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中学数学中涉及的思想方法很多,其中“数形结合”是很重要的一种.华罗庚教授说“数”缺少“形”时,少直观;“形”缺少“数”时,难入微.可见“数形结合”在数学中的地位.某些不等式若采用“数形结合”的思想方法来解,将事半而功倍. 例 1 已知f(x)=ax2-c,且-4≤f(1)≤-1,-1≤f(2)≤5,求f(3)的范围. 分析本题若采用通法求解,很容易出错;而用高中数学新教材(试验本)第二册(上)中所讲的“线性规划”,采用“数形结合”来求解,将令人赏心悦目.
There are many ideas and methods involved in middle school mathematics, of which the combination of number and shape is a very important one. Professor Hua Luogeng said that when “number” lacks “shape”, it is less intuitive; when “shape” lacks “number,” it is difficult to see. The position of “number and shape combination” in mathematics. If some inequality is solved by the “numeric combination” thought method, it will achieve half the effort. Example 1 It is known that f(x)=ax2-c, and -4 ≤ f(1) ≤ -1, -1 ≤ f(2) ≤ 5, find the range of f(3). It is easy to make mistakes if you use the general method to solve this problem, but use the new high school mathematics textbook (test book) The “linear programming” in the second volume (above), which is solved using the “numerical form combination”, will be pleasing to the eye.