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在数学解题过程中,遇到用常规方法很难或不能解决的问题时,不妨尝试构造数学模型来解题。下面笔者略举数例,便可见其巧妙之处。1构造点模型解题例1求函数:y=(x~2+4)~(1/2)+(x~2-8x+25)~(1/2)的值域。解析:将原函数变形,则y=((x-0)~2+(0-2)~2)~(1/2)((x-4)~2+(0+3)~2)~(1/2),构造两个定点A(0,2),B(4,-3),一个动点P(x,0),所求的问题表示:求点P到A,B两点的距离之和的最大值和最小值。利用三角形两边之和大于第三边的性质可得:y_(min)=(41)~(1/2),而无最大
In the process of solving mathematical problems, encountered in the conventional method is difficult or can not solve the problem, try to construct a mathematical model to solve the problem. Here I give a few examples, we can see its cleverness. 1. Construct Point Model Solution Example 1 Find the function: y = (x ~ 2 + 4) ~ (1/2) + (x ~ 2-8x + 25) ~ (1/2). Analysis: The original function deformation, then y = ((x-0) ~ 2 + (0-2) ~ 2) ~ (1/2) (x- 4) ~ 2 + (0 +3) ~ 2) ~ (1/2), we construct two fixed points A (0,2), B (4, -3) and a moving point P (x, 0) The maximum and minimum of the sum of the distances of the points. Using the property that the sum of the two sides of the triangle is larger than the third one, y min = (41) ~ (1/2) without maximum