Spiral Approach To Introductory Calculus
583743
Edwin Moise, perhaps, needs no introduction. Many are familiar with his outstanding Geometry textbook:  Elementary Geometry From An Advanced Standpoint ; this one, dealing with Calculus, seems to be lesser known. Given that Apostol, Courant, Spivak seems to have won the day with their respective (excellent all) Calculus tomes, what will one discover that is 'new' , or different, in this Calculus offering ?
Preface: "...presents, more thorough than is customary, material covered in one-year college introductions," "...in this book, certain theorems are proved long before being stated..." and  "problem sets come in various guises. " Some are theoretical, some are puzzles, some analytic and visual, and, some are discovery problems which anticipate later material. Also, " the viewpoint of differential equations has been stressed throughout." (Volume Two--Preface).
The preface laid bare, let us now  peruse the contents:
(1) Ten Chapters span: Inequalities, Analytic Geometry, Functions--Derivatives and Integrals, Trigonometry and Exponential Functions, Variation of Continuous Functions, Techniques of Integration. Definite Integral, Conic Sections, Paths and Vectors in the Plane, finally, Number Theory and Partial Fractions.
(2) Appendices A to J: provide twenty-five pages of ancillary topics (Proofs of the epsilon-delta sort). Of particular pedagogic utility is
the "Spiral" approach, whereby one meets the same concept at various stages of  increasing abstraction. (Caveat: I am partial to the "Spiral" approach, thus, already I'm a believer). Section 4.8, " Exponentials and Logarithms investigated by the Method of Wishful Thinking," is a treat. The section following fills in the gaps and presents the proofs. Thus, a one-two punch. Another delight, sections 5.4 & 5.5, the use of existence theorems and functional equations as shortcuts, is rather uncommon at this level. Next up,
(3) completing the Real Number system: we meet the wonderful nested-interval postulate and least-upper bound postulate. Again, discourse worth perusal. Next, most of your standard integration methods are here (Substitution,By-Parts,Partial Fractions, to name a few).
Arc Length, Limit of Sums, Volumes ,Surfaces of Revolution, Improper Integrals--these occupy the chapter on definite integration.
Of particular note is the fine discussion which follows  the "...integrability of continuous functions." (Pages 345-351).
Thoughtful questions, such as: "Is it possible for a function to be uniformly continuous on an open interval ? Why or Why Not ?
(4) Conics (that is: Second-Degree Equations, Ellipse, Hyperbola, Parabola) given thorough treatment. Vectors discussed from viewpoint of velocity, acceleration, curvature. The problems delve more deeply into the subject at hand (Linear Independence). The crown jewel is the final chapter of Number Theory and Partial Fractions. It being a glimpse into further terrain.
(5) Answers are provided for odd-numbered problems.
In conclusion: a worthy companion for first-year Calculus students.
It won't supplant Apostol, Courant, or Spivak, but, it will enlighten your outlook.
3083211
This book's primary value is as a source of solved calculus problems. Although a concept discussion precedes each set of solved and supplemental problems, these discussions are extremely short, sometimes just a brief paragraph before problems are presented. The primary value of these discussions will be as a concise review for those already familiar with the areas covered.

Each of the fifty-nine chapters includes: 1) brief concepts discussions, 2) solved problems with fully-worked solutions, and 3) supplementary problems. The solved problems are excellent, providing a wide and comprehensive coverage of undergraduate calculus.  However, the supplementary problems only provide answers, not fully worked solutions. Thus, readers who do not get matching answers may find it difficult to determine how the problem should be solved, Even if correct answers are obtained readers cannot be certain they approached the problem solution in the most appropriate manner.  Thus, the supplemental problems provide little additional value beyond the traditional odd-numbered problems in typical class texts, for which answers without detailed solution steps are usually provided.

As with some other books of calculus problems, there are deficiencies in the solution approach to word problems involving dimensions. For example, in Chapter 20 although dimensions are given in the problem statements, the authors change to 'pure numbers' during the problem solution phase, leaving off dimensions until the final solution. This means there's no check that dimensions are being properly used during the problem solving process. The use of dimensionless numbers can make work 'cleaner', but it may also mask wrong answers. It always means that an additional check on the validity of a solution is missing. Also, as is common with many paperback review books, the paper appears to be low quality; some of my Schaum's editions less than a decade old are already starting to brown.

If your need is for first year calculus, you may also want to look at 'AP Calculus' preparation books. These typically have somewhat more detailed discussions, as well as fully-worked solutions to all problems.

Considering the relatively minor changes in this edition compared to the previous fourth edition, it may be worth considering that edition as an option, if you can obtain it at significantly reduced cost.

In summary, this edition is a good source of solved calculus problems at a quite reasonable price, with only some minor deficiencies and omissions.
2566857
Larson's Calculus book is a revelation. My colleagues and I have used it for years. It covers all the main topics (for both AB and BC Calc in high school, or first year Calc in college). It has all the major theorems, and tons of worked example problems.

