The second round review method of liberal arts mathematics in college entrance examination

Analytical method and comprehensive method

I. Learning objectives

The core of mathematical ability is thinking ability, and the forms of thinking are various, such as observation, comparison, analysis, induction and synthesis. In the process of thinking, we should be good at spreading our wings, that is, analytical method and comprehensive method. The so-called analytical method is to trace back the original facts that make the conclusion valid, and the "cause" must be linked with topics, theorems, axioms and formulas. That is, "cause is caused by result". The so-called synthesis method is "from cause to effect", that is, continuous calculation and reasoning according to existing conditions. The direction of deduction is "conclusion" and "desired result". These two methods must be fully staggered and properly used in the process of solving problems. Cause and effect are closely linked. Frequent use of these two methods can solve contradictions and come naturally. Otherwise, the blind are riding blind horses, running left and right, and solving problems is confusing. Even an infarction will not help. Whether it is a proof problem, a calculation problem or an application problem.

Second, the case analysis

[Example 1] Let the function be derivable at point x0, and try to find the following limit values.

Thinking analysis:

In the definition of derivative, the increment Δ x has many forms, but no matter which form Δ x chooses, Δ y must also choose the corresponding form. Using the condition that the function is differentiable at point x0, the given limit identity can be transformed into the structural form defined by derivative.

Answer:

[Example 2] It is proved that if the function is differentiable at point x0, then the function is continuous at point x0.

Thinking analysis

To seek transformation methods and strategies from known and proven problems, it is necessary to prove continuity at x0 point. Because the function is differentiable at the point x0, according to the definition of the function at the point x0, two transformations are gradually realized, one is the transformation of the trend, and the other is the transformation of the form (into the derivative definition form).

Solution:

The function is continuous at point x0.

[Example 3] Find the change δy and dy of the function when x changes from 1 to 1.0 1.

explain

feedback

It is easy to find that when Δ x→ 0, that is, the differentiation of a function at a point is a linear approximation of the increment of the function Δ y ≈ dy, this is the application of differentiation-for approximate calculation.

Third, exercise questions

(1) multiple-choice questions (only one of the four options given in each question meets the requirements of the topic).

1. There is no inverse function in the following functions.

A.y=x2-2x+3(x≤0)

2. Let N={ the first or fourth quadrant angle}, then

Morning =N

D. none of the above relationships are established.

3. The function f(x) defined on R satisfies: f(2+x)=f(2-x). If the equation f(x)=0 has only three unequal real roots, and 0 is one of them, then the other two roots of the equation must be.

A.-2,2

B.2,4

C. 1，- 1

D.- 1,4

4. On the complex plane, point A corresponds to the complex number 2, point B corresponds to the complex number-1+i, and rotate the vector 90 clockwise around point A to get the vector, and point C corresponds to the complex number as follows.

A.3+3i

1+3i

C. 1-3i

D.- 1+i

5. In the infinite geometric series {an} in which all terms are positive numbers, if the first term a 1= 1, the common ratio q≠ 1, a2, a3 and a5 become arithmetic progression, then the sum of the terms of {an} is

6. The number of intersections between circle C: x2+y2+2x-6y- 15 = 0 and straight line L: (1+3m) x+(3-2m) y+4m-17 = 0 is

A: 0

B. 1

C.2

D. this number is related to the value of m.

7. In a triangular prism A1B1C1-ABC, A1B1:AB =1:3, and point M is the midpoint of side A1A.

Volume ratio of

8. Let y=f(x) be a function defined on a real number set, then the images of function y=f(x-2) and function y=f(4-x) are related.

The line with A.x=0 is symmetrical.

B. the straight line x= 1 is symmetrical.

C. the straight line x=2 symmetry

D. the straight line x=3 symmetry

9. There are points M and N on the straight lines x-y=0 and y=0, respectively, so that M, N and A (3, 1) satisfy |AM|+|MN|+|NA|. The coordinates of the points m and n with the minimum value are respectively

A.( )

C.( 1,3),(2,0)

10. If the definition domain of the function f(x)= is the real number set r, then the value domain of the real number A is

A. Rare

1 1.n ∈ n, the largest coefficient in the binomial (a+b)2n expansion must be

A. odd numbers

B. even numbers

C. Not necessarily an integer

D is an integer, but whether it is odd or even depends on the value of n.

