Excellent lecture notes on the maximum value of function 1 1. Talking about teaching materials
(a) Status and importance
The maximum value of function is the content of the first semester of senior high school mathematics, and it is an important part of the basic properties of function. In the process of solving practical problems, after establishing the functional relationship between variables, seeking the maximum value cultivates students' ability to study specific problems by using basic theories, which is also one of the purposes of learning mathematics. The teaching of maximum function can not only cultivate students' mathematical thought of combining numbers with shapes, but also cultivate students' study habit of rigorous thinking. Function thought is an important mathematical thought, which embodies the viewpoint of movement change and unity of opposites. This lesson plays a connecting role in the connection of junior and senior high school knowledge. Combining the maximum problem of function with the knowledge of inequality, equation, parameter range, analytic geometry, etc., it can often compile new comprehensive questions and comprehensively examine students' ability to analyze and solve problems by using function knowledge, thus becoming a high-level answer in college entrance examination and one of the hot spots in college entrance examination.
(B) Teaching objectives
Knowledge and ability goal: master the collocation method, a common method to find the maximum value of quadratic function, and cultivate students' mathematical thinking of combining numbers with shapes and their ability to study and solve specific problems by using basic theories.
Emotional goal: to experience the process of mathematics activities and the role of mathematics in real life, to stimulate students' enthusiasm for learning mathematics knowledge and to establish confidence in learning mathematics well.
Process goal: to cultivate students' cooperative communication through classroom learning activities, and to develop students' thinking habits of expression, abstraction and summary in the process of mutual communication, so as to gain successful experience.
Scientific research goal: under the guidance of teachers, students experience and experience the method of inquiry process.
(C) Teaching focus and difficulties
Key points: Find the maximum value of quadratic function by combining matching method with number and shape.
Difficulty: the maximum value of quadratic function in closed interval.
Second, talk about teaching methods and learning methods.
In junior high school, students have learned the knowledge of quadratic function. According to the content of this course and the actual level of students, this course mainly adopts the inquiry teaching method and the combination of teaching and practice. The teaching process is also the process of students' active construction. Teachers cannot ignore students' existing experience and try to force new knowledge into students' minds from the outside. Instead, we should take students' existing knowledge and experience as the growth point of new knowledge, and guide students to "grow" and discover new knowledge and experience from the original knowledge and experience. In the study of this course, students play the main role, actively think about the optimal strategy to solve the maximum value, and sum up their own problem-solving methods, actively incorporate knowledge into the constructed knowledge system, and truly "learn to learn".
Third, talk about the teaching process
(a) Presentation
For example, the zoo will build two rectangular panda houses with the same area against the wall. If the material that can be used to build the fence is 30 meters long, what is the maximum width of the panda room? What is the largest area of the Panda Pavilion?
Through this example, let students feel the necessity of solving the problem of maximum function in practical problems, thus stimulating their interest in learning this section.
Teaching means: use PPT to show the topic.
The teacher guides the students to discuss the answers, and gives guidance on individual answers, collects students' answers, picks out some answers, displays them on the physical projector, and makes comments. The students' solutions are mainly the function maximum method and the method of finding the maximum value by using basic inequality, which is evaluated by students and lays the foundation for the teaching of the maximum value of quadratic function in closed interval.
Teaching means: physical projector.
Excellent lecture notes on the maximum value of functions 2 i. Concepts of the maximum value and minimum value of functions.
By solving the maximum value of cited examples, students are guided to explain the concepts of maximum and minimum values of functions.
Generally speaking, let the function value of the function be: if the inequality holds for any one of the domains, it is called the minimum value of the function, and it is recorded as; If this inequality holds for any one of the domains, it is called the maximum value of this function.
Second, practice with examples.
Example 1. Find the maximum or minimum of quadratic function: teachers and students work together to complete an example. Senior one students should develop standardized writing formats and habits, and the rest of the questions should be completed by students. According to students' existing ability and experience, students can get the answers manually, and teachers will comment on them. Mind you, when you take any value, the function takes the maximum value.
