Representation of sets: description, enumeration, interval, (Venn icon)
Prove simple logic by reducing to absurdity
1. For the set problem, determine which set (number set, point set or graph row set) it belongs to.
Note: Angle sets and angle sets cannot be represented by intervals, and intervals can only represent number sets.
2. Simplify the set operation before calculating.
3. The set problem with parameters should be treated according to the anisotropy of the elements in the set, and sometimes it should be discussed in categories and combined with numbers and shapes.
4. Set questions are mostly related to functions, equations and inequalities, and should be used together with other knowledge.
5. Pay attention to some sentences in the set questions, such as: all but not all, and some at will.
Note: even function+even function = even function (proved by definition)
Supplement: (1) An empty set is a subset of any set and a proper subset of any non-empty set.
(2) Any set is a subset of itself.
(3)Cu(A∪B)=(CuA)∪( CuB)Cu(A∪B)=(CuA)∪( CuB)
Theme dual function
Three elements of function: domain, range and corresponding rules.
Methods of expressing functions: tables, images and analytical expressions.
Prove monotonicity and parity of function by definition
Odd function is symmetric about the origin and even function is symmetric about y.
The properties of functions include: domain, parity, monotonicity and periodicity.
* concrete analysis of concrete problems of abstract functions
The range problem of 1. function often boils down to the problem of finding the maximum value of the function, and attention should be paid to using basic inequalities, quadratic functions and monotonicity of functions. When looking for the range of function values, we should pay attention to the role of correspondence law, especially the restrictive role of definition range, and also pay attention to other restrictive conditions, that is, we should pay special attention to the whole: the given interval of quadratic function
2. Analysis method:
(1) introduces appropriate variables, which are applicable to practical problems (application problems), that is, ````` modeling''.
(2) undetermined coefficient method
(3) Alternative methods
(4) Equation solving method: According to the known equation, construct other equations, form equations, and find out f(x).
3. Judging monotonicity:
(1) definition method
*(2) increase+increase = increase, decrease-decrease = decrease.
Increase-decrease = increase-decrease-increase = decrease.
(3) Odd sameness and even antithesis
(4) Reciprocal functions have the same monotonicity.
(5) If f(-x) is an increasing (decreasing) function in the interval d, then f(x) is also an increasing (decreasing) function in any subinterval of d..
(6) The same increase but different decrease
Judgment equivalence
(1) Mining the periodicity and parity of functions in solving problems provides convenience for solving problems.
(2) Simplify the definition of function reuse.
f(-x)= f(x)←→f(-x)f(x)= 0←→f(-x)/f(x)= 1[f(x)≠0]
☆ Note: If it is odd function, it is either (0,0) or (0,0).
When solving function problems, if you encounter difficulties, you can consider the following two methods:
(1) If it is difficult, it will be reversed.
(2) Separation of variables
The idea of mutual transformation of quadratic function, quadratic equation and quadratic inequality is used to solve various comprehensive problems such as extreme value problem, root distribution problem, inequality problem and application problem.
1. For the function f(x)=a(x-h)? For the maximum value of +k (a > 0) x ∈[p, q], it is best to use the mirror image method, especially in the case of ```fixed variable axis interval' and ``variable axis interval change', both of which use the mirror image as a reference. It is found that discussion is the standard of classification. In order to solve this situation, the image can be omitted. if
2. The problem that f(x)≥0 is constant in the interval [p, q] is equivalent to the problem of the minimum value of f(x) in [p, q]. The classic way to solve this problem is to separate variables.
3. When the quadratic coefficient is negative, it should be converted into positive. When solving a quadratic equation with one variable, images are often used when one of the roots has restrictive conditions.
4. Pay attention to the inverse problem when solving the unary quadratic inequality, especially the equivalent proposition when the solution set of the unary quadratic inequality is empty and R: ax? The solution set of +bx+c > 0 is r ←→ {a > o, △ < 0 or {a = b = o, c > 0.ax? The solution set of +bx+c < 0 is r ←→ {a
Some problem-solving skills:
When solving quadratic inequality: the first step is to consider delta, symmetry axis and sometimes vieta theorem.
The second step is to observe the constraints of the domain and the value domain.
Note: When there are real roots in an interval, the product of two values in the interval is negative, which is called interval method.
When there are parameters in the letter, pay attention to the classified discussion.
When several solutions are found, it is necessary to verify whether each solution meets the requirements.
When exponential function and logarithmic function appear, pay attention to the special requirements for letters, and sometimes method of substitution can be used (note that parity and monotonicity can be used to judge).
Attention should also be paid to details in topic setting.
The base of 1. exponential function is greater than 0 and not equal to 1. This is an implicit condition.
