequality group an ≥0 an+ 1 ≤0; When a 1
48. Important properties of geometric series: (1) If, then; (2),, into a geometric series
49. Have you found that when applying geometric series to find the sum of the first n terms, it needs to be discussed in different categories. When?)
50. A geometric series summation formula: let the sum of the first n terms of the geometric series be, and the common ratio be, then
.
5 1, a property of arithmetic progression: Let it be the sum of the first n terms of a sequence, and it is a arithmetic progression if and only if
(a, b are constants) The tolerance is 2a.
52. Do you know how to use the method of "dislocation subtraction" when summing series? (If, among them, arithmetic progression and geometric progression are the sum of the top n items)
53. Have you ever noticed that when you use the general formula to find the sequence?
54. Do you remember the summation of split terms? (for example. )
Fourth, permutation and combination, binomial theorem
55. The basis for solving the problem of permutation and combination is: classification addition, step-by-step multiplication, orderly arrangement and disorderly combination.
56. The laws to solve the permutation and combination problem are: adjacent problem binding method; Interpolation method for non-adjacent problems: single-line method for multi-line problems; Positioning problem priority method; Classification of multivariate problems; Ordered distribution problem method; Select a question first, and then return; At most, at least indirectly. Do you remember when the zoning method was used?
57. The formula of permutation number is: the formula of combination number is: the relationship between permutation number and combination number is:
Combined number attribute: =+= =
Binomial theorem:
General formula of binomial expansion:
Verb (abbreviation of verb) solid geometry
58. The proof of parallelism and verticality is mainly proved by the transformation of line-plane relationship: line//line//plane//plane, line ⊥ line ⊥ plane ⊥ plane, vertical common vector.
59. What is the main method to make the plane angle of dihedral angle? Three verticality method: a plane, two vertical lines and three diagonal lines, the projection is visible.
60. The dihedral angle solutions mainly include: right triangle, cosine theorem, projective area method and normal vector.
6 1. What is the conventional method to find the distance from a point to a surface? (direct method, equal volume transformation method, normal vector method)
62. Do you still remember the three vertical theorems and their inverse theorems?
63. The solution of the spherical distance between two points on a sphere is mainly to find the angle of the center of the sphere, which is often associated with latitude and longitude. Do you remember the meaning of longitude and latitude? (Longitude is the face angle; Latitude is the angle between a line and a plane)
64. Remember the Euler formula of simple polyhedron? (V+F-E=2, where V is the number of vertices, E is the number of edges, and F is the number of faces), do you still remember the two algorithms of edges? (① If every face of a polyhedron is an N polygon, then E =；; (2) If every vertex of a polyhedron has m edges, then E=)
Six, analytic geometry
65. When setting the linear equation, the slope of the straight line can generally be set to K. Have you noticed that when the straight line is perpendicular to the X axis, the slope K does not exist? (For example, if a straight line passes through a point and the chord length cut by a circle is 8, find the equation of the straight line where this chord is located. Pay attention to this problem and don't miss the solution of x+3=0. )
66. What is the coordinate formula of fractional points? (You can specify the starting point, midpoint, equinox and value)
Coordinate formula of fixed point of line segment
Let P(x, y), P 1(x 1, y 1), P2(x2, y2), and then
Midpoint coordinate formula
If so, the coordinates of the center of gravity g of △ABC are.
67. Did you notice when solving problems with fixed fractions?
68. In analytic geometry, when studying the positional relationship between two straight lines, it is possible that the two straight lines coincide, while in solid geometry, the two straight lines can generally be understood as non-coincidence.
69. Several forms of linear equations: point inclination, oblique section, two-point section, generality and its limitations (for example, point inclination is not applicable to straight lines without slope).
70. For two non-overlapping straight lines, there are
; .
7 1, the cross section of the straight line on the coordinate axis can be positive, negative or 0.
72. The intercept of a straight line on two coordinate axes is equal, and the equation of a straight line can be understood as, but don't forget that when a=0, the intercept of a straight line y=kx on two coordinate axes is 0 and equal.
