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所在平台: Udemy |
课程主页: https://www.udemy.com/course/calculus1-keypoints/
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课程名称：微积分 I：关键点与技巧 课程概述：本课程旨在强调微积分 I 的核心概念、关键计算方法和基本技巧。我们将聚焦于重要的内容，省略不必要的细节、简单主题和非核心的定理证明。完成本课程后，您将对微积分 I 的所有基础主题有坚实的理解，为未来的学习打下良好的基础，并为期末考试做好充分准备。每节课结束时都会布置练习题，这是课程的重要组成部分，旨在帮助您更好地理解和掌握材料。这些练习题简短易做，请务必完成。 课程内容组织如下： 1. 极限的评估方法：极限法则；洛必达法则；因式分解；比较无穷大的阶数；有理化；夹逼定理；三角函数的极限；单侧极限。 2. 连续性与不连续点：连续性的定义；可去不连续点；阶梯不连续点；无穷大不连续点；振荡不连续点；中间值定理；水平、垂直和斜渐近线。 3. 导数和微分规则：导数的定义；基本微分公式；加法和减法规则；乘法和除法法则；链式法则；隐函数微分；对数微分；反函数的导数；切线和法线；高阶导数；线性近似与微分。 4. 导数的应用：单调性；凹向上和凹向下；局部和全局极大值与极小值；拐点；曲线草图；相关速率；优化；牛顿法；平均值定理。 通过本课程，学习者将系统掌握微积分的基本知识，为进一步的数学学习做好准备。
This course is designed to emphasize the core concepts, key computational methods, and essential techniques of Calculus I. We will streamline our focus by skipping trivial details, overly elementary topics, and non-essential theorem proofs.By the end of this course, you will have a solid grasp of all the fundamental topics in Calculus I, establishing a strong foundation for future studies and ensuring you are well-prepared for the final exam.Practice exercises are assigned at the end of each lesson as an essential part of the course. They are designed to help you better understand and master the material. The exercises are concise and won't take much time to complete, so please make an effort to work through them.The course content is organized as follows:1. Methods to evaluate limits: limit laws; l'Hospital's rule; factoring; compare the order of infinity; rationalization;squeeze theorem; limits with trigonometric functions; one-sided limits.2. Continuity and discontinuous points: definition of continuity; removable discontinuous points; step discontinuous points; infinity discontinuous points; oscillating discontinuous point; intermediate value theorem; horizongtal , vertical and slant asymptotes.3. Derivative and defferential rules: definition of derivative; basic differential formulas; summation and subtraction rule; product and quotient rule; chain rule; implicit differentiation; logarithm differentiation; derivative for inverse functions; tangent and normal line; higher order derivatives; linear approximation and differential.4. Applications of derivative: increasing and decreasing; concave up and concave down; local and global maximum and minimum; inflection points; curve sketching; related rates; optimization; Newton's method; mean value theorem.
