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根据我們領导学生数学小組的經驗,利用导数来研究中学課程中一些已知的事实,引起了学生們很大的兴趣。让我們来举一些几何学方面的例子。在导出了整数指数冪的求导数公式以后,我們就可以对比下列已知公式: K=πR~2,c=2πR; V_(圆柱)=πR~2H,S_侧=2πRH; V_球=4/3πR~3,S_(球面)=4πR~2。学生們不难发現,在右边一列中半径R的函数,是从左边一列中相应的函数进行微分而得到的。下面,我們只須說明这些公式之間发生联系的原因。我們仅仅指出关于圓的公式K=πR~2与c=2πR之間发生这种联系的原因(对于圓柱体和球的討論是类似的)。
Based on our experience in leading student mathematics groups, using derivative to study some known facts in the middle school curriculum has aroused students’ great interest. Let us give some examples of geometry. After deriving the derivative formula of integer exponentiation, we can compare the following known formulas: K=πR~2,c=2πR; V_(cylindrical)=πR~2H, S_side=2πRH; V_sphere= 4/3πR~3, S_(spherical)=4πR~2. It is not difficult for students to find that the function of radius R in the right column is derived from the corresponding function in the left column. In the following, we only need to explain why these formulas are related. We only point out the reason for this connection between the formulas K=πR~2 and c=2πR for circles (the discussion of cylinders and balls is similar).