Monotonicity Rules in Calculus
Monotonicity Rules in Calculus
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DOI：
10.1080/00029890.2006.11920367
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发表时间：
2006-11
期刊：
影响因子：
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通讯作者：
G. D. Anderson;M. Vamanamurthy;M. Vuorinen
中科院分区：
文献类型：
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作者：
G. D. Anderson;M. Vamanamurthy;M. Vuorinen
1.单调性规则。在微积分的第一学期，学生学习如果函数f在区间[a，B]上连续，并且在（a，B）上有正（负）导数，则f在[a，B]上增加（减少）。利用拉格朗日中值定理很容易得到这个结果。学生用这种方法证明单调性的函数通常是多项式、有理函数或其他初等函数。如果试图建立两个函数的商的单调性，人们经常会发现商的导数相当混乱，过程繁琐。一些作者已经开发了改进的这种方法证明单调性的连续性。我们所知道的第一个这样的改进是M.格罗莫夫[11，第42页]，这出现在他的工作中微分几何（格罗莫夫的证明只使用单调性和基本性质的积分）：
1. RULES FOR MONOTONICITY. In the first semester of calculus a student learns that if a function f is continuous on an interval [a, b] and has a positive (negative) derivative on (a, b), then f is increasing (decreasing) on [a, b]. This result is obtained easily by means of the Lagrange mean value theorem. The functions that the student proves monotone in this way are usually polynomials, rational functions, or other elementary functions. If one is attempting to establish the monotonicity of a quotient of two functions, one often finds that the derivative of the quotient is quite messy and the process tedious. Several authors have developed refinements of this method for proving monotonicity of quotients. The first such refinement of which we are aware is the following one by M. Gromov [11, p. 42], which appears in his work in differential geometry (Gromov’s proof uses only monotonicity and elementary properties of integrals):