Outstanding Lecture Notes on the Maximum Value of a Function

(a) the status and importance

The function of the most value of the function of the first semester of the first year of High School Mathematics is the content of the first year of high school mathematics, is the function of the basic properties of an important part. In the process of solving practical problems, after establishing the functional relationship between variables, finding the most value cultivates students' ability to use basic theories to study specific problems, which is also one of the purposes of learning mathematics. The teaching of the function of the most value of the students in the cultivation of mathematical ideas of the combination of mathematics and shapes, the mathematical idea of induction can also make the students develop the habit of learning rigorous thinking. The idea of function is an important mathematical idea, which embodies the movement of change and the unity of opposites point of view, this section of the lesson on the connection between the knowledge of the middle and high school played a role in the beginning and the end. The function of the maximum value of the problem and inequalities, equations, parameter ranges and analytical geometry and other knowledge integrated together can often compile a comprehensive and strong new topics, you can comprehensively examine the application of knowledge of the function of students to analyze the ability to solve the problem, so as to become a high-grade answer to the high school entrance examination, is one of the hot spots of the college entrance examination test.

(B) teaching objectives

Knowledge and ability goals: to master the quadratic function of the most common methods - matching method, to train students to combine mathematical ideas and the use of basic theories of mathematical thinking and the ability to solve specific problems.

Emotional goals: experience and experience the process of mathematical activities and the role of mathematics in real life, to stimulate students to learn mathematics knowledge, and establish confidence in learning mathematics.

Process goals: through classroom learning activities to cultivate students' cooperation and communication with each other, and in the process of communicating with each other to develop students' habits of expression, abstraction, summarization of thinking, and then get the experience of success.

Scientific research objectives: under the guidance of the teacher, students experience and experience the method of the inquiry process.

(C) teaching key points

Key points: matching method, the combination of number and shape of the quadratic function of the most value.

Difficulties: quadratic function on the closed interval of the most value.

Second, the teaching and learning method

In junior high school students have learned the knowledge of the quadratic function, according to the content of this lesson and the actual level of students, this lesson is mainly used in the exploration method of teaching and teaching method. The teaching process is also a process of active construction of students, teachers can not ignore the students' existing experience, trying to force new knowledge from the outside into the minds of students, but to the students' existing knowledge and experience as a new knowledge of the growth point, to guide the students from the original knowledge and experience of "growth" and discover new knowledge and experience. In this class, the students play the main role, active thinking to explore the optimal strategy to solve the optimal value, and summarize their own solution, the knowledge of the initiative into the knowledge system has been constructed, and really do "learn to learn".

The teaching process

(a) the introduction of the subject

Example: the zoo to build a wall of the same area of the two rectangular panda room, if the construction of the enclosure materials available is 30 meters long, then the width of how many meters in order to make the construction of the panda room area is the largest? What is the maximum area of the panda room in square meters?

Through this example, students feel the need to solve the function of the most value of the problem in real problems, which triggered the interest in learning the content of this section.

Teaching tools: PPT presentation of the topic.

Teachers guide students to discuss the answer, and individual Q&A, pointing, collecting students' solutions, picking out a number of answers on the physical projector for display and comment. Students' solutions are mainly function of the method of the most value and the use of basic inequalities to find the most value, by the students to evaluate the two methods, for the closed interval on the quadratic function of the most value of the teaching.

Teaching tools: physical projector.

The function of the most value of the excellent lecture notes 2

I. The concept of the maximum and minimum value of the function

By citing the example of the most value of the solution, to guide the students to elaborate the concept of the maximum value of the function and the minimum value of the function.

In general, set the function at the function value is, if for any of the definition domain, the inequality holds, then it is called the minimum value of the function, notation; if for any of the definition domain, the inequality holds, then it is called the maximum value of the function notation.

Second, examples of practice

Example 1, the maximum or minimum value of the quadratic function: teachers and students *** with the completion of an example, the first year of high school students to develop a standardized writing format and habits, and the rest of the topics on the board of students. Students according to the existing ability and experience, hands on the answer, the teacher comments. Remind attention to when taking what value, the function takes the most value.