73. The distance formula of the sum of two straight lines is D = ——————————
74. Do you remember the direction vector of a straight line? What is the relationship between the direction vector of a straight line and its slope? When the direction vector of the straight line L =(x0, y0), the slope of the straight line K = —————; When the slope of a straight line is k, the direction vector of the straight line = —————
75. When will the formula of azimuth angle and included angle be used?
76. There are two ways to deal with the positional relationship between a straight line and a circle: (1) the distance from a point to a straight line; (2) Linear equation and circular equation are simultaneous and discriminant. Generally speaking, the former is simpler.
77. To deal with the positional relationship between circles, we can use the relationship between the center distance and radius of two circles.
78. In a circle, pay attention to the right triangle composed of radius, half chord length and chord center distance, and think more about the geometric properties of the circle.
79. When using conic to define and solve problems in a unified way, did you notice the order of numerator and denominator in the definition? The two definitions are often used together, which is sometimes very helpful for us to solve problems. It may be more convenient to use the second definition for the problem of focus chord. (formula of focal radius: ellipse: | pf1| = ————; | PF2 | =————； Hyperbola: | pf1| = ————; | pf2 | = —— (where F 1 is the left focus and f2 is the right focus); Parabola: |PF|=|x0|+)
80. When solving a conic curve and a straight line at the same time, we should pay attention to whether the coefficient of the quadratic term in the equation obtained after elimination is zero. Limitations of discriminant. (Finding the intersection point, chord length, midpoint, slope, symmetry and existence are all carried out below).
8 1, in an ellipse, the relationship between a, b and c is-; Eccentricity e = ———; The alignment equation is ————; The distance from the focus to the corresponding directrix is-in hyperbola, the relationship between A, B and C is-; Eccentricity e = ———; The alignment equation is ————; The distance from the focus to the corresponding directrix is-
The path is the shortest chord of all focus chord of a parabola.
83. Do you know? The key to solving analytic geometry problems is to algebra the geometric conditions in the topic, especially some obscure conditions, which sometimes play a key role, such as points on curves, intersections, * * * lines, circles passing through a point with the diameter of a line segment, angles, perpendicularity, parallelism, midpoint, bisector of angles, midpoint chords and so on. Don't forget the parametric equations of circles and ellipses, sometimes it is very convenient to solve problems. The combination of numbers and shapes is an important way of thinking to solve several problems. Remember to draw and analyze!
84. Have you noticed? There is a difference between finding trajectory and finding trajectory equation. Don't forget to find the range when solving the trajectory equation!
85. When solving the application problem of linear programming, there are the following steps: first, find the constraints, make the feasible region and define the objective function. The key is to find out the geometric meaning of the objective function, and pay attention to changing the coefficient of y in the linear equation to a positive value when finding the feasible region. Seek 2.
Seven, vector
86. Remember the condition that two vectors are parallel lines or * * * lines? There are two forms to express it? Attention is a necessary and sufficient condition for vector parallelism. (Definition and coordinate representation)
87. Vector can solve problems such as included angle, distance, parallelism and verticality. Remember the following formula: || 2 =,
cosθ=
88. Using vector parallelism or verticality to solve the problems of parallelism and verticality in analytic geometry does not need to discuss the situation that slope does not exist. It should be noted that the vector included angle is a necessary condition, but not a sufficient condition.
89. The operation of vectors should be different from the operation of real numbers: if one vector cannot be omitted on both sides, the multiplication of vectors will not satisfy the associative law, that is, remember that two vectors cannot be divisible.
90. Remember the geometric meaning of the basic theorem of vectors? Its essence is that any vector on the plane can be expressed linearly by two vectors of any line on the plane. Do you know the meaning and solution of its coefficient?
9 1, the sum of vectors formed by the end-to-end connection of closed graphs is zero, which is a natural condition in the topic. Pay attention to the application. For a vector equation, we can shift the term, multiply the two sides by a real number, and take the modulus at the same time, multiply the two sides by a vector, but we can't divide the two sides by a vector.
