必修課程介紹

• 預備知識：高中數學
• 適合年級： 大一
• 課程簡介：

微積分（Calculus）最早在拉丁文是「小鵝卵石」的意思，是一種是用來計算的工具。隨著時空的轉移，幾百年的發展，讓微積分漸趨成熟與完整。尤其上個世紀中，在許多領域，如物理、工程、化學、生物甚至社會學等，都有重要發展，一大部分要歸功於這個「小鵝卵石」。

我們存在的世界充滿了變化與運動，地球繞行太陽、石頭往上丟會落回地面、世界的人口一直在增加、車子加速與減速的運動等等。這些物理現象，無時無刻都在變化，難以用定量的方式來分析與了解。而微積分主要的目的，就在於分析處理這個瞬息萬變、令人目不暇幾的花花世界。

拜笛卡兒與費瑪之賜，他們引進函數的觀念，我們才能透過函數清楚地描述變量之間的關係。例如自由落體運動中，時間與速度的關係。微積分學分成兩個部份，其中微分學就是研究函數變化的數學，而積分學則是討論變量的累積。無論是微分學也好，積分學也好，都須要面對「極限」的問題。因此，微積分的不二法門便是「先算有限、再取極限」。

那麼積分與微分的關係是什麼呢？一般說來，一個函數先積分後再微分，或先微分再積分，都會回到原來的函數，這也就「微積分基本定理」。舉例來說，距離的微分是速度，而速度的積分又是距離。藉由這個基本定理，積分的技巧才得以有了重要的進展。

除了微分積分之方法與計算之外，這門課的學習重點在於基本數學觀念的理解。例如中間值定理、平均值定理、極值定理等。這些定理不僅本身有其基本應用的價值，背後也有它們數學的涵義與想法。瞭解這些想法，一方面可以推廣這些定理，另一方面當我們面臨更複雜的問題時，解決問題初步的試探可以以這些基本數學想法做為基礎或做為類比。這是數學做為一種科學思考的價值。

在過去的二千多年，微積分在無數學者的努力下，一點一滴的醞釀，而在牛頓與萊布尼茲催生下，哇哇落地。這樣一個美麗的數學王國，在十八世紀成長，而在十九世紀培養了穩固的基礎，於是才有今日完整而多樣的風貌。微積分不僅是人類偉大的理論成就，也是獨一無二的計算工具，同時更是後續課程如微分方程，機率，工程數學的重要基礎，值得我們好好地思考與體會。

• 課程大綱：

1. Limits and Continuity:
   The Basic (Intuitive) Concepts of Limit, The Definition of Limit, Evaluating Limits and Rules for Finding Limits, Some Limit Theorems (with proofs), One-Sided Limits, Continuity.
2. Differentiation:
   Definition of Derivative, Basic Differentiation Rules, and Higher-Order Derivatives, The Chain Rule, Differentiatingthe Trigonometric Functions, Implicit Differentiation.
3. Applications of Differentiation:
   Extreme Value Theorem, The Role's Theorem and the Mean-Value Theorem (with proofs), Increasing and Decreasing Functions and the First Derivative Test for Local Extreme Values, Concavity and the Second Derivative Test, Limits at Infinity, Asymptotes and Summary of Curve Sketching, Linearization and Differentials, Optimization Problems.
4. Integration:
   AntiderivativesandIndefinite Integrals,Riemann Sums and Definite Integrals, Properties and the Mean-Value Theorem for Integrals, The Fun- damental Theorem of Calculus (with proof), Integral by Substitution: Change of Variables.
5. Transcendental Functions:
   Inverse Functions and Their Derivatives, The Natural Logarithm Function: Differentiation and Integration, The Exponential Functions: Differ- entiation and Integration, Logarithm and Exponential Functions with Bases Other Than e , Indeterminate Forms and Rule, Inverse Trigonometric Functions: Differentiation and Integration, (Optional) Hyperbolic Functions: Differentiation and Integration.
6. Techniques of Integration:
   Basic Integration Formulas, Integration by Parts, Partial Fractions, (Optional) Trigonometric Integrals, Trigonometric Substitutions, Improper Integrals.
7. Applications of Integrals:
   Area Problems: Area of a Region Between Two Curves, Volumes of Solids of Revolution, Area of Surface of Revolution, Lengths of Plane Curves, Solutions of Differentiation Equations, and Growth/Decay Problems.
8. Infinite Series:
   Sequences, Theorems for Calculating Limits of Sequences, Infinite Series and Convergence, The Integral Test, The Comparison Test, Alter- nating Series, Absolute and Conditional Convergence, The Ratio and Root Tests, Power Series, Taylor and Maclaurin Series Approxima- tions of Functions and Error Estimates.
9. Parametrized Curves and Analytic Geometric in Space:
   Parametrizations of Plane Curves, Calculus with Parametrized Curves, (Optional) Polar Coordinates and Polar Graphs, (Optional) Cylindrical and Spherical Coordinates.
10. Vector-Valued Functions:
    Review of Vectors in Space, Dot Products and Cross Products, Vector-Valued Functions and Space Curves, Differentiation and Integration of Vector-Valued Functions, Tangent Vectors and Normal Vectors.
11. Multivariable Functions and Partial Derivatives:
    Introduction to Functions of Several Variables, Limits and Continuity, Partial Derivatives, Directional Derivatives and Gradients, Differentiability and Differentials, The Mean-Value Theorem and Chain Rules for Multivariable Functions, Tangent Plane and Normal Lines, Extreme Values and Saddle Points, Lagrange Multipliers, Taylor s Formula.
12. Multiple Integrals:
    Double Integrals as the Limit of Riemann Sums, Evaluation of Double Integrals, Some Applications of Double Integrals: Volumes and Surface Areas, Triple Integrals, Change of Variables: Jacobians, (Optional) Double Integrals in Polar Coordinates, and Triple Integrals in Cylindrical and Spherical Coordinates.
13. Integration in Vector Field:
    Vector Field, Line Integrals, Conservative Vector Field and Path Independence, Green's Theorem, Parametrized Surfaces, Surface Integrals, The Divergence Theorem, The Stokes's Theorem and a Unified Theory.

• 參考書目：

1. Calculus (Early Transcendentals), James Stewart, 6th Edition

• 其他：

1. 微積分教學小組