How to solve the type questions of high school functions

By definition, there is (sinx).'=lim[sin(x+ x)-sinx] ( x), where x 0, will sin(x+ x)-sinx, that is, sinxcos x+cosxsin x-sinx, since x 0, so cos x 1, thus sinxcos x+cosxsin x-sinx cosxsin x, then (sinx)'=lim(cosxsin x) x, here an important limit must be used, when x 0, lim( sin x) x=1, then (sinx)'=cosx.

Similarly, (cosx)'=lim[cos(x+ x)-cosx] x, where x 0And at this time cos(x+ x)-cosx cosxcos x-sinxsin x-cosx -sinxsin x,(cosx)' lim(-sinxsin x) x=-sinx.

lnx)’=lim[ln(x+△x)-lnx]/△x, △x→0.ln(x+ x)-lnx=ln(1+ x x), and a limit is also needed here: when t 0, ln(1+t) t

So we have (lnx)'=lim[ln(1+ x x)] x=( x x) ( x)=1 x

And with the formula logax=lnx lna=(loga e)lnx, we have found (lnx)'=1 x, so [logax]'=[(loga e)lnx]'=(loga e) x

The derivation of these formulas requires the use of some important limits that are not mentioned in secondary school textbooks, so the textbooks do not derive formulas and directly write the results. That's all for my answer, if you don't understand anything, please continue to discuss.

The types of mathematical functions and problem-solving skills in the first year of high school include: substitution method, monotonicity method, undetermined coefficient method, commutation method, and construction equation method.

First, the substitution method

There are two main ways to substitute the method, one is to appear in the multiple-choice question, that is, to directly bring the answer options of the question into the question for verification, which is also a relatively fast method, and the other is to find the symmetry function of the known function about a certain point or a straight line, and bring in the expression formula of the function or the nature of the function, and the direct solution of the problem is usually suitable for fill-in-the-blank questions, and the difficulty is not too great.

2. Monotonicity method

Monotonicity is a very common method of solving problems when solving functions or the most worthwhile time, and the monotonicity of functions is a particularly important property of functions, and it is also the focus of the annual college entrance examination. However, due to the lack of understanding of basic concepts, many students are not clear about the examination questions, and they are prone to make mistakes when answering such questions. The following is an explanation of the matters that need to be paid attention to when doing this kind of question, so as to attract the attention of students.

3. Pending coefficient method

The key to solving the problem with the undetermined coefficient method is to correctly list the equations or equations based on the functional relationships between known variables. To determine whether a problem is solved by the undetermined coefficient method, the main thing is to see whether the mathematical problem solved has a certain definite mathematical expression, and if so, it can be solved by the undetermined coefficient method.

The basic steps of using the undetermined coefficient method to solve the function problem are: 1. First, determine the analytic formula of the problem containing the undetermined coefficient; 2. According to the condition of identity in the problem, list a set of equations with undetermined coefficients; 3. Solve the system of equations with the basic properties of the function or eliminate the pending coefficients, so that the problem can be solved.

Fourth, the exchange method

The commutation method is mainly used to solve compound function problems, and the form of a small function expression expressed by a variable is called commutation method, which can reduce the difficulty of the problem and facilitate observation and understanding.

5. Structural equation method

No matter what kind of functional necrosis, the equation of the function can undoubtedly reduce the difficulty of solving the problem in the application, so the equation of the constructor is also a kind of problem-solving skill that is often used, especially in the finale of the college entrance examination solution question, the step of the constructor can also achieve a high score, so we must pay attention to the skill of the constructor method.

The high school math function question types and problem-solving skills are as follows:

1. Establish a library of basic question types and basic problem solutions. The knowledge structure and content have been sorted out and memorized, and we are going to carry out actual combat. As with the knowledge points, each module is divided into several basic question types and several special topics.

2. For a type of question, you must read the example questions or understand the teacher's explanation, and then do the same type of questions according to the teacher's solution. Don't be innovative, or stick to your biased approach to solving problems and don't give up. I don't object to the tactics of the sea of questions, but you have to make the audition accurately, which question type will not drill into the corresponding sea of questions, and the question type that you are already very proficient in will practice less.

That is, the so-called targeting, and the focus should be highlighted. And in the process of doing, you must constantly summarize and reflect, otherwise you will not improve even if you swim into the Pacific Ocean. If you don't master a type of question, you will practice it repeatedly, and you won't know five questions in one way, and you won't know ten questions in five questions.

Mathematical functions

A mathematical function is a relationship that causes each element in one set to correspond to a unique element in another (possibly identical) set. The function consists of an independent variable and a dependent variable, the dependent variable changes with the change of the independent variable, and when the independent variable takes a unique value, the dependent variable has and only a unique value corresponding to it.

Make an image of the function y=x-1.

