Lecture notes on mathematics in senior two: the concept of derivative

The following is about "Lecture Notes on Mathematics in Senior Two: The Concept of Derivative" for your reference!

I. teaching material analysis

The concept of derivative is the content of the first chapter 1. 1.2 in the new high school textbook 2-2. On the basis of the physical average speed, instantaneous speed and average change rate learned in the last class, this paper expounds the relationship between the average change rate and instantaneous change rate, and draws the concept of derivative from examples, which lays a foundation for better learning the geometric meaning and application of derivative in the future.

The new textbook has changed a lot in dealing with this problem. The difference between it and the old textbook is that it starts with the average change rate and defines the derivative with an intuitive "approximation" method.

Question 1 average inflation rate of balloon-→ instantaneous inflation rate

Question 2: Average speed of high platform diving-→ instantaneous speed-→

According to the analysis of the structure and content of the above textbooks, based on the students' cognitive level, the following teaching objectives, key points and difficulties are formulated.

Second, the teaching objectives

1, knowledge and skills:

Through the analysis of a large number of examples, we have experienced the transition from the average rate of change to the instantaneous rate of change, and learned the actual background of the concept of derivative, knowing that the instantaneous rate of change is the derivative.

2, process and method:

① Cultivate students' abilities of observation, analysis, comparison and induction through hands-on calculation.

(2) Through the exploration of problems, we can realize the mathematical thinking method of approaching, analogizing and exploring the unknown with the known, and from the special to the general.

3. Emotions, attitudes and values:

Understanding the connotation of derivative from the viewpoint of movement makes it no longer difficult for students to master the concept of derivative, thus stimulating students' interest in learning mathematics.

Iii. Key Points and Difficulties

Emphasis: the formation of the concept of derivative and the understanding of its connotation.

Difficulties: On the basis of the average change rate, explore the instantaneous change rate, deeply understand the connotation of derivative, and guide students to break through the difficulties by approaching observation.

Four, teaching ideas (specific in the following table)

Teaching content, teacher-student interaction design ideas, creating scenarios, and introducing new lesson slides.

Review the thinking questions left over from last class:

In platform diving, there is a functional relationship between the height h (unit: m) of the athlete relative to the water surface and the time t (unit: s) after taking off. h(t)=-4.9t 2+6.5t+ 10。 Calculate the average speed of athletes during this period and think about the following questions:

(1) Are the athletes still here these days?

(2) Do you think there is anything wrong with using the average speed to describe the athletes' state of motion?

First of all, review the thinking questions left over from last class:

On the basis of students' discussion on the exchange results, it is proposed that the average speed of athletes during this period is "0", but we know that athletes are not "static" during this period. Why is this happening?

Arouse students' curiosity and realize that the average speed can only roughly describe the motion state of an object in a certain period of time. In order to describe the motion of an object more accurately, it is necessary for us to study the speed at a certain moment, that is, the instantaneous speed.

Let students walk into the classroom with questions, stimulate students' curiosity, explore and show their connotation.

According to students' cognitive level, the formation of concepts is divided into two levels:

Combined with the diving problem, the definition of instantaneous velocity is clear.

Question 1: Please think about how to find the instantaneous speed of athletes, such as the instantaneous speed when t=2.

Put forward the first question, organize students' discussion, guide students to naturally think of choosing a specific moment, such as t=2, study the average speed change near it, find the train of thought of the problem, and thus make the abstract problem concrete.

Understanding the connotation of derivative is the key and difficult point in this course. By setting up different levels of doubts, students will be pushed to the center of the problem, so that students can operate and feel intuitively, highlight key points and break through difficulties.

Question 2: Please keep thinking. What value do you want to calculate when δ t takes different values?

δt

δt

-0. 1 0. 1

-0.0 1 0.0 1

-0.00 1 0.00 1

-0.000 1 0.000 1

-0.0000 1 0.0000 1

……….….…….…

Students' cognition of concepts needs a lot of intuitive data, so I ask students to use calculators in groups to complete question 2.

Help students understand the mathematical thinking method of starting from the average speed and "exploring the unknown with the known", and cultivate students' practical ability.

Question 3: When Δ t tends to 0, what is the trend of average speed?

δt

δt

-0. 1 - 12.6 1 0. 1 - 13.59

-0.0 1 - 13.05 1 0.0 1 - 13. 149

-0.00 1 - 13.095 1 0.00 1 - 13. 1049

-0.000 1 - 1300995 1 0.000 1 - 13. 10049

-0.0000 1 - 13.09995 1 0.0000 1 - 13. 100049

……….….…….…

On the one hand, we discuss in groups, perform on stage and show the calculation results. At the same time, we say: at the moment of t=2, when δ t tends to 0, the average speed tends to a certain value-13. 1, that is, the instantaneous speed, and we will approach this idea for the first time; On the other hand, with the help of animation, guide students to observe, analyze, compare and summarize through multiple channels, and experience the idea of approaching for the second time. For the convenience of expression, simple symbols are used in mathematics, namely

The combination of numbers and shapes clears students' thinking obstacles, breaks through teaching difficulties and experiences the beauty of simplicity in mathematics.

Question 4: How to express the instantaneous speed of athletes at a certain moment?

Guide students to continue thinking: how to express the instantaneous speed of athletes at a certain moment? The students realized that it would replace 2.

Compared with the old textbooks, the concept of limit is not mentioned here, but the instantaneous speed of the moment is defined by vivid approximation, which is more in line with the students' cognitive law, improves their thinking ability and embodies the special to general thinking method.

Use other examples to abstract the concept of derivative.

