1. Understanding the meaning of elements in a set is the key to solving the set problem: is it clear that elements are the values of independent variables in functional relationships? Or the value of the dependent variable? Or a point on the curve? … ;
2. The combination of numbers and shapes is a common method to solve the set problem: when solving the problem, make use of tools such as number axis, rectangular coordinate system or Wayne diagram as much as possible to concretize and visualize the abstract algebraic problem, and then use the thinking method of combining numbers and shapes to solve it;
3. When sets A and B are known, have you noticed the "extreme" situation: or; Did you forget to find a subset of the set?
For example: (1) When finding the planting range of a for all constants, do you discuss the case of A = 2?
(2) If the set is known, the value range of the real number p is. ( )
4. For a finite set m with n elements, the number of subsets, proper subset, nonempty subset and nonempty proper subset is as follows.
5. Inverse law:,.
6. It is a subset of any set and a proper subset of any non-empty set.
7. The negation of "P and Q" is "non-P or non-Q"; The negation of "P or Q" is "non-P and non-Q"
8. The negation of proposition only negates the conclusion; No proposition is that conditions and conclusions are negative.
9. Several important properties of functions:
If a function has everything, then the image of the function is symmetrical about a straight line? Is an even function;
(2) If there are both, then the image of the function is symmetrical about a straight line; Functions and images of functions are symmetrical about a straight line; Special case: function and image of function are symmetrical about a straight line.
③ If a function has everything, then the function is a periodic function, and t = 2a.
If the function has everything, then the image of the function is symmetric about the point ().
⑤ Function and function image are symmetrical about a straight line; Functions and images of functions are symmetrical about a straight line; The function and the image of the function are symmetrical about the coordinate origin;
6. If odd function is the increasing function in the interval, it is also the increasing function in the interval; If an even function is a increasing function on an interval, it is a subtraction function on an interval;
⑦ Shift the image 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 function +a is obtained by translating the auxiliary image by one unit along the Y axis; The image of function +a is obtained by translating the auxiliary image down the Y axis by one unit.
⑨ Stretch the image of the function to the original image along the X axis to obtain the image of the function;
Attending the image of the function is obtained by stretching the image of the function to the original A times along the Y axis.
10. When finding the analytic expression of the function and the inverse function of the function, is the domain of the function marked?
1 1. Did you notice the value range of x when finding the maximum value of quadratic function?
Example: Given (x+2)2+ = 1, find the range of x2+y2. (because (x+2) 2 = 1 of (x+2) 2 =1≤1,∴-3≤x≤- 1, x2+y2 is in x =-/kloc-0. The range of x2+y2 is [1,])
12. A useful conclusion between a function and its inverse function: the intersection of the original function and the inverse function image is not all on y=x (for example:); It can only be understood as the function value at x+a.
13. If the original function monotonically increases in the interval, there must be an inverse function, and the inverse function also monotonically increases; But functions have inverse functions and are not necessarily monotonous. 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? Special circumstances:
14. What is the canonical format when the monotonicity of a function is proved by definition? (value, there is a difference, judging positive and negative. ) When studying monotonicity of functions with derivatives, we must pay attention to ">; 0 (or
15. Do you know the monotone interval of the function? (This function monotonically increases in or; This is a widely used function! Please review its special case "check function"
16. Remember that odd function y=f(x) defined on r must pass through the origin.
17. Monotonicity and parity of abstract functions must be solved by the definitions of monotonicity and parity. At the same time, we should understand the important method to prove the equation by using the inequality relation and the monotonicity of the function: f(a)≥b and f(a)≤b? f(a)=b .
18. When solving the logarithmic function problem, did you notice the restrictions on real numbers and bases? (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.
For example, if the range of the function is r, the range of the value of is. ( )
19. Have you mastered the logarithmic formula and its deformation? ( )
20. Do you remember the logarithmic identity? ( )
2 1 "A quadratic equation with real coefficients has a real number solution" is transformed into "",have you noticed the necessity? When a=0, "the equation has a solution" cannot be transformed into. If the original question does not point out that it is a "quadratic" equation, function or inequality, do you consider that the coefficient of the quadratic term may be zero? For example, have you discussed how to find the value range of A for everything when A = 2?
Example: (1) If the real number is constant, then "AND" is a necessary and sufficient condition for "arbitrary and having".