The process of cultivating students' ability to explain, analyze and understand concepts and introducing the concept of maximum is designed according to the cognitive law of knowing the unknown from the known. The research of modern educational psychology holds that effective concept teaching is based on students' existing knowledge structure, so teachers must pay attention to finding the anchor point of new concepts in students' existing knowledge structure in the process of designing teaching, and guide students to master new concepts through assimilation or adaptation, so as to improve the knowledge structure. Let students start with the maximum value of practical problems and get the method of finding the maximum value (minimum value) of quadratic function from the familiar vertex characteristics of quadratic function images. Highlight the students' dominant position, give full play to the leading role of teachers, cultivate the rigor and transformation ability of thinking, and let students fully feel the solution of the maximum value of quadratic function through the change of interval. The relationship between symmetry axis and given interval should be discussed.
Teaching method: combining lectures with practice.
Example 2. Find the maximum and minimum values of a function under conditions.
Teachers guide students to think deeply step by step;
1. What is the relationship between the domain and the maximum value of the function?
2. What is the function to be studied after the transformation?
The objective function is to further deduce the combination of number and shape of the objective function and pay attention to the rigorous way of thinking, and further realize the interactive relationship between the domain, the range and the maximum value.
Teaching methods: students' independent inquiry
Third, summary.
ring
teaching process
design instruction
summary
1, the concept of maximum and minimum values of functions.
2. The relationship between the definition range and value range of the function and the maximum value of the function.
3. Matching method is more suitable for finding the maximum value (minimum value) of quadratic function, especially paying attention to the maximum value of function in closed interval can be solved by combining numbers and shapes.
4. Mathematical thinking: through the combination of numbers and shapes, changing ideas, summarizing methods and ideas, students' ability to analyze and solve problems is improved, which is helpful for subsequent problem learning and research.
Teaching method: student exchange summary.
Fourth, classroom exercises.
ring
teaching process
design instruction
class exercise
Find the maximum value of the following function
Theme design objectives:
1, see the basic content of this section.
2. Investigate the concept of quadratic function and students' transformation ability.
Teaching method: Let students perform on the blackboard.
Verb (abbreviation for verb) assignment
1, find the maximum 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 65,438+000 guest beds, and each bed can be fully occupied when the daily charge is 65,438+00 yuan. If the daily cost of each bed is increased, 65,438+00 guest beds will be rented out. In this way, in order to reduce investment and gain more profits, how much should the daily charge for each bed be increased?
Homework can not only reflect students' understanding and mastery of this section of knowledge, but also be a process of consolidating knowledge. Therefore, the design of homework is one of the keys to improve the quality of classroom teaching. The content should be close to the textbook and integrate the knowledge learned, which is a further improvement of ability.
Maximum value of function Excellent lecture notes 3 Teaching objectives
Master the maximum value of quadratic function and its solution.
focus
The maximum value of quadratic function and its solution.
difficulty
The maximum value of quadratic function and its solution.
First of all, introduce.
Maximum value of quadratic function:
Second, the case analysis:
Example 1: Find the maximum value of a quadratic function and the value when finding the maximum value.
Variable theme 1:
Variant 2: Find the maximum value of the function.
Variant 3: Find the maximum value of the function.
Example 2: Given that the maximum value is 3 and the minimum value is 2, find the range of values.
Example 3: If is the minimum of two real roots of quadratic equation.
Third, practice in class:
1. If there is a minimum value in the function, the maximum value is 2. If yes, then = _ _ _ _ _ _ _ _ _ _ _ _ _ _.
2. It is known that the minimum value is () for two real roots of a quadratic equation.
a、0 B、 1 C 、- 1 D、2
3. Find the maximum value of the function in the interval.
Fourth, review summary
This lesson includes the following:
1, the maximum value of quadratic function and its solution.
Homework after class
Class: () Class name _ _ _ _ _ _ _
First, the basic question:
1, function
A, there are six at most? B.is there a minimum value of 6? C. Is there a maximum value of 10? D, there are at most two
2, the maximum value of the function is 4, when =2, =5, then = _ _ _ _, = _ _ _ _.
Second, improve the problem:
3. Try to find the maximum value of the function, about the third year of high school.
4. When the function is known, the maximum value is 2, and the value is realistic.
5. What is known is the maximum and minimum of the two real roots of the equation.
Third, the question:
A function is known, in which the maximum and minimum values of the function are found, and the values of the independent variables corresponding to the maximum and minimum values of the function are found.