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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/single-variable-calculus
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课程名称：微积分：单变量部分1 - 函数 课程概述：微积分是人类思想的一项伟大成就，它解释了从行星轨道到城市的最佳规模再到心跳的周期性等诸多现象。本课程快速覆盖单变量微积分的核心概念，强调理解和应用，非常适合工程、物理和社会科学领域的初学者。课程的独特之处包括：1) 从一开始引入并使用泰勒级数与近似；2) 将离散与连续微积分形式进行新颖的综合；3) 强调概念理解而非计算；4) 提供清晰、动态且统一的教学方法。 在这第一部分（五部分中的第一部分），您将扩展对泰勒级数的理解，复习极限，学习l'Hopital法则背后的理由，最重要的是，学习描述函数增长和衰减的新语言：大O符号。 课程大纲： 1. **介绍**：欢迎来到微积分：单变量！以下将提供课程的诊断考试，您可以选择参加。尽管没有最低分数要求，但如果分数较低，您可能需要调整期望：这是一门非常困难的课程。 2. **函数复习**：本模块将复习您的（预）微积分基础，为后续课程奠定基础，并探讨问题：*究竟什么是* 指数函数？ 3. **泰勒级数**：这一模块是整门课程的核心：泰勒级数，它提供了一个函数的近似，表现为一个级数或“长多项式”。您将学习泰勒级数的定义以及如何计算它。别担心！符号可能不熟悉，但其实就是处理多项式的问题。 4. **极限与渐近性**：泰勒级数是否收敛，取决于其极限（或“渐近”）特性。实际上，泰勒级数是理解大大小小极限的完美工具，使得l'Hopital法则等方法变得有意义。为了巩固这些新技能，我们引入“大O符号”的语言，以界定渐近项的大小。该语言将在后续的微积分章节中使用。
Name:Introduction
Description:Welcome to Calculus: Single Variable! below you will find the course's diagnostic exam. if you like, please take the exam. you don't need to score a minimal amount on the diagnostic in order to take the course. but if you do get a low score, you might want to readjust your expectations: this is a very hard class...
Name:A Review of Functions
Description:This module will review the basics of your (pre-)calculus background and set the stage for the rest of the course by considering the question: just what is the exponential function?
Name:Taylor Series
Description:This module gets at the heart of the entire course: the Taylor series, which provides an approximation to a function as a series, or "long polynomial". You will learn what a Taylor series is and how to compute it. Don't worry! The notation may be unfamiliar, but it's all just working with polynomials....
Name:Limits and Asymptotics
Description:A Taylor series may or may not converge, depending on its limiting (or "asymptotic") properties. Indeed, Taylor series are a perfect tool for understanding limits, both large and small, making sense of such methods as that of l'Hopital. To solidify these newfound skills, we introduce the language of "big-O" as a means of bounding the size of asymptotic terms. This language will be put to use in future Chapters on Calculus.
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this first part--part one of five--you will extend your understanding of Taylor series, review limits, learn the *why* behind l'Hopital's rule, and, most importantly, learn a new language for describing growth and decay of functions: the BIG O.