(2) Find the range of function y=
Solution: y = (y-1) x = 2y+1∴ y≠ 1 and x = ≦- 3, and the original function value of y ≠1is obtained.
(3) The equation 2kx2+(8k+ 1)x+8k=0 about x has two unequal real roots, so the value range of k is: k >；; -116 and k≠ 0
22 The important nature of arithmetic progression: If so, then; Arithmetic
Important properties of geometric series: if yes, then; Equal proportion.
Do you find that it is necessary to discuss the sum of the first n terms in different categories when using geometric series? ) Did you pay attention to the geometric series?
A property of arithmetic progression: Let it be the sum of the first n terms of a sequence, and it is arithmetic progression if and only if (a, b are constants), that is, Sn is a quadratic form of n without constant terms, and its tolerance is 2a.
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)
When using the formula to find the general term of a series, an is usually in the form of a segment, right? Have you noticed?
Do you remember the summation of split terms? (for example)
Superposition method:
Iterative multiplication:
Can you find the general formula of simple recursive sequence?
For example: (1) known, search;
(2) knowing and seeking;
(3) Unity of knowledge and action.
When solving trigonometric problems, have you noticed the definition domains of tangent function and cotangent function? Notice the boundedness of sine and cosine functions? At △ABC, Sina & gt Simbo? A>b, is that right? Example: Assuming that the straight line is the symmetry axis of the function image, the value of is. ( )
3 1 Generally speaking, if sine and cosine functions are added with absolute values or squares, their periods will be halved. (For example, the period of is 0, but the period of is 0). Note: the period of is 0.
Is the 32 function a periodic function? (Neither)
Do you know the symmetry axis and center of sine curve, cosine curve and tangent curve?
Do you know how much 1 equals in a triangle? (
These are collectively called 1 substitutions), and various substitutions of the constant "1" are widely used.
In the constant deformation of the triangle, we should pay special attention to various transformations of angles.
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)
Do you remember the formula of induction? (even if it changes suddenly, the symbol looks at the quadrant. What does parity mean? How to treat the quadrant where the angle is located? )
Do you remember the general method of triangle simplification? (Analyze the differences from three aspects: function name, angle and operation. The common skills are: cutting strings, multiplying the power formula, and transforming with trigonometric formula to generate special angles. Different angles are the same, different names are the same, high order and low order. )
Do you know some trigonometric function values of special angles?
( )
Remember the formula of arc length and sector area under arc system? ( )
4 1 auxiliary angle formula: It plays an important role in finding the maximum value and simplifying the corresponding angle.
The inverse trigonometric function is used to express the inclination angle of a straight line, the included angle between two vectors, 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 dihedral angles range from:
② The range of inclination angle, arrival angle and included angle of straight line is:
③ The included angle range of the vector is [0, π].
Example: Let the included angle satisfied by the vector be 600. If the angle between the vector and is obtuse, the range of the real number is.
What is the necessary and sufficient condition of "if, then"?
How to find the modulus of a vector? What is the projection in the direction?
45 If the angle with θ is obtuse, then cos θ
Do you remember the translation formula? This is the most basic method of translating problems. It can also be concluded that if the image with y=f(x) moves |h| units to the left and |k| units to the upward, the translation vector is =(-|h|, |k|).
What is the standard writing of the solution set of inequality 47? (generally written as an expression of a set)
What is the general solution of 48 fractional inequality? (Comprehensive score of transfer project)
Pay attention to the relationship between the solution set of inequality and the root of the corresponding equation.
How to remove the absolute value of an inequality with two absolute values? (Double-sided square or classified discussion)
When 5 1 uses important inequalities and variants to find the maximum value of a function, have you noticed the conditions when a, b (or a, b are nonnegative) and "equal sign holds"? Should the product ab or sum a+b be fixed values?
Example: It is known that the minimum value of is. ( )
How to discuss when solving inequalities with parameters? (especially the bottom sum of exponents and logarithms) After the discussion, it should be written: To sum up, the solution of the original inequality is ………………………………………………………………………….
The general method of solving inequality with parameters is "domain as the premise, function increase and decrease as the basis, and classification discussion as the key."
The usual solution of constant inequality problem: monotonicity of corresponding function, and its main skills are combination of numbers and shapes, separation of variables and method of substitution.