2. Exponential function A > has a cardinality of 1, which is an increasing function. When 0
*3. When comparing the sizes of two exponential powers:
(1) Transform the same basis or parameter: When the basis and parameter are different, the same exponential function is constructed, and the ratio is large.
When the same basis is different, two exponential functions are constructed and the mirror ratio is used.
(2) By finding the intermediate ratio.
4. When solving simple exponential inequality, when the base is a parameter and the size of the base and 1 is uncertain, it should be discussed in different categories.
5. The basic method for comparing the sizes of two logarithms is:
(1) to construct the corresponding logarithmic function.
[(2) Establish the same base as the exchange base ㏒ab=㏒eb/㏒ea]
(3) Pay attention to the comparison with 0 or 1
6. Pay attention to the use of separation variables
7. The basic idea of solving logarithmic equation is to transform it into algebraic equation. Its common types are:
The form of (1) is logaf (x) = logag (x) (a >; 0, a≠ 1) can be solved by f(x)=g(x).
(2) Substitution method is applied to the equation of F(logax)=0.
(3) Use the exponential formula [f(x)∧c=g(x) to solve the logf(x)g(x)=c equation.
Note: Sometimes it is helpful to convert exponential equation and logarithmic equation to each other.
*8. Pay attention to the root test when solving logarithmic equations.
9. When solving exponential and logarithmic equations with parameters, pay attention to converting the original equations into mixed groups equivalently, simplify the solution under the principle of equivalent conversion, and discuss the parameters.
10. Exponential and logarithmic equations belong to transcendental equations, so we should pay attention to transformation.
The inflection point is the change of running trend or running speed in the development of things.
In mathematics, it refers to the connection point between convex curve and concave curve! !
When a point on the function image makes the second derivative of the function zero and the third derivative non-zero, this point is the inflection point of the function.
In life, inflection point is often used to indicate that a certain situation continues to rise for a period of time and then begins to decline or fall back.
Therefore, with the economic turning point, lower the long turning point and the stock market turning point.
If the function y=f(x) is derivable at point C, and point C is convex on one side and concave on the other, then C is called the inflection point of function y=f(x). In addition, if c is an inflection point, there must be f''(c)=0 or f''(c) does not exist; On the contrary, it is not established; For example, if f (x) = x 4, there is f''(0)=0, but both sides of 0 are convex, so 0 is not the inflection point of the function f (x) = x 4.
Solution of inflection point: We can judge the inflection point of continuous curve y=f(x) in interval I according to the following steps:
(1) Find f'' (x);
(2) Let f''(x)=0, find the real root of this equation in interval I, and find the point where f''(x) does not exist in interval I;
(3) For each point x0 where there is no real root or second derivative, check the symbols f''(x) on the left and right sides of x0, then when the symbols on both sides are opposite, the point (x0, f(x0)) is an inflection point, and when the symbols on both sides are the same, the point (x0, f(x0)) is not an inflection point.
The connection point of the bump is the inflection point.
Functions with inflection points are inflection points.
If the analytic expression of a function contains an absolute value sign, the function can be turned into a piecewise function. The common solution is to treat the functions on each segment as independent functions, find out their monotonous intervals respectively, and then integrate them together, but it should be noted that the monotonous intervals of piecewise functions must be within their definition domain.
When judging the maximum value of the function by intuition of the quadratic function image, it is necessary to determine whether the open direction and symmetry axis of the quadratic function fall within the interval.
Solving the problem of function application is generally divided into the following four steps:
(1) examination: clarify the meaning of the question, analyze the conditions and conclusions, and straighten out the quantitative relationship;
(2) Modeling: transforming written language into mathematical language, and using mathematical knowledge, establishing corresponding mathematical model;
(3) solving: solving the mathematical model to get a mathematical conclusion;
(4) Reduction: the conclusion is reduced to the meaning of the actual question, that is, the answer.
There are eight ways to find the function range:
Method 1: Observation This method is suitable for answering multiple-choice questions and fill-in-the-blank questions.
Method 2: Inequality method This method is suitable for solving comprehensive problems.
Method 3: Inverse function method This method has a narrow scope of application and is most suitable for the case where x is once.
Method 4: Separation constant method
Method 5: discriminant method This method is suitable for the case that X is quadratic.
Method 6: Image method This method is most suitable for multiple-choice questions and fill-in-the-blank questions. Draw a sketch of the function, and the problem becomes straightforward.
Method 7: the intermediate variable method is extremely narrow and needs to be mastered flexibly.
Method 8: Matching method This method needs to be mastered flexibly and can often achieve unexpected results.
Function is an important part of high school mathematics, and inverse function is an important part of function, which is also one of the difficulties in learning function. Inverse function also occupies a certain proportion in the college entrance examination over the years. The properties of the inverse function are summarized as follows.
