1 from special to general, from concrete to abstract, grasp the "principal contradiction", cultivate students' ability to summarize and improve their ability to solve general problems.
There are several extremely important concepts in advanced mathematics, all of which start from solving practical problems, such as derivative.
Example 1 Let a point move along a straight line, let the position function of the moving point at time t be s=s(t), and find the instantaneous velocity of the moving point at time t0. Turn "unknown" into "known". First, the average velocity from time t0 to time t is: v=■=■, but the precise concept of moving point velocity at time t0 needs to be t→t0, that is, v =■■■.
Example 2 Let curve C be a graph with function y=f(x), and find the slope of curve C at (x0, y0). Firstly, the slope of secant is found and the definition of tangent is analyzed. The limit of secant slope is the slope of tangent, and k=■■
The essence of advanced mathematics lies in the mathematical thinking method of solving problems, which is often realized through the process of infinite change (taking the limit), which is also the difference between advanced mathematics and elementary mathematics. Aside from the concrete problems of the two, the concept of function derivative is obtained from their * * * properties in quantitative relations. Derivative is the limit of a special mode, which is the ratio of function increment to independent variable increment. Starting with "special problems", we get "general problems". As Carker said, "Generalization and abstraction are the most important functions of mathematics. It is precisely because of generalization and abstraction that mathematics can be so effective. " In daily life, too, we should grasp the main contradictions of things, summarize and generalize them when things happen, and improve our ability to solve general problems.
2 from the integral transformation that "wisdom lies in transformation"
What is wisdom? The solution to seemingly unsolvable problems is wisdom. "Cao Chong calls an elephant" and transforms the elephant into a stone. The weight of the stone is the total weight of the elephant. As the Book of Changes says, "Poverty leads to change, change leads to communication, and general rules last for a long time". Wisdom lies in change, not directly but indirectly, so it is flexible, magical and ingenious. Although indefinite integral has certain methods and skills, the method of reduction is flexible. Through the following examples, I realize that wisdom lies in transformation.
Example 3 Finding ■■dx
Solution1:■■ dx =■■ dx =■■ dx-■ secxtanxdx = tanx-secx+c
Solution 2: ■■ DX =■■ DX = ■■ DX = -■ Sec2 (■ -■) D (■ -■) =-Tan (■ -■)+C.
Solution 3: ■■ DX = ■■ DX = ■■ DX = 2 ■■ D (1+tan ■) =-■+C.
Solution 1, multiply by numerator and denominator1-sinx; Solution 2 uses the formula cos2x =■variant；; Scheme 3: skillfully use sin2x+cos2x= 1 deformation. Although the results are in different forms, they are all correct.
Example 4 Finding ■■dx
Solution1:■■ dx =■■ dx =■1dx-■■ dx = x-ln (1+ex)+C.
Solution 2: ■■ dx = ■■ dx = -■■ d (e-x+1) =-ln (e-x+1)+c.
=x-ln( 1+ex)+C
Solution 3: let 1+ex=t, x=ln(t- 1), dx=■dt.
■■dx =■■dt =■(■-■)dt = ln(t- 1)-lnt+C = x-ln( 1+ex)+C
Example 5 Find ■■.
Solution1:■■ =■■ dx =■■■■■ d (x10+1) = lnx-■ ln (x/kloc-0+1)+c
Solution 2: ■■ =■■ =■■ (■ -■) dx10 =■ [lnx10+ln (x10+1)]+c = lnx-■ ln (x/).
Solution 3: ■■ =■■ = -■■■ = -■ ln (1+x-10)+c = lnx-■ ln (x10+1)+c.
Different ways of thinking, different angles of thinking, different methods and different forms of results. Therefore, indefinite integral can be regarded as a way to exercise thinking mode, flexibly deform and innovate thinking.
3 Do problems-do things-be a man
Wei Yi pointed out: "Rigidity is to mathematicians what morality is to people." After learning the important limit ■■= 1, and the product of bounded function and infinitesimal is infinitesimal. The following four limits: (1)■■■, (2) ■, (3)■xsin■, (4)■xsin■, students often make mistakes. (1)(4) is an important limit and the result is1; (2)(3) Using the property of infinitesimal, the result is 0. Another example is: (1) ■■■■, (2)■■■, (3) ■■■ DX, (4) ■■■ DX, (5) ■■■, (6)■■ They are similar in form, but they are different in usage, and they will make mistakes if they are not careful. Learning knowledge should "know what it knows, not what it doesn't know, but what it knows", be down-to-earth and not sloppy. Missing is thousands of miles away. In the study of advanced mathematics, don't "seem" or "almost", otherwise "it is right at first sight and wrong at once". The same is true of being a man and doing things.