Introduction to Calculus with Infinitesimals
Document information
| Author | Dan Sloughter |
| School | Furman University |
| Major | Mathematics |
| Year of publication | 2007 |
| Place | Greenville |
| Document type | thesis |
| Language | English |
| Number of pages | 147 |
| Format | |
| Size | 3.88 MB |
- Calculus
- Hyperreal Numbers
- Non-standard Analysis
Summary
I. Introduction
The document 'Introduction to Calculus with Infinitesimals' serves as a foundational text for understanding calculus through the lens of the hyperreal number system. The author, Dan Sloughter, emphasizes the use of infinitesimal and infinite numbers as essential tools in calculus. Unlike traditional calculus texts that rely on the real number system, this work does not provide a logical justification for the hyperreals, which may challenge readers unfamiliar with these concepts. The text is designed for those with a basic understanding of calculus, algebra, and trigonometry, making it accessible yet demanding. The preface sets the stage for a deeper exploration of calculus, inviting readers to engage with the material critically. The author suggests consulting foundational texts such as Abraham Robinson’s Non-standard Analysis for those interested in the theoretical underpinnings of the hyperreal system. This approach highlights the document's unique perspective on calculus, focusing on the practical applications of infinitesimals in mathematical analysis.
II. Derivatives
The section on derivatives introduces key concepts such as the arrow paradox, which illustrates the complexities of motion and velocity. Zeno's paradox raises questions about the nature of movement, particularly when considering the concept of zero divided by zero. This paradox challenges the reader to think critically about the definitions of velocity and distance in the context of infinitesimals. The author explains that infinitesimals provide a framework for understanding ratios of quantities with zero magnitude, which is crucial for developing the principles of differential calculus. Furthermore, the document explores the implications of adding zeroes and the indeterminate nature of products involving infinity and zero. These discussions are vital for grasping the foundational ideas of calculus, as they lay the groundwork for understanding rates of change and the behavior of functions. The significance of this section lies in its ability to connect abstract mathematical concepts with real-world applications, demonstrating how calculus can be used to model and analyze dynamic systems.
2.1 The Arrow Paradox
Zeno's arrow paradox serves as a critical entry point into the discussion of motion and velocity. The paradox posits that an arrow in flight is at rest at every instant of time, leading to the conclusion that it cannot move. This assertion raises fundamental questions about the nature of motion and the mathematical definitions that underpin it. The author uses this paradox to illustrate the challenges of defining velocity when considering the concept of zero divided by zero. The exploration of this paradox not only highlights the limitations of traditional calculus but also emphasizes the need for a more nuanced understanding of motion through the lens of infinitesimals. By addressing these philosophical and mathematical dilemmas, the document encourages readers to reconsider their assumptions about calculus and its applications.
2.2 Rates of Change
The concept of rates of change is central to calculus and is thoroughly examined in this document. The author discusses how infinitesimals allow for a more precise understanding of how quantities change over time. This section emphasizes the importance of derivatives in capturing the instantaneous rate of change of a function. By employing the hyperreal number system, the author provides a framework for analyzing functions that may not conform to traditional definitions. The practical applications of these concepts are vast, ranging from physics to economics, where understanding rates of change is crucial for modeling dynamic systems. The document effectively illustrates how the principles of calculus can be applied to real-world scenarios, reinforcing the value of infinitesimals in mathematical analysis.
III. Continuous Functions
The exploration of continuous functions is another significant aspect of the document. Continuous functions are essential in calculus as they allow for the application of various mathematical techniques, including limits and derivatives. The author discusses the properties of continuous functions, highlighting their importance in ensuring that small changes in input lead to small changes in output. This section also delves into the implications of continuity for the behavior of functions, such as the existence of limits and the application of the Intermediate Value Theorem. Understanding these properties is crucial for students and practitioners of calculus, as they form the basis for more advanced topics such as integration and optimization. The document emphasizes that a solid grasp of continuous functions is necessary for effectively applying calculus in various fields, including engineering, physics, and economics.
3.1 Properties of Continuous Functions
The properties of continuous functions are explored in depth, focusing on their significance in calculus. The author outlines key characteristics, such as the ability to be graphed without lifting the pencil, which illustrates the concept of continuity. Additionally, the document discusses the implications of continuity for the existence of limits and the behavior of functions at critical points. The author emphasizes that understanding these properties is essential for applying calculus effectively in real-world scenarios. The discussion includes practical examples that demonstrate how continuous functions can be used to model various phenomena, reinforcing the document's value as a resource for students and professionals alike.
Document reference
- Non-standard Analysis (Abraham Robinson)
- Lectures on the Hyperreals (Robert Goldblatt)