About this course: 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.
Created by: University of Pennsylvania
Taught by: Robert Ghrist, Professor
Mathematics and Electrical & Systems Engineering
| Commitment | 6-8 hours/week |
| Language | English |
| How To Pass | Pass all graded assignments to complete the course. |
Syllabus
WEEK 1
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 videos, 2 readings, 2 practice quizzes
- Reading: Your Guide to Getting Started in this Course
- Reading: How Grading Works
- Video: Sequences
- Video: BONUS!
- Video: Differences
- Practice Quiz: Challenge Homework: Differences
- Video: Discrete Calculus
- Practice Quiz: Challenge Homework: Discrete Calculus
Graded: Core Homework: Sequences
Graded: Core Homework: Differences
Graded: Core Homework: Discrete Calculus
WEEK 2
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 videos
- Video: Numerical O.D.E.s
- Video: Numerical Integration
Graded: Core Homework: Numerical O.D.E.s
WEEK 3
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 videos, 2 practice quizzes
- Video: Infinite Series
- Practice Quiz: Challenge Homework: Infinite Series
- Video: Convergence Tests I
- Practice Quiz: Challenge Homework: Convergence Tests I
- Video: Convergence Tests II
- Video: Absolute & Conditional
Graded: Core Homework: Infinite Series
Graded: Core Homework: Convergence Tests I
Graded: Core Homework: Convergence Tests II
Graded: Core Homework: Absolute & Conditional
WEEK 4
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 videos, 2 practice quizzes
- Video: Power Series
- Practice Quiz: Challenge Homework: Power Series
- Video: Taylor Series Redux
- Practice Quiz: Challenge Homework: Taylor Series Redux
- Video: Approximation and Error
- Video: BONUS!
Graded: Core Homework: Power Series
Graded: Core Homework: Taylor Series Redux
Graded: Core Homework: Approximation and Error
WEEK 5
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 videos, 2 readings
- Video: Calculus Redux
- Video: BONUS!
- Video: Foreshadowing
- Video: Credits
- Reading: About the Chapter 5 Exam
- Reading: About the 5-CHAPTER GREAT BIG FINAL EXAM!
Graded: Chapter 5: Discretization - Exam
Graded: 5-CHAPTER GREAT BIG FINAL EXAM!
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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.
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