The monotonicity of a function is to study whether its function y increases or decreases when the independent variable x increases. For example, the monotonic increase of function is characterized by "Y increases with the increase of X". Different from the parity of a function, the parity of a function is to study whether Y becomes an inverse number when X becomes an inverse number, that is, the symmetric nature of the function.
The monotonicity of a function is similar to the extreme value of a function, which is a local property of the function and may not be available in the whole domain. This is different from the parity of function, the maximum and minimum of function, which are the properties of function in the whole domain.
The research method of function monotonicity also has typical significance, which embodies the general method of function research, that is, strengthening the combination of "number" and "shape", from intuition to abstraction; From the special to the general, firstly, with the help of observation, analysis and induction of function images, the intuitive characteristics of function increase and decrease are found, and then the numerical characteristics of increase and decrease are further quantified, so as to be further described by mathematical symbols.
The concept of function monotonicity is the basis of studying the monotonicity of specific functions, and it has important applications in studying the range, definition, maximum value and minimum value of functions. It also has important applications in the research of solving inequalities, proving inequalities, and the properties of sequence. It can be seen that the monotonicity of function has important applications both inside and outside the function, so it has a core position in mathematics.
The key point of teaching is to guide students to make an abstract symbolic description of the feature that the function "Y increases (or decreases) with the increase of X" in the interval (a, b): take x 1, x2 arbitrarily in the interval (a, b), and when X 1 < X2, there will be F (X2) > F (X
Two. Target and target analysis
This lesson requires students to understand the monotonous meaning of a function in a certain interval, and master the methods (steps) to prove that a simple function has a certain monotonicity in a certain interval with the definition of monotonicity of a function.
1. Use specific examples to illustrate whether a function is a increasing function or a subtraction function in a certain interval;
2. It can be illustrated that the function is monotonous in the subset (interval) of the definition domain, but not necessarily in the whole definition domain, which shows that the monotonicity of the function is the local property of the function;
3. For a specific function, we can use the definition of monotonicity to prove whether it is a increasing function or a subtraction function: arbitrarily take x 1, x2, let X 1 < X2, find the difference F (X2)-F (X 1), and then judge whether the difference is positive or negative, thus proving whether the function is in the interval.
Three. Diagnosis and analysis of teaching problems
The existing cognitive basis of students is that they have studied the concept of function in junior high school and initially realized that function is a mathematical concept that describes the quantitative relationship of some movement changes; After entering high school, I further studied the concept of function and realized that function is the corresponding relationship between two groups of numbers. Students also learned that there are three ways to express functions, especially with the help of images, which can intuitively examine the characteristics of functions. In addition, several simple and concrete functions, such as linear function, quadratic function and inverse proportional function, were studied, and their images and properties were studied. It is particularly noteworthy that students have the experience of comparing two numbers by using the properties of functions.
"The image is rising and the function is monotonically increasing; The image is declining and the function is monotonically decreasing. " It is not difficult for students to intuitively describe the monotonicity of functions only from the perspective of images. The difficulty lies in abstracting the monotonicity of concrete and intuitive functions and describing it with mathematical symbolic language. That is to say, for "any x 1 < X2, there is f", the special property of "with the increase of x, y increases in a certain interval" (monotonically increasing) is expropriated.
In teaching, through the study of the image and numerical change characteristics of specific functions such as linear function and quadratic function, it is concluded that "image is rising", and correspondingly, "with the increase of X, Y also increases", and the theory of monotonic increase is preliminarily put forward. Students try to describe the general situation through discussion and communication, and put forward that "in a certain interval, if there is f (x 1 < x2) for any x1< x2, the function has the characteristics of" image rising "and" y increasing with x ". Furthermore, the definition of monotonicity of function is given, and then students can understand this concept through analysis and practice.
It may be unrealistic to try to complete students' real understanding of monotonicity of functions in one class. In the future study, students can judge the monotonicity of the function, find the monotonicity interval of the function, use the monotonicity of the function to solve specific problems, and gradually understand this concept.