The Calculus Lifesaver added January 22, 2011

Adrian Banner is an unusual teacher; his skill in teaching mathematical concepts is readily evident, and he effectively presents complex concepts in a way that delights students. His Calculus Lifesaver video lectures and this companion book are invaluable resources for working through material typically covered by courses identified as Calculus I and Calculus II.

I used it as a reference in my studies of single-variable calculus, and frequently found it more useful than the official textbook of the courses. The video lectures of Adrian Banner are entertaining and enlightening. The Princeton University Press page for the book contains more details.

These foundational concepts of calculus are dealt with in the text:

• functions, graphs, and lines
• a review of Trigonometry
• an introduction to limits
• how to solve limit problems involving polynomials
• the concepts of continuity and differentiability
• implicit differentiation and related rates
• exponentials and logarithms
• inverse functions and inverse trigonometric functions
• the derivative and graphs
• sketching graphs
• optimization and linearization
• L'Hôpital's Rule and an overview of limits
• an introduction to integration
• the fundamental theorems of calculus
• techniques of integration
• basic concepts of improper integrals
• basic concepts of sequences and series
• how to solve series problems
• Taylor polynomials, Taylor series, and power series
• how to solve estimation problems
• how to solve Taylor and power series
• parametric equations and polar coordinates
• complex numbers
• volumes, arc length, and surface areas
• differential equations
• limits and proofs
• estimating integrals

While there is much more to learn about mathematics than the material discussed in The Calculus Lifesaver, the topics presented are valuable and should be carefully studied by every student of mathematics. It is a book I plan to regularly reference for many years. I used it extensively in preparing for undergraduate calculus exams, and my academic experience of working through the courses was greatly aided by it.

Lessons from working through this book:

• Simple illustrations can make abstract concepts much easier to grasp.
• Pedagogic skill is very important for developing materials of benefit to students.
• The beauty of mathematics is thrilling, and books like this one serve to motivate and inspire.
• Watching companion lectures and observing an instructor working through problems solidifies understanding.
• Personally writing out examples found in mathematics texts fosters increased comprehension.