Single Variable Calculus

University of Pennsylvania

About this Course

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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.

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Approx. 14 hours to complete

English

Subtitles: French, Portuguese (Brazilian), Russian, English, Spanish

Instructor

Instructor rating

4.53/5 (15 Ratings)

Robert Ghrist

Professor

Mathematics and Electrical & Systems Engineering

129,602 Learners

5 Courses

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Approx. 14 hours to complete

English

Subtitles: French, Portuguese (Brazilian), Russian, English, Spanish

Offered by

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.

Syllabus - What you will learn from this course

Week

Week 1

4 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 hours to complete

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: Sequences30m

Core Homework: Differences30m

Challenge Homework: Differences30m

Core Homework: Discrete Calculus30m

Challenge Homework: Discrete Calculus30m

Week

Week 2

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.

1 hour to complete

2 videos (Total 34 min)

2 videos

Numerical O.D.E.s17m

Numerical Integration16m

1 practice exercise

Core Homework: Numerical O.D.E.s30m

Week

Week 3

4 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 hours to complete

4 videos (Total 65 min)

4 videos

Infinite Series16m

Convergence Tests I16m

Convergence Tests II16m

Absolute & Conditional14m

6 practice exercises

Core Homework: Infinite Series30m

Challenge Homework: Infinite Series30m

Core Homework: Convergence Tests I30m

Challenge Homework: Convergence Tests I30m

Core Homework: Convergence Tests II30m

Core Homework: Absolute & Conditional30m

Week

Week 4

3 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.

3 hours to complete

4 videos (Total 59 min)

4 videos

Power Series16m

Taylor Series Redux15m

Approximation and Error17m

BONUS!9m

5 practice exercises

Core Homework: Power Series30m

Challenge Homework: Power Series30m

Core Homework: Taylor Series Redux30m

Challenge Homework: Taylor Series Redux30m

Core Homework: Approximation and Error30m

Frequently Asked Questions

When will I have access to the lectures and assignments?
Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:
The course may not offer an audit option. You can try a Free Trial instead, or apply for Financial Aid.
The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.
What will I get if I purchase the Certificate?
When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
Is financial aid available?
Yes, Coursera provides financial aid to learners who cannot afford the fee. Apply for it by clicking on the Financial Aid link beneath the "Enroll" button on the left. You’ll be prompted to complete an application and will be notified if you are approved. Learn more.
Will I earn university credit for completing the Course?
This Course doesn't carry university credit, but some universities may choose to accept Course Certificates for credit. Check with your institution to learn more. Online Degrees and Mastertrack™ Certificates on Coursera provide the opportunity to earn university credit.

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Single Variable Calculus

Single Variable Calculus