Calculus: Early Transcendentals

Reviewed by Susan Aydelotte, Professor, College of Western Idaho on 9/12/22

Comprehensiveness rating: 5 see less

The first two chapters contain review material, with perhaps one exception being hyperbolic functions. I liked the use of set notation and the examples used
to illustrate concepts. I might include more graphs to help visual learners. The order of the concepts in Chapter 4 is a bit different than I am used to (which is
to say the text discusses the derivative as a function before discussing tangent lines), but all the essentials are included. The text has an effective index and
answers to selected exercises at the end of the book.

Content Accuracy rating: 5

The content is accurate. The examples in Chapters 1 and 2 are excellent for preparing students for subsequent chapters. Students will need to read the text
carefully (as with most texts) to recognize all the details necessary for understanding the concepts and notation.

Relevance/Longevity rating: 5

The content is classic Calculus material. The text does not refer to any graphing tools (such as TI-84 graphing calculator, desmos.com, and geogebra.org) nor
instructions on how to use the tools.

Clarity rating: 5

The author writes in a very accessible, clear style. I found the context for the technical terminology more than adequate.

Consistency rating: 5

The formatting of theorems and examples gave a nice consistent layout to the textbook. Terminology usage is internally consistent and standard.

Modularity rating: 5

Sections within each chapter divide the textbook well. Occasionally, there was a reference to a prior theorem or example without a page number
provided. On page 92 the author refers to L'Hopital's Rule, which is not explained until Chapter 5 on page 192. I don't believe the goal of easily reorganizing
the subunits should apply to most mathematical texts, since concepts tend to build from one to the next.

Organization/Structure/Flow rating: 5

The organization of the concepts was slightly different yet refreshing, and topics were presented in a logical and clear way.

Interface rating: 5

The pdf version of the text was free of issues. It would be nice to have a more interactive online version with additional links for references and topics in the
index.

Grammatical Errors rating: 5

No grammatical errors were found.

Cultural Relevance rating: 5

The text and examples are inclusive and not insensitive or offensive.

Comments

I really appreciated the thoughtful explanations in the text and the inclusion of multiple methods to solve a given problem. The language and reasoning used
in the text resonated with me.

Reviewed by Richard McBride, Mathematics Instructor, Lane Community College on 1/12/21

Comprehensiveness rating: 4 see less

The text is very comprehensive, and covers all topics in a typical three semester calculus sequence. The writing is very concise, but on the other hand does not provide a lot of context or applications interwoven throughout the sections. The exercise sections could be a bit more robust.

Content Accuracy rating: 5

The book seems to be very accurate. I did not notice any errors, either mathematical or typographical.

Relevance/Longevity rating: 5

As mentioned previously, the writing is concise and does not have much context. This brevity is a positive from the perspective of keeping the material relevant, as there is not much that can become out of date. Also, this brevity allows the text to be very adaptable to individual instructor or departmental needs and objectives. All relevant techniques that students are likely to need in subsequent classes are covered.

Clarity rating: 5

Overall the writing is very clear. In this regard, the brevity is a strength.

Consistency rating: 3

The book lacks a bit in consistency. For example, its treatment of applications of derivatives is very thorough, providing good explanations and examples, while some of the treatment on rules of finding derivatives is a bit sparse (e.g. only one example provided for using the quotient rule.)

Modularity rating: 5

The author does a nice job in partitioning the material into sections (for example the chapter on basics of integration is rather short, but then individual techniques of integration receive a thorough and individualized treatment in the next chapter.)

Organization/Structure/Flow rating: 5

The text is organized very well. There is a nice balance between sections being self-contained (or as self-contained as feasible), with a gradual progression and accumulation of ideas.

Interface rating: 5

The overall appearance of the text is clean and streamlined, making for a very comfortable interface. Chapter 14 (Multiple Integration) is particularly well laid out.

Grammatical Errors rating: 5

There were no grammatical errors noticed.

Cultural Relevance rating: 4

The text seems to be written with inclusivity in mind, in particular with its frequent use of multiple ways of explaining/examining a particular problem or concept.

