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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/calculus-through-data-and-modelling-integration-applications
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课程名称：通过数据与建模学习微积分：积分应用 课程概述：本课程延续微积分的学习，着重于积分的应用。本部分的应用有许多共同特征。首先，每个应用都是通过评估定积分计算得出的量的示例。其次，该应用的公式是从黎曼和推导而来的。与我们之前的微分学中测量变化率不同，定积分使我们能够测量某个区间内量的累积。这种累积的概念可以应用于多种量，包括金钱、人口、重量、面积、体积和空气污染物。本课程中的概念也适用于传统数学以外的许多学科。 我们将扩展数据集的平均值概念，允许无穷值的存在；开发弧长和曲率的公式；推导速度、加速度以及曲线之间面积的公式。通过实例和项目，我们将应用本课程的工具来分析和建模现实世界的数据。 课程大纲： 模块1：函数的平均值 描述：在本模块中，我们将平均值的概念推广到（有限）点集。你是否曾想过如果存在无限多个温度读数，我们如何计算一天的平均温度？或者如何计算平均降雨量？本模块中的概念将使我们能够扩展平均值的想法，以便在连续区间上计算（无限）值的平均数。 模块2：弧长与曲率 描述：我们所说的曲线的弧长是什么意思？我们可能会想象用一根绳子沿着曲线进行测量，然后再用尺子测量这根绳子。然而，当处理复杂曲线时，这一点是困难的。在本模块中，我们发展了在xy平面和空间中精确定义曲线的弧长和曲率的概念。 模块3：速度与加速度 描述：在本模块中，我们展示了切向量和法向量的思想如何应用于物理学，以研究物体的运动，包括速度和加速度，但现在我们专注于三维空间中的曲线。这里开发的技术使我们能够研究更高级函数的变化率。 模块4：曲线之间的面积 描述：找到两条曲线之间的面积不仅是从几何角度看定积分的一个有趣应用，当使用适当的函数时，还在经济学、商业甚至医学上具有应用价值。
Name:Module 1: Average Value of a Function
Description:In this module, we generalize the notion of the average value of a (finite) set of points. Did you ever wonder how we compute the average temperature during the day if infinitely many temperature readings are possible? Or how the average rainfall is calculated? The notions in this module will allow us to expand the idea of an average value to compute averages with (infinite) values over a continuous interval.
Name:Module 2: Arc Length and Curvature
Description:What do we mean by the arc length of a curve? We might think of fitting a piece of string to the curve and then measuring the string against a ruler. But this is difficult to do when working with a complicated curve. In this module we develop the precise notion of the length and curvature of an arc of a curve in both the xy plane and in space.
Name:Module 4: Velocity and Acceleration
Description:In this module, we show how the ideas of tangent and normal vectors can be used in physics to study the motion of an object, including its velocity and acceleration, but now we focus on curves in three dimensional space. The techniques developed here then allow us to study the rates of change for more advanced functions.
Name:Module 4: Areas Between Curves
Description:Finding the area between two curves is not just an interesting application of definite integrals from a geometric view, but when working with the appropriate functions, has applications in economics, business, and even medicine.
This course continues your study of calculus by focusing on the applications of integration. The applications in this section have many common features. First, each is an example of a quantity that is computed by evaluating a definite integral. Second, the formula for that application is derived from Riemann sums. Rather than measure rates of change as we did with differential calculus, the definite integral allows us to measure 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. We will expand the notion of the average value of a data set to allow for infinite values, develop the formula for arclength and curvature, and derive formulas for velocity, acceleration, and areas between curves. Through examples and projects, we will apply the tools of this course to analyze and model real world data.