But most important to me, the book has an extraordinary collection of beautiful, varied nuanced problems for each chapter. Most chapters have between 100 and 200 problems. These range from straightforward computational problems using the new techniques, to logical brain twisters examining theorems in depth, to an amazing collection of word problems and applications.
523517
I am really impresed with this calculator.  It is great for anyone in a Calculus class--both high school and college.  Graph functions, derivatives, integrals, etc.
2218208
For those who are in college or in university beginning with calculus, or for those who want to look back and see how the machinery of calculus really works this vivid introduction to calculus may be an eye opening experience.
Of course, after that you should seriously consider other books on calculus like the one by Morris Kline (An Intuitive and Physical Approach, 2ed.), or by Michael Spivak (Calculus, 4th ed.)
2434482
A solid resource for either an introduction to calculus or to reacquaint yourself with the subject. It is easy to understand, follow, and it breaks information and examples down nicely. Overall, I am pleased with my purchase and refer to it often.
2890992
This book is intended to assist upper level undergraduate and graduate students in their understanding of differential geometry, which is the study of geometry using calculus.  Usually students study differential geometry in reference to its use in relativity. I personally have a rather oddball application for the subject - modeling of curved geometry for computer graphics applications. The fundamental concepts are presented for curves and surfaces in three-dimensional Euclidean space to add to the intuitive nature of the material.

The book presumes very little in the way of background and thus starts out with the basic theory of vectors and vector calculus of a single variable in the first two chapters. The following three chapters discuss the concept and theory of curves in three dimensions including selected topics in the theory of contact.

Great care is given to the definition of a surface so that the reader has a firm foundation in preparation for further study in modern differential geometry. Thus, there is some background material in analysis and in point set topology in Euclidean spaces presented in chapters 6 and 7. The definition of a surface is detailed in chapter eight. Chapters 9 and 10 are devoted to the theory of the non-intrinsic geometry of a surface. This includes an introduction to tensor methods and selected topics in the global geometry of surfaces. The last chapter of the outline presents the basic theory of the intrinsic geometry of surfaces in three-dimensional Euclidean space.

Exercises are primarily in the form of proofs, and there are plenty of worked examples. Since the examples are kept to no more than three dimensions, the outline contains plenty of good instructive diagrams that illustrate key concepts. This Schaum's outline has quite a bit of instruction in it past the bare required minimum, but you might still want a good primary textbook. My personal favorite is Pressley's "Elementary Differential Geometry". Overall I find this to be a very good outline and source of solved problems on the subject and I highly recommend it.
4523580
Seeing the fundamental concepts of calculus elucidated without the use of functions is a spectacle everyone in mathematics should have to experience. I worked through the first book the summer after taking my first course in undergraduate analysis and this really changed my perspective on the subject. Seeing the concept of the limit get done without functions, existence of tangent lines to curves without functions, everything you know about elementary calculus gets done without functions. It makes it clear how sophisticated a piece of technology functions really are, and how they can really obscure ideas that would otherwise seem intuitive from a geometric point of view. But all of this is about Newton's work. I must say if I had tried to work through this without Densmore's guidance, explanation, and insights it would have been much more difficult than it needs to be as the language and concepts Newton uses are sometimes quite archaic or vague. Densmore guides you through the proofs and guides you through the physics, if you're into that sort of thing. I found this book in a library and was so impressed by it I had to get my own copy. One of the best purchases I've ever made in terms of advancing my understanding of mathematics.
2667086
My friend and I have gone through more than a dozen calculus books looking to understand the subject.  This was the first one to really explain itself  well. It starts with the algebraic basics and works up to the calculus at  whatever rate one sees fit. Now, a few weeks later, we're both fluent in  differential calculus and working our way through integral calculus -- all  from a basic algebra background.
3783432
The main public of the book is the undergrad (or grad) physics student who wants to make a tour of these functions. The approach, at least during the first 7 chapters is mainly through Series. The required knowledge is minimal (math 201) and whatever is needed is nicely introduced at the begining.

The book is well written and  easy to read

In the first chapter, the author shows how to solve linear second order differential equation by power series and obtain directly the power serie expansion of the solution.

Second Chapter is devoted to the differential equation defining hypergeometric function and its solution in power series. Generalizing this expression gives rise to "generalized hypgeos" and "confluent hypgeos".

The next 3 chapters are variations around this theme, begining with various equations of mathematical physics (penduluum, harmonic oscillators of quantum mechanics, wave equation with cylindric symetry, ...), obtaining the power serie solutions and the defining functions (elliptic, hermit, bessel,...) and show their relation to heyper gemotric functions.