Comments

Overall, this is a strong textbook for a traditional Calculus sequence. It is clearly and concisely written, and topics/sections are modularized well. It would also be very easy to adapt to individual needs. The only real weaknesses would the depth of coverage is somewhat inconsistent in spots, and several sections could use more exercises.

Reviewed by Olusegun Otunuga, Assistant Professor, Marshall University on 5/13/20

Comprehensiveness rating: 4 see less

1. Section 9.2: It would be nice if the author can include telescoping series examples, showing how limit of partial sum lim s_{n} =s (convergence of series)

2. Section 8.3: Give a general formula (in form of a theorem) for finding the volume of the solid generated (using DISK METHOD) when area under curves y=f(x) and y=g(x), f(x)>=g(x), a<=x<=b, is rotated about x-axis, OR y-axis, OR about a line x=a or y=b

3. Section 8.3: It is also important to show students how to use the method of CYLINDRICAL SHELLS to find volume of solid formed by rotating regions under curve.

4. Sections 8.4 & 8.5: The same applies to writing formulas (in form of theorems) of how to find average value of a function, f(x), a<=x<=b

5. Section 9.5: You stated the Direct-Comparison test. Can you also state the Limit-Comparison Test, with examples?

6. Section 9.2, Series: After definition of partial sum, it would be nice to show formal limit definition of convergence of series in form of s_{n}

7. Section 4.1: Can you include how to find "instantaneous rate of change as limit of average rate of change"?

8. Also give two different formula for finding derivative of a function at a point (instead of just giving formula for finding derivative function)

9. Section 2.3: Before discussing exponential function, can you discuss linear functions and give couple examples of word problems that can be solved using linear functions

Content Accuracy rating: 5

1. Typo in Definition 5.7, Page 169. Replace "f(x_{0}>=f(x)" with "f(x_{0})>=f(x)"

2. Example 5.43 should be retitled "Second Derivative Test"

3. Page 265: Can you replace the statement "du=sec x tan x +sec^{2} dx" with "du=(sec x tan x +sec^{2}) dx"

4. Theorem 9.19 Page 347: The starting and ending of the index on the summation should be clearly stated. Like \sum_{i=0}^{\infty} k x^{i}

5. Title of Section 4.2.1 "Differentiable" should be revised to either "Derivative" or "Differentiation"

Relevance/Longevity rating: 5

Content is up-to-date and updates will be relatively easy and straightforward to implement..

Clarity rating: 5

On page 257, while explaining the substitution method, you showed students how NOT TO SOLVE THE DEFINITE INTEGRAL. Can this be shown in RED (or catching) color to catch students' attention that they are not supposed to solve that way?

Consistency rating: 5

The text is internally consistent in terms of terminology and framework.

Modularity rating: 5

The text is easily and readily divisible into smaller reading sections that can be assigned at different points within the course

Organization/Structure/Flow rating: 5

The topics in the text are presented in a logical, clear fashion.

Interface rating: 5

The text is free of significant interface issues

Grammatical Errors rating: 4

1. Typo in Definition 5.7, Page 169. Replace "f(x_{0}>=f(x)" with "f(x_{0})>=f(x)"

2. Example 1.24 Page 21: Can you modify the statement "To find x-intercept(s) of the line y=2x-3 we set..." as "To find x-intercept(s) of the line y=2x-3, we set..."

Cultural Relevance rating: 5

The text is not culturally insensitive or offensive in any way.

Comments

1. Section 9.4, Theorem 9.32, can you simply write decreasing instead of non-increasing?

2. Can you solve couple of examples using Alternating Series Test

3. Section 9.5: You stated the Direct-Comparison test. Can you also state the Limit-Comparison Test, and solve couple of examples on both Direct Comparison and Limit Comparison Test?

4. Section 9.8: Power Series: Can you solve example where one (or both) of the endpoint(s) is(are) included in the interval of convergence?

