Calculus 1 – Differentiation and Integration: Over 1,900 Solved Problems

Book Preface

Similar to the previous books published by the Hamilton Education Guides, the intent of  this book is to build a strong foundation by increasing student confidence in solving mathematical  problems. To achieve this objective, the author has diligently tried to address each subject in a  clear, concise, and easy to understand step-by-step format. A great deal of effort has been made to  ensure that the subjects presented in each chapter are explained simply, thoroughly, and  adequately. It is the authors hope that this book can fulfill these objectives by building a solid  foundation in pursuit of more advanced technical concepts.

The scope of this book is intended for educational levels ranging from the 12th grade to  adult. The book can also be used by students in home study programs, parents, teachers, special  education programs, tutors, high schools, preparatory schools, and adult educational programs,  including colleges and universities as a main text, a thorough reference, or a supplementary book.
A thorough knowledge of algebraic concepts in subject areas such as linear equations and  inequalities, fractional operations, exponents, radicals, polynomials, factorization, non-linear and quadratic equations is required.

“Calculus I” is divided into five chapters. Sequences and series are introduced in Chapter 1.  How to compute and find the limit of arithmetic and geometric sequences and series including  expansion and simplification of factorial expressions is discussed in this chapter. Derivatives and its  applicable differentiation rules using the Prime and ddx notations are introduced in Chapter 2. In addition, use of the Chain rule in solving different types of equations, the implicit differentiation  method, derivative of functions with fractional exponents, derivative of radical functions, including  the steps for solving higher order equations is discussed in this chapter. Differentiation of  trigonometric functions, exponential and logarithmic functions, hyperbolic functions, and inverse  hyperbolic functions is discussed in Chapter 3. Furthermore, evaluation of expressions referred to as  indeterminate forms using a general rule known as L’Hopital’s Rule is discussed in Chapter 3. The  subject of integration is introduced in Chapter 4. Integration using basic integration formulas and  methods such as the substitution method is discussed in this chapter. Additionally, integration of  trigonometric functions, inverse trigonometric functions, exponential and logarithmic functions is  addressed in Chapter 4. Other integration techniques such as integration by parts, integration using  trigonometric substitution, and integration by partial fractions is introduced in Chapter 5. The steps  in integrating hyperbolic functions is also discussed in this chapter. Finally, detailed solutions to the  exercises are provided in the Appendix. Students are encouraged to solve each problem in the same  detailed and step-by-step format as shown in the text.

In keeping with our commitment of excellence in providing clear, easy to follow, and  concise educational materials to our readers, I believe this book will again add value to the Hamilton  Education Guides series for its clarity and special attention to detail. I hope readers of this book  will find it valuable as both a learning tool and as a reference. Any comments or suggestions for  improvement of this book will be appreciated.

With best wishes,
Dan Hamilton