Seeking the summary of high school mathematics knowledge points (full version)

1. For the set, we should grasp the "certainty, mutual difference and disorder" of the representative elements and elements of the set.

What is the element?

Pay attention to the set problem with the help of number axis and venn diagram.

An empty set is a subset of all sets and the proper subset of all non-empty sets.

3. Please note the following attributes:

(3) De Morgan's Law:

4. Can we solve the problem with the idea of complement set? (Exclusion method, indirect method)

The value range of.

6. What are the four forms of propositions and their relationships?

A proposition with reciprocal negation is an equivalent proposition. )

Both the original proposition and the negative proposition are true and false; Whether it is an inverse proposition or not, a proposition is the same as true or false.

7. Do you know the concept of mapping? Mapping F: A → B, have you noticed the arbitrariness of elements in A and the uniqueness of corresponding elements in B? What kind of correspondence can form a mapping?

(one-to-one, many-to-one, allowing the elements in B to have no original image. )

8. What are the three elements of a function? How to compare whether two functions are the same?

(Definition domain, corresponding rule, value domain)

9. What are the common types of finding function domain?

10. How to find the domain of composite function?

The meaning domain is _ _ _ _ _ _ _ _.

1 1. When finding the analytic expression of the function or the inverse function of the function, is the domain of the function marked?

12. What are the conditions for the existence of the inverse function?

(One-to-one correspondence)

Have you mastered the steps of finding the inverse function?

(① Inverse solution x; ② exchange x and y; (3) indicate the domain name)

13. What are the properties of the inverse function?

① the image with reciprocal function is symmetrical about the straight line y = x;

(2) keeping the monotonicity and odd function of the original function;

14. How to prove the monotonicity of a function by definition?

(Take values, make differences, and judge positive and negative)

How to judge the monotonicity of composite function?

∴……)

15. How to judge the monotonicity of a function with derivatives?

The value is ()

A. come on 1C。 2D。 three

The maximum value of ∴a is 3)

16. What are the necessary (insufficient) conditions for the function f(x) to have parity?

(f(x) domain is symmetric about the origin)

Please note the following conclusions:

(1) 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 even function and odd function is odd function.

17. Are you familiar with the definition of periodic function?

Function, t is a period. )

For example:

18. Have you mastered the common image changes?

Note the following "flip" transformation:

19. Are you familiar with the images and properties of common functions?

The hyperbola of.

Application: ① the relationship between "three quadratic" (quadratic function, quadratic equation, quadratic inequality) and quadratic equation.

② Find the maximum value on the closed interval [m, n].

(3) Find the maximum value of the interval (motion) and the symmetry axis (motion).

④ Distribution of roots of quadratic equation in one variable.

Remember nature through images! (Pay attention to the limit of cardinality! )

What is the difference between using monotonicity to find the maximum value and using mean inequality to find the maximum value?

20. Do basic operations often go wrong?

2 1. How to solve the abstract function problem?

(assignment method, structural transformation method)

22. Have you mastered the common methods of finding the function value domain?

(Quadratic function method (collocation method), inverse function method, method of substitution, mean value theorem method, discriminant method, function monotonicity method, derivative method, etc. )

Find the maximum value of the following function:

23. Do you remember the definition of radian? Can you write the formula of arc length and sector area with central angle α and radius R?

24. The definition of memory trigonometric function and the definition of trigonometric function line in the unit circle.

25. Can you draw a graph of sine, cosine and tangent functions quickly? And write monotone interval, symmetry point and symmetry axis from the image?

(x, y) make an image.

27. When finding the angle in trigonometric function, we should pay attention to two aspects-first find the value of a trigonometric function, and then determine the value range of the angle.

28. When solving problems with sine and cosine functions, have you noticed the boundedness of using functions?

29. Have you mastered the trigonometric function image transformation?

(Translation transformation, expansion transformation)

Translation formula:

Image?

