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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/calculus-through-data-and-modelling-series-and-integrals
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课程名称：通过数据与建模学习微积分：级数与积分 课程概述：本课程在微积分的学习基础上，继续引入级数、数列和积分的概念。这些基础工具使我们能够发展微积分第二大工具——积分的理论与应用。与测量变化率不同，积分提供了一种测量某一数量在某个输入值区间上的积累的方法。这种积累的概念可以应用于金钱、人口、重量、面积、体积和空气污染等不同的数量。课程中的理念适用于许多传统数学以外的学科。通过项目，我们将应用本课程的工具来分析和建模现实世界的数据，并对政策进行批判性分析。 我们会遵循导数的模式，开发几种重要的积累计算方法。课程开始时将学习微积分基本定理，这是一个深刻而重要的结果，揭示了微分和积分操作之间的关系。如果你有兴趣学习更高级的数学，这门课程非常适合你。 课程大纲： 1. 模块1：数列与级数 - 描述：微积分分为两部分：微分与积分。该模块介绍了积分的过程。我们将首先看到如何用定积分找到曲线下的面积，然后探讨微分与积分之间的反演关系，最后学习不定积分及其计算策略。 2. 模块2：定积分 - 描述：该模块引入了黎曼和的概念，这是对积分的一种近似，通过有限求和来实现。常见的应用包括近似函数或图形上线的面积、曲线长度等。近似面积的累积将引导我们了解定积分及其后续应用。 3. 模块3：微积分基本定理 - 描述：我们将介绍学习的第一个主要工具——微积分基本定理。该定理将函数的微分与积分概念联系起来，包括两个部分，第一部分暗示了连续函数的反导数的存在，第二部分在实际应用中发挥着更大的作用。该定理的美丽与实用性使我们能够避免数值积分，从而提供更好的数值准确性。 4. 模块4：不定积分 - 描述：在本模块中，我们专注于发展查找反导数的能力，或者更一般地说，查找反导数的家族。微积分中的反导数通过不定积分表示，而求解反导数的过程称为反微分，这与微分相反，完成了我们对微积分两大工具的知识。 5. 模块5：计算器和表格的积分 - 描述：尽管寻找反导数的技术很有用，但某些函数的反导数可能过于复杂，因此需要更精确的数值方法来近似定积分。在本模块中，我们介绍了两种求解复杂积分的技术：使用技术工具或积分表，以及估算技术。我们还将应用我们所学知识来分析随机事件应用于决策理论的策略。
Name:Module 1: Sequences and Series
Description:Calculus is divided into two halves: differentiation and integration. In this module, we introduce the process of integration. First we will see how the definite integral can be used to find the area under the graph of a curve. Then, we will investigate how differentiation and integration are inverses of each other, through the Fundamental Theorem of Calculus. Finally, we will learn about the indefinite integral, and use some strategies for computing integrals.
Name:Module 2: The Definite Integral
Description:In this module, we introduce the notion of Riemann Sums. In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum, named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. This notion of approximating the accumulation of area under a group will lead to the concept of the definite integral, and the many applications that follow.
Name:Module 3: The Fundamental Theorem of Calculus
Description:We now introduce the first major tool of our studies, the Fundamental Theorem of Calculus. This deep theorem links the concept of differentiating a function with the concept of integrating a function. The theorem will consists of two parts, the first of which implies the existence of antiderivatives for continuous functions and the second of which plays a larger role in practical applications. The beauty and practicality of this theorem allows us to avoid numerical integration to compute integrals, thus providing a better numerical accuracy.
Name:Module 4: The Indefinite Integral
Description:In this module, we focus on developing our ability to find antiderivatives, or more generally, families of antiderivatives. In calculus, the general family of antiderivatives is denoted with an indefinite integral, and the process of solving for antiderivatives is called antidifferentiation. This is the opposite of differentiation and completes our knowledge of the two major tools of calculus. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
Name:Integration with Calculators and Tables
Description:While the technique of finding antiderivatives is useful, there are some functions that are just too difficult to find antiderivatives for. In cases like these, we want to have a numerical method to approximate the definite integral. In this module, we introduce two techniques for solving complicated integrals: using technology or tables of integrals, as well as estimation techniques. We then apply our knowledge to analyze strategies and decision theory as applied to random events.
This course continues your study of calculus by introducing the notions of series, sequences, and integration. These foundational tools allow us to develop the theory and applications of the second major tool of calculus: the integral. Rather than measure rates of change, the integral provides a means for measuring the accumulation of a quantity over some interval of input values. This notion of accumulation can be applied to different quantities, including money, populations, weight, area, volume, and air pollutants. The concepts in this course apply to many other disciplines outside of traditional mathematics. Through projects, we will apply the tools of this course to analyze and model real world data, and from that analysis give critiques of policy. Following the pattern as with derivatives, several important methods for calculating accumulation are developed. Our course begins with the study of the deep and significant result of the Fundamental Theorem of Calculus, which develops the relationship between the operations of differentiation and integration. If you are interested in learning more advanced mathematics, this course is the right course for you.
