The course covers topics such as limits, derivatives, integration, and applications of calculus to real-world problems.
Throughout the course, students will learn how to apply these concepts to solve problems in fields such as physics, engineering, and economics. The course also includes interactive quizzes, problem sets, and video lectures to help students practice and reinforce what they have learned.
By the end of the course, students should have a strong understanding of calculus and be able to apply its concepts to solve real-world problems. They should be able to calculate derivatives and integrals, understand the relationship between the two, and use these concepts to analyze functions and data.
This course is suitable for students with a background in algebra and trigonometry who are interested in learning calculus. Students who have completed this course will be well-prepared for further study in calculus or other areas of mathematics and science.
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.
The Single Variable Calculus course by Robert Ghrist on Coursera is divided into 5 modules, with each part covering a different aspect of single variable calculus. The course includes a total of 18 video lectures, along with interactive quizzes and problem sets to help students master the material.
Here is a detailed breakdown of the course content:
Module 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 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 2 readings 2 quizzes
4 videosTotal 53 minutes
- Sequences 14 minutes Preview module
- BONUS! 5 minutes
- Differences 17 minutes
- Discrete Calculus 16 minutes
2 readingsTotal 20 minutes
- Your Guide to Getting Started in this Course 10 minutes
- How Grading Works 10 minutes
2 quizzesTotal 60 minutes
- Challenge Homework: Differences 30 minutes
- Challenge Homework: Discrete Calculus 30 minutes
Module 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 approximate solutions to differential equations?" and "how do we approximate definite integals?" Perhaps unsurprisingly, Taylor expansion plays a pivotal role in these approximations.
2 videosTotal 34 minutes
- Numerical O.D.E.s 17 minutes Preview module
- Numerical Integration 16 minutes
Module 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 our attention for this module. I hope you haven't forgotten your big-O notation --- you are going to need it!
4 videos 2 quizzes
4 videosTotal 64 minutes
- Infinite Series 16 minutes Preview module
- Convergence Tests I 16 minutes
- Convergence Tests II 16 minutes
- Absolute & Conditional 14 minutes
2 quizzesTotal 60 minutes
- Challenge Homework: Infinite Series 30 minutes
- Challenge Homework: Convergence Tests I 30 minutes
Module 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 quizzes
4 videosTotal 59 minutes
- Power Series 16 minutes Preview module
- Taylor Series Redux 15 minutes
- Approximation and Error 17 minutes
- BONUS! 9 minutes
2 quizzesTotal 60 minutes
- Challenge Homework: Power Series 30 minutes
- Challenge Homework: Taylor Series Redux 30 minutes
Module 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
4 videosTotal 39 minutes
- Calculus Redux 14 minutes Preview module
- BONUS! 8 minutes
- Foreshadowing 13 minutes
- Credits 2 minutes
2 readingsTotal 20 minutes
- About the Chapter 5 Exam 10 minutes
- About the 5-CHAPTER GREAT BIG FINAL EXAM! 10 minutes
The Single Variable Calculus course by Robert Ghrist on Coursera has received high praise from students and educators for its innovative teaching methods and engaging content. The course is designed to be accessible to a wide range of learners, and provides a strong foundation in calculus concepts.
One of the key strengths of the course is its use of video lectures, which are clear, concise, and engaging. Ghrist has a knack for explaining complex calculus concepts in a way that is easy to understand, making the material less intimidating for students who may be new to the subject.
The course also features interactive quizzes and assignments, which help students to apply the concepts they have learned and reinforce their understanding. This approach is highly effective, as it allows students to practice problem-solving and critical thinking skills in a supportive and structured environment.
Another strength of the course is its emphasis on real-world applications of calculus concepts. Ghrist uses examples from physics, engineering, and other fields to illustrate the practical value of calculus, which helps students to see the relevance of the material to their own lives and interests.
One potential downside of the course is that it can be challenging for students who do not have a strong foundation in algebra and trigonometry. The course assumes some prior knowledge of these subjects, and students who are struggling may need to seek additional support or review materials to keep up with the pace of the course.
