Practical Calculus: Math for Gifted Students

This book is not intended to replace an $1000$-page textbook which gives detailed lectures on every aspect of calculus. Instead, it is intended to explain core concepts in an intuitive way and to focus on practical perspectives of learning calculus.First, this book emphasizes on computational skills. The vast majority of students will use calculus as a computational tool in various scientific areas such as mathematics, physics, engineering, computer science, quantitative finance, and so on. Therefore, being able to correctly and quickly calculate derivatives, integrals etc is extremely useful and important to these students. Many calculus theorems are also covered. However, those overly abstract and academical details will be kept to a minimal. Such contents are very important to those students who are interested in pursuing pure mathematics. However, focusing too much on these topics may be a distraction to most other students.Another feature of this book is its focus on relations between calculus and various high school computational techniques such as trigonometric identities, polynomial transformation, and so on. Proficiency in using these techniques plays a vital role in solving many calculus problems. While these methods and formulas will be included in this book when they are needed, it is strongly recommended that readers should review these topics before attempting this book. Discussion of these techniques can be found in some Math All Star series books such as \textit{Power Calculation}, \textit{Trigonometry}, \textit{Competition Algebra}, and so on.Meanwhile, this book also includes many examples and applications in various subjects which use calculus. These include continuous compounded interest in finance, root finding algorithm in computer science, optimization in machine learning, and of course various applications in physics. This will help prepare students for applying their calculus skills in their future study and career. Finally, some selective historical competition problems are also included. These exercises provide students at different levels with an opportunity to test their problems solving skills that suit their individualized learning objective. Full solutions with detailed explanations are included.