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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/calculus-through-data-and-modelling-applying-differentiation
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课程名称：通过数据与建模的微积分：应用微分 课程概述：此课程重点介绍导数作为变化率的重要性，提供了对图形形状的信息。我们将应用导数来寻找单变量和多变量函数的线性近似，这为评估复杂函数提供了直接的方法。此外，还将利用导数定位函数的最大值和最小值，这些优化技术对于自然科学和数据分析等各个领域至关重要。课程内容适用于多种现实应用，如机器学习、成本最小化或利润最大化。 课程大纲： 1. 线性近似与切平面 - 描述：在单变量微积分中，导数计算切线的斜率，用于在某一点创建切线方程，以便为复杂函数提供准确的估计。该理论推广到空间中的切平面，在此模块中，我们将通过公式和应用探索这些概念。 2. 单变量函数的最大值与最小值 - 描述：微分计算的一些重要应用是优化问题，其目标是找到最佳解决方案。诸如市场营销、经济学、库存分析、机器学习和商业中的问题都关注于寻找最佳解决方案。本模块探讨如何利用导数找到函数的最大或最小值。 3. 多变量函数的最大值与最小值 - 描述：随着模型变得更加复杂，描述它们的函数也变得更加复杂。许多函数需要多个输入来描述其输出，这些多变量函数同样包含我们希望利用微积分工具找到的最大值和最小值。 4. 拉格朗日乘子法 - 描述：在数学优化中，拉格朗日乘子法是用于寻找在等式约束下的局部最大值和最小值的策略。本模块深入发展此强大工具的理论，并通过实例说明如何将受限问题转化为非受限问题的形式，从而应用导数测试。 5. 期末项目 - 优化 - 描述：在真实问题中应用我们所学的理论与实践，建立与施工项目相关的成本模型，以寻找最佳价格点。该项目具有挑战性，答案可能因您所做的假设而略有不同。在报告中清晰阐述您在过程中的假设。
Name:Linear Approximations and Tangent Planes
Description:In single variable calculus, the derivative computes the slope of the tangent line where defined. This is then used to create the equation of the tangent line at a point, which can be used as an accurate estimation tool for complicated functions. This theory generalizes to lines in space which are used to create tangent planes. In this module, we work through the formulas and applications of these notions, using our developed theory of derivatives and partial derivatives.
Name:Maxima and Minima of Single-Variable Functions
Description:Some of the most important applications of differential calculus are optimization problems in which the goal is to find the optimal (best) solution. For example, problems in marketing, economics, inventory analysis, machine learning, and business are all concerned with finding the best solution. These problems can be reduced to finding the maximum or minimum values of a function using our notions of the derivative.
Name:Maxima and Minima of Multivariable Functions
Description:As models become more complicated, the functions used to describe them do as well. Many functions require more than one input to describe their output. These multivariable functions also contain maximum and minimum values that we seek to find using the tools of calculus. In this module, we will extend our optimization techniques to multivariable functions.
Name:Lagrange Multipliers
Description:In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. It is named after the mathematician Joseph-Louis Lagrange. In this module, we develop the theory and work through examples of this powerful tool which converts a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a usually easier reformulation of the original problem.
Name:Final Project - Optimization
Description: We now put all our theory and practice to use in a real world problem to model the costs associated to a construction project in an effort to find the best possible price point. This project is challenging and answers may vary slightly based on the assumptions you use. Be thoughtful and clear in your report about any assumptions you make along the way.
As rates of change, derivatives give us information about the shape of a graph. In this course, we will apply the derivative to find linear approximations for single-variable and multi-variable functions. This gives us a straightforward way to estimate functions that may be complicated or difficult to evaluate. We will also use the derivative to locate the maximum and minimum values of a function. These optimization techniques are important for all fields, including the natural sciences and data analysis. The topics in this course lend themselves to many real-world applications, such as machine learning, minimizing costs or maximizing profits.
