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.
Single Variable Calculus
4.8
(126 ratings)
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von
University of Pennsylvania
Lehrplan - Was Sie in diesem Kurs lernen werden
Woche
12 Stunden zum Abschließen
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 (Gesamt 53 min), 2 Lektüren, 5 Quiz
2 Lektüren
Your Guide to Getting Started in this Course10m
How Grading Works10m
5 praktische Übungen
Core Homework: Sequences14m
Core Homework: Differences12m
Challenge Homework: Differences8m
Core Homework: Discrete Calculus10m
Challenge Homework: Discrete Calculus4m
Woche
21 Stunde zum Abschließen
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 (Gesamt 34 min), 1 Quiz
1 praktische Übung
Core Homework: Numerical O.D.E.s4m
Woche
32 Stunden zum Abschließen
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 (Gesamt 65 min), 6 Quiz
6 praktische Übungen
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
Woche
42 Stunden zum Abschließen
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 (Gesamt 59 min), 5 Quiz
5 praktische Übungen
Core Homework: Power Series12m
Challenge Homework: Power Series8m
Core Homework: Taylor Series Redux8m
Challenge Homework: Taylor Series Redux4m
Core Homework: Approximation and Error10m
Top-Bewertungen
This course is tricky and also excellent. I am a computer science student from germany, and it took me quite some time and effort to pass it. The course is well structured, and can be done in time.
Feeling really great after finishing the course. Learnt a lot and gained deeper insights into the calculus.\n\nLooking forward to Multi Variable Calculus.
Dozent
Robert Ghrist
ProfessorMathematics and Electrical & Systems Engineering
Über 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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