The definition domain and value domain of the property 1 original function are the definition domain and value domain of the inverse function respectively.
If we can make full use of this property, it will be of great help to solve the inverse function of the original function, the definition of the inverse function and the related problems of the value domain.
There are two basic methods to master function images: description method and image transformation method (trigonometric function has five points)
Image transformation: translation, rotation, symmetry and expansion.
Images and properties of special trigonometric functions
1. Use the unit circle, trigonometric function image and the number axis (the number axis is commonly used when finding the intersection of intervals, which is simpler than the coordinate system) to find the definition domain of trigonometric function.
2. Common methods for finding the range of trigonometric functions:
(1) discriminant, important inequality, monotonicity
★(2) Transform the given trigonometric function into a quadratic function, and find the domain by matching method. For example: into: y=asin? x+bsinx+c
(3) Use the boundedness (maximum and minimum) of SINX sinx and cosx to estimate the domain.
(4) Alternative methods
When using method of substitution to find the range of trigonometric functions, we should pay attention to the equivalence before and after (after exchange, the range is equal, and the defined range remains unchanged)
3. Determination of monotonicity of trigonometric function: Generally, the function is converted into the standard form of trigonometric function first, and then it is solved by deformation or combination of numbers and shapes. If the function is described and drawn, the solution can be obtained through the intuition of the graph.
4. To judge the parity of a function, we must first judge the symmetry of its domain.
5. Solving the minimum positive period of trigonometric function: it is mainly transformed into the basic trigonometric function type through identities, such as:
Y=Asin(ωx+φ), but pay attention to the equivalence before and after deformation. There are also image method and definition method.
In short, to find the monotone interval, period and parity of a function, we should pay attention to the application of the idea of reduction. For example, the increase and decrease of functions y=sin(-x) and y=sin(x) in the same interval are opposite, because sin(-x)=-sin(x).
6. Trigonometric image transformation is a variable change rather than an angle change.
7. Give the analytical formula of image judgment: y=Asin(ωx+φ). Sometimes the first point (-φ/ω, 0) in the five-point method is taken as the breakthrough point, and the position of the first zero point should be accurately found from the periodic fluctuation.
Additional formula:
1. and angle formula
sin(x+y)= sinx cosy+cosx siny(Sx+y)
cos(x+y)= cosx cosy-sinx siny(Cx+y)
tan(x+y)= tanx+tany/ 1-tanxtany(Tx+y)
2. Differential angle formula
sin(x-y)= sinxcosy-cosx siny(Sx-y)
cos(x-y)= cosx cosys+inx siny(Cx-y)
tan(x-y)= tanx-tany/ 1+tanx tany(Tx-y)
3. Double angle formula
sin2x=2sinxcosx
cos2x=(cos^2)x-(sin^2)x=2(cos^2)x- 1= 1-2sin^2x
tan2x=2tanx/ 1-(tan^2)x
sin3x=3sinx-4(sin^3)x
cos3x=4(cos^3)x-3cosx
tan3x=3tanx-(tan^3)x/ 1-3(tan^2)x
4. Power reduction formula
(sin^2)x= 1-cos2x/2
(cos^2)x=i=cos2x/2
1. General formula
Let tan(a/2)=t
sina=2t/( 1+t^2)
cosa=( 1-t^2)/( 1+t^2)
tana=2t/( 1-t^2)
2. Double angle formula
sin2x = 2 sinxcosx cos2x=cos^2x-sin^2x=2cos^2x- 1= 1-2sin^2x tan2x = sin2x/cos2x
3. Triple angle formula
sin(3a)=3sina-4(sina)^3
cos(3a)=4(cosa)^3-3cosa
tan(3a)=[3tana-(tana)^3]/[ 1-3(tana^2)]
4. Sum and difference of products
Sina * cosb =[sin(a+b)+sin(a-b)]/2
cosa * sinb =[sin(a+b)-sin(a-b)]/2
cosa * cosb =[cos(a+b)+cos(a-b)]/2
Sina * sinb =-[cos(a+b)-cos(a-b)]/2
5. Sum-difference product
Sina+sinb = 2 sin[(a+b)/2]cos[(a-b)/2]
Sina-sinb = 2sin[(a-b)/2]cos[(a+b)/2]
cosa+cosb = 2cos[(a+b)/2]cos[(a-b)/2]
cosa-cosb =-2 sin[(a+b)/2]sin[(a-b)/2]
The problem after class involves the triple angle formula and introduces the universal formula. There is also a half-angle formula, which is actually a variant of the double-angle formula.
There are many variants in trigonometric function, which can be accumulated in doing problems.
