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微積分作業演算(一) Calculus Exercise I

課程綱要

  • 微積分作業演算(一) Calculus Exercise I

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.

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