Teaching type: review class

Teaching objectives

1. inequality (group) will be used to represent inequality;

2. Be familiar with the properties of inequality and be able to apply the properties of inequality to solve the "range problem", which will be used as a relatively large difference method;

3. Know the solution of unary quadratic inequality, and be familiar with the relationship between unary quadratic inequality, unary quadratic equation and quadratic function;

4. The plane area that can be expressed by binary linear inequalities (groups) can solve simple linear programming problems;

5. Make clear the mean inequality and its establishment conditions, and flexibly apply the mean inequality to prove or solve the maximum value.

Teaching focus

The application of inequality properties, the solution of univariate quadratic inequality, the representation of plane area by binary linear inequality (group), the optimal solution of linear objective function under linear constraints, and the application of basic inequality.

Teaching difficulties

Using inequality addition rule and multiplication rule to solve problems, find the optimal solution of objective function, and apply basic inequalities.

teaching process

1. Knowledge structure of this chapter

2. Knowledge carding

(A) inequality and unequal relations

1, using inequality (group) to represent inequality;

The main properties of inequality:

(1) symmetry:

(2) transitivity:

(3) the law of addition:

(4) the law of multiplication:

(5) Reciprocity rules:

(6) the law of power:

(7) Prescription rules:

2. Compare the sizes of two real numbers by using the properties of inequality:

difference method

3. Prove the essence of inequality.

(B) One-dimensional quadratic inequality and its solution

Solution of quadratic inequality in one variable

Solution set of unary quadratic inequality;

Let the two roots of the corresponding quadratic equation of one variable be 0, then the solution of the inequality is as follows: (Let students complete the table on page 86 of the textbook independently)

quadratic function

Image of ()

monadic quadratic equation

There are two different real roots.

There are two equal real roots.

There is no real root

rare

(3) Linear programming

1, and the plane area is expressed by binary linear inequality (group)

The binary linear inequality AX+BY+C > 0 represents a plane region composed of all points on the side of straight line Ax+By+C=0 in a plane rectangular coordinate system. (The dashed line indicates that the area does not include boundary straight lines. )

2. The determination method of which plane region is represented by binary linear inequality.

Since all points () on the same side of the straight line Ax+By+C=0 are substituted into Ax+By+C, the signs of the real numbers obtained are the same, so we only need to take a special point (x0, y0) on one side of the straight line, and we can judge which side of the straight line AX+BY+C > 0 represents from the positive and negative of AX0+By0+C..

3. Related concepts of linear programming:

① Linear constraints: In the above problems, the inequality group is a set of constraints of variables X and Y, both of which are linear inequalities about X and Y, so it is also called linear constraints.

② Linear objective function:

The linear expression z =2x+y about x and y is an analytical expression involving variables x and y that reach the maximum or minimum value, which is called linear objective function.

③ Linear programming problem:

Generally speaking, the problem of finding the maximum or minimum value of linear objective function under linear constraints is called linear programming problem.

④ Feasible solution, feasible region and optimal solution:

The solution (x, y) satisfying the linear constraint is called a feasible solution.

The set of all feasible solutions is called feasible domain.

The feasible solution to maximize or minimize the objective function is called the optimal solution of linear programming problem.

4. The step of finding the optimal solution of the linear objective function under the linear constraint:

(1) Find the linear constraint condition and linear objective function;

(2) The plane region represented by binary linear inequality constitutes a feasible region;

(3) Find the optimal solution of the objective function in the feasible region.

Basic inequality

1. If A and B are positive numbers, then

2. The geometric meaning of the basic inequality is "the radius is not less than half a chord"

3. Typical examples

1, and inequality is represented by inequality.

Example 1. A computer user plans to buy single-chip software and boxed software with unit prices of 60 yuan and 70 yuan respectively with no more than 500 yuan. If necessary, buy at least three softwares and at least two boxes of disks, and write the inequalities that satisfy the above inequalities.

Example 2: The coffee shop has prepared two kinds of drinks. The first kind of milk powder, coffee and sugar are 9g, 4g and 3g respectively. The milk powder, coffee and sugar used in the second beverage are 4g, 5g and 5g respectively. It is known that the daily raw materials are 3600g milk powder, 2000g coffee and 3000g sugar. Write all the inequalities satisfied by the theory of the number of cups for preparing two kinds of drinks.

relative dimension

Example 3 (1) (+) 26+2;

(2)( - )2 ( - 1)2;

(3) ;

(4) when a > b > 0, record a log b.

(5 ) (a+3)(a-5) (a+2)(a-4)

(6)

Using the properties of inequality to evaluate the domain

Example 4 If,, then

The range of (1) is, and the range of (2) is,

The value range of (3) is, and the value range of (4) is.

Example 5 Known function, if,, then

The value range of is.

[Thinking expansion] The range of values known and sought. ()

Solve a quadratic inequality in one variable

Example 6 Solving inequality: (1); (2)

Example 7 It is known that the equation about x (k-1) x2+(k+1) x+k+1= 0 has two different real roots, and the value range of realistic number k is sought.

Binary Linear Equation (System) and Plane Region

Example 8 Draw the plane region represented by the inequality group.

Finding the optimal solution of linear objective function under linear constraints

Example 9 Given that X and Y satisfy inequality, find the minimum value of Z = 3x+Y. ..

[Thinking expansion] It is known that x and y satisfy the inequality group. Try to find the coordinates of the whole point when the maximum value of z =300x+900y, and the corresponding maximum value of z..

Prove inequality with basic inequality

Example 8 Verification

Using basic inequality to find the maximum value

Example 9 if x > 1, then 0, y>0, find the minimum value of xy.

[Thinking expansion] Find the minimum value of (x>5).

Step 4 evaluate the design

On page 1 15 of the textbook, review the reference group [A] questions 1, 2, 3, 4, 5, 6, 7 and 8.

blackboard-writing design