关于平均速度的观测

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-01-11 DOI:10.1080/00029890.2022.2159265

S. Weintraub

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引用次数: 0

摘要

假设一个粒子以位置s（t）和速度v（t）移动，对于t∈[0，t]。如果s（t）在[0，t]上是连续的，在（0，t）上是可微的，那么（几乎）所有微积分文本中的中值定理断言，在（O，t）中有一个点，粒子的速度等于其在[0、t]上的平均速度。但很自然地会问，是否存在长度为1的某个子区间，其平均速度等于[0，T]上的平均速度。这个问题（几乎）没有微积分文本，答案如下：

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An Observation on Average Velocity

Suppose that a particle moves with position s(t) and velocity v(t) for t ∈ [0, T ]. If s(t) is continuous on [0, T ] and differentiable on (0, T ), then the mean value theorem, which is in (almost) all calculus texts, asserts that there is some point in (0, T ) at which the particle’s velocity is equal to its average velocity on [0, T ]. But it is natural to ask whether there is some subinterval of length 1 on which its average velocity is equal to its average velocity on [0, T ]. The answer to this question, which is in (almost) no calculus texts, is as follows:

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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