30. Have you mastered the trigonometric function relation and inductive formula?

"Odd" and "even" mean that k is odd or even.

A. Positive or negative B. Negative C. Non-negative D. Positive

3 1. Have you mastered the formulas of sum, difference, multiplication and power of two angles and their reverse application?

Understand the relationship between formulas:

Apply the above formula to simplify trigonometric function. (Simplification requirements: the number of items is the least, the types of functions are the least, the denominator does not contain trigonometric functions, and those that can be evaluated should be evaluated as much as possible. )

Specific methods:

(2) Name change: chorus or intercept.

(3) the transformation of degrees: the formula of ascending and descending power.

(4) Shape transformation: unify the function form and pay attention to the application of algebraic operation.

32. Do you remember the expressions of sine and cosine theorems? How to realize the transformation of edges and angles and solve oblique triangles?

(Application: Know the included angle between two sides and find the third side; It is known to find the trihedral angle. )

33. Pay attention to the value range of the angle when using the inverse trigonometric function to represent the angle.

34. What is the nature of inequality?

Answer: c

35. Use the average inequality:

Value? (One positive, two positive, three-phase and so on. )

Please note the following conclusions:

36. Have you mastered the basic method of inequality proof?

Comparison, analysis, synthesis, mathematical induction, etc. )

And pay attention to the application of simple scaling method.

(Move the term into general division, factorize the numerator and denominator, and the coefficient of x becomes 1. Find the result by the through-axis method. )

38. Use the "through-axis method" to solve the higher inequality-"odd penetration, even tangent", starting from the upper right of the largest root.

39. The solution of inequality with parameters should pay attention to the discussion of letter parameters.

40. How to solve the inequality with two absolute values?

(Find the zero point, discuss in sections, remove the absolute value sign, and finally take the union of the sections. )

Prove:

(Scaling in unequal directions)

42. What are the commonly used methods to deal with the problem of constant inequality? (Can be converted into a maximum problem, or a "△" problem)

43. Definition and nature of arithmetic progression.

Quadratic function of 0)

Item, namely:

44. Definition and properties of geometric series

46. Are you familiar with the common methods of finding the general term formula of series?

For example: (1) difference (quotient) method

Solution:

[practice]

(2) Iterative method

Solution:

(3) Arithmetic recurrence formula

[practice]

(4) Proportional recurrence formula

[practice]

(5) Reciprocity method

47. Are you familiar with the common methods of finding the sum of the first n items in a series?

For example, (1) split item method: the items in a series are split into the sum of two or more items, so that the pairs of items with opposite numbers appear.

Solution:

[practice]

(2) Dislocation subtraction:

(3) Addition in reverse order: Write the sequence of the series in reverse, and then add it with the sequence of the original series.

[practice]

48. Do you know anything about savings and loans?

△ Calculation model of lump-sum deposit and lump-sum withdrawal of principal and interest (simple interest);

If the principal of each installment is P yuan, and the interest rate of each installment is R, after N installments, the sum of principal and interest is:

△ If it is compound interest, such as loan problem-calculation model of mortgage loan repayment in each installment (mortgage loan-loan type that repays principal and interest in equal installments)

If the loan (bank loan) is RMB P, it will be repaid in equal installments. From the date of loan, the first installment (such as one year) is the first repayment date, and so on, and the nth repayment is made. If the interest rate of each period is R (calculated by compound interest), then each period should also be X yuan.

P- loan times, R- interest rate, N- repayment periods.

49. The basis for solving the problem of permutation and combination is: classified addition, step-by-step multiplication, orderly arrangement and disorderly combination.

(2) Arrangement: randomly select m(m≤n) elements from n different elements and arrange them in a certain order.

(3) Combination: randomly select m(m≤n) elements from n different elements to form a group and call them from n different elements.

50. The law to solve the problem of permutation and combination is:

Adjacent topic binding method; Interpolation method of phase interval problem: positioning problem priority method; Classification of multivariate problems; At most, at least the problem indirect method; The same elements can be grouped by division method, and the results can be discharged one by one when the number is small.

