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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/discrete-calculus
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课程概述：单变量微积分 微积分是人类思维的一项伟大成就，解释了从行星轨道到城市的最佳大小，再到心跳的周期性等各种现象。本课程快速覆盖了单变量微积分的核心概念，强调概念理解和实际应用，特别适合工程、物理和社会科学的学生。课程的独特之处包括：1）从一开始引入和使用泰勒系列及其近似；2）将离散和连续的微积分形式进行新颖的综合；3）强调概念而非计算；4）采用清晰、动态和统一的方法。 课程内容包括五个部分，最后一部分覆盖序列、数值方法、级数及收敛测试、幂级数和泰勒级数，并以期末考试结束课程。参加本课程的学习者可以通过注册Coursera的认证证书项目并通过期末考试获得证书。 课程大纲： 1. 序列的微积分：重新构建离散输入函数的微积分，介绍离散微积分的工具和术语。 2. 数值方法简介：探索数值分析的基础，学习如何对微分方程和定积分进行近似计算，泰勒展开在这些近似中扮演关键角色。 3. 级数和收敛测试：研究关于无限和的收敛性，尤其是在离散微积分中的应用，强调大O记法的重要性。 4. 幂级数和泰勒级数：通过已掌握的测试和工具，深入理解泰勒展开的意义，讨论一般的幂级数。 5. 单变量微积分总结：回顾学习内容，为未来的学习打下基础，反思我们一起走过的旅程。 通过这个课程，学习者将深入理解单变量微积分的基本理论和应用，为更加复杂的数学概念奠定基础。
Name:A Calculus for Sequences
Description:It's time to redo calculus! Previously, all the calculus we have done is meant for functions with a continuous input and a continuous output. This time, we are going to retool calculus for functions with a discrete input. These are sequences, and they will occupy our attention for this last segment of the course. This first module will introduce the tools and terminologies for discrete calculus.
Name:Introduction to Numerical Methods
Description:That first module might have seemed a little...strange. It was! In this module, however, we will put that strangeness to good use, by giving a very brief introduction to the vast subjects of numerical analysis, answering such questions as "how do we approximate solutions to differential equations?" and "how do we approximate definite integals?" Perhaps unsurprisingly, Taylor expansion plays a pivotal role in these approximations.
Name:Series and Convergence Tests
Description:In "ordinary" calculus, we have seen the importance (and challenge!) of improper integrals over unbounded domains. Within discrete calculus, this converts to the problem of infinite sums, or series. The determination of convergence for such will occupy our attention for this module. I hope you haven't forgotten your big-O notation --- you are going to need it!
Name:Power and Taylor Series
Description:This course began with an exploration of Taylor series -- an exploration that was, sadly, not as rigorous as one would like. Now that we have at our disposal all the tests and tools of discrete and continuous calculus, we can finally close the loop and make sense of what we've been doing when we Talyor-expand. This module will cover power series in general, from we which specify to our beloved Taylor series.
Name:Concluding Single Variable Calculus
Description:Are we at the end? Yes, yes, we are. Standing on top of a high peak, looking back down on all that we have climbed together. Let's take one last look down and prepare for what lies above.
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 fifth part--part five of five--we cover a calculus for sequences, numerical methods, series and convergence tests, power and Taylor series, and conclude the course with a final exam. Learners in this course can earn a certificate in the series by signing up for Coursera's verified certificate program and passing the series' final exam.
