摘? 要:微積分從誕生之初就因為無窮小量(?x)到底是不是0的問題,產(chǎn)生過糾紛,被貝克萊大主教攻擊,從而引發(fā)了數(shù)學史上第二次危機。直到極限的提出和嚴格定義,才基本解決了這一問題。該文通過定義一個完整的計算過程中,所有的0大小都相等,所有的無窮小量大小都相等,也就是說,在計算過程中,0也是有大小的,0加0應該等于2個0,0減0應該等于0個0,它們和0也就是1個0都是不同的,保證了所有的0大小一致,就可以得到0/0=1。然后從導數(shù)定義和微積分基本定理出發(fā),通過完善四則混合運算,在0做除數(shù)的時候,采用除法優(yōu)先的原則,證明了某些連續(xù)函數(shù)上的尖點也有導數(shù),從另一角度探討了導數(shù)計算時?x是不是0的問題。
關鍵詞:0可以做除數(shù)? 除法優(yōu)先? 一一對應? 導數(shù)的重新定義? 0除以0等于1
中圖分類號:O13? ? ? ? ? ? ? ? ? ? ? ? ? ?文獻標識碼:A文章編號:1672-3791(2021)05(b)-0230-03
Abstract: Since the beginning of the birth of calculus, because of the infinite small quantity (?x) is zero or not in the end, there has been a dispute, was attacked by the Archbishop of Berkeley, which led to the second crisis in the history of mathematics. It was not until the limit was put forward and strictly defined that the problem was basically solved. In this paper, by defining a complete computation, all zeros are equal in magnitude, all infinitesimal quantities are equal in magnitude, that is, in the computation, 0 also has magnitude, 0 plus 0 should be equal to two 0's, 0 minus 0 should be equal to zero 0's, they are not the same as 0, that is, one 0.I make sure that all zeros are the same size, and I get it. Then, starting from the definition of derivative and the fundamental theorem of calculus, by improving the four mixed operations, when 0 is the divisor, using the principle of division first, it is proved that some of the sharp points on continuous functions also have derivatives, from another point of view, the problem of ?x is not 0 when the derivative is calculated is discussed.
Key Words: 0 can be a divisor; Division is preferred; One to one correspondence; The redefinition of the derivative; 0 divided by 0 is 1
3? 結語
綜上所述,連續(xù)函數(shù)在尖點處沒有導數(shù)的結論是錯誤的,只不過是因為平滑處的導數(shù)不會因?x的取值而改變,而尖點處會改變。而導數(shù)計算中?x是不是0呢?該文認為應該是0,而不是一個無限趨近于0而不等于0的數(shù),因為只有?x=0導數(shù)的值才具有唯一性,同時,像也只有在?x=0的情況下,才具備邏輯的嚴密性。
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