Overall, the Single Variable Calculus course by Robert Ghrist is an excellent resource for anyone looking to improve their understanding of calculus. It is well-designed, engaging, and highly effective, and has received high praise from students and educators alike.
At the time, the couse has an average rating of 4.6 out of 5 stars based on over 295 ratings.
What you'll learn:
After completing the Single Variable Calculus course by Robert Ghrist on Coursera, students should have developed a deep understanding of calculus and the skills to apply it to a wide range of problems. Specifically, they will have gained:
Mastery of calculus concepts: Students will have a solid understanding of calculus concepts such as limits, derivatives, and integrals, and how to apply them in different situations. They will have learned how to evaluate limits using algebraic and graphical techniques, differentiate functions using the power rule, chain rule, and product rule, and integrate functions using the substitution rule, integration by parts, and other techniques.
Problem-solving skills: Through the course's problem sets and interactive quizzes, students will have developed their ability to apply calculus to solve a variety of problems. They will have learned how to use calculus to optimize functions, solve related rates problems, and apply calculus in physics and engineering contexts.
Modeling and real-world problem-solving: The course's focus on modeling and applying calculus to real-world problems will have prepared students to analyze and solve problems in a wide range of fields, including physics, engineering, and economics. They will have learned how to use calculus to model and solve problems such as finding maximum or minimum values of functions, determining the rate of change of physical quantities, and calculating areas and volumes of complex shapes.
Function analysis and graphing: Students will have learned how to analyze functions using calculus, including understanding their behavior, identifying critical points and inflection points, and sketching their graphs. They will have gained a deeper understanding of the properties of functions such as continuity, differentiability, and concavity.
Multidisciplinary skills: The skills developed in this course are valuable not just for further study in mathematics, but also in a wide range of fields. Students will have gained skills that are relevant to physics, engineering, economics, and other sciences.
Overall, completion of the Single Variable Calculus course will equip students with the skills and knowledge necessary to tackle advanced topics in calculus, as well as to apply calculus in a variety of contexts.
Robert Ghrist is a mathematician and professor at the University of Pennsylvania. He received his Ph.D. in Mathematics from the University of California, Berkeley in 1994, and has since made significant contributions to the fields of algebraic topology, applied algebraic topology, and computational topology.
Ghrist has authored numerous research papers and several books, including the widely-used textbook "Elementary Applied Topology" which is considered to be a seminal work in applied topology. He is also a co-author of the book "Topology and Robotics", which explores the relationship between algebraic topology and robotics.
In addition to his research, Ghrist is known for his innovative teaching style and commitment to online education. He has developed several online courses, including the "Single Variable Calculus" course on Coursera, and has received several awards for his contributions to education, including the NSF CAREER Award and the American Society for Engineering Education's Frederick Emmons Terman Award.
Ghrist's work in topology and its applications has been widely recognized by the mathematics community. He has been awarded the National Science Foundation's Presidential Faculty Fellowship, the Sloan Research Fellowship, and the Institute of Mathematics and Its Applications Prize in Mathematics and Applications.
Overall, Robert Ghrist is a highly accomplished mathematician with a strong background in topology and its applications. His work in research and teaching has had a significant impact on the field of mathematics, and his dedication to online education has made calculus accessible to a wider audience.
The requirements for the Single Variable Calculus course by Robert Ghrist on Coursera are:
A strong foundation in algebra and trigonometry: Students should have a good understanding of algebraic and trigonometric functions, including their properties and graphs.
Basic knowledge of calculus concepts: Students should have some familiarity with calculus concepts such as limits, derivatives, and integrals.
Access to a computer and internet: The course is delivered entirely online, so students will need access to a computer and a reliable internet connection.
Time commitment: The course consists of 10 weeks of lectures, assignments, and quizzes, with an estimated workload of 4-6 hours per week.
Motivation and willingness to learn: Like any college-level course, the "Single Variable Calculus" course requires students to be motivated and committed to learning. Students should be prepared to dedicate time and effort to the course, and be willing to engage with the material and ask questions when necessary.
Overall, the course is designed to be accessible to a wide range of learners, from high school students to adult learners. However, students should have a strong foundation in algebra and trigonometry, as well as some familiarity with calculus concepts, in order to succeed in the course.