Sum and difference of products
Sina * cosb =[sin(a+b)+sin(a-b)]/2
cosa * sinb =[sin(a+b)-sin(a-b)]/2
cosa * cosb =[cos(a+b)+cos(a-b)]/2
Sina * sinb =-[cos(a+b)-cos(a-b)]/2
Sum difference product
Sina+sinb = 2 sin[(a+b)/2]cos[(a-b)/2]
Sina-sinb = 2sin[(a-b)/2]cos[(a+b)/2]
cosa+cosb = 2cos[(a+b)/2]cos[(a-b)/2]
cosa-cosb =-2 sin[(a+b)/2]sin[(a-b)/2]
The theme is flat and oriented to quantity.
[Edit this paragraph] The concept of vector
A quantity with both direction and size is called a vector (called a vector in physics), and a quantity with only size and no direction is called a quantity (called a scalar in physics).
[Edit this paragraph] Geometric representation of vector
A directed line segment is called a directed line segment, and a directed line segment with a starting point and AB ending point is recorded as AB. (AB is printed, that is, bold letters, and writing is to add a → to it)
The length of the directed line segment AB is called the module of the vector, and it is denoted as |AB|.
A directed line segment contains three factors: starting point, direction and length.
Equal vector, parallel vector, * * * line vector, zero vector and unit vector:
Vectors with the same length and direction are called equal vectors.
Two nonzero vectors with the same or opposite directions are called parallel vectors.
Vectors a and b are parallel, which is marked as a//b, and zero vector is parallel to any vector, that is, 0//a,
In the vector, the * * * line vector is a parallel vector (this is different from a straight line, which is the same straight line, and the vector * * * line refers to two parallel vectors).
A vector with a length equal to 0 is called a zero vector and recorded as 0.
The direction of the zero vector is arbitrary; A zero vector is perpendicular to any vector.
A vector with a length equal to 1 unit length is called a unit vector.
[Edit this paragraph] Vector operation
Add operation
AB+BC = AC, this calculation rule is called triangle rule of vector addition. (End to end, end to end connection, pointing to the end)
It is known that the two vectors OA and OB starting from the same point O are parallelogram OACB, and the diagonal OC starting from O is the sum of the vectors OA and OB. This calculation method is called parallelogram rule of vector addition.
For zero vector and arbitrary vector a, there are: 0+a = a+0 = a.
|a+b|≤|a|+|b| .
The addition of vectors satisfies all the laws of addition.
subtraction
AB-AC=CB, this calculation rule is called triangle rule of vector subtraction. (* * * starting point, even end point, direction pointing to the minuend)
The vector with the same length and opposite direction as A is called the inverse quantity of A, -(-a) = A, and the inverse quantity of zero vector is still zero vector.
( 1)a+(-a)=(-a)+a = 0(2)a-b = a+(-b).
multiply operation
The product of real number λ and vector A is a vector, and this operation is called vector multiplication, which is denoted as λa, | λ A | = | λ| | A |. When λ > 0, the direction of λ A is the same as that of A. When λ
Let λ and μ be real numbers, then: (1) (λ μ) a = λ (μ a) (2) (λ+μ) a = λ a+μ a (3) λ (ab) = λ a λ b (4) (-λ) a =-(.
The addition, subtraction and multiplication of vectors are collectively called linear operations.
[Edit this paragraph] Vector product
Given two nonzero vectors a and b, then |a||b|cos θ is called the product or inner product of a and b, and is denoted as a? B, θ is the included angle between A and B, and |a|cos θ(|b|cos θ) is called the projection of vector A in direction B (B is in direction A). The product of zero vector and arbitrary vector is 0.
Answer? Geometric meaning of b: quantity product a? B is equal to the product of the length of a |a| and the projection of b in the direction of a |b|cos θ.
The product of two vectors equals the sum of the products of their corresponding coordinates.
Properties of scalar product of vectors
( 1)a=∣a∣^2≥0
(2)a b=b a
(3)k(ab)=(ka)b=a(kb)
(4)a (b+c)=a b+a c
(5)a b=0？ a⊥b
(6)a=kb？ a//b
(7)e 1？ e2=|e 1||e2|cosθ=cosθ
[Edit this paragraph] The basic theorem of plane vector
If e 1 and e2 are two non-linear vectors on the same plane, then there is only one pair of real numbers λ and μ for any vector A on this plane, so that A = λ * E 1+μ * E2.
A quantity with both direction and size is called a vector (called a vector in physics), and a quantity with only size and no direction is called a quantity (called a scalar in physics).
A directed line segment is called a directed line segment, and a directed line segment with a starting point and AB ending point is recorded as AB. (AB printing, writing with A →)
The length of the directed line segment AB is called the module of the vector, and it is denoted as |AB|.
A directed line segment contains three factors: starting point, direction and length.
A vector with a length equal to 0 is called a zero vector and recorded as 0; A vector with a length equal to 1 unit length is called a unit vector.