Reviewed by Peng Lu, Professor, University of Oregon on 5/15/19

Comprehensiveness rating: 5 see less

1. Chapter 1 helps students to warm up to the course which usually starts from Chapter 2.
2. Section 3.5.3 and 3.5.4 is not in many other calculus book.
3. The Taylor polynomial as approximation (5.4.3) is nice, my teaching experience is some students
does not appreciate the value of linear approximation enough, so this subsection could help they
to absorb the idea of approximation as an first exposure to the idea of numerical analysis.
4. Chapter 9 on Sequence and series is not substantial enough. At our university this chapter takes about
8 week out of 10 week quarter. This is a difficult topic for many students, but it is also valuable in forcing students
to think (e.g., which test methods to use). I do not support the idea of under-weight this topics. The topics before
here are more of computational nature. The author may consider to use J. Stuart's book to add some more substance,
note that 9.8 and 9.9 are laughable short

Content Accuracy rating: 5

The book is carefully written. I thought I found one typo, but now I can not even find it from my notes.

Relevance/Longevity rating: 5

The choice of the material, for most part, are very well in my judgement.
There are a few places I want to raise some concerns for the sole purpose for the author to ponder.
1. Is 3.5.4 over done (too much material, a comprehensive treatment), should it limit to fewer pages and just give some
cases so that students know the idea.
2. 5.6.5 and its exercise. Should it be light covered further in the selection of examples and exercise. In the many years I taught
calculus I do not see many students can graph well. In light of the calculator graphics we may consider to turn the hard excises around, given them the graph and ask students to figure out some feature in detail (e.g., matching graph, the questions in 5.6
before 5.6.5)

Clarity rating: 5

Yes. I have one issue to raise which needs the teaching from the book and talking to students to settle.
There are several places with this long monologue argument which would be nice when a lecturer presents
it orally in class, however I would worry about whether students will have the strength to understand them
by reading alone.
p.54-55, bottom of p.77-top of p.79, bottom of p.98 to p.99 (just list this many).

Consistency rating: 5

Yes, it is very consist, the book overall is very readable.

Modularity rating: 5

There are sections where the length is not even enough, like 3.5 (11pp), 5.4(10pp),
and several sections are of 2-3pages (9.6).
The text layout including the graphical view are nice.
The quick click from TOC to regular text is very helpful.

Organization/Structure/Flow rating: 5

I like most part of Chapter 5 in this aspect.

Interface rating: 5

Not see any

Grammatical Errors rating: 5

Not see any

Cultural Relevance rating: 5

I need to be educated how calculus and culturally insensitivity can meshed well.

Comments

It is a difficult job to write a good calculus book, in particular, in the choice of the topics and its presentation,
partly because there are diversified bodies of students, and there are different requirement for calculus
from different institutes.
Overall the book did not a good job, like the dealing of infinity limits (not give a comprehensive list of all
possibility, but give enough example that students know what to expect). But I still think there is a gap
between this book's quality and the book by J. Stuart. Stuart's book is too hard/deep for today's use and
it is concise, lucid, and well-balanced. For this book if you look at the homework for 3.3, you will know what
I aim, you can not only give exercises with only one graph. To me, three different graphs would be better,
just my two cents. Too few questions or examples here will make the lecturer have a difficult time to pick questions
to lecture and assign homework even from webwork.
I kudos the author's effort, hopefully in next round we see a much better book

Reviewed by Wei Wei, Associate Professor, Metropolitan State University on 1/7/16

Comprehensiveness rating: 5 see less

This textbook contains 10 chapters and selected exercise answers. The book covers the review of algebra and functions, limit, derivatives, derivative tests, graphing using the derivative tests, L'Hopital Rule, optimization problem, integration, techniques of integration, calculation of area between curves and volume, first and second order differential equations, polar coordinates and parametric equations. This book contains all the materials needed for Calculus I and most materials for Calculus II. The book offers great explanations of concepts, formula and theories. In particular, it contains some mnemonic keys to help students memorize important rules in the review chapter (first chapter). I really like the author pointing out the misconceptions that students usually have throughout the book and listing the detailed steps of problem solving. I also like the author using color contrasts to label definitions, theorems, and examples. This book does not have index or glossary at the end.

Content Accuracy rating: 4

This book has been modified and some exercises and examples are taken from the other book Elementary Calculus: An Approach Using Infinitesimals. Therefore, I believe the majority of the materials should be accurate and unbiased. I haven't used this text yet myself, so I can't identify whether the solutions to the exercises are accurate. The graphs and tables are clean and accurate. The solutions to examples are laid out nicely with detailed steps. The only mistake that I found is on page 133, example 4.40, the right hand side for the third step should be "1-y2x" instead of "-y2x".

