Calculus through Data & Modelling: Series and Integration

Johns Hopkins University via Coursera

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

Introduction

**Course Review: "Calculus through Data & Modelling: Series and Integration" on Coursera** **Overview** If you seek to deepen your understanding of calculus, the course “Calculus through Data & Modelling: Series and Integration” offers an essential pathway to develop your skills in this critical field. This course is designed to help learners transition from basic principles of differentiation to more complex concepts encompassing series, sequences, and the integral. Through this innovative approach, students will gain vital tools that are not only foundational to advanced mathematics but also applicable in real-world situations, such as measuring growth patterns, financial accumulation, and other cumulative data scenarios. **Course Structure** The syllabus is thoughtfully organized into five distinct modules that build upon one another, ensuring a comprehensive understanding of calculus' two main features: differentiation and integration. - **Module 1: Sequences and Series** Building the groundwork for integration, this module introduces the concept of the definite integral and how it relates to the area beneath a curve. The Fundamental Theorem of Calculus is examined, showing how differentiation and integration are interlinked, thereby enriching your conceptual grasp of calculus. - **Module 2: The Definite Integral** This module offers insights into Riemann Sums, a crucial technique for approximating integrals through finite sums. Understanding this concept is essential for visualizing and calculating areas under curves, laying a solid foundation for the impending discussions on definite integrals. - **Module 3: The Fundamental Theorem of Calculus** Dissecting the Fundamental Theorem of Calculus brings to light the relationship between differentiation and integration. Students discover how this theorem enables the finding of antiderivatives and renders numerical integration unnecessary for many practical applications, thus enhancing accuracy and comprehension. - **Module 4: The Indefinite Integral** Antidifferentiation is the focus here, where learners will work on finding families of antiderivatives. This is a critical step in mastering calculus as it ties back into both definite integrals and the Fundamental Theorem, fostering a well-rounded understanding of these crucial concepts. - **Integration with Calculators and Tables** This concluding module navigates complexities through numerical methods and technological assistance for tackling difficult integrals. The application of estimation techniques not only cements theoretical knowledge but also prepares students for real-life applications in decision-making scenarios. **Recommendations** **Who Should Enroll?** This course is highly recommended for students pursuing degrees in engineering, economics, data science, or any field where a strong grasp of calculus is pertinent. Even those embarking on self-study or professional development in mathematics or related disciplines will find considerable value. **Why Take This Course?** - **Structured Learning**: The course provides a clear progression through concepts, making them increasingly manageable for students at all levels. - **Real-World Applications**: It emphasizes practical applications of calculus, which can be beneficial for students looking to apply mathematics in various fields, particularly in modeling and data analysis. - **Interactive Elements**: Expect exercises and quizzes that reinforce the material, ensuring that you can apply what you've learned effectively. **Conclusion** “Calculus through Data & Modelling: Series and Integration” is an excellent choice for anyone interested in mastering calculus' critical aspects, especially integration and series. With its well-organized modules, emphasis on real-world applications, and robust foundational approach, this course equips learners with the skills necessary for academic and professional success. If you're ready to deepen your understanding of calculus in a structured and practical way, this course on Coursera is highly worth your investment!

Syllabus

Module 1: Sequences and Series

Calculus is divided into two halves: differentiation and integration. In this module, we introduce the process of integration. First we will see how the definite integral can be used to find the area under the graph of a curve. Then, we will investigate how differentiation and integration are inverses of each other, through the Fundamental Theorem of Calculus. Finally, we will learn about the indefinite integral, and use some strategies for computing integrals.

Module 2: The Definite Integral

In this module, we introduce the notion of Riemann Sums. In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum, named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. This notion of approximating the accumulation of area under a group will lead to the concept of the definite integral, and the many applications that follow.

Module 3: The Fundamental Theorem of Calculus

We now introduce the first major tool of our studies, the Fundamental Theorem of Calculus. This deep theorem links the concept of differentiating a function with the concept of integrating a function. The theorem will consists of two parts, the first of which implies the existence of antiderivatives for continuous functions and the second of which plays a larger role in practical applications. The beauty and practicality of this theorem allows us to avoid numerical integration to compute integrals, thus providing a better numerical accuracy.

Module 4: The Indefinite Integral

In this module, we focus on developing our ability to find antiderivatives, or more generally, families of antiderivatives. In calculus, the general family of antiderivatives is denoted with an indefinite integral, and the process of solving for antiderivatives is called antidifferentiation. This is the opposite of differentiation and completes our knowledge of the two major tools of calculus. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

Integration with Calculators and Tables

While the technique of finding antiderivatives is useful, there are some functions that are just too difficult to find antiderivatives for. In cases like these, we want to have a numerical method to approximate the definite integral. In this module, we introduce two techniques for solving complicated integrals: using technology or tables of integrals, as well as estimation techniques. We then apply our knowledge to analyze strategies and decision theory as applied to random events.