The two chapters of "straight line and circle" and "conic curve" in the textbook reflect that the essence of analytic geometry is to study the geometric properties of graphics by algebraic method.
Several forms of linear equations: point-oblique type, oblique section type, two-point section type, general formula and its limitations (for example, point-oblique type is not suitable for straight lines with no slope, so when setting point-oblique type or oblique section type equations, we should first consider the situation with no slope).
When setting the linear equation, the slope of the straight line can generally be set to K. Have you noticed that the slope K does not exist when the straight line is perpendicular to the X axis? (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. )
When solving the feasible region of simple linear programming problem, we should pay attention to whether the region represented by inequality is above and below the corresponding straight line, including the points on the boundary. Judging by special points).
59 pairs of two straight lines that do not coincide,, have
; .
The cross section of a straight line on the coordinate axis can be positive, negative or 0. Resolutely crack down on the argument that intercept is distance! )
6 1 lines have equal intercepts on the two coordinate axes, and the equation of the lines can be understood as, but don't forget that when a=0, the intercepts of the lines y=kx on the two coordinate axes are both 0 and equal.
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) The linear equation and the square equation are simultaneous, and the discrimination method is adopted. Generally speaking, the former is simpler.
To deal with the positional relationship between circles, we can use the relationship between the center distance and radius of two circles.
In a circle, pay attention to the right triangle composed of radius, half chord length and chord center distance.
What is the coordinate formula of 65 fixed fraction point? (The starting point, midpoint, equinox and numerical value should be clear) Did you notice when you solved the problem with the fixed point?
In analytic geometry, when studying the positional relationship between two straight lines, it is possible that these two straight lines overlap; Generally speaking, the two straight lines mentioned in solid geometry can be understood as non-coincidence (the two planes are also non-coincidence by default, but the lines are not coincident in the plane and cannot be ignored); Vector lines are parallel.
Do you know the equation of curve system? Linear system equation? Equation of circle system? The focus of the elliptic system and the asymptote of the hyperbolic system?
The common chord equation obtained by the intersection of two circles is obtained by subtracting the quadratic term from the equation of two circles. X0x+y0y=r2 represents the tangent of the point (x0, y0) on the circle x2+y2=r2. If the point (x0, y0) is outside the known circle, what does x0x+y0y=r2 stand for? (Tangent chord)
The three parameters A, B and C in the elliptic equation satisfy a2+b2=c2, right? What relation should the three parameters in hyperbolic equation satisfy?
Note that the length of the major axis in the ellipse is 2, not.
In an ellipse, pay attention to the right triangle consisting of the focus, the center of the circle and the endpoint of the short axis.
Remember the focal radius formula of ellipse and hyperbola?
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.
When solving problems with the unified definition of conic section, did you notice the order of numerator and denominator in the definition?
When solving with conic curve and straight line at the same time, we should pay attention to the equation obtained after elimination: Is the coefficient of quadratic term zero? Limitations of discriminant (finding intersection, chord length, midpoint, slope, symmetry and existence are all carried out below).
Path 75 is the shortest chord of all focus chord of parabola.
76 parabola y2 = 2px(p & gt；; 0) If the chord parabola of the focus is a (x 1, y 1) and b (x2, y2), then the focal radius formula | ab | = x1+x2+p.
77 If A (X 1, Y 1) and B (X2, Y2) are the two endpoints of the chord of the quadratic curve C: F (X, Y) = 0, then F(x 1, y 1)=0 and F (X2 When the midpoint and slope of a chord are involved, the point difference method is often used as f (x 1, y 1)-f (x2, y2) = 0 to find the relationship between the midpoint coordinates of chord AB and the slope of chord AB.
78 what are the main methods to make the plane angle of dihedral angle (definition method, three perpendicular theorem method, vertical plane method)
Do you know what is the key of the three vertical theorems? There are four straight lines on one side, and the vertical line is the key, so it is called three vertical lines.
What is the conventional method to find the distance from a point to a surface? (direct method, volume transformation method, vector method)
8 1 The key to finding the spherical distance between two points is to find the spherical central angle.
Some common conclusions in solid geometry: the height of a regular tetrahedron with a side length of V=.