For example, the test scores of four students with student numbers 1, 2, 3 and 4.

Then all the possible situations of these four students' test scores are ()

A.24B。 15C。 12D。 10

Analysis: It can be divided into two categories:

(2) The middle two scores are equal.

The same two numbers are taken as 90,965,438+0,92 respectively, and the corresponding arrangement can be counted, including 3,4,3 kinds and ∴ 65,438+00 kinds respectively.

∴ * * has 5+ 10 = 15 (species)

5 1. binomial theorem

Nature:

(3) Maximum value: when n is even, n+ 1 is odd, and the binomial coefficient of the middle term is the largest, which is the first.

Representative)

52. Are you familiar with the relationship between random events?

The sum of (and).

(5) mutually exclusive events: "A and B can't happen at the same time" is called mutual exclusion of A and B.

(6) Opposing events (reciprocal events):

(7) Independent events: whether A occurs has no influence on the probability of B. Such two events are called mutually independent events.

53. The solution of the event probability:

What we need to distinguish is (1) and the probability of other possible events (permutation and combination methods are often used, that is,

(5) If the probability that A occurs in one trial is p, then A occurs in n independent repeated trials.

For example, suppose that there are 4 defective products and 6 genuine products in 10, and find the probability of the following events.

(1) Two of them are defective;

(2) Only 2 out of any 5 pieces are defective;

(3) There are at least 2 defective products in any 3 returned products;

Analysis: the tape was put back for 3 times (each time 1 piece), ∴ n = 103.

At least two defective products are "exactly two defective products" and "three defective products"

(4) Take five pieces from them in turn, and just two pieces are defective.

Parse: ∫ Extract one by one (in order)

Distinguish (1), (2) is a combination problem, (3) is a repeated arrangement problem, and (4) is a non-repeated arrangement problem.

54. Sampling methods mainly include: simple random sampling (lottery method, random number table method) is often used when the population is small, and its characteristic 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 feature of stratified sampling is stratified proportional sampling, which is mainly used for obvious differences among people. Their similarity is that the probability of each individual being drawn is equal, which reflects the objectivity and equality of sampling.

55. Estimation of population distribution —— Use the frequency of sample occurrence as the probability of the population, and use the expectation (average) and variance of the sample to estimate the expectation and variance of the population.

Familiar with the method of sample frequency histogram:

(2) determine the interval and the number of groups;

(3) Deciding on separation;

(4) Column frequency distribution table;

(5) draw a frequency histogram.

For example, 6 girls from 10 and 5 boys from10 will participate in the competition. If randomly sampled by gender, the probability of forming this team is _ _ _ _ _ _ _ _.

56. Do you know the concept of vector?

(1)Vector- a quantity with both magnitude and direction.

Under this rule, the vector can move in parallel in the plane (or space) unchanged.

(6) Parallel vector-a vector with the same or opposite direction.

Specifies that the zero vector is parallel to any vector.

(7) The addition and subtraction of vectors are as follows:

(8) Basic theorem of plane vector (decomposition theorem of vector)

A set of substrates.

(9) Coordinate representation of vector

Express delivery.

57. The product of the number of plane vectors

Geometric meaning of quantity product;

(2) Quantity product algorithm

[practice]

Answer:

Answer: 2

Answer:

58. The demarcation point of the line segment

Can you distinguish the center of gravity, center of gravity, outer heart, inner heart and their nature of a triangle? ※?

59. Is the idea of proving the relationship between parallelism and verticality clear in solid geometry?

The proof of parallelism and verticality mainly uses the transformation of line-plane relationship;

Determination of parallelism between straight line and plane;

Properties of parallel lines and planes:

Three vertical theorems (and inverse theorems);

Vertical lines and planes:

Face to face vertical:

60. Definition and solution of three angles

The angle θ formed by (1) straight lines on different planes is 0.