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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/introduction-to-calculus
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课程名称：微积分入门 课程概述：微积分入门课程专注于数学在科学、工程和商业中的应用基础。课程强调微积分的关键概念和历史动机，同时在理论与应用之间保持平衡，帮助学生掌握基础数学中的关键概念。 本课程的学习目标包括： - 熟悉微积分前的关键概念，涵盖基本方程和初等函数的操作（前两周）。 - 掌握切线和极限的初步方法，以及导数的定义（第三周）。 - 发展应用微分计算的方法并进行实践（第四周）。 - 学习和实践积分计算的方法（第五周）。 课程大纲： 1. 预备微积分（背景介绍）：本模块研究实数线上的不同数类型、小数扩展和近似，接着探讨方程和不等式的操作、符号图及笛卡尔平面的使用。 2. 函数（有用的重要工具）：介绍函数的概念，以准确捕捉不同量或测量之间的联系，涵盖二次、三次及一般幂和多项式函数，指数和对数函数，以及与周期行为相关的三角函数，并通过组合和反演创建新函数。 3. 微分计算导论：介绍微分计算的技术，关注平均变化率如何转化为瞬时变化率，引入导数的概念，利用切线的微分技巧，使用莱布尼茨符号获取函数导数信息及其应用。 4. 导数的性质和应用：继续发展微分计算，介绍函数的一阶和二阶导数，利用导数符号图进行曲线草图绘制，并引入复杂函数的导数求法（链式法则、乘法法则和商法则）及优化问题的解法。 5. 积分计算导论：介绍积分计算，研究切线的斜率和曲线下的面积，探讨微积分基本定理，使用速度曲线下的面积估计位移，并通过下上矩形近似理解极限，求得圆的面积及抛物线下的面积，以及利用黎曼和和定积分准确计算曲线下的面积，最后讨论奇偶函数的性质及与指数增长相关的逻辑斯蒂函数。 本课程为期五周，旨在为学生打下坚实的数学基础，支持未来在相关领域的学习和应用。
Name:Precalculus (Setting the scene)
Description:This module begins by looking at the different kinds of numbers that fall on the real number line, decimal expansions and approximations, then continues with an exploration of manipulation of equations and inequalities, of sign diagrams and the use of the Cartesian plane.
Name:Functions (Useful and important repertoire)
Description:This module introduces the notion of a function which captures precisely ways in which different quantities or measurements are linked together. The module covers quadratic, cubic and general power and polynomial functions; exponential and logarithmic functions; and trigonometric functions related to the mathematics of periodic behaviour. We create new functions using composition and inversion and look at how to move backwards and forwards between quantities algebraically, as well as visually, with transformations in the xy-plane.
Name:Introducing the differential calculus
Description:This module introduces techniques of differential calculus. We look at average rates of change which become instantaneous, as time intervals become vanishingly small, leading to the notion of a derivative. We then explore techniques involving differentials that exploit tangent lines. The module introduces Leibniz notation and shows how to use it to get information easily about the derivative of a function and how to apply it.
Name:Properties and applications of the derivative
Description:This module continues the development of differential calculus by introducing the first and second derivatives of a function. We use sign diagrams of the first and second derivatives and from this, develop a systematic protocol for curve sketching. The module also introduces rules for finding derivatives of complicated functions built from simpler functions, using the Chain Rule, the Product Rule, and the Quotient Rule, and how to exploit information about the derivative to solve difficult optimisation problems.
Name:Introducing the integral calculus
Description:This fifth and final module introduces integral calculus, looking at the slopes of tangent lines and areas under curves. This leads to the Fundamental Theorem of Calculus. We explore the use of areas under velocity curves to estimate displacement, using averages of lower and upper rectangular approximations. We then look at limits of approximations, to discover the formula for the area of a circle and the area under a parabola. We then develop methods for capturing precisely areas under curves, using Riemann sums and the definite integral. The module then introduces indefinite integrals and the method of integration by substitution. Finally, we discuss properties of odd and even functions, related to rotational and reflectional symmetry, and the logistic function, which modifies exponential growth.
The focus and themes of the Introduction to Calculus course address the most important foundations for applications of mathematics in science, engineering and commerce. The course emphasises the key ideas and historical motivation for calculus, while at the same time striking a balance between theory and application, leading to a mastery of key threshold concepts in foundational mathematics. Students taking Introduction to Calculus will: • gain familiarity with key ideas of precalculus, including the manipulation of equations and elementary functions (first two weeks), • develop fluency with the preliminary methodology of tangents and limits, and the definition of a derivative (third week), • develop and practice methods of differential calculus with applications (fourth week), • develop and practice methods of the integral calculus (fifth week).
