Calculus: Single Variable Part 1 - Functions

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

Introduction

### Course Review: Calculus: Single Variable Part 1 - Functions on Coursera In the vast landscape of mathematics, calculus stands as a monumental pillar that has shaped our understanding of the world. If you're intrigued by its beauty and potential applications, then "Calculus: Single Variable Part 1 - Functions" on Coursera is an excellent starting point. Developed to demystify the concepts of single-variable calculus, this course offers a rigorous yet approachable journey through foundational topics that are essential for aspiring engineers, scientists, and social scientists. #### Overview of the Course The course begins with a clear articulation of the importance of calculus, bridging connections to real-world phenomena—from the intricacies of planetary orbits to the nuances of human heartbeat patterns. Designed with a focus on conceptual understanding and practical applications, it promises to equip students with the necessary tools to navigate through the complexities of calculus. **Key Features:** 1. **Taylor Series Introduction**: One of the distinctive features of this course is its early introduction to Taylor series, a powerful method for approximating functions. This topic serves as a cornerstone, providing students with a foundational tool that will be highly beneficial in higher mathematics. 2. **Comprehensive Coverage**: The syllabus encompasses essential concepts such as limits, asymptotic behavior, and the "big-O" notation, ensuring students grasp the transformative nature of calculus. #### Detailed Syllabus Breakdown - **Introduction**: The course kicks off with a diagnostic exam to assess your readiness. While it is optional, completing it can offer insights into your understanding of pre-calculus concepts, setting the tone for your learning journey. - **A Review of Functions**: In this module, students revisit the basics of functions, with a special focus on the exponential function. This establishes a solid foundation necessary for understanding the complexities that follow. - **Taylor Series**: Here, the course delves deep into Taylor series, presenting them as a series of polynomials that approximate functions. While the notation may initially seem daunting, the module effectively breaks down these concepts, making them accessible and engaging. - **Limits and Asymptotics**: This crucial module connects the dots between Taylor series and limits. Students learn about the convergence of Taylor series and explore l'Hôpital's rule, which aids in calculating limits. The introduction of "big-O" notation serves as a practical tool for students to bound asymptotic terms, equipping them with a language that will serve them in more advanced calculus. #### Why You Should Enroll This course is particularly recommended for: - **Beginners in Engineering, Physical, and Social Sciences**: The material is tailored for those just embarking on their calculus journey, making it an ideal introduction to the subject. - **Learners Seeking Conceptual Understanding**: The course emphasizes understanding over memorization, focusing on how calculus applies to real-life scenarios. - **Students Looking for Practical Applications**: With its emphasis on application, students can expect to learn not just how to perform calculations, but why they matter in various fields. In conclusion, "Calculus: Single Variable Part 1 - Functions" on Coursera stands out for its thoughtful structure, engaging content, and the emphasis on understanding the core principles of calculus. Whether you're starting your academic journey or looking to brush up on your skills, this course will serve as an invaluable resource. Prepare yourself for an intellectually rewarding experience as you unlock the wonders of calculus!

Syllabus

Introduction

Welcome to Calculus: Single Variable! below you will find the course's diagnostic exam. if you like, please take the exam. you don't need to score a minimal amount on the diagnostic in order to take the course. but if you do get a low score, you might want to readjust your expectations: this is a very hard class...

This module will review the basics of your (pre-)calculus background and set the stage for the rest of the course by considering the question: just what is the exponential function?

This module gets at the heart of the entire course: the Taylor series, which provides an approximation to a function as a series, or "long polynomial". You will learn what a Taylor series is and how to compute it. Don't worry! The notation may be unfamiliar, but it's all just working with polynomials....

A Taylor series may or may not converge, depending on its limiting (or "asymptotic") properties. Indeed, Taylor series are a perfect tool for understanding limits, both large and small, making sense of such methods as that of l'Hopital. To solidify these newfound skills, we introduce the language of "big-O" as a means of bounding the size of asymptotic terms. This language will be put to use in future Chapters on Calculus.