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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/calculus-through-data-and-modelling-imits-derivatives
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课程名称：通过数据与建模学习微积分：极限与导数 课程概述：该课程是关于单变量微积分概念的第一门课程，将介绍函数的极限概念以定义函数的导数。在数学中，导数用于衡量函数对变化的敏感性。例如，一个物体的位置相对于时间的导数就是物体的速度：这衡量了在时间推进时物体位置变化的速度。这个基本概念将通过数据的建模和分析得到应用。 课程大纲： 1. **函数的极限**：学习分析变化率和运动的基本概念，包括极限和导数的核心思想。通过多种方法，视觉和代数方式计算函数的极限，并将其应用于定义可微分的核心概念——导数。 2. **极限法则**：介绍极限的代数性质——极限法则，通过这些法则以准确的方式计算函数的极限，进而推导定理用于研究更复杂函数的行为。 3. **连续性**：定义连续函数，讲解当x接近某个数时，如何利用极限来定义函数的连续性，并解释数学上的连续性与日常语言中该词的含义之间的联系。 4. **无穷极限**：探讨当x趋于正无穷或负无穷时函数的行为，介绍水平渐近线的正式定义以及对某些类型函数的端行为进行分类的方法。 5. **导数**：讲解切线的斜率和物体瞬时速度的计算，理解导数的概念，并探讨其在自然科学、社会科学或工程学中作为变化率的应用。 6. **最终项目**：通过分析数据趋势，运用可微分微积分的工具和语言进行数据建模和分析，聚焦于世界各地区教育成就的性别比例变化趋势。 该课程旨在帮助学习者掌握微积分的基本概念，并能够将这些概念应用于实际数据分析中。
Name:The Limit of a Function
Description:One of the goals in studying Calculus is to analyze rates of change and movement. In this module, we introduce the central ideas which will help us achieve this goal: the notions of the limit and the derivative. Rather than evaluating a function at a single point, the limit allows for the study of the behavior of a function in an interval around that point. In this module, you will find limits of functions by a variety of methods, both visually and algebraically. Finally, we will apply limits to define the key idea of Differentiable Calculus, the derivative.
Name:The Limit Laws
Description:Using calculators or graphs is an imprecise way to find the limit of a function. In this module, we will state and use algebraic properties of limits, called the Limit Laws, to calculate the exact values of limits. A solid understanding of these laws will allow us to derive theorems which in turn can be used to study the behavior of more advanced functions.
Name:Continuity
Description:In the last module, there were several types of functions where the limit of a function as x approaches a number could be found by simply calculating the value of the function at the number. Functions with this property will be called continuous and in this module, we use limits to define continuity. We will see that the mathematical definition of continuity will correspond closely with the English meaning of the word continuity used in every day language.
Name:Limits at Infinity
Description:In this module, we allow for x to become arbitrarily large in the positive or negative direction to understand the end-behaviors of functions. This will allow for the formal definition of a horizontal asymptote and to provide classifications of end-behavior of certain types of functions.
Name:Derivatives
Description:The problem of finding the slope of the tangent line to a curve and the problem of finding the instantaneous velocity of an object both involve finding the same type of limit. This special type of limit is called the derivative and in this module, we will see that this notion of the derivative can be interpreted as a rate of change in any of the natural or social sciences or engineering.
Name:Final Project
Description:In this final project, we will apply the tools and language of differentiable calculus to analyze trends in data. This project will focus on modelling and analyzing gender ratios in educational attainment over time in several regions of the world.
This first course on concepts of single variable calculus will introduce the notions of limits of a function to define the derivative of a function. In mathematics, the derivative measures the sensitivity to change of the function. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. This fundamental notion will be applied through the modelling and analysis of data.
