單變量微積分（01）- Introduction

1. Introduction

單變量微積分是MIT的開放課程18.01 Single Varialbe Calculus，該課程是理工科學生在MIT的第一年上半學期的學習內容，只需要高中的代數與三角相關的知識。
這門課程主要講解一元函數的微分與積分，以及它們的應用，主要知識點包括：

• Concepts of Function, Limits and Continuity
• Differentiation Rules, Application to Graphing, Rates, Approximations, and Extremum Problems
• Definite and Indefinite Integration
• The Fundamental Theorem of Calculus
• Applications to Geometry: Area, Volume, and Arc Length
• Applications to Science: Average Values, Work, and Probability
• Techniques of Integration
• Approximation of Definite Integrals, Improper Integrals, and L’Hôspital’s Rule

2. Goals

學完本課程，學生應該對單變量微積分的概念有個清楚的認識，並且可以用微分與積分的實際意義去解決分析現實問題。

兩個基本概念：

1. Derivatives as rates of change, computed as a limit of ratios
2. Integrals as a “sum,” computed as a limit of Riemann sums

下面是對學完該課程的知識要求：

• Use both the limit definition and rules of differentiation to differentiate functions.
• Sketch the graph of a function using asymptotes, critical points, the derivative test for increasing/decreasing functions, and concavity.
• Apply differentiation to solve applied max/min problems.
• Apply differentiation to solve related rates problems.
• Evaluate integrals both by using Riemann sums and by using the Fundamental Theorem of Calculus.
• Apply integration to compute arc lengths, volumes of revolution and surface areas of revolution.
• Evaluate integrals using advanced techniques of integration, such as inverse substitution, partial fractions and integration by parts.
• Use L’Hospital’s rule to evaluate certain indefinite forms.
• Determine convergence/divergence of improper integrals and evaluate convergent improper integrals.
• Determine the convergence/divergence of an infinite series and find the Taylor series expansion of a function near a point.

3. Course Structure

課程主要包括了以下幾個部分：

1. Differentiation
2. Applications of Differentiation
3. The Definite Integral and its Applications
4. Techniques of Integration
5. Exploring the Infinite