Introduction to Calculus

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

Introduction

# Course Review: Introduction to Calculus on Coursera Calculus is often regarded as the cornerstone of higher mathematics, making it an essential subject for anyone interested in fields such as science, engineering, and commerce. The **Introduction to Calculus** course available on Coursera provides a comprehensive grounding in this fundamental area of mathematics. In this review, we'll explore the course structure, content, and overall value to help you determine if it's the right fit for your academic or professional needs. ### Course Overview The **Introduction to Calculus** course focuses on laying the groundwork for understanding how calculus is applied in various domains. The curriculum emphasizes crucial concepts and historical motivations tied to these mathematical ideas. By striking a careful balance between theoretical principles and practical applications, the course aims to facilitate a deep understanding of essential calculus concepts. ### Learning Outcomes By the end of this course, students will: - Gain familiarity with key ideas of precalculus, setting a solid foundation for calculus concepts. - Understand functions and how they relate to mathematics. - Comprehend the principles of differential calculus, including derivatives and their applications. - Explore integral calculus, including the Fundamental Theorem of Calculus and methods for calculating areas under curves. ### Course Syllabus The course is structured into five comprehensive modules, each designed to build on previous concepts: 1. **Precalculus (Setting the Scene)**: - This module introduces students to various types of numbers, decimal expansions, and basic manipulation of equations. It sets the stage for understanding functions and their graphical representations. 2. **Functions (Useful and Important Repertoire)**: - Here, the concept of functions is explored in depth, covering a variety of function types (polynomials, exponential, logarithmic, and trigonometric). The focus on composition and inversion of functions equips students with tools to navigate relationships between quantities effectively. 3. **Introducing the Differential Calculus**: - This module dives into the core idea of derivatives, emphasizing how average rates of change evolve into instantaneous rates. The introduction to Leibniz notation simplifies the process of differentiation and sets the groundwork for more complex applications. 4. **Properties and Applications of the Derivative**: - Building on the principles introduced in the previous module, students examine the first and second derivatives, learning to sketch curves systematically and apply derivative rules to solve optimization problems. 5. **Introducing the Integral Calculus**: - The final module ties together the knowledge gained from the previous ones by introducing the integral. Students learn about areas under curves, Riemann sums, and the Fundamental Theorem of Calculus, which is pivotal in understanding mathematical analysis. ### Pros and Cons #### Pros: - **Well-Structured Content**: The modules are logically sequenced to facilitate gradual learning. - **Balance of Theory and Application**: Each concept is supported with applications, making it easier to grasp the practical necessity of calculus. - **Clear Explanations**: The course does a commendable job breaking down complex ideas into understandable segments, which is particularly helpful for beginners. - **Access to Resources**: Coursera often provides valuable supplementary materials and community support. #### Cons: - **Format Limitations**: While video lectures are beneficial, some learners may prefer interactive or hands-on learning experiences. - **Pace of Learning**: Some students may find the course pace challenging, especially if they struggle with precalculus concepts. ### Conclusion and Recommendation The **Introduction to Calculus** course on Coursera is an excellent resource for anyone looking to strengthen their mathematical foundations, especially those who plan to pursue careers in science, technology, engineering, or math (STEM) fields. Its comprehensive approach, from precalculus concepts to fundamental calculus principles, ensures that students are well-prepared for more advanced studies. I highly recommend this course for high school students preparing for college, university undergraduates seeking a solid refresher, or professionals in need of calculus for their work. The course provides a conducive environment for building confidence in mathematics, which can open doors to numerous academic and career opportunities. Enroll today to embark on your journey into the world of calculus!

Syllabus

Precalculus (Setting the scene)

This module begins by looking at the different kinds of numbers that fall on the real number line, decimal expansions and approximations, then continues with an exploration of manipulation of equations and inequalities, of sign diagrams and the use of the Cartesian plane.

This module introduces the notion of a function which captures precisely ways in which different quantities or measurements are linked together. The module covers quadratic, cubic and general power and polynomial functions; exponential and logarithmic functions; and trigonometric functions related to the mathematics of periodic behaviour. We create new functions using composition and inversion and look at how to move backwards and forwards between quantities algebraically, as well as visually, with transformations in the xy-plane.

This module introduces techniques of differential calculus. We look at average rates of change which become instantaneous, as time intervals become vanishingly small, leading to the notion of a derivative. We then explore techniques involving differentials that exploit tangent lines. The module introduces Leibniz notation and shows how to use it to get information easily about the derivative of a function and how to apply it.

This module continues the development of differential calculus by introducing the first and second derivatives of a function. We use sign diagrams of the first and second derivatives and from this, develop a systematic protocol for curve sketching. The module also introduces rules for finding derivatives of complicated functions built from simpler functions, using the Chain Rule, the Product Rule, and the Quotient Rule, and how to exploit information about the derivative to solve difficult optimisation problems.

This fifth and final module introduces integral calculus, looking at the slopes of tangent lines and areas under curves. This leads to the Fundamental Theorem of Calculus. We explore the use of areas under velocity curves to estimate displacement, using averages of lower and upper rectangular approximations. We then look at limits of approximations, to discover the formula for the area of a circle and the area under a parabola. We then develop methods for capturing precisely areas under curves, using Riemann sums and the definite integral. The module then introduces indefinite integrals and the method of integration by substitution. Finally, we discuss properties of odd and even functions, related to rotational and reflectional symmetry, and the logistic function, which modifies exponential growth.