Chapter 7 and 8 introduce complexe numbers and contour integration, which allows the author to come back and define the various functions by definitie integrals, rodriguez formula, generating functions, recursion formula and so on.

In conclusion, this is a pretty complete overview of special functions and easy to read.
3236123
As a non-traditional student of 35 returning to undergraduate university, I had tested into Calculus but as the semester progressed, it bacame apparent that the intervening years had taken their toll on my math basics.  It was clear that even as I struggled to understand the new material, I  had either forgotten my basics or never learned them well.  This lack was  seriously impeding learning the new material.  I finally took a review  course with this book, re-took the calculus, graduated cum laude, went on  for Masters degrees, and am now working on my doctorate.  This book cut  through all the "math-ese" to understanding the basics of the  calculations, not just regurgitating the material. The authors also seem to  be able to focus on exactly where people routinely have problems in a very  understandable language. It is not an exaggeration to say a single book can  be pivotal in one's success.
644386
If you are a student that doesn't like wasting time, then you will love this book! It is really concise and straight to the meat of why we take the class in the first place. If you want to learn the material the right way and not have to sift through a giant paper weight, you have found the right book. No fluff will be found here. I am so glad to have used this book, it cut down my study time and yet I still learned all the material at a really deep level. I loved this book and you will to. Buy it, you won't be sorry.

*Note* I asked the authors to send me the table of contents to include in the review. It is included below.

