Single Variable Calculus

University of Pennsylvania

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.

100% online course

Start instantly and learn at your own schedule.

Approx. 18 hours to complete

Suggested: 6-8 hours/week

English

Subtitles: English

100% online course

Start instantly and learn at your own schedule.

Approx. 18 hours to complete

Suggested: 6-8 hours/week

English

Subtitles: English

Syllabus - What you will learn from this course

1

Section

2 hours to complete

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 discrete input. These are sequences, and they will occupy our attention for this last segment of the course. This first module will introduce the tools and terminologies for discrete calculus....

4 videos (Total 53 min), 2 readings, 5 quizzes

4 videos

Sequences14m

BONUS!5m

Differences17m

Discrete Calculus16m

2 readings

Your Guide to Getting Started in this Course10m

How Grading Works10m

5 practice exercises

Core Homework: Sequences14m

Core Homework: Differences12m

Challenge Homework: Differences8m

Core Homework: Discrete Calculus10m

Challenge Homework: Discrete Calculus4m

2

Section

1 hour to complete

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 numerical analysis, answering such questions as "how do we approximate solutions to differential equations?" and "how do we approximate definite integals?" Perhaps unsurprisingly, Taylor expansion plays a pivotal role in these approximations....

2 videos (Total 34 min), 1 quiz

2 videos

Numerical O.D.E.s17m

Numerical Integration16m

1 practice exercises

Core Homework: Numerical O.D.E.s4m

3

Section

2 hours to complete

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 series. 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 (Total 65 min), 6 quizzes

4 videos

Infinite Series16m

Convergence Tests I16m

Convergence Tests II16m

Absolute & Conditional14m

6 practice exercises

Core Homework: Infinite Series10m

Challenge Homework: Infinite Series4m

Core Homework: Convergence Tests I16m

Challenge Homework: Convergence Tests I4m

Core Homework: Convergence Tests II6m

Core Homework: Absolute & Conditional4m

4

Section

2 hours to complete

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 (Total 59 min), 5 quizzes

4 videos

Power Series16m

Taylor Series Redux15m

Approximation and Error17m

BONUS!9m

5 practice exercises

Core Homework: Power Series12m

Challenge Homework: Power Series8m

Core Homework: Taylor Series Redux8m

Challenge Homework: Taylor Series Redux4m

Core Homework: Approximation and Error10m

5

Section

2 hours to complete

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 (Total 40 min), 2 readings, 2 quizzes

4 videos

Calculus Redux14m

BONUS!8m

Foreshadowing13m

Credits2m

2 readings

About the Chapter 5 Exam10m

About the 5-CHAPTER GREAT BIG FINAL EXAM!10m

2 practice exercises

Chapter 5: Discretization - Exam20m

5-CHAPTER GREAT BIG FINAL EXAM!36m

Instructor

Robert Ghrist

Professor

About 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. ...

Frequently Asked Questions

When will I have access to the lectures and assignments?
Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.
What will I get if I pay for this course?
If you pay for this course, you will have access to all of the features and content you need to earn a Course Certificate. If you complete the course successfully, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. Note that the Course Certificate does not represent official academic credit from the partner institution offering the course.
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Single Variable Calculus