A Problem-Solving Approach to Single-Variable Calculus
Volume II
- 2026
- Book
- Author
- Farzin Asadi
- Publisher
- Springer Nature Switzerland
About this book
Single-variable calculus is a fundamental mathematical framework essential for students across diverse fields—including the basic sciences, engineering, and economics—because it enables the precise modeling and understanding of dynamic systems. It introduces two indispensable tools: the derivative, which quantifies instantaneous rates of change, and the integral, which measures accumulation over time or space. These concepts form the foundation of nearly all quantitative disciplines.
To truly master the concepts and techniques of single-variable calculus, practice is paramount. "A Problem-Solving Approach to Single-Variable Calculus" is your essential guide, offering a comprehensive, two-volume set filled with plenty of meticulously solved, step-by-step problems designed to build your skills and deepen your understanding. This book empowers you to confidently tackle any single-variable calculus challenge, transforming them into triumphs.
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Table of Contents
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Frontmatter
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1. Definite Integrals
Farzin AsadiThis chapter delves into the fundamental concept of definite integrals, their properties, and essential formulas. It covers the calculation of definite integrals for various functions, including even and odd functions, and provides practical examples and exercises. The text also explores improper integrals, detailing their types and methods of evaluation. Additionally, it includes solved problems and exercises to reinforce understanding. The chapter concludes with examples of calculating areas bounded by curves, using both analytical and graphical methods. This comprehensive guide is ideal for those seeking to deepen their understanding of integrals and their applications.AI Generated
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AbstractThe definite integral is a fundamental concept in calculus, representing the signed area beneath a function’s curve over a specific interval \(\left[ {a,b} \right]\). It is formally defined through the limit of Riemann Sums and evaluates to a single, precise numerical value. This chapter explores the fundamental principles and applications of definite integrals. -
2. Polar Coordinates
Farzin AsadiThis chapter delves into the world of polar coordinates, a system that uses distance from the origin and angle of rotation to define points in a two-dimensional plane. It provides a comprehensive overview of converting between polar and rectangular coordinates, calculating areas using polar coordinates, and solving practical problems. The chapter includes numerous examples and exercises, making it an invaluable resource for professionals seeking to understand and apply polar coordinates in their work. The text also explores the use of polar coordinates in describing shapes with circular or rotational symmetry, such as spirals and orbits, and provides step-by-step solutions to problems involving these shapes. The chapter concludes with a discussion on the use of polar coordinates in calculus and physics, highlighting their importance in simplifying complex problems.AI Generated
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AbstractPolar coordinates offer an alternative way to locate a point in a two-dimensional plane, moving beyond the familiar rectangular \(\left( {x,y} \right)\) system. Instead of measuring horizontal and vertical distance from the origin, the polar system uses a point’s distance from the origin and its angle of rotation. This chapter examines the fundamentals of polar coordinates and their use in solving calculus problems. -
3. Numeric Differentiation and Integration
Farzin AsadiThis chapter delves into the essential techniques of numeric differentiation and integration, providing a robust toolkit for approximating derivatives and definite integrals when analytical methods fall short. It begins by introducing numeric differentiation formulas, highlighting the Central Difference method for its superior accuracy. The chapter then explores numeric integration formulas, emphasizing Simpson’s 1/3 Rule for its high efficiency. Through a series of solved problems, readers will learn to apply these methods to approximate the first and second derivatives, as well as to compute definite integrals using various techniques. The chapter also demonstrates the practical application of these methods through MATLAB code examples, showcasing how to integrate discrete data points and calculate derivatives symbolically. By the end, readers will gain a comprehensive understanding of numeric differentiation and integration, equipped with the skills to tackle complex engineering problems with confidence.AI Generated
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AbstractIn many real-world and complex engineering problems, functions are often represented by discrete data points (like values recorded by a sensor) or are too complicated to integrate or differentiate analytically (using standard calculus formulas). This chapter introduces Numeric Differentiation and Numeric Integration, powerful techniques that allow us to approximate derivatives and definite integrals using only numerical methods. -
4. Applications of the Integration