TABLE OF CONTENTS

Chapter 1:  The Difference Quotient, Limits, and the Derivative  1
Section 1.1:  Introduction 1
Section 1.2:  Slope of the Linear Functions 1
Section 1.3:  Slope of a Curvilinear Functions 2
Section 1.4:  The Difference Quotient 3
Section 1.5:  Limit of a Function 8
Section 1.6:  The Mathematical Definition of the Limit 11
Section 1.7:  The Case of InfinityUnbounded Increasing Functions, Functions
  Approaching Infinity, Functions Without Limits 13
Section 1.8:  The Derivative 18
Section 1.9:  Requirements for the Existence of the Derivative and Continuity 20
Section 1.10:  In Retrospect 25
Chapter 2:  Differentiation Rules 29
Section 2.1:  Introduction 29
Section 2.2:  Derivatives of Powers, Products, and Quotients 29
Section 2.3:  Derivatives of Composite Functions 36
Section 2.4:  Derivatives of Exponential and Logarithmic Functions 39
Section 2.5:  Logarithmic Differentiation 43
Section 2.6:  Inverse Functions 46
Section 2.7:  Implicit Differentiation 48
Chapter 3:  Optimization 55
Section 3.1:  Introduction 55
Section 3.2:  Optimum Values of a Function  56
Section 3.3:  Candidates for Local Optima 60
Section 3.4:  First Derivative Test 63
Section 3.5:  Absolute and Local Maxima and Minima 68
Chapter 4:  Higher Order Derivatives and Their Applications 73
Section 4.1:  Introduction 73
Section 4.2:  Higher Order Derivatives 73
Section 4.3:  Concavity and Points of Inflection 75
Section 4.4:  Second Derivative and Local Optima 84
Section 4.5:  Curve Sketching 88
Chapter 5:  Partial Derivatives 96
Section 5.1:  Introduction 96
Section 5.2:  Derivatives of Multivariate Functions 96
Section 5.3:  Optimum Values of Multivariate Functions of two Independent Variables 104
Section 5.4:  Saddle Point 112
Section 5.5:  Optimum Values of Multivariate Functions of More Than two Independent VariablesOptional  113
Section 5.6:  Constrained OptimaThe Method of Substitution and the Method of Lagrange Multipliers 114
Chapter 6:  Integral Calculus 132
Section 6.1:  Introduction 132
Section 6.2:  The Definite Integral 132
Section 6.3:  Relationship Between Integral and Differential Calculus 136
Section 6.4:  The Fundamental Theorem of Integral Calculus 138
Section 6.5:  Properties of the Definite Integral 141
Section 6.6:  Variable Limit of Integration 143
Chapter 7:  Methods of Integration 148
Section 7.1:  Introduction 148
Section 7.2:  Rules of Integration 148
Section 7.3:  Integration of Exponential and Logarithmic Functions 151
Section 7.4:  Integration by u-Substitution 153
Section 7.5:  Integration by Parts 158
Section 7.6:  Integration by Table of Integrals 162
Chapter 8:  Applications of Integral Calculus 168
Section 8.1:  Introduction 168
Section 8.2:  Area 168
Section 8.3:  Probability 172
Section 8.4:  The Expected Value and Variance of Random Variables and the Tailor-Made Probability Distributions 196
Section 8.5:  Cost Theory and Capital Formation (Optional) 183
Section 8.6:  Capital Formation and Continuous Financial Processes (Optional) 184
Section 8.7:  Introduction to Differential Equations 186
Section 8.8:  Mathematical Sociology 188
Chapter 9:  Multiple Integrals 194
Section 9.1:  Introduction 194
Section 9.2:  Iterated Double Integrals 194
Section 9.3:  Geometric Interpretation of Double Integrals 200
Section 9.4:  Iterated Triple Integrals 203
Section 9.5:  Improper Multiple Integrals 204
23950
The authors have written a superb text.  New material is introduced intuitively and graphically and then the proofs are given.  It makes math fun.  This book is the best introduction to calculus that I have read.  The  clearity of exposition, the natural transitions from easier to more  difficult material is seamless.  A real joy to see math taught so well.  The layout of the book is thoughtfully designed, especially important in  math books where it is easy to clutter up a page with too many math  symbols.  If you want to see math presented in the finest way possible,  this is the book to get.  Especially good for self instruction.  Has a  student's solution book - purchased separately.
2218639
Who says Calculus ever needed to be "reformed" ?
I refer to the 1987 Colloquium: Calculus For A New Century (Mathematical Association Of America).
Extant are many excellent (pre-reform) Calculus Texts, including: Courant, Artin, Spivak, Apostol, Hille, Lax, Moise and--yes-- Kuratowski !
( Certain it is that I've forgotten a few, certain too that there remain a few which I have yet to peruse).
There is, and always has been, an abundance of excellent texts from which to learn Calculus. Where does Kuratowski fit in ?
Fewer applications than some (say, Courant or Lax),yet more theoretical than others (say, Artin or Moise). Kuratowski  reminds me of the beautiful (post-reform !) terse text by Hijab: Introduction To Calculus and Classical Analysis (1997, first edition) !
My suggestion: study first the 1957 notes of Emil Artin. Then, proceed to Kuratowski.
Artin plus Kuratowski comprises a total pagination of 458 (126 plus 332, respectively). I've reviewed the notes of Artin's course elsewhere.
Contents of Kuratowski's "Introduction":
(1) Eleven chapters, beginning with Sequences and Series (eighty pages):
Induction, Bolzano-Weierstrass, arithmetic and geometric means, absolute convergence...to name a few topics,
(2) Functions and Limits (sixty pages): monotone, inverse, continuity, uniform convergence....an interlude on Logic
awaits (Pages 129-136).... to name but a few topics,
(3) Derivatives (sixty pages): elementary functions, Rolle Theorem, power series, differentials...to name a few topics,
(4) Integrals (one hundred pages): indefinite and definite integrals, elliptic integrals, area, arc-length, curvature, Wallis
 and Stirling, Riemann and Darboux, improper integrals and their connection to infinite series....to name a few topics,
(5) Each individual chapter has its own exercise set, often accompanied by hints to their solution ! . For example:
" Prove that every function is a sum of an even function and an odd function." (Page 102) and  " Characterize the Lipschitz condition geometrically." (Page 117).
(6) Supplementary Exercises conclude the text (seventeen pages). Example: " Give a geometrical interpretation of the Euler Constant."
As with his text, Introduction To Set Theory And Topology (I've reviewed elsewhere), the writing is crisp and much of this material is
off-the-beaten-path. For instance, glance at: Limit and number 'e' (Page 95) or, deducing Wallis' formula (Page 261).
As is my preference, nothing surpasses Apostol's Volume One (beginning, as he does with integrals), but, Kuratowski
( beginning, as he does, with sequences and series) surely scores high points.
As with any textbook, I compel the reader to glance at the Problems first. Yes, read the problem sets first !
Then, return to the body of the text. Keep those student exercises in mind while studying the text.
Another fine introductory text--Mathematics Of Physics And Modern Engineering, by Sokolnikoff and Redheffer--
begins (their chapter one) with sequences and series. Sound pedagogy. Kuratowski will amplify their discussion.
Learning Calculus, whether it be through Apostol, Artin, Courant, Hille, Kuratowski, Lax, Moise, or Spivak
(alphabetically listed), can be--and is--exciting and rewarding.
Whether a mathematics or physics student, Kuratowski opens doors.
Compare and contrast with Hijab's fine " Introduction " !
Study one, or, study them all ( as did I ).
3223584
This topically organized, colorful (yet not cluttered) presentation of trig and analytic geometry is a book I will enjoy using as a teacher.  BJU is a solid secondary math program.  The last 2 chapters even include an introduction to differential calculus.
1880098
The difference between this older book and the newer texts is mostly in the way it explains where the concepts originated as it goes through the standard litany of calculus.  For the type of person who learns more holistically (rather than by rule sets, as most things are taught these days), this is a huge improvement.  This is a good secondary source to reinforce and expand on the concepts introduced by the instructor and text in your university calculus series.
4612427
I was not very optimistic about getting this book for my calculus class, but I did anyway and must say it was worth it. The authors did a great job of making it easy to understand the concepts with lots of practice problems.