Single Variable Calculus

Go to Course: https://www.coursera.org/learn/discrete-calculus

Introduction

### Course Review: Single Variable Calculus on Coursera #### Overview Calculus stands as a cornerstone of mathematics and a critical tool across various fields, including engineering, physical sciences, and social sciences. The **Single Variable Calculus** course offered on Coursera invites students to navigate the intricate landscape of calculus by emphasizing not just the procedural aspects but also the conceptual understandings and real-world applications. This course is designed for newcomers to the subject, providing a solid foundation for those intending to delve deeper into technical fields. #### Course Features One of the distinguishing features of this course is its innovative approach to calculus through the introduction of **Taylor series** and approximations. This foundational concept is key in bridging the gap between theoretical mathematics and applied problem-solving. #### Syllabus Breakdown 1. **A Calculus for Sequences** - This module cleverly redefines the calculus framework by focusing on discrete functions—sequences. It introduces essential tools and terminologies that prepare participants for the discrete calculus landscape, paving the way for deeper understanding. 2. **Introduction to Numerical Methods** - Here, learners encounter the exciting world of numerical analysis. This module addresses fundamental questions such as how to approximate solutions to differential equations and definite integrals. The Taylor expansion becomes a vital part of these approximations, showcasing its significance within broader mathematical contexts. 3. **Series and Convergence Tests** - Transitioning from integrals to infinite sums, this segment explores the critical topics of series and their convergence. The module revisits big-O notation and tests for convergence, reinforcing skills necessary for tackling complex calculus problems. 4. **Power and Taylor Series** - This module provides a more rigorous examination of the Taylor series, tying together the insights gleaned from previous modules. By the end of this section, students will have a comprehensive understanding of power series, connecting theoretical concepts with practical applications. 5. **Concluding Single Variable Calculus** - As the course wraps up, this final module encourages reflection on the material covered. Students will consolidate their understanding and prepare for more advanced topics in calculus, equipped with the tools to tackle future challenges. #### Recommendation The **Single Variable Calculus** course on Coursera is an exceptional resource for anyone seeking to build a robust foundation in calculus. Its unique focus on both discrete and continuous aspects of calculus makes it suitable for a diverse range of learners. The course skillfully balances theory with practical application, which is invaluable for students in engineering, physical sciences, and social sciences. **Pros:** - Clear and structured content - Emphasis on conceptual understanding - Real-world applications of calculus principles - An introduction to advanced topics like Taylor series in an accessible manner **Cons:** - The pace may be brisk for those without a prior background in mathematics - Some concepts may require additional resources for a complete grasp Overall, whether you’re a student gearing up for a career in a STEM field or simply someone interested in enhancing your mathematical understanding, I highly recommend this course. By fostering both theoretical knowledge and practical skills, this course offers a comprehensive approach to mastering single-variable calculus—a pivotal step in many academic and professional journeys.

Syllabus

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.

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.

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!

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.

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.