Parse. x-1，（x≥1）

x21, first of all, the positive and negative of the percolation x11 is discussed, 1-x, (x<1).

Remove the absolute value. /y=x-1

y=1-x segmentation function of the image segmentation.

x<>

Law. y=if(x)lImage drawing:

Y=f(x) retains the image of the upper part of the x-axis.

The image below the x-axis is folded above the x-axis.

It is said that high school functions are difficult, and the ruler of functions is so difficult to learn because it is varied, and there may be many different variations or combinations of the same formula principle and the same method.

Many students memorize formulas, some fixed functional properties or images, and do not use them comprehensively. It's like giving an ordinary person a toolbox, but he can't assemble machines and equipment as skillfully as a mechanic. Why?

The principle is the same, not understanding, lack of practice, incorrect method of practice, and lack of mastery of relevant skills and methods.

The combination of functional knowledge will produce many changes, but this change is usually regular, and we can only grasp its laws through in-depth analysis and research.

Many students find functions difficult to learn because they can't adapt to the changes in functions and are not good at grasping the invariance in the changes.

There are multiple-choice questions, fill-in-the-blank questions, and the last question of the solution question, which are basically the application of the knowledge points of the function, and the multiple-choice questions and fill-in-the-blank questions are the types of questions with strong skills.

The skills that can be used to solve function problems are as follows: substitution method, monotonicity method, undetermined coefficient method, commutation method, and system of equation construction method.

Substitution method. There are two main ways to substitute the method, one is to appear in multiple-choice questions, that is, to directly bring the answer options of the question into the question for verification, which is also a relatively fast method.

The other is to find the symmetry function of a known function about a certain point or a certain line, bring in the expression formula of the function or the properties of the function, and solve the problem directly, which is usually suitable for fill-in-the-blank questions, and the difficulty is not too great.

Function question type: Solve the analytical formula of the number of ballasts. The common methods for finding the analytic formula of the function are the coefficient method, the commutation method, the matching method, and the system of equations method.

The ancient Chinese word "Han" and the word "contain" are common, and both have the meaning of "contain". The definition given by Li Shanlan is: "Heaven is contained in the formula, which is a function of heaven." "In ancient China, the four characters heaven, earth, people, and things were used to represent four different unknowns or variables.

The implication of this definition is: "Whenever a formula contains the variable x, then the formula is called a function of x." So "function" means that the formula contains variables.

Let the domain of the function f(x) be d, and the interval i is contained in d. If for any two points x1 and x2 on the interval and when x1 is for any two points x1 and x2 on the interval i, when x1f(x2), then the function f(x) is said to be monotonically decreasing on the interval i. Monotonically increasing and monotonically decreasing functions are collectively referred to as monotonic functions.

First of all, we should briefly review the various properties of functions (monotonicity, maximum and minimum values, periodicity, parity, etc.), and then review various elementary functions (quadratic functions, exponential functions, logarithmic functions, power functions, etc., focusing on mastering the properties of quadratic functions, because the properties of quadratic functions are often used, especially the distribution of its roots must be mastered), and then we should review the zero point theorem and the derivative of functions, the derivative function is a very important tool to solve functional problems, We must master how to find its monotonicity and the most value, and finally enter the actual combat, and constantly summarize a variety of different function question types and their solutions in the actual combat, about this it is best to do the questions about the function in the college entrance examination questions in previous years, and if possible, you can also do the college entrance examination questions in other provinces. According to my own summary and the college entrance examination questions of each year, the question type of function in high school is generally placed in the position of the penultimate or third largest question, and the difficulty is generally not very large, and if it is placed in the last question, the difficulty will increase. Generally speaking, there are three main types of function questions, the first of which is generally to find the monotonic interval of the function (note:

First of all, it is necessary to define the domain (generally direct derivation is sufficient), which is the first principle of doing function questions, otherwise you are very prone to make mistakes! The second question might be to find the extreme or the maximum, or to find the range of a certain parameter (pay attention to the use of numbers and shapes to discuss the use of the idea of classification). The third question is generally a proof inequality, which is generally a constant proof problem (Method:

Function method or variable separation method, specific problems are analyzed), of course, the second and third questions may be reversed! In short, the function is the main line that runs through the whole high school, and it occupies a very important position, so you must master it! Finally, I would like to emphasize that the mind must be flexible in doing the question type here, and it is necessary to analyze it according to the specific problem, and it is best to accumulate and summarize the question types in this aspect!

Well, that's all for now, I hope it helps you! I wish you success in the college entrance examination!

To grasp the main contradictions, the first is to define the domain, then to the individual characteristics, such as monotonicity, parity, symmetry, periodicity, concave and convex cases, etc., and then to classify and discuss the combination of numbers and shapes, and finally to do more exercises, practice makes perfect.

"Dragon Gate Feature" I feel pretty good.