(2) Fill in the blanks (fill in the answers on the horizontal lines in the questions).

12.

13.(2 x2+4x+3)6 = A0+a 1(x+ 1)2+A2(x+ 1)4+…+A0(x+ 1)65438+。

14. if the function f(x)=(x+a)3, for any t∈R, there is always f( 1+t)=-f( 1-t), then f(2)+f(-2).

15. As shown in the figure, in the known right-angle ABCD, AB= 1, BC=a, PA⊥ plane ABCD, if there is only one point Q on the side of BC and PQ⊥DQ, then

a=。

(3) solving problems (the process or calculation steps of explaining and proving the solution).

16. In △ABC, BC=a, AC=b, AB=C, and the side length C is the largest. It is known that AC COSA+BC COSB < 4 s (S is the area of ABC). Verification: △ABC is an acute triangle.

17. If the three inner corners A, B and C of △ABC become arithmetic progression, then verify:.

18. Solving inequalities about x

19. when α∈R is known, the inequality about x (1+sinα+cos α) x2-(1+2sinα) x+sinα > 0 is constant, so we find α.

20. Prove that the image of a function is the locus of a point whose absolute value of the distance difference between two points on a plane is constant.

2 1. With point A as the center, there is a point B on the circle of 2cosθ (0 < θ).

(1) When θ takes a certain value, what curve is the trajectory p of point M?

(2) point m is the moving point on the trajectory p, and point n is the moving point on QA. Let the maximum value of |MN| be f(θ), and find the range of f(θ) (without proof).

Reference answer

1—5 B D B A A 6— 1 1 C D D D B D B

12、

13、

14, answer: -26

Description: f (1+t)+f (1-t) = 0 (1+t+a) 3+(1-t+a) 3 = 0.

∴ 1+a=0, A =- 1, F (x) = (x- 1) 3, then f(2)+f(-2)=-26.

15、

16、

17、

18、

19、

20、

2 1、

Analogy and reduction thinking method

I. Executive summary

In the long-term mathematical practice, people have established many concepts, many problem models, and mastered many fixed general methods (solving linear and quadratic equations and inequalities, finding the range of some basic elementary functions, finding conic equations, etc.). ). In the face of objective problems, we sometimes use the methods of association and analogy to restore or inject new problems into a mathematical model, and then solve them by conventional methods. The above is the thinking method of number comparison and reduction, and it is also a common thinking strategy in mathematics. For example, when calculating the volume of a polyhedron, it is often divided into several pyramids, prisms or truncated cones, and they are calculated separately; Solving a complex inequality often boils down to solving a quadratic inequality. To sum an unknown sequence, we can first analyze the general term formula, and then get it by the sum formula of arithmetic (ratio) sequence or the split term method respectively. Metaphor is intended to observe and discover, not to "achieve a bright future" (success). It is not good to return for the sake of "returning". This volume aims to train and test candidates in this field.

Second, the case analysis

[Example 1] Write the following propositions in the form of "If p is q", and write their inverse propositions, negative propositions and negative propositions:

(1) When x=2, x2-3x+2 = 0;

(2) the vertex angles are equal;

(3) The last digit is an integer of 0, which can be divisible by 5.

Thinking analysis:

Write according to the definition of four propositions.

Answer: (1) Original proposition: If x=2, then x2-3x+2=0.

Inverse proposition: If x2-3x+2=0, then x=2.

No proposition: if x≠2, then x2-3x+2≠0.

Negative proposition: If x2-3x+2≠0, then x≠2.

(2) Original proposition: If two angles are antipodal angles, they are equal.

Inverse proposition: If two angles are equal, it is antipodal angle.

There is no proposition: if two angles are not diagonal, they are not equal.

Negative proposition: If two angles are not equal, they are not antipodal angles.

(3) Original proposition: If the last digit of an integer is 0, the integer can be divisible by 5.

Inverse proposition: If an integer is divisible by 5, then the last digit of the integer is 0.

