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 discrete input. These are sequences, and...

4 videos, 2 readings, 2 readings

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 numerical analysis, answering such questions as "how do we ...

2 videos

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 series. The determination of convergence for such will occupy ...

4 videos, 2 readings

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 se...

4 videos, 2 readings

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

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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