2. It is easy to ignore the principle of domain priority when solving problems related to functions.

3. When judging the parity of a function, it is easy to ignore whether the domain of the function is symmetrical about the origin.

4. What is the canonical format when the monotonicity of a function is proved by definition? Take whatever you want, do whatever you want, and judge whether it is positive or negative.

5. When finding the monotonicity of a function, it is easy to add the symbols "∨" and "or" between multiple monotonous intervals by mistake.

6. Monotone interval cannot be expressed by set or inequality. Two monotonous intervals should be connected by commas.

7. When using the mean value theorem to find the maximum value (or range), it is easy to ignore the condition of verifying "one positive, two definite, three equal".

8. The function (whose image image in the first quadrant is "√", specially named: symbolic function, the symbolic function is odd function, and the image is symmetrical about the origin) is monotonically increasing; Monotonously decreasing in the world)

9. Monotone interval of function: increasing monotonously in the world; Odd number function, the image is symmetrical about the origin.

10. Restriction conditions of real number and base of logarithmic function: real number is greater than zero, and base greater than zero is not equal to 1, and letter base needs to be discussed.

1 1. When solving problems by substitution, it is easy to ignore the equivalence before and after substitution, that is, the range of independent variables after substitution.

12. When judging the number of solutions (or the number of intersections) of an equation by discriminant, it is easy to ignore whether the coefficient of quadratic term is 0, especially when a straight line intersects a conic curve.

13. Important properties of arithmetic progression: If m+n=p+q, then; (otherwise)

14. Important properties in geometric series: If m+n=p+q, then. (otherwise)

15. it is easy to ignore the common ratio q = 1 when summing with the summation formula of equal ratio series.

16. It is easy to ignore the case of n = 1.

17. A property of arithmetic progression: Let it be the sum of the first n terms of the sequence {0}, and the necessary and sufficient conditions for {0} to be arithmetic progression are: (a, b are constants) and its tolerance is 2a.

18. "Dislocation Subtraction" for the summation of series-if {} is arithmetic progression and {} is geometric progression, find the sum of the first n items of {}.

19. "Split Sum" of Series Sum (for example)

20. When solving trigonometric functions, pay attention to the definition domain of tangent function and cotangent function, pay attention to the boundedness of sine function and cosine function, and always pay attention to the angle range when solving trigonometric functions.

2 1. The general methods of triangle simplification (chord cutting, power reduction and angle expansion, special angles, homonyms and synonyms with the same name generated by triangle formula transformation).

22. Are the arc length formula and the sector area formula under the arc system? —— )

23. The substitution of "1" in trigonometric functions is widely used.

24. Unlike the real number 0, the modulus of 0 is not without direction, but with uncertain direction. It can be considered to be parallel to any vector, but not perpendicular to any vector.

25. Then, but you can't get it or have it.

26. sometimes, yes Otherwise, it cannot be started.

27. Generally speaking, there is no allocation rate for vector operations.

28. In China,

29. When sine theorem is used, forgetting ratio is equal to 2R. Homogeneous substitution

30. When solving the solution set, definition domain and value domain of inequality, the result must be expressed by set or interval; Can't be expressed by inequality.

3 1. When multiplying two inequalities, we must pay attention to the same direction and the same time to multiply, that is, we can multiply in the same direction; At the same time, we should pay attention to "reciprocal of the same number"

That is, a > b > o, a < b < o.

32. The general idea of solving fractional inequality is to move the term into a general point and then divide it into zero points.

33. The solution refers to the monotonicity of exponential function and logarithmic function, and the real number of logarithm is greater than zero. So it refers to the square solution of inequality.

34. When solving inequalities with parameters, we must discuss them, especially the bottom or of exponent and logarithm.

35. Write after discussion: To sum up, the solution of the original inequality is ... This article is used for all major mathematical problems.

36. Common scaling techniques:

37. The main idea of analytic geometry is to study the properties of graphs by algebraic method. The main method is coordinate method.

38. It is easy to ignore the fact that the slope does not exist when establishing the equation of a straight line with oblique points or oblique points.

39. The range of inclination angle, arrival angle and included angle of a straight line is.

40. The translation formulas of images of functions, equations and points are confusing:

4 1. For two non-coincident straight lines, there are

; (When solving problems, use slope and intercept after discussion)

42. The intercept of a straight line on the coordinate axis can be positive, negative or 0.

43. 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.

44. To deal with the positional relationship between circles, we can use the relationship between the center distance and radius of two circles.

45. In a circle, pay attention to the right triangle composed of radius, half chord length and chord center distance.

46. The meaning of a, b, c, p,,, in the conic equation.

47. The relationship between eccentricity and curve shape (roundness and mouth size) The eccentricity of equilateral hyperbola is root number 2.

48. When solving conic curves and straight lines simultaneously, we should pay attention to whether the coefficient of the quadratic term in the equation obtained after elimination is zero and the limitations of the discriminant (finding the intersection point, chord length, midpoint, slope, symmetry, existence and other issues are all carried out below).

49. In the ellipse, pay attention to the right triangle composed of the focus, the center of the circle and the endpoint of the short axis. (A, B, C)

50. The path is the shortest chord in all focus chord of parabola. Think about the conclusion in the hyperbola? )

5 1. Difference between A, B and C in Elliptic and Hyperbolic Standard Equations

52. If the straight line is parallel to the asymptote of hyperbola, the straight line and hyperbola have only one intersection point; If a straight line is parallel to the axis of a parabola, there is only one intersection between the straight line and the parabola. At this time, the two equations are simultaneous, and after elimination, they are linear equations.

