单变量微积分（01）- Introduction

1. Introduction

单变量微积分是MIT的开放课程18.01 Single Varialbe Calculus，该课程是理工科学生在MIT的第一年上半学期的学习内容，只需要高中的代数与三角相关的知识。
这门课程主要讲解一元函数的微分与积分，以及它们的应用，主要知识点包括：

• 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

学完本课程，学生应该对单变量微积分的概念有个清楚的认识，并且可以用微分与积分的实际意义去解决分析现实问题。

两个基本概念：

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

下面是对学完该课程的知识要求：

• 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

课程主要包括了以下几个部分：

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