Relevance/Longevity rating: 4

This text doesn't contain many real-life examples. But the knowledge included are up-to-date. The structure of the text is definition-theorem-examples, so it will be relatively easy to add real-life examples and implement.

Clarity rating: 5

The text is written clearly. The explanations of theorem are precise and in details. The solutions to examples are step by step. The graphs help students to understand the materials. This book doesn't contain enough real-life related examples, so it could be mathematical and technical to non-math major students.

Consistency rating: 5

The author did a great job to keep the consistency in terms of language and structure of the book. The book keeps the same color scheme for definitions, theorem and examples. The structure for each chapter each section are consistent.

Modularity rating: 5

The text follows the standard Calculus sequence. Each section are 3-6 pages. The book contains enough subheadings and subsections and can be easily divided into smaller reading sections. The text is not overly self-referential.

Organization/Structure/Flow rating: 5

The organization of the book is excellence. Each section starts with an opening of a problem, and then the author introduces related definitions or theorem, followed by examples. The text spends two chapters to review algebra and function which is necessary for non-math majored students. It introduces asymptotes of graphs in the chapter of applications of derivatives, which is different from the traditional approach that introduces asymptotes in the chapter of limit. This way, students can have a complete knowledge of graphing a function. This text includes linear approximation at the beginning of the chapter of applications of derivatives and introduces related rates at the very end of that chapter. This is also different from other textbooks. This design separates the introduction of derivative and applications of derivative completely. Related rates is a difficult topic in Calculus I. Introducing related rates at the end of derivatives helps students to practice more before getting into it.

Interface rating: 4

This text has no navigation issues. Whenever it refers to previous examples or definitions, the author uses internal hyperlink taking you to the right place. But it would be nice if there is a way to return back to where the reader was after reviewing the referred materials. Some of the figures are not labeled, but they are referred from the text right above.

Grammatical Errors rating: 5

I didn't notice any grammar mistakes

Cultural Relevance rating: 5

The text has not many life-related examples. Majority of the examples are mathematical. So the text is not culturally insensitive or offensive in any way.

Comments

I personally like this book. It has a very nice structure and detailed explanations to examples. It also points out the misconceptions that students usually have which helps students to grasp the knowledge accurately.

