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在讲《用二分法求方程的近似解》这堂课时,有同学提问:“为什么用二分法求函数零点时,每次都要取已有闭区间的中点?如果每次都取某个三等分点,循环n次后所得精确度会不会相对更高一些?”我们试从概率统计观点出发,通过计算最后含零点区间长度的数学期望,证明通常的二等分法实际上是最优的.最后,我们给出这个结论在第二十届(2017年3月)北京市高中数学知识应用竞赛决赛第五题中的应用.
In speaking about “use dichotomy to find the approximate solution to the equation” this class, some students ask: “Why use the dichotomy function zero, each time you have to take a closed interval of the middle point? A third of the points, the cycle after n times the accuracy will be relatively higher? ”We try from the probability of statistical point of view, by calculating the length of the last containing zero mathematical expectations, to prove that the usual dichotomy practice Finally, we give the conclusion of this conclusion in the fifth (March 2017) Beijing High School Mathematical Knowledge Application Contest fifth application.