關於此課程：Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this fifth part--part five of five--we cover a calculus for sequences, numerical methods, series and convergence tests, power and Taylor series, and conclude the course with a final exam. Learners in this course can earn a certificate in the series by signing up for Coursera's verified certificate program and passing the series' final exam.
製作方： University of Pennsylvania
教學方： Robert Ghrist, Professor
Mathematics and Electrical & Systems Engineering
| 承諾學習時間 | 6-8小时/周 |
| 語言 | English |
| 如何通過 | 通過所有計分作業以完成課程。 |
授課大綱
第 1 週
A Calculus for Sequences
It's time to redo calculus! Previously, all the calculus we have done is meant for functions with a continuous input and a continuous output. This time, we are going to retool calculus for functions with a <i>discrete</i> input. These are <i>sequences</i>, and they will occupy our attention for this last segment of the course. This first module will introduce the tools and terminologies for <b>discrete calculus</b>.
4 個視頻, 2 個閱讀材料, 2 個練習測驗
- 阅读: Your Guide to Getting Started in this Course
- 阅读: How Grading Works
- 視頻: BONUS!
- 視頻: Differences
- 练习测验: Challenge Homework: Differences
- 視頻: Discrete Calculus
- 练习测验: Challenge Homework: Discrete Calculus
已評分: Core Homework: Sequences
已評分: Core Homework: Differences
已評分: Core Homework: Discrete Calculus
第 2 週
Introduction to Numerical Methods
That first module might have seemed a little...strange. It was! In this module, however, we will put that strangeness to good use, by giving a very brief introduction to the vast subjects of <b>numerical analysis</b>, answering such questions as <i>"how do we approximate solutions to differential equations?"</i> and <i>"how do we approximate definite integals?"</i> Perhaps unsurprisingly, Taylor expansion plays a pivotal role in these approximations.
2 個視頻
- 視頻: Numerical Integration
已評分: Core Homework: Numerical O.D.E.s
第 3 週
Series and Convergence Tests
In "ordinary" calculus, we have seen the importance (and challenge!) of improper integrals over unbounded domains. Within discrete calculus, this converts to the problem of infinite sums, or <b>series</b>. The determination of convergence for such will occupy our attention for this module. I hope you haven't forgotten your big-O notation --- you are going to need it!
4 個視頻, 2 個練習測驗
- 练习测验: Challenge Homework: Infinite Series
- 視頻: Convergence Tests I
- 练习测验: Challenge Homework: Convergence Tests I
- 視頻: Convergence Tests II
- 視頻: Absolute & Conditional
已評分: Core Homework: Infinite Series
已評分: Core Homework: Convergence Tests I
已評分: Core Homework: Convergence Tests II
已評分: Core Homework: Absolute & Conditional
第 4 週
Power and Taylor Series
This course began with an exploration of Taylor series -- an exploration that was, sadly, not as rigorous as one would like. Now that we have at our disposal all the tests and tools of discrete and continuous calculus, we can finally close the loop and make sense of what we've been doing when we Talyor-expand. This module will cover power series in general, from we which specify to our beloved Taylor series.
4 個視頻, 2 個練習測驗
- 练习测验: Challenge Homework: Power Series
- 視頻: Taylor Series Redux
- 练习测验: Challenge Homework: Taylor Series Redux
- 視頻: Approximation and Error
- 視頻: BONUS!
已評分: Core Homework: Power Series
已評分: Core Homework: Taylor Series Redux
已評分: Core Homework: Approximation and Error
第 5 週
Concluding Single Variable Calculus
Are we at the end? Yes, yes, we are. Standing on top of a high peak, looking back down on all that we have climbed together. Let's take one last look down and prepare for what lies above.
4 個視頻, 2 個閱讀材料
- 視頻: BONUS!
- 視頻: Foreshadowing
- 視頻: Credits
- 阅读: About the Chapter 5 Exam
- 阅读: About the 5-CHAPTER GREAT BIG FINAL EXAM!
已評分: Chapter 5: Discretization - Exam
已評分: 5-CHAPTER GREAT BIG FINAL EXAM!
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製作方
University of Pennsylvania
The University of Pennsylvania (commonly referred to as Penn) is a private university, located in Philadelphia, Pennsylvania, United States. A member of the Ivy League, Penn is the fourth-oldest institution of higher education in the United States, and considers itself to be the first university in the United States with both undergraduate and graduate studies.
