This book was written by a couple of former mathematics students with the intent of helping their fellow students to make a successful transition from high-school to undergraduate mathematics. Of course, the book can be used also by other groups for other purposes: undergraduate students, graduate students (in their first years), high-school teachers, even mathematics faculty (to get ideas on making their courses more accessible to students).
The book covers an array of twenty topics, from simple inequalities to series, and is easy to read, which makes it ideal for self-study. Each chapter has exercises at the end (with answers in the back of the book!), but starts with a “test yourself” (so that students who can convince themselves they know that particular topic… can move on).
The book also has a chapter called Extension Questions, followed by Worked Solutions to Extension Questions, for those interested in a few challenges. However, there is no bibliography or suggestions for further study, probably because the authors knew the courses their intended readers (high-school graduates getting ready for college) will have to take as college undergraduates.
Although the book is written in a very accessible way, which makes it attractive for young students, it does not go into much detail. Most of the problems included are not difficult. An extensive list with formulae, identities, etc for each topic are included at the end of the book.
I do not see this book used as a textbook; it is ideal, instead, for students entering college to review topics. It is an easy and entertaining read. It can also be a good book to have for any student taking mathematics courses, as well as for any teacher teaching mathematics courses. There are approaches and examples in this book that can be used from time to time to understand or to explain things better.
Preface.- 1. Inequalities.- 2. Trigonometry, Differentiation and Exponents.- 3. Polar Co-Ordinates.- 4. Complex Numbers.- 5. Vectors.- 6. Matrices.- 7. Matrices as Maps.- 8. Separable Differential Equations.- 9. Integrating Factors.- 10. Mechanics.- 11. Logic, Sets and Functions.- 12. Proof Methods.- 13. Probability.- 14. Distribution.- 15. Making Decisions.- 16. Geometry.- 17. Hyperbolic Trigonometry.- 18. Motion and Curvature.- 19. Sequences.- 20. Series.- A. Appendix.- Extension Questions.- Worked Solutions to Extension Questions.- Solutions to Exercises.- Index.