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

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