Calculus through Data & Modeling: Limits & Derivatives

Go to Course: https://www.coursera.org/learn/calculus-through-data-and-modelling-imits-derivatives

Introduction

### Course Review: Calculus through Data & Modeling: Limits & Derivatives In the realm of mathematics, understanding calculus is not just a prerequisite for advanced studies; it is a critical skill that influences various fields, including engineering, economics, and data science. Coursera’s **Calculus through Data & Modeling: Limits & Derivatives** serves as a fundamental introduction to single-variable calculus, focusing specifically on limits and derivatives. This course provides an excellent starting point for both newcomers to calculus and those looking to refresh their knowledge. #### Course Overview **Course Name:** Calculus through Data & Modeling: Limits & Derivatives **Platform:** Coursera **Target Audience:** Beginners in calculus, students needing a refresher, data enthusiasts The course begins with a comprehensive exploration of the concept of limits. It sets the stage for defining derivatives, which measure how a function responds to changes in its input. For example, understanding an object's velocity as related to its position over time is a practical application of derivatives. This foundational knowledge is essential not only for mathematics but also for practical applications in various fields. #### Course Syllabus Breakdown 1. **The Limit of a Function:** This initial module introduces the critical concepts of limits and derivatives, framed within the context of rates of change. Students will learn to evaluate limits using both visual and algebraic approaches, paving the way for a deeper understanding of differentiable calculus. 2. **The Limit Laws:** Here, the course delves into algebraic properties known as Limit Laws. Mastery of these laws is paramount as they allow students to compute limits with precision—an indispensable skill for analyzing more complex functions. 3. **Continuity:** This module clarifies the concept of continuity and defines how limits relate to it. The content emphasizes how mathematical continuity aligns with everyday interpretations of the term, enhancing students' grasp of both the concept and its implications. 4. **Limits at Infinity:** Students will explore the behavior of functions as inputs grow large or small. This module introduces horizontal asymptotes and the end-behavior of functions, enriching students' understanding of functions in various contexts. 5. **Derivatives:** Perhaps the cornerstone of this course, this module teaches students how to find the derivative, linking it to real-world interpretations like instantaneous velocity and slopes of tangent lines. This connection to everyday applications makes the theory relatable and engaging. 6. **Final Project:** The course culminates in a practical project where students will apply their learned skills to analyze gender ratios in educational attainment over time across different regions. This project not only reinforces theoretical concepts but also connects them to real data analysis, a key skill in today’s data-driven world. #### Course Experience Coursera’s platform provides a user-friendly interface with a blend of video lectures, quizzes, and interactive assignments that cater to various learning styles. As the course progresses, learners are encouraged to actively engage with the material through practical problems and discussions, making the learning experience more dynamic and less passive. #### Recommendation **Who Should Take This Course?** This course is highly recommended for anyone interested in strengthening their mathematics foundation, particularly those pursuing careers in science, technology, engineering, and mathematics (STEM). It is also suitable for data enthusiasts who wish to incorporate mathematical analysis into their skill sets. **Why You Should Enroll:** - **Foundational Knowledge:** Gain a solid grasp of the basics of calculus that is applicable across a multitude of fields. - **Practical Applications:** The course’s focus on modeling and data will allow you to see the relevance of calculus in the real world. - **Flexible Learning:** Being an online course, it provides the flexibility to learn at your own pace, making it ideal for busy professionals or students. In conclusion, **Calculus through Data & Modeling: Limits & Derivatives** is an outstanding course provided by Coursera. It strikes a balance between rigorous theoretical foundations and practical applications, making it a wise investment for anyone looking to thrive in an increasingly data-centric world. Whether you’re entering the workforce or continuing your education, this course will not only enhance your mathematical acumen but also prepare you for more advanced studies in calculus and beyond.

Syllabus

The Limit of a Function

One of the goals in studying Calculus is to analyze rates of change and movement. In this module, we introduce the central ideas which will help us achieve this goal: the notions of the limit and the derivative. Rather than evaluating a function at a single point, the limit allows for the study of the behavior of a function in an interval around that point. In this module, you will find limits of functions by a variety of methods, both visually and algebraically. Finally, we will apply limits to define the key idea of Differentiable Calculus, the derivative.

Using calculators or graphs is an imprecise way to find the limit of a function. In this module, we will state and use algebraic properties of limits, called the Limit Laws, to calculate the exact values of limits. A solid understanding of these laws will allow us to derive theorems which in turn can be used to study the behavior of more advanced functions.

In the last module, there were several types of functions where the limit of a function as x approaches a number could be found by simply calculating the value of the function at the number. Functions with this property will be called continuous and in this module, we use limits to define continuity. We will see that the mathematical definition of continuity will correspond closely with the English meaning of the word continuity used in every day language.

In this module, we allow for x to become arbitrarily large in the positive or negative direction to understand the end-behaviors of functions. This will allow for the formal definition of a horizontal asymptote and to provide classifications of end-behavior of certain types of functions.

The problem of finding the slope of the tangent line to a curve and the problem of finding the instantaneous velocity of an object both involve finding the same type of limit. This special type of limit is called the derivative and in this module, we will see that this notion of the derivative can be interpreted as a rate of change in any of the natural or social sciences or engineering.

In this final project, we will apply the tools and language of differentiable calculus to analyze trends in data. This project will focus on modelling and analyzing gender ratios in educational attainment over time in several regions of the world.