本文將對 Putnam 數學競賽中微積分題型之題目進行探討，第二章介紹函數求極限的方法，及連續函數最重要的兩個定理---中間值定理和極值定理。第三章介紹導數的基本性質，及從導數變化而來的應用問題，包含切線與速率及導數在幾何上的意義，以及導數中重要的均值定理。第四章介紹積分的意義與應用，除了微積分基本定理外，還有弧長、面積、體積及物理中常見的質矩與質心。第五章針對積分的函數可能的形式，介紹了各形式的積分的技巧，藉此使得積分變為較容易計算。除了一般常見的變數變換，還介紹了如何使用萊布尼茲法則求積分。第六章除了介紹如何找出數列的通項與極限，也介紹無窮級數求和與判斷斂散性的問題。在數列部分又分為具有明確知道項的形式與沒有明確形式的數列，若碰到這些類型的問題又該如何解題，在本文中將分別介紹。

This paper investigates methods of solving calculus
problems in Putnam Mathematical Competition.Chapter 2 presents the methods of finding limits, and the most important theorems of continuity---Intermediate Value Theorem and Extreme Value Theorem. Chapter 3 introduces to the properties of derivatives, and the application problems change from the basic problems of derivative. It
contains the tangent line and the rate and the meaning of derivative on the geometry.In this chapter also includes the most important theorem---Mean Value Theorem---in derivatives.
Chapter 4 introduces to the properties of integral, and the application problems change from the basic problems of integral. There are the Fundamental Theorem of Calculus,
Arc length, area, volume and the mass moment and centroid of physical.
Chapter 5 investigates the integral techniques of the various forms of possible form for the integral function, to take the integral becomes relatively easy to calculate.
In addition to the common variable transformation, also describes how to use the Leibniz Rule for solving integrating.
In Chapter 6, it presents that how to determine terms of sequence and its limit, and introduces the infinite summation and to determine convergence or divergence of series.

摘要 ii
Abstract iii
表次 vi
圖次 vi
第一章介紹 1
第二章函數的極限與連續性 5
2.1 極限的介紹 5
2.2 由畫圖與數值方法求極限 7
2.3 由分析計算極限 7
2.4 約分與有理化 9
2.5 夾擠定理 10
2.6 羅必達法則 11
2.7 泰勒餘式定理 13
2.8 極限不存在 15
2.9 連續單變數函數 15
2.10 中間值定理 16
2.11 極值定理 18
第三章導數與導數應用 21
3.1 導數 21
3.2 切線 25
3.3 速率改變 27
3.4 一階與二階導數的幾何意義 28
3.5 極值 31
3.6 洛爾定理與均值定理 33
第四章積分與積分應用 37
4.1 黎曼和與微積分基本定理 37
4.2 面積、弧長與體積 39
4.3 質矩、質心和形心 46
4.4 積分均值定理 50
第五章積分技巧 52
5.1 變數變換 53
5.2 部分分式 56
5.3 分部積分 57
5.4 萊布尼茲積分法則 58
5.5 二重積分與積分變數交換 59
5.6 極座標變換 60
5.7 柱座標與球座標 62
第六章數列與級數 64
6.1 數列的極限 64
6.2 夾擠定理 67
6.3 柯西數列 68
6.4 遞迴數列 69
6.5 無窮級數 74
6.6 第n 項檢定 75
6.7 積分檢定 76
6.8 p 級數與調和級數 77
6.9 級數比較 78
6.10 交錯級數檢定 81
6.11 比例檢定與根式檢定 82
6.12 對消級數 85
6.13 微分和積分求和 86
6.14 泰勒定理 89
參考文獻 95

Alexanderson, G. L., Klosinski, L. F. and Larson, L. C. (1985). The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965-1984. Washington, DC: The Mathematical Association of America.

Gleason, A. M., Greenwood, R. E. and Kelly, L. M. (1980). The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938-1964. Washington, DC: The Mathematical Association of America.

Kedlaya, K. S., Poonen, B and Vakil, R (2002). The William Lowell Putnam Mathematical Competition Problems and Solutions: 1985-2000. Washington, DC: The Mathematical Association of America.

Larson, R and Edwards, H. B. (2010). Calculus, 9th Edition. Monterey, CA: Brooks/Cole, Cengage Learning.

Putnam Mathematical Competition (2012). ”William Lowell Putnam Mathematical Competition.” The Mathematical Association of America.
http://math.scu.edu/putnam/
http://amc.maa.org/a-activities/a7-problems/putnamindex.shtml