92. Cartesian coordinate operation of vector
That is all right
Let A=, B=,
Then-=
Eight. derivative
93, the geometric meaning of derivative is the slope of the tangent of the curve at this point, learn to define all kinds of deformation.
94. Derivatives of several important functions: ①, (c is a constant) ②.
Four algorithms of derivative
95. The monotonicity of a function can be proved or judged by using derivatives. Note that when f '(x)≥0 or f '(x)≤0, there is an equal sign.
96.(x0)=0 is an insufficient and unnecessary condition for the extreme value of function f(x) at x0. What is the necessary and sufficient condition for f(x) to take the extreme value at x0?
97. Steps to seek the maximum value of derivative: (1) Seek the derivative (2) Seek the root of equation =0.
(3) Calculate extreme value and endpoint function value.
(4) Determine the maximum and minimum values according to the above values.
98. The method of finding the extreme value of a function: first find the domain, then find the derivative, find the boundary point of the domain, and find the extreme value according to monotonicity. Telling the extreme value of a function is equivalent to giving two conditions: ① the derivative value of the function at this point is zero, and ② the value of the function at this point is fixed.
Nine, probability statistics
99. How to find the probability of an event: convert the required event into an equal probability of possible events (often using the knowledge of permutation and combination), into the probability of one of several mutually exclusive events events, and convert the probability of opposing events into the probability of simultaneous occurrence of independent events, which is regarded as the probability that an event happens exactly k times in n experiments, but we should pay attention to the conditions of using the formula.
1) If events A and B are mutually exclusive events, then
P(A+B)=P(A)+P(B)
(2) If events A and B are independent events, then
P(A B)=P(A) P(B)
(3) If events A and B are opposite events, then
P(A)+P(B)= 1
Generally speaking,
(4) If the probability of an event occurring in an experiment is p, then it happens exactly k times in n independent repeated experiments.
100, sampling methods mainly include: simple random sampling (lottery method, random sampling table method) is often used when the population is small, and its main feature is to extract one by one from the population; Systematic sampling is often used when the total number is large, and its main feature is that it is evenly divided into several parts, and only one copy is taken from each part; The main characteristics of stratified sampling and stratified proportional sampling are mainly used for people with obvious differences. The same feature of them is that the probability of each individual being drawn is equal.
10 1. The method of estimating samples according to the population is to take the frequency of samples as the probability of the population.
X. Methods and skills to solve problems
102, the overall test-taking strategy: easy first, then difficult, generally do multiple-choice questions first, then fill in the blanks, and finally do big questions. Multiple choice questions strive to ensure speed and accuracy and save time for the big questions behind, but accuracy is the premise. For the fill-in-the-blank problem, it seems that there is no idea or the calculation is too complicated to give up. For big problems, try not to leave blank. It is possible to score points by transforming the conditions in the questions into algebra, and learn to give up and get rid of them in the exam.
103. What's the special way to answer multiple-choice questions? (Forward deduction method, estimation method, special case method, feature analysis method, intuitive selection method, reverse deduction method, combination of numbers and shapes, etc.). )
104. What should I pay attention to when answering the fill-in-the-blank questions? (specialization, diagram, equivalent deformation)
105. What are the most basic requirements when answering application questions? (Examining questions, identifying keywords in questions, setting unknowns, listing functional relationships, substituting initial conditions, indicating units, and answering)
106, when answering open-ended questions, you need to think extensively and comprehensively, and penetrate knowledge vertically and horizontally.
107, when answering knowledge questions, thoroughly understand the new information in the questions, which is the premise of accurate problem solving.
108, when solving the multi-parameter problem, the key is to extract the parameter variables properly and try to get rid of the entanglement of the parameter variables. Among them, the strategy of separation, concentration, elimination, substitution and anti-subjectivity of parameter variables seems to be the general method to solve this kind of problem.
109. Learn the skills of scoring jump shots. The first question can't be answered, but the second question can be answered. When using the first question, you can directly use the conclusion of the first question. You should learn to connect with such languages as "from what is known", "from the meaning of the problem" and "from the knowledge of plane geometry". Once you want to come, you can write "supplementary proof" at the back.