微積分（一）

Abstract

課程首頁

本課程是由交通大學應用數學系提供。

本課程影片於99學年暑假課程錄製，配合99年上學期網路課程調整安排課程內容。

課程概述 本課程除了微分積分之方法與計算之外，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。

課程章節

章節

課程內容

第一章

Functions and Model

1.5 Exponential Functions 1.6 Inverse Functions and Logarithms

第二章

Limits and derivatives

2.2 The Limit of a Function 2.3 Calculating Limits Using the Limit Laws 2.5 Continuity 2.6 Limits at Infinity; Horizontal Asymptotes 2.7 Derivatives and Rates of Change 2.8 The Derivative as a Function

第三章 Stochastic Integrals

3.1 Derivatives of Polynomials and Exponential Functions 3.2 The Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 The Chain Rule 3.5 Implicit Differentiation 3.6 Derivatives of Logarithmic Functions 3.7 Rates of Change in the Natural and Social Sciences 3.9 Related Rates 3.10 Linear Approximations and Differentials 3.11 Hyperbolic Functions

第四章

Applications of Differentiation

4.1 Maximum and Minimum Values 4.2 The Mean Value Theorem 4.3 How Derivatives Affect the Shape of a Graph 4.4 Indeterminate Forms and L’Hospital’s Rule 4.5 Summary of Curve Sketching 4.7 Optimization Problems 4.8 Newtons Method 4.9 Antiderivatives

第五章

Integrals

5.1 Areas and Distances 5.2 The Definite Integral 5.3 The Fundamental Theorem of Calculus 5.4 Indefinite Integrals and the Total Change Theorem 5.5 The Substitution Rule

第六章

Applications of Integration

6.1 Areas between Curves 6.2 Volumes 6.3 Volumes be Cylindrical Shells 6.5 Average Value of a Function

第七章 Techniques of Integration

7.1 Integration by Parts 7.2 Trigonometric Integrals 7.3 Trigonometric Substitution 7.4 Integration of Rational Functions by Partial Fractions 7.7 Approximate Integration 7.8 Improper Integrals

第八章

Further Applications of Integration

8.1 Further Applications of Integration 8.2 Area of a Surface of Revolution 8.3 Applications to Physics and Engineering 8.4 Applications to Economics and Biology 8.5 Probability

第十章 Parametric Equations and Polar Coordinates

10.1 Curves Defined by Parametric Equations 10.2 Calculus with Parametric Curves 10.3 Polar Coordinates 10.4 Areas and Lengths in Polar Coordinates

課程書目

Calculus(Early Transendentals), James Stewart, 6th Edition.

評分標準

項目 百分比

Practice Quizes 5%

學習歷程檔案 & 討論區 5%

小考 40%

期中考 20%

期末會考 30%

授課對象：大學一年級學生

預備知識：高中數學