微積分甲(一)

(1) 本課程之目標在於學習微分積分之方法與計算,以及更重要的是基本數學觀念的理解。同時希望大家學到自律、表達、合作與互動、獨立思考與自我學習。

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• Limits and derivatives

  2.2 The Limit of a Function 2.3 Calculating Limits Using the Limit Laws 2.4 The Precise Definition of a Limit 2.5 Continuity 2.6 Limits at Infinity; Horizontal Asymptotes 2.7 Derivatives and Rates of Change 2.8 The Derivative as a Function

• Differentiation Rules

  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 3.8 Exponential Growth and Decay 3.9 Related Rates 3.10 Linear Approximations and Differentials 3.11 Hyperbolic Functions

• 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 Newton's Method 4.9 Antiderivatives

• 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

• Applications of Integration

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

• 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

• Further Applications of Integration

  8.1 Arc Length 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

• Parametric Equations and Polar Coordinates

  10.1 Curves Defined by Parametric Equations 10.2 Calculus with Parametric Curves

• Functions and Model

  1.5 Exponential Functions 1.6 Inverse Functions and Logarithms

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