No proposition: If the last digit of an integer is not 0, then the integer cannot be divisible by 5.

Negative proposition: If an integer is not divisible by 5, then the last digit of the integer is not 0.

It is proved that two angles in a triangle cannot be right angles.

Thinking analysis:

It is difficult to prove this problem by direct method, so we can consider using reduction to absurdity.

It is proved that if two possible angles are right angles, let A = 90 and B = 90, then A+B+C = 90+90+C >180 is the same as A+B+C =

180 is contradictory and the assumption is wrong, so two angles in a triangle can't be right angles.

[Example 3] Summarize the basic nature of inequalities learned in junior high school.

Answer: Basic properties of inequality:

Description:

1, the name of each attribute above is expressed in brackets, which is intended to help you deepen your understanding and memory. These properties in high school.

Senior two needs systematic study. If you master the basic properties of inequality in senior one, your whole math study will be

Make fewer mistakes.

2. The modern language symbols ""and ""are used above, and the necessary and sufficient conditions will be studied later. Now ""is translated into "push out".

And "A B" means "A B", which translates into "equivalence". If you skillfully use these symbols earlier, it will promote your math study.

Third, the test questions

1. If the set A = {1, 2, 3, 4, 5} B = {6, 7, 8} and F: A → B, then the condition f (1) ≥ f (2) ≥ f (3) ≥ f is satisfied.

A.3 B.6 C. 12 D.2 1

2. If five of the six sides of a tetrahedron have a length, the maximum volume of the tetrahedron is

A.B. C. D。

3. Given 0≤x≤, the minimum and maximum values of the function f(x)=3sin are respectively

A.B.3，c，3 D，

4. let the complex number Z=2+ai(a∈R), then the minimum value of | Z+ 1-I |+Z- 1+I | is

A. 4 D BC

5. Given sequence {an} satisfies: Sn=, then the value is.

A.- 1 B. 1 C.-2 D.2

6. When x∈[0, π], the increasing interval of y=|sinx|+|cosx| is

A.B.[] C. D

7. Known as a real number, the point set corresponding to the complex number z can be

A.x axis B. Y axis C. X axis or Y axis D. A circle with the origin as the center and a radius of.

8. Let the function f(x) = x4-4x3+6x2-4x+1(x ≤1), then the inverse function f- 1(x) of f (x) is

A.B. C. D。

9. It is known, so the maximum value of y=2sinx+2cosx+2sin2x- 1 is

A.+ 1 B.- 1 C. D

10. Given a and b∈R+, which of the following is correct?

a . cos 2θLGA+sin 2θlgb & gt； lg(a+b)

b . cos 2θLGA+sin 2θlgb & lt； lg(a+b)

The minimum value is 1 1. θ∈ (0,2 π) is

In 4 BC.

12. As shown in the figure, in the polyhedron ABCD-A 1 b1c1d1,the bottom ABCD is a polygon with sides of1,A 1A, b/.

A. D.4 in 2 BC

13. The function f(x) defined on R is a odd function and a periodic function with a period of 2, so f(200 1)=.

14. The known point p is on the ellipse. If the distance from P to its right directrix is exactly the arithmetic average of the distances from the two focal points of the ellipse, the abscissa of point P is.

When 15.x∈( 1 2) inequality (x- 1) 2 < Logax is constant, then the range of a is.

16. If the item containing x is the sixth item, let (1-x+2x2) n = A0+a1x+a2x2+…+a2nx2n, then a 1+a2+…+a2n=.

17.

18. ellipse (a >；; An endpoint B (0, 1) of the short axis is the vertex of a right angle, and the ellipse is inscribed with an isosceles right angle △ABC. Does this triangle exist? If so, how many can you do at most?

19.

20. Equation 3x2-(6m-1) x+m2+1= 0, where two are α and β, and |α|+|β|=2, find the value of the number m..

2 1. Let a>0, a≠ 1, function f(x)=loga.

(1) discusses and proves the monotonicity of f(x).

(2) let g(x)= 1+loga(x-3), and if the equation f(x)=g(x) has two unequal real roots, find the range of a.

22.

Answer:

1、D

2. A.

3. A.

4、B

5、D