First, keep pace with the times to understand the "double basis", put the basis and essence of "derivative" into practice, and improve students' mathematical thinking ability.
A development of the new curriculum standard in the curriculum concept and goal is to emphasize the understanding and understanding of the essence of mathematics in mathematics learning and teaching. Whether it is basic knowledge, basic skills, mathematical reasoning and argumentation, or the application of mathematics, we must firmly grasp this main line. In the teaching of "derivative", through the re-study of the nature of function, we can once again enhance our understanding of the concept and essence of function. By comparing and solving problems, let students feel the superiority of derivative method. For example, 05 Shandong College Entrance Examination: it is known that x= 1 is an extreme point of the function f (x) = mx3-3 (m+1) x2+NX+1,where m, n∈R and m < 0 (i) find the relationship between m and n. (ii) Find the monotone interval of f(x). From f ′ (1) = 0, we get n=6+3m, and substitute it into the original formula to get f (x) = mx3-3 (m+1) x2+(6+3m) x+1. In teaching practice, we must repeatedly train to find the monotone interval of the function with parameters and the maximum value of the function in the closed interval, so that practice makes perfect. You can also compile a certain amount of judgment questions and analysis questions, so that students can give appropriate counterexamples and cultivate their critical and profound thinking. You can also use derivative summation through explanation: Sx =1+2x+3x2+...+nxx-1to cultivate students' flexibility in thinking, and to enjoy the fun of thinking at any time. At the same time, we should also guide students to treat the derivative method dialectically, give up the root and get rid of the end, and prescribe the right medicine. For example, if f(x)=(x- 1)2 and g(x)=x2-2 are known, the monotone interval of f [g (x)] can be judged by using the images of two quadratic functions and the monotonicity law of the composite function, without being constrained by the derivative method. The average theorem of two available numbers and the problems solved need not be constrained by the derivative method. Make students learn to analyze specific problems and deal with math problems flexibly.
Second, pay attention to the teaching of practical application problems and cultivate students' application consciousness and innovative thinking.
In addition to learning some basic knowledge and improving some basic operation skills, the chapter "Derivative" is largely the embryonic form of "Applied Mathematics". It has become the fundamental purpose and eternal theme of "derivative" learning to solve the problems left over from the study of senior one function to produce practical problems in life. Therefore, in order to better implement the concept of "new curriculum standard", it is necessary to explain in detail the paper-cut folding box problem, the maximum value of the rectangle inscribed in the circle and the time-saving problem of transportation, so that students can have a clear thinking path about the maximum value of unimodal function in the open interval. Students can also realize that calculus is active water through the repeated training of tangent problem in kinematics and υ = s ′ (t) a = υ ′ (t). Really realize that mathematics comes from life and in turn serves the true meaning of life. At the same time, it can also compile insurance problems, tuition payment problems, construction cost problems and gasoline saving problems. Make the application problems keep pace with the times. Really cultivate students' ability to use mathematics. Make students develop an outlook on life that cares about life and the national economy and people's livelihood, and truly reflect the importance of "education is life". The cultivation of mathematics application consciousness can not be solved by doing more application problems. It is closely related to the openness, activity and democracy of peacetime teaching. It is not the vigorous noise in the classroom, but the education of "sneaking into the night with the wind and moistening things silently".
Three, the correct use of modern information technology, strengthen the integration of modern information technology and "derivative" content.
The extensive application of modern information technology is profoundly affecting the content of mathematics topics, mathematics teaching methods and learning methods. The advantages of information technology in teaching mainly lie in: fast calculation function, rich graphic presentation and production function, large data processing function and interactive learning and research environment. In teaching, we should properly use modern information technology and give full play to its advantages. Helping students to understand the basic concept of derivative and the extensive function of derivative method will undoubtedly play a role in giving timely help and icing on the cake. In the teaching of "derivative", there are many examples of using modern information technology to stimulate students' various senses and fully acquire colorful perceptual knowledge. For example, when explaining the definition of tangency of a point on a curve, we can use the animation effect of the geometric sketchpad to demonstrate how the secant of the curve becomes tangency, so as to understand the formation process of the limit thought. When explaining the extreme value of a function, you can draw an image of y=x3-4x+4 with the geometric sketchpad, so that students can feel the relationship between the monotonicity of the function and the extreme point. Let students realize that function is no longer the abstract "master of the world", but actually stands out in front of them. When explaining the wide application of derivatives, we can use multimedia to introduce topics. For example, the scene of rocket launch, the lens of product packaging and the speed of object movement make students truly realize that the application of derivatives is vast and inseparable from real life. Finally, when explaining the background and historical significance of the establishment of calculus, photos of Newton and Leibniz, the inventors of calculus, can be displayed by computer. Subtitles can also be used to show the arduous exploration of Descartes, Barrow, Fermat, Newton and Leibniz and the process of germination, development and establishment of calculus, so that students can realize that the creation of calculus is not the inspiration of one person, but the result of unremitting efforts of several generations and the crystallization of collective wisdom. So as to encourage students to make continuous progress and climb the scientific peak.
Fourth, attach importance to mathematical culture, guide derivative teaching with dialectics, and promote the formation of students' scientific outlook.
Mathematics is full of contradictions and dialectics. Contradictions are everywhere and dialectics is everywhere. The essence of extreme thought process is an excellent example of qualitative change caused by quantitative change. When explaining the idea of limit, students can really realize that the definition of high school tangent is no longer an extension of middle school tangent, but the limit of secant, instantaneous speed is the limit of average speed, acceleration is the rate of change of speed to time, angular velocity is the rate of change of angle to time, current is the rate of change of electricity to time and so on. Let students realize that extreme thinking and derivative methods are indeed a panacea to deal with "changeable" mathematics. It can also explain to students that the introduction of derivative has brought a revolution in mathematics. The mathematical basis of the whole classical mechanics is calculus in essence, which enables students to form a scientific view of loving, learning and respecting science. In teaching, we can also differentiate and analyze similar concepts such as continuous point, discontinuous point, derivable point, extreme point, extreme point and point with derivative of 0, so that students can develop a rigorous academic style. We can also appreciate the outstanding wisdom of ancient mathematicians in China and the outstanding talent in dealing with mathematical problems through the germination of the extreme thought of "one foot pestle, half a day is inexhaustible" in ancient times and the excellent example of Liu Hui's successful application of extreme thought in the Three Kingdoms period. At the same time, by introducing the outstanding contributions of Newton, Leibniz, Barrow and Fermat in the process of establishing calculus, students can think: Why did China only have sporadic seeds of calculus, while western scientists were able to establish a complete theoretical system of calculus? So as to realize the differences between Chinese and western cultural and scientific views, and then stimulate students' determination to be diligent and inquisitive and serve the motherland.
In a word, derivative, as an important part of high school mathematics content, is of great significance to teachers and students, both in the face of college entrance examination and in their later lives. I believe that the infiltration of the new curriculum standard concept will definitely inject a clear spring into derivative teaching and inspire endless vitality.