Question 5: How does the volume represent the instantaneous inflation rate of the balloon?

By analogy with the instantaneous velocity problem learned before, guide students to get the expression of instantaneous inflation rate.

Active teacher-student interaction can help students see the connection between knowledge points, help to reorganize and transfer knowledge, and find out the instantaneous change rate of mathematics in different practical backgrounds, which has different practical significance for different practical problems.

Question 6: If the functions in these two rate of change problems are expressed as, what is the instantaneous rate of change of the functions?

Under the foreshadowing of the first two problems, we further put forward the instantaneous rate of change of the function we study here, that is, the derivative of the function, and write it as follows

(It can also be written as)

Guide students to abandon the practical significance of specific problems, get the definition of derivative abstractly, from easy to difficult, from special to general, and help students complete the leap of thinking; At the same time, the background of derivative is mentioned, so that students can feel the influence of mathematical culture and feel that mathematics comes from life and serves life.

Step by step, extend

Extended example 1: crude oil needs to be cooled and heated when it is refined into different products such as gasoline, diesel oil and plastics. If the crude oil temperature (unit:) is x h

(1) Calculate the instantaneous change rate of crude oil temperature at the 2nd and 6th hours, and explain its significance.

(2) Calculate the instantaneous change rate of crude oil temperature at the 3rd and 5th hours, and explain its significance.

Steps:

① Enlighten students to sum according to the definition of derivative.

② Since we have obtained that the instantaneous rate of change of crude oil temperature is -3 and 5 for 2h and 6h, respectively, can you explain its meaning?

(3) Can you solve problem 2 in the same way?

(4) Teachers and students use induction, and the derivative is the instantaneous rate of change, which can reflect the speed of object change.

Ask questions step by step to guide students to explore the connotation of derivative.

Cultivating students' application consciousness is one of the important ideas advocated by senior high school mathematics curriculum standards. Taking specific problems as the carrier in teaching can deepen students' understanding of the connotation of derivatives and experience the application of mathematics in real life.

Variant exercise: It is known that the displacement (m) of the object motion satisfies the relationship with time, t(s) =-2t2+5t (1), and the instantaneous velocity of the object in the 5th and 6th seconds is found.

(2) Find the instantaneous velocity of the object at time t..

(3) Find the motion acceleration of the object at time t and judge what motion the object is making.

Students finish independently, perform on stage and experience the idea of approaching for the third time.

The purpose is to let students learn to look at the physical model from the perspective of mathematics, establish the relationship between disciplines, grasp the changing law of things more deeply, and summarize and internalize knowledge.

1, the concept of instantaneous velocity

2. The concept of derivative

3. Thinking method: "Explore the unknown with the known", method, analogy, from special to general.

Guide the students to discuss, complement each other and then answer. The teacher comments and gives them slides.

Let the students sum up by themselves, not only knowledge, but also mathematical thinking methods. This is a process of knowledge reorganization, multi-dimensional integration and high-level self-cognition, which can help students build their own knowledge system, clarify the context of knowledge and develop good study habits.

Assignment, blackboard writing design (required reading) Page 10 Exercise Group A, Questions 2, 3 and 4.

(Optional): Thinking 1/page Exercise Group B1is the feedback of students' information, which can find and make up for the shortcomings in teaching, while paying attention to individual differences and teaching students in accordance with their aptitude.

The attached blackboard book is clear and neat in design, which is convenient for highlighting knowledge objectives.

Verb (abbreviation of verb) learning method and teaching method

Learning methods and teaching tools

Studying law:

(1) Cooperative learning: guide students to discuss in groups, cooperate and communicate, and discuss problems together. (Such as the handling of question 2)

(2) Autonomous learning: through personal experience, guide students to participate in mathematics activities through speaking, thinking and hands-on. (Such as the handling of question 3)

(3) Inquiry learning: guide students to exert their subjective initiative and actively explore new knowledge. (such as the handling of instances)

Teaching tools: computer, multimedia, calculator.

Teaching methods: The whole class revolves around the teaching principle of "all for students' development", highlighting ① teacher-student interaction and * * * exploration. (2) guidance-teacher guidance, step by step.

(1) new curriculum introduction-asking questions to stimulate students' curiosity.

(2) Understand the connotation of derivative-the combination of numbers and shapes, calculate by hands, organize students to explore independently, and get the definition of derivative.

(3) Example handling-always start from the problem and ask questions at different levels, so that they can enjoy the knowledge in the exploration.

(4) Variant exercise-deepen the understanding of the connotation of derivatives and consolidate new knowledge.

Evaluation and analysis of intransitive verbs

This course shows a complete mathematical inquiry process from average speed to instantaneous speed and then to derivative. Ask questions, calculate and observe, discover laws and give definitions, so that students can experience the process of knowledge rediscovery and promote personalized learning.

From the old textbooks, the starting point of derivative concept learning is limit, that is, from the limit of sequence, to the limit of function, and then to derivative. This concept construction method is logical and systematic, but it is difficult for students to understand the formal definition of limit, which also affects their understanding of the essence of derivative.

The new textbook does not introduce the formal definition of limit and related knowledge, but defines derivative in an intuitive way.

By calculating the list, we can intuitively grasp the changing trend of the function (including the descriptive definition of the limit), which is convenient for students to understand;

The advantages of defining derivatives in this way are:

1. Avoid the contradiction between students' cognitive level and knowledge learning;

2. Pay more attention to the understanding of the essence of derivative;

3. Students have a rich intuitive foundation and a certain understanding of approximation thought, which is conducive to learning a strict definition of limit in the primary stage of university.

(Attached) Blackboard Design