This book grew out of lectures the author delivered at China Agricultural University in the Spring of 2018 and is addressed to science students of various levels, as well as educators or people with an interest in mathematics. The main purpose of this book is to fill the gaps left in standard elementary calculus courses with a particular focus on the clarity, accuracy and completeness of analytical and logical arguments. The author also aims to foster mathematical maturity by maintaining a high level of rigor throughout the book.