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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/applications-calculus
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课程名称：微积分：单变量第4部分 - 应用 课程概述：微积分是人类思维的伟大成就之一，能够解释从行星轨道到城市最佳规模再到心跳的周期性等各种现象。本课程快速讲解单变量微积分的核心概念，强调理解和应用，适合工程、自然科学和社会科学领域的初学者。课程的独特之处包括：1) 从一开始引入泰勒级数及其近似；2) 将离散与连续形式的微积分进行新颖的综合；3) 强调概念理解而非单纯的计算；4) 提供清晰、动态、统一的方法论。 在本课程的第四部分（共五部分）中，我们将讨论面积和体积的计算、其他几何应用、物理应用以及平均值和质量的相关内容。同时，我们还将介绍概率的基础知识。 课程大纲： 1. 计算面积和体积：回顾经典的积分动机，介绍微分元素的核心概念，通过计算面积和体积元素，探索解决复杂几何问题的方法。 2. 其他几何应用：深入研究高维几何，从长度和面积的角度回到3D世界，重点构建适当的微分元素进行积分。 3. 物理应用：探索积分在物理和金融等领域的广泛应用，涉及功、力、扭矩、质量以及现值和未来值的计算。 4. 平均值和质量：探讨积分在统计方面的应用，特别是计算平均值，结合物理问题的动机，如质心和惯性矩。 5. 概率简介：本模块为概率的简要介绍，基于已有的积分和微分元素知识，定义概率密度函数及相应的概率元素，提供有关期望、方差和标准差的独特视角。 本课程通过强调概念理解和应用，帮助学生建立对微积分的深入认识。
Part: 1
Title:Computing Areas and Volumes
Description:Having seen some calculus before, you may recall some of the motivations for integrals arising from area computations. We will review those classical applications, while introducing the core idea of this module -- a differential element. By computing area and volume elements, we will see how to tackle tough geometry problems in a principled manner.
Part: 2
Title:Other Geometric Applications
Description:There's more to geometry than just area and volume! In this module, we will take things "to the next level", ascending to higher dimensions. Coming back to the 3-d world, we will return to problems of length and area, but this time in the context of curves and surfaces. As always, the emphasis will be on how to construct the appropriate differential element for integrating.
Part: 3
Title:Physical Applications
Description:There is so much more to applications of integrals than geometry! So many subjects, from physics to finance, have, at heart, the need for setting up and computing definite integrals. In this short but intense module, we will cover applications including work, force, torque, mass, and present & future value.
Part: 4
Title:Averages and Mass
Description:There is a statistical aspect to integrals that has not yet been brought up in this course: integrals are ideal for computing averages. Motivated by physical problems of mass, centroid, and moments of inertia, we will cover applications of integrals to averages.
Part: 5
Title:An Introduction to Probability
Description:This capstone module gives a very brief introduction to probability, using what we know about integrals and differential elements. Beginning with common-sense uniform probabilities, we move on to define probability density functions and the corresponding probability element. Building on the physical intuition obtained from centers of mass and moments of inertia, we offer a unique perspective on expectation, variance, and standard deviation.
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 fourth part--part four of five--we cover computing areas and volumes, other geometric applications, physical applications, and averages and mass. We also introduce probability.
