The Calculus Lifesaver: All the Tools
You Need to Excel at Calculus
(Princeton Lifesaver Study Guides)
by Adrian Banner
An Essential Companion For Calculus Study
For many students, calculus can be the most mystifying and frustrating
course they will ever take. The Calculus Lifesaver provides students with
the essential tools they need not only to learn calculus, but to excel at it.
All of the material in this user-friendly study guide has been proven to get
results. The book arose from Adrian Banners popular calculus review
course at Princeton University, which he developed especially for students
who are motivated to earn As but get only average grades on exams. The
complete course will be available for free on the Web in a series of
videotaped lectures. This study guide works as a supplement to any
single-variable calculus course or textbook. Coupled with a selection of

exercises, the book can also be used as a textbook in its own right. The
style is informal, non-intimidating, and even entertaining, without sacrificing
comprehensiveness. The author elaborates standard course material with
scores of detailed examples that treat the reader to an inner monologue--
the train of thought students should be following in order to solve the
problem--providing the necessary reasoning as well as the solution. The
books emphasis is on building problem-solving skills. Examples range from
easy to difficult and illustrate the in-depth presentation of theory.
The Calculus Lifesaver combines ease of use and readability with the
depth of content and mathematical rigor of the best calculus textbooks. It is
an indispensable volume for any student seeking to master calculus.
Serves as a companion to any single-variable calculus textbook Informal,
entertaining, and not intimidating Informative videos that follow the book--a
full forty-eight hours of Banners Princeton calculus-review course--is
available at Adrian Banner lectures More than 475 examples (ranging from
easy to hard) provide step-by-step reasoning Theorems and methods
justified and connections made to actual practice Difficult topics such as
improper integrals and infinite series covered in detail Tried and tested by
students taking freshman calculus
Personal Review: The Calculus Lifesaver: All the Tools You
Need to Excel at Calculus (Princeton Lifesaver Study Guides) by
Adrian Banner
I purchased this book as part of a summer math review before starting a
PhD program. As mentioned in a couple of reviews above, this book does
not contain actual problem sets. However, it does contain the most
thorough and understandable explanations that I have seen to date.
Most math textbooks, including the ones I purchased to accompany MIT's
OpenCourseWare, are inconsistent in their explanations. More often than
not, the author will present a solution with little to no work, leaving the
student to puzzle out how the author reached it (which can be especially
difficult if the author simplified his equation differently or used methods that
he considers "common knowledge"). Banner specifically sets out to
counteract this approach, taking the extra space and time needed to
elaborate on sample problem solutions and tie those solutions to methods
covered elsewhere in the text. The result is an extremely readable, easy
to follow guide to calculus. For good measure, Banner also throws in
some nice tricks that students can use to tackle complex problems.
The lectures that accompany this book on the website are also very
valuable, but I love this book in part because it requires no lectures to be
understood. Banner does not punt on any topic in the book, unless he
feels that the material is beyond the scope (and even then, there is a good
deal of complementary explanation in the Appendices).

My only complaint is that my calculus studies will take me beyond the
Lifesaver. If Banner ever wants to write a book that presents some
Calculus II material that is useful for graduate study (partial derivatives, for
instance) or Linear Algebra, I will happily buy it.
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