Single Variable Calculus

We wrote this calculus textbook for students seriously considering a career in STEM: Science, Technology, Engineering, and Mathematics. Having taught calculus for over thirty years at universities, we have gained invaluable insights into making the material more accessible and less daunting through the use of simple, direct language. Thus, even for students with weak mathematical foundations or for those who are approaching calculus for the very first time, we believe that this textbook can facilitate their comprehension. Numerous examples and problem sets were designed so that the major concepts of calculus could be understood, then solved, and finally applied.

The contents of this book are as follows. Chapter 1 introduces the definitions of functions and deals with inverse and composite functions. Specifically, exponential functions, logarithmic functions, and trigonometric functions?all crucial parts in science and engineering?are thoroughly examined. Chapter 2 deals with the limits of functions and the concept of continuity. Special attention is paid to the limits of transcendental functions and the limits at infinity, as well as the mean value theorem through continuity. Chapter 3 introduces derivatives and deals with tangent and normal line equations, as well as various differential calculations using slope. Chapter 4 examines the close relationship between mathematics and its practical applications in real life via local maxima, local minima, and optimization problems. Chapter 5 introduces indefinite integrals and definite integrals, focusing on the relationship between the derivative and the integral through the fundamental theorem of calculus. Chapter 6 introduces the application of definite integrals. Mainly examined are the areas between curves, the volume of revolution, the motion of curves, the length of arcs on a plane, and improper integrals. This chapter expands the concepts of derivatives or integrals from single-variable calculus to solve problems related to maximum or minimum values of multivariate functions or volumes of solids, helping students understand physical phenomena using vector functions. Chapter 7 deals with differential equations.

We received a lot of help from Jungbin Yoon in writing this book. In reviewing the text, examples, and exercises, they offered creative solutions and multiple-choice questions, providing student perspectives that allowed us to anticipate potential difficulties in the material. Among the many examples that come to mind, while calculating the limit of a trigonometric function, instead of obtaining the derivative first, Jungbin Yoon suggested an easier method of obtaining the graph by transforming the given trigonometric function. It was through their diligence that, in the process of substitution integration, they organized the solution themselves to solve a first-order linear differential equation, thereby greatly improving the quality of this book.

In addition, Jungbin Yoon’s careful review of the text and scope of this book, correcting formulas as well as English grammar, allowed us to devote our energies to the mathematical concepts and the overarching whole of the project. Finally, we would like to extend our special gratitude to our publisher, Freedom Academy. Any corrections or updates identified after publication will be made available in the Resources section of the Freedom Academy website (www.freeaca.com) for reference. We hope that this textbook can help students develop a better understanding and appreciation of calculus and the benefits it will have in their professional lives.