发表机构
Washington and Lee University; Williams College; Texas A&M University（华盛顿与李大学; 威廉姆斯学院; 德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用实分析工具（极限、连续性、导数、介值定理）自包含地证明了欧拉数 $e$ 的斜率定义（存在唯一底数使 $y=b^x$ 在 $(0,1)$ 处斜率为 1）的合理性，面向本科生及教师。
AI 中文摘要
欧拉数 $e$ 通常通过复利、无穷级数、面积或指数函数的斜率来引入。最后一种观点尤其吸引人：选择底数 $b\in (2,3)$，使得 $y=b^x$ 的图像在 $(0,1)$ 处的切线斜率为 $1$。然而，好奇的学生可能会提出几个简短的介绍必然隐含的问题。当 $x$ 为无理数时，$b^x$ 是什么意思？为什么 $b^x$ 在 $x=0$ 处可导？如何证明对应于底数 $2$ 和 $3$ 的斜率位于斜率 $1$ 的两侧？最后，为什么存在唯一的底数使其斜率恰好为 $1$？本文利用实分析入门课程中的工具（包括极限、连续性、导数和介值定理）来回答这些问题。其主要受众是熟悉入门分析课程中标准概念、希望应用所学知识探索早期课程中遗留问题的本科生。次要受众是教师，他们可以在证明入门课程中选用部分内容作为补充材料。数学成分是经典的；我们的目标是将它们组织成自包含的阐述，并明确标准处理中常常压缩的逻辑步骤。
英文摘要
Euler's number $e$ is often introduced through compound interest, an infinite series, an area, or the slope of an exponential function. The last viewpoint is especially attractive: choose the base $b\in (2,3)$ so that the graph of $y=b^x$ has tangent slope $1$ at $(0,1)$. However, a curious student may raise several questions that a brief introduction necessarily leaves implicit. What does $b^x$ mean when $x$ is irrational? Why is $b^x$ differentiable at $x = 0$? How can one show that the slopes corresponding to bases $2$ and $3$ lie on opposite sides of slope $1$? Finally, why is there a unique base whose slope is exactly $1$? The paper provides an answer using tools from a first course in real analysis, including limits, continuity, derivatives, and the Intermediate Value Theorem. Its primary audience is undergraduate students familiar with standard concepts from an introductory analysis course who wish to apply what they have learned to explore questions left open in earlier courses. A secondary audience is instructors who may use selected portions as supplementary material in an introduction-to-proofs course. The mathematical ingredients are classical; our aim is to organize them into a self-contained exposition and to make explicit the logical steps that are often compressed in standard treatments.
Comments19 pages, 2 figures