Farzin AsadiThis chapter delves into the practical applications of integral calculus, starting with the average value of a function and the Mean Value Theorem for Integrals. It explains how to calculate the average value of a continuous function over a closed interval and the significance of the Mean Value Theorem. The chapter then explores the calculation of arc length for continuous functions, providing examples and exercises to illustrate the concepts. It also covers the surface area of revolution, detailing the formulas for rotating curves around the x-axis and y-axis, and includes solved problems to demonstrate the application of these formulas. Additionally, the chapter discusses the volume of solids of revolution, explaining the Disk Method and the Washer Method with practical examples. The chapter concludes with applications in finding the center of gravity for both discrete systems and continuous regions, providing a comprehensive overview of how integral calculus can be applied to solve real-world problems.AI Generated
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AbstractWith a foundation established in the techniques of integration, this chapter transitions to exploring some of the most significant applications of the integral in various fields. -
5. Series
Farzin AsadiThis chapter delves into the world of mathematical series, exploring their definition, types, and crucial aspects like convergence and divergence. It begins by defining series as the sum of terms in a sequence, highlighting their importance in calculus and analysis. The text then discusses absolute and conditional convergence, providing clear explanations and examples. Essential convergence tests such as the nth term test for divergence, ratio test, root test, integral test, and alternating series test are thoroughly explained, with practical examples to illustrate their application. The chapter also covers arithmetic and geometric progressions, power series, and their convergence intervals. Additionally, it includes solved examples using MATLAB, demonstrating the practical application of these concepts. The text concludes with exercises and further examples, reinforcing the understanding of series and their convergence tests. Readers will gain a comprehensive understanding of series, their behavior, and the tools to determine their convergence or divergence, making this chapter a valuable resource for professionals in mathematics and related fields.AI Generated
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AbstractA series in mathematics is essentially the sum of the terms of a sequence. While a sequence is a list of numbers in a specific order, a series is the value obtained by adding those numbers together. Series are fundamental to calculus and analysis because they provide a way to represent complex functions as infinite sums. The most crucial aspect of studying a series is determining its convergence or divergence: if the sum approaches a finite, fixed number, the series converges (like a geometric series where the ratio is less than one); otherwise, it diverges. Understanding series is essential for applications like approximating transcendental functions (using Taylor or Maclaurin series), analyzing power consumption, and solving differential equations. This chapter examines the essential properties and behaviors of mathematical series. -
6. Power Series Expansions (Taylor and Maclaurin Series)
Farzin AsadiThis chapter delves into the world of power series expansions, focusing on Taylor and Maclaurin series. It begins by defining power series and their general form, highlighting their importance in approximating functions and solving differential equations. The text then explores the Taylor series, providing a detailed derivation and explaining its application to various functions. Special attention is given to Maclaurin series, a specific case of Taylor series centered at zero, and its important formulas. The chapter also discusses the behavior of Taylor series for odd and even functions, as well as the concept of removable singularities. Practical examples and exercises are included to illustrate the concepts and reinforce understanding. The chapter concludes with a discussion on the convergence of series operations and the radius of convergence, providing a comprehensive overview of this crucial mathematical tool.AI Generated
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AbstractA power series (in one variable) is an infinite series of the form \(\mathop \sum \nolimits_{n = 0}^{\infty } c_{n} (x - x_{0})^{n}\), where \(c_{n}\) are coefficients, \(x\) is a variable, and \(x_{0}\) is a constant called the center of the series. These series provide a powerful tool for approximating functions, solving differential equations, and understanding the behavior of complex functions. By representing functions as power series, we can often simplify calculations, analyze their properties more easily, and even extend their domains of definition. In this chapter, we explore Taylor and Maclaurin series, fundamental power series that provide a bridge between transcendental functions and simple polynomial approximations. -
7. Introduction to Ordinary Differential Equations
Farzin AsadiThis chapter delves into the world of Ordinary Differential Equations (ODEs), focusing on their classification based on order, degree, and linearity. It begins by defining ODEs and explaining their general form, followed by a detailed exploration of how to determine the order and degree of various ODEs through practical examples. The chapter also distinguishes between linear and nonlinear ODEs, providing clear examples of each. Furthermore, it addresses the existence and uniqueness of solutions, demonstrating how to verify if a given function satisfies a differential equation. The chapter concludes with a series of examples showcasing how to solve initial value problems using computational tools like MATLAB, offering a hands-on approach to understanding and solving ODEs. Readers will gain insights into the theoretical aspects of ODEs and their practical applications, making this chapter a comprehensive guide for professionals seeking to enhance their understanding of differential equations.AI Generated