Reviewed by Bruce Aubertin, Instructor, Langara College on 8/15/13

Comprehensiveness rating: 5 see less

The text reviewed here is a version (May 2013) of the single variable portion (chapters 1 -11; 318 pages) of the full text, Calculus: Early Transcendentals by Guichard et al, which includes both single and multivariable calculus and can be found at: http://www.whitman.edu/mathematics/multivariable/ The single variable edition is a complete course presented in a traditional sequence, except that differential equations do not appear at all until chapter 17 of the multivariable text. Separable DEs, mandatory in the BC curriculum, are treated in 17.1, so adopters/adapters of the text may wish to have this section appended or inserted as an additional section in Chapter 9, Applications of Integration. The index could certainly be improved, particularly for those using a print copy and unable to do a search as a pdf file would allow. For example, section 8.8 on numerical integration covers the trapezoidal rule and Simpson's rule with formulas for the errors (incidentally, the midpoint rule, standard in most texts, is not discussed here), however, there is no mention in the index or in the table of contents of "trapezoidal rule." The term "related rates" appears in both the index and table of contents, while "Newton's method" appears in the table of contents and not in the index. The term "cycloid" does appear in the index (one of my tests of an index in any calculus text!), in addition to "hypercycloid"and "hypocycloid", pointing to nice exercises in the text expanding the discussion of parametric equations in 10.4. While the economies taken in the body of the text in producing such a wonderfully readable and complete text in about half the number of pages of a typical commercial text are much appreciated, the index is a different story, and 300+ pages demands a good index! Diagrams in the text are relatively few and far between, though are used effectively when present. The book does favour algebraic/analytical reasoning working from definitions over graphical arguments, and limits (including one-sided limits) receive the full epsilon-delta treatment. But there are instances/acknowledgement of graphical reasoning in the text! To illustrate, the Squeeze Theorem presented in 4.3 is followed by the comment, "This theorem can be proved using the official definition of limit. We won’t prove it here, but point out that it is easy to understand and believe graphically." There follows immediately the classic example of (x^2)sin(pi/x) as x approaches 0, complete with a diagram that illustrates the theorem perfectly. Another example appears in 4.7, where the derivative of ln(x) is derived graphically using the fact that the derivative of exp(x) is itself. I liked to see this very much! This reviewer never waits for implicit differentiation, as most texts do, before demonstrating that if dy/dx exists at (a, b) and is not zero, then dx/dy also exists at this point, and equals 1/(dy/dx). (If I am running twice as fast as you at a certain instant, then at this instant you are running half as fast as me!) The graphical derivation in the text is then followed by: "We have discussed this from the point of view of the graphs, which is easy to understand but is not normally considered a rigorous proof" it is too easy to be led astray by pictures that seem reasonable but that miss some hard point. It is possible to do this derivation without resorting to pictures, and indeed we will see an alternate approach soon. Left and right continuity are not mentioned in the text (unusual), and nor are one-sided derivatives (usual). But these are not seen as omissions; instructors with any text will want occasionally to amplify or draw diagrams, and expand/extend concepts. For example, a question such as (one of my favourites) "What is the slope of the graph of cos(sqrt(x)) at x=0?"would not be at all out of place in this book. Conversely, of course, an instructor using this text may wish not to follow the rigorous epsilon-delta approach to limits. For this reviewer, in first year, I take the limit laws to be all intuitively obvious, and no use at all on all of the "interesting" limits (what I call the indeterminate forms!) The exercises at the end of each section are well chosen and numerous enough in applications such as optimization and related rates where they need to be. They range from routine practice to more challenging questions, and most have short answers in the back of the book. These could be supplemented using the open-source online homework system WeBWorK http://webwork.maa.org/ (This reviewer has currently only had experience with the commercial systems WebAssign and MathXL). Overall, I like this book a lot. It is very well written and friendly to read, without the usual clutter of sidebars, footnotes and appendices! It moves quickly through all the important definitions and theorems of calculus with many examples and also a certain amount of just-in-time precalculus (for example, with the exponential and logarithm functions). There is appropriate rigour throughout, though the book is not at all in the style of Rudin's classic graduate text, "Principles of Mathematical Analysis!" It is much more conversational, and suited even for self-study. Maybe slightly too much so, as sometimes definitions or important formulas appear in the flow of the discourse and are not highlighted for easy visual reference for the student. Most are numbered, but the conversion formulas for switching from polar to rectangular coordinates in 10.1 would be a case in point.

Content Accuracy rating: 5

The text appears to be remarkably free of errors of any kind, and any question of bias in the sense intended here not applicable. I did notice somewhere a period missing at the end of a sentence. Also, in the remark in parentheses at the end of Example 1.4 in section 1.3, which reads: "(You might think about whether we could allow 0 or (minimum of a and b) to be in the domain. They make a certain physical sense, the term "(minimum of a and b)" should be replaced with "min(a, b)/2." I did also notice, in the discussion immediately following Theorem 11.17 in section 11.2, that the constant c was not taken to be nonzero explicitly as it should have been. Of course there are natural biases expected in terms of style, rigour, choice of definitions etc., and these are mostly very agreeable to this reviewer. For example, it is refreshing to see the function 1/x declared continuous, following the definition of continuity given in section 2.5 - Adjectives for Functions. Though I may continue to say that there is an infinite discontinuity at x=0. It does go slightly against the grain however, to allow as the book does, the endpoints of an interval [a, b] to be local extrema. I like the book's treatment in 6.5 of the Mean Value Theorem (MVT), or Motor Vehicle Theorem as I call it, and let me contrast it with that given in Stewart's Calculus - Concepts and Contexts, another admirable text with which this reviewer is familiar and has taught from for some time. Both texts state the theorem and illustrate its usefulness and interpretation with respect to motion. The text under review fully proves it from Rolle's Theorem, which in turn is proved from the (unproved) Extreme Value Theorem. There is no diagram in this section, and the function g(x) = f(x)-m(x-a)-f(a) used to derive the MVT from Rolle's Theorem appears pulled out of a hat and is not explained. By contrast, Stewart does not mention Rolle's Theorem or prove the MVT, but does provide diagrams making it seem plausible. Annoyingly, however, the hypothesis in Stewart's MVT is that f(x) is differentiable on the closed interval [a, b], making it not applicable, for example, to the square root function on the interval [0, A].