Cultivate the ability of students to elaborate, analyze and understand the concept of the introduction of the concept of the maximum value of the process is to follow the known to understand the unknown law of understanding the design of the modern study of educational psychology that effective conceptual teaching is based on students' existing knowledge structure, so teachers must pay attention to the process of designing the teaching of the students' existing knowledge structure to find new concepts of the fixed point, guide students through the assimilation or assimilation of the new concepts. Therefore, in the process of designing the teaching, the teacher must pay attention to find the new concepts in the students' existing knowledge structure, and guide the students to master the new concepts through assimilation or conformity, and then improve the knowledge structure. Let the students start from the maximum value of the actual problem, from the familiar quadratic function graph of the vertex has the characteristics of the quadratic function, to get the maximum value of the quadratic function (minimum value) method. Highlight the student's main position, play the leading role of the teacher, cultivate the rigor of thinking as well as the ability to transform, through the change of the interval so that students fully feel the quadratic function of the maximum value of the solution to discuss the relationship between the axis of symmetry and the given interval.

Teaching method: combination of lecture and practice

Example 2, under the conditions of the maximum and minimum values of the function.

The teacher guides the students to think deeper and deeper step by step:

1, what is the relationship between the domain of definition and the maximum value of the function?

2. What is the function to be studied after the transformation?

The target function for the further introduction of the target function of the combination of numerical and morphological at the same time pay attention to the rigorous way of thinking, to further recognize the interactive relationship between the definition domain and the value domain, the most value.

Teaching Method: Students' Independent Inquiry

III. Summarization

Circumstances

Teaching Process

Design Notes

Summary

1. The concepts of the maximum and minimum of a function.

2. The relationship between the definition domain and value domain of a function and the maximum value of a function.

3, the matching method is more suitable for the maximum value of the quadratic function (minimum), in particular, we should pay attention to the closed interval on the function of the maximum value of the combination of digital and physical solution.

4, mathematical ideas: the combination of number and shape ideas, the idea of transformation, through the method, the idea of the summary of the students to analyze, the ability to solve problems has been improved, which will help the subsequent problem learning and research.

Teaching methods: students exchange summaries.

Fourth, classroom practice

Ring

Teaching process

Design description

Classroom practice

Finding the maximum value of the following functions

Topic design objectives:

1. Check the learning mastery of the basic content of this section.

2. Examine the concept of quadratic function and students' ability to transform.

Teaching methods: ask students to board

V. Assignment

1, the most value of the function.

2, known, find the maximum value of the function.

3. Find the . Maximum and minimum values.

4. Find the maximum and minimum values of the function.

5, a hotel has 100 guest beds, each bed daily fee of 10 yuan can be all full, if the daily fee for each bed every day to raise 2 yuan, it will be reduced by 10 guest beds rented out. In this way, in order to reduce the input more profitable, how many dollars per bed per day charges should be raised?

Homework can reflect both the students' understanding and mastery of the knowledge of this section, but also a process of consolidation of knowledge, so the design of the homework is to improve the quality of classroom teaching one of the keys, the content should not only be close to the textbook but also to synthesize the knowledge learned, it is the ability to further improve.

Teaching Objectives

Skillfully master the most value of quadratic function and its method of finding.

Focus

The most value of a quadratic function and its solution.

Difficulties

The most value of a quadratic function and how to find it.

I. Introduction

The maximum value of a quadratic function:

Second, the analysis of examples:

Example 1: To find the maximum value of a quadratic function, and the value at which the maximum value is obtained.

Variation 1:

Variation 2: Find the maximum value of the function.

Variation 3: Find the maximum value of a function.

Example 2: The maximum value of the function is known to be 3 and the minimum value is known to be 2, find the range of values.

Example 3: If, is the quadratic equation of two real roots, find the minimum value.

Third, with practice:

1, if the function has a minimum value on the maximum value of 2, if, then = ________, = ________.

2, it is known that, is the two real roots of the quadratic equation about, then the minimum value is ()

A, 0 B, 1 C, -1 D, 2

3, find the maximum value of the function on the interval.

Fourth, review the summary

This lesson has been the following:

1, the maximum value of the quadratic function and its method of finding.

After-class homework

Class: () class name __________

I. Basic questions:

1. Function

A, has a maximum value of 6? B, has a minimum value of 6? C, has a maximum value of 10? D, has a maximum value of 2? maximum value of 2

2, The maximum value of the function is 4, and when = 2, = 5, then = ______, = _______.

2. Improvement questions:

3. Try to find the maximum value of the function about on, senior year.

4, it is known that the function at that time, take the maximum value of 2, find the value of real numbers.

5, is known to be the two real roots of the equation, the maximum value and minimum value.

Three, the question:

Known function, which, seek the maximum and minimum value of the function, and find the function to take the maximum and minimum value of the corresponding value of the independent variable.