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AbstractAn Ordinary Differential Equation (ODE) is an equation that contains one or more derivatives of an unknown function of a single independent variable. This chapter introduces ODE’s and establishes the criteria for their classification. -
8. First Order Differential Equations
Farzin AsadiThis chapter delves into the world of first-order differential equations, which are crucial in various scientific and engineering fields. It begins by explaining the general form of these equations and their significance in modeling phenomena like population growth and radioactive decay. The chapter then provides a summary of essential formulas needed to solve these equations, including linear differential equations and important integrals. Through a series of solved examples, it demonstrates how to find general solutions and verify them using MATLAB. The chapter also covers specific types of first-order equations, such as separable, linear, and exact equations, and provides practical exercises for further practice. By the end of the chapter, readers will have a solid understanding of how to solve first-order differential equations and apply these methods in their own research or projects.AI Generated
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AbstractDifferential equations are a deep subject typically requiring dedicated study. However, the specific differential equations covered in this chapter can be solved using the integration techniques already introduced. Therefore, working through these problems serves as an excellent review of those prior integration methods. -
9. Essentials of MATLAB®
Farzin AsadiThis chapter introduces the essentials of MATLAB, a powerful programming and numerical computing platform designed for engineers and scientists. It covers the MATLAB environment, basic operations with complex numbers, matrix manipulations, and advanced functions like polynomial calculations, plotting, and accessing help. The text provides detailed examples and step-by-step instructions for various operations, including defining variables, performing basic arithmetic, and plotting data. It also explains how to use commands like roots, polyval, and integral for solving mathematical problems. The chapter concludes with instructions on accessing help within MATLAB, making it a comprehensive guide for those looking to enhance their MATLAB skills.AI Generated
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AbstractMATLAB® (short for MATrix LABoratory) is a programming and numerical computing platform created specifically for engineers and scientists. It is used to analyze and design the systems and products that are transforming our world. This chapter covers the foundations of MATLAB. -
10. Curve Fitting in MATLAB
Farzin AsadiThis chapter delves into the process of curve fitting using MATLAB's Curve Fitting Toolbox, focusing on linear, quadratic, and custom equation fitting. It begins with a practical example of fitting a linear model to a dataset, demonstrating how to use the toolbox to find the best-fit line and interpret the results. The chapter then explores quadratic curve fitting, providing another example to illustrate the process. Finally, it covers fitting data to a custom mathematical equation, with a step-by-step guide on how to enter and fit a custom equation. Throughout the chapter, the importance of statistical measures like R-squared and Sum of Squared Errors is emphasized, providing a comprehensive guide to curve fitting in MATLAB.AI Generated
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AbstractCurve fitting is the process of creating a mathematical function that best represents a series of data points. This chapter will guide you through solving curve fitting problems using MATLAB’s Curve Fitting Toolbox. -
11. Calculator Skills and Formula Writing in Microsoft® Word®
Farzin AsadiThis chapter delves into essential calculator skills and formula writing in Microsoft Word, crucial for engineering professionals. It begins with basic calculator operations, including exponents, radicals, logarithms, and complex number conversions, providing accurate results for various calculations. The chapter also covers finding roots of polynomials and non-linear equations, evaluating derivatives and integrals, and understanding trigonometric ratios and inverse trigonometric functions. The second part of the chapter focuses on writing formulas in Microsoft Word using the Equation Editor. It guides users through inserting an equation box, using the toolbar to input mathematical expressions, and accessing additional tools for a wider range of mathematical symbols and functions. By mastering these skills, readers can significantly improve their problem-solving abilities and achieve better results in their engineering studies and professional tasks.AI Generated
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AbstractCalculators have become an essential tool for engineers, significantly enhancing their efficiency and accuracy in a wide range of tasks. Calculators play an important role in university exams as well. So, it is important to learn how to work with your calculator correctly. By mastering your calculator, you can significantly improve your problem-solving skills and achieve better results in your engineering studies. -
Backmatter
- Title
- A Problem-Solving Approach to Single-Variable Calculus
- Author
-
Farzin Asadi
- Copyright Year
- 2026
- Publisher
- Springer Nature Switzerland
- Electronic ISBN
- 978-3-032-13949-8
- Print ISBN
- 978-3-032-13948-1
- DOI
- https://doi.org/10.1007/978-3-032-13949-8
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