53. When finding the angle formed by two straight lines on different planes, the angle formed by straight lines and planes, and the dihedral angle, if the angle is 90, don't forget that there is another way to find the angle, that is, to prove that they are vertical.

54. The judgment theorem and property theorem of line-plane parallelism are three conditions in application, but these three conditions are easy to be confused; The judgment theorem of plane-to-plane parallelism is easy to record the condition as "two intersecting straight lines in one plane are parallel to two intersecting straight lines in another plane", which leads to too big steps in the proof process.

55. The main methods to determine the plane angle of dihedral angle are definition method, three perpendicular lines method and vertical plane method.

Three perpendicular lines method: a plane, two perpendicular lines and three diagonal lines, the projection is visible.

56. The conventional methods for finding the distance from point to surface include direct method, equal volume method, change point method and vector method.

57. The conventional methods for calculating the volume of polyhedron are filling and excavation method and equal product method.

58. Angle range formed by two straight lines on different planes: 0.

The range of the angle formed by the straight line and the plane: 0o≤α≤90.

The plane angle range of dihedral angle is 0 ≤α≤ 180.

59. The order of A and B in the binomial expansion formula remains unchanged.

60. The binomial coefficient is easily confused with the expansion coefficient. The binomial coefficient of r+ 1 is.

The maximum binomial coefficient of 6 1. is easily confused with the maximum binomial coefficient in the expansion. The maximum binomial coefficient is one or two in the middle. The solution of the maximum coefficient term in the expansion is to determine R by solving the inequality group.

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

63. The law to solve the permutation and combination problem is: the adjacent problem binding method; Interpolation method for non-adjacent problems: single-line method for multi-line problems; Positioning problem priority method; Double reduction method of scheduling problem: classification of multivariate problems; Ordered distribution problem method; Select a question first, and then return; At most, at least the problem is indirect or considered as several coincidences.

64. The general formula of binomial expansion, the probability of event A occurring k times in n independent repeated tests, and the distribution table of binomial distribution are all easy to remember.

General formula: (it is r+ 1 term instead of r term).

Probability of event a happening k times:.

Where k = k=0, 1, 2, 3, …, n …, n and 0.

65. The derivative formula of common functions:

; ; ; .

. . . .

,

If a series starts from the second term, the difference between each term and its previous term is equal to the same constant. This series is called arithmetic progression, and this constant is called arithmetic progression's tolerance, which is usually represented by the letter D.

Arithmetic progression's general formula is:

an = a 1+(n- 1)d( 1)

The first n terms and formulas are:

Sn=na 1+n(n- 1)d/2 or Sn=n(a 1+an)/2 (2).

It can be seen from the formula (1) that an is a linear function (d≠0) or a constant function (d=0) of n, and (n, an) is arranged in a straight line. According to formula (2), Sn is a quadratic function (d≠0) or a linear function (d =

The relationship between any two am and an is:

an=am+(n-m)d

It can be regarded as arithmetic progression's generalized general term formula.

From arithmetic progression's definition, general term formula and the first n terms formula, we can also deduce that:

a 1+an = a2+an- 1 = a3+an-2 =…= AK+an-k+ 1，k∈{ 1，2，…，n}

If m, n, p, q∈N*, m+n=p+q, then there is.

am+an=ap+aq

Sm- 1=(2n- 1)an，S2n+ 1 =(2n+ 1)an+ 1

Sk, S2k-Sk, S3k-S2k, …, Snk-S(n- 1)k… or arithmetic progression, and so on.

If a series starts from the second term and the ratio of each term to the previous term is equal to the same constant, this series is called geometric series. This constant is called the common ratio of geometric series and is usually represented by the letter Q.

The general formula of geometric series is:

an=a 1？ qn- 1

The first n terms and formulas are:

sn = a 1( 1-qn)/ 1-q

In geometric series, the median term of a and b is g = (ab) 1/2.

And the relationship between any two am and an is an=am? qn-m

If the common ratio q of geometric progression satisfies 0 < ∣ q < ∣ 1, then this series is called infinite recursive geometric progression, which has various kinds.

The formula for the sum of items (also called the sum of all items) is:

From the definition of geometric series, the general term formula and the first n terms formula, we can deduce that:

a 1？ an=a2？ an- 1=a3？ An -2=…=ak? an-k 1，k∈{ 1，2，…，n}

If m, n, p, q∈N*, there are:

ap？ aq=am？ Ann,

Write πn=a 1? Aortic second sound ... Ann, yes.

π2n- 1=(an)2n- 1，π2n 1 =(an 1)2n 1

In addition, each term is a geometric series with positive numbers, and the same base number is taken to form a arithmetic progression; On the other hand, taking any positive number c as the cardinal number and a arithmetic progression term as the exponent, a power energy is constructed, which is a geometric series. In this sense, we say that a positive geometric series and an arithmetic series are isomorphic.

What matters is not only the definitions, properties and formulas of the two basic sequences; Moreover, the mathematical thinking and wisdom contained in the process of summation are extremely precious, such as "inverted addition" (arithmetic progression) and "dislocation subtraction" (geometric progression).

There are two main problems in series, one is to find the general term formula of series, and the other is to find the sum of the first n terms of series.