83 area projective theorem, in which both the projective area and the original area are expressed.
When using "translation method" to solve the angle formed by straight lines on different planes, we must pay attention to whether the angle obtained after translation is an angle or its complement.
Pay attention to the "invariance" and "invariance" of geometric elements before and after folding and unfolding.
When is the projection of the prism vertex on the bottom surface the inner center, outer center, vertical center and center of gravity of the bottom surface?
The laws to solve the permutation and combination problem are: element analysis, position analysis-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; First use the back row method to select the topic; At least the most problems, indirect method.
Are the term with the largest coefficient, the maximum coefficient of term and the maximum binomial coefficient of term the same concept in binomial theorem?
Can we use "assignment method" and "transformation method" when solving the algebraic sum of the coefficients of binomial expansion, and can we use "general method" and "structural analysis method" when solving specific items?
90 pay attention to some characteristics of binomial (such as; )。
9 1 To master the derivative, monotonicity, extreme value and maximum value of polynomial function.
What are the applicable conditions of formulas P(A+B)=P(A)+P(B) and P(AB)=P(A)P(B)?
The similarity between simple random sampling and stratified sampling is that the probability of each individual being drawn is equal.
94 =0 is a necessary and sufficient condition for the function y=f(x) to have an extreme value at x=x0.
Note that the derivative value of a point on the curve is the slope of the tangent. (Geometric meaning of derivative)
Understand variance and standard deviation.
97. Remember the common probability formula?
Example 1: Throw two dice and find the probability that the sum of points is 6.
When the sum of points is 6, there are (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) * *, so the probability of "the sum of points obtained is 6" is p =.
Example 2: A's shooting percentage is 0.8, B's shooting percentage is 0.7, and each person throws the ball three times. What is the probability that both people hit twice?
Misunderstanding that "A hits twice" is event A and "B hits twice" is event B, then hitting both twice is event A+B, and p (a+b) = p (a)+p (b);
The wrong reason for analyzing this problem is that independent simultaneous events are regarded as mutually exclusive events, and both of them are understood as the sum of "A hits twice" and "B hits twice".
Correct answer: Let "A hits twice" be event A, and "B hits twice" be event B. A and B are independent of each other, so both of them hit twice as event A? B, then
P(A？ B)=P(A)×P(B)=。
Example 3: When someone is at home, the probability of an incoming call being answered is 0. 1 at the first ring, 0.3 at the second ring, 0.4 at the third ring and 0. 1 at the fourth ring. What is the probability that the phone will be answered within the first four rings?
Wrong solutions are recorded as events A 1, A2, A3, A4 and P(A 1)=0. 1.
P (A2) = 0.3, P (A3) = 0.4, P (A4) = 0. 1, so the probability that the phone is answered within the first four rings is P=P(A 1)? P(A2)？
P(A3)？ p(A4)= 0. 1×0.3×0.4×0. 1 = 0.00 12。
The reason for the wrong solution of this question is that mutually exclusive events is regarded as an independent and simultaneous event. According to the experience in real life, whether each call in the first four rings is mutually exclusive or not, so P=P(A 1) ten p (a2)+p (a3)+p (a4) = 0.1+0.3+0.
What is the special solution to multiple-choice questions? (Forward deduction method, estimation method, special case method, feature analysis method, intuitive selection method, reverse deduction method, etc. )
What should we pay attention to when solving the fill-in-the-blank problem? (specialization, diagram, equivalent deformation)
100 what are the most basic requirements for answering application questions? (Examining questions, identifying keywords in questions, setting unknowns, listing functional relationships, substituting initial conditions, indicating units, and answering)
10 1 when answering open-ended questions, we need to think extensively and comprehensively, and penetrate knowledge vertically and horizontally.
102 When answering informative questions, a thorough understanding of the new information in the questions is the prerequisite.
103 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, separation strategy, concentration strategy, elimination strategy, substitution strategy and anti-objectivity strategy are the general methods to solve this kind of problem. )
104 The common methods for solving trajectory equations are: direct method, undetermined coefficient method, definition method, transfer method (correlation point method), parameter method, etc.
105 because the college entrance examination is marked by computer, we must try our best to make the handwriting neat and the paper clean, and remember to answer the questions in the designated area.
106 keeping a good attitude is the key to playing the college entrance examination well!