Relevance/Longevity rating: 5

The content in a mainstream calculus text such as this is relatively timeless. The book is regularly being updated by the author, taking into account feedback from users of the text. I will leave it to other reviewers more familiar with manipulating source code to comment on the ease of editing the text.

Clarity rating: 5

The writing of this text is exemplary.

Consistency rating: 5

The text uses standard mathematical terminology throughout.

Modularity rating: 5

The text is structured in a standard and traditional sequence for a calculus text.

Organization/Structure/Flow rating: 5

The organization and flow of this text is exemplary.

Interface rating: 5

There are no significant interface issues with this text. The internal hyperlinks in the pdf version of the text are a very nice feature, taking you instantly to a referenced diagram, definition, or solution of an exercise. However, it would be nice if there was a way to return to the exact previous position in the text with a single click, after viewing the reference, rather than having to navigate back using the bookmarked pages or sections of the text. I did find that clicking on the external links labeled (AP) that are attached to many of the diagrams resulted only in "page not found." I don't know why, but it can't be serious.

Grammatical Errors rating: 5

I did not notice any grammatical errors in the text.

Cultural Relevance rating: 5

The text is culturally neutral.

Comments

It has been a real pleasure reading this book.

This review originated in the BC Open Textbook Collection and is licensed under CC BY-ND.

Reviewed by Sam Pimentel, Assistant Professor of Mathematical Sciences, Trinity Western University on 8/7/13

Comprehensiveness rating: 4 see less

The textbook covers all the topics necessary for a Calculus 1 course. The entire textbook, chapters 1-11, cover material for a Calculus 2 course, with the exception that the current copy received for review doesn't include a section/chapter on first-order separable differential equations. (A chapter on differential equations is made mention of in the small print on the inside front cover, but does not appear in the contents).

Content Accuracy rating: 5

No major inaccuracies were discovered.

Relevance/Longevity rating: 5

Updated versions of the textbook are made available on the website. The TeX files used to generate the textbook are freely available as well, thus allowing users to update and edit the text themselves, if required. Some familiarity with LaTeX is required, in this regard, simply downloading the TeX files and using LaTeX to generate a pdf textbook won't work without some tinkering with the various options on offer.

Clarity rating: 5

A conversational writing style makes the text very readable and the presentation of material has a natural flow.

Consistency rating: 5

No major issues found.

Modularity rating: 5

Section and subsection labeling are used well. Definitions, Theorems, Examples, and Exercises are helpfully numbered.

Organization/Structure/Flow rating: 5

The textbook has a sensible ordering of chapters and sections that, for the most part, follows the usual structure of other introductory calculus textbooks. Organization of the material that is perhaps slightly unusual includes introducing the derivative before introducing continuity, leaving limits at infinity until later (Section 4.10), introducing integration by parts and integration of rational functions using partial fractions before any applications of integration. The partition between a Calculus 1 and a Calculus 2 course is often such that some integral applications are required as part of the Calculus 1 syllabus, but that integration by parts and integration using partial fractions is not encountered until Calculus 2. Again, having the tex files allows for rearranging and omitting certain material as required for particular course offerings.

Interface rating: 4

Some figures contain so-called "AP" links to interactive applets, these were broken in the copy under review. This is only relevant for the pdf of the textbook.

Grammatical Errors rating: 5

U.S. spelling is used throughout e.g. center rather than centre. This could be quickly and easily changed, if desired, by running a Canadian English spell check through the textbooks .tex files.

Cultural Relevance rating: 3

By the natural of the textbook in question issues of cultural relevance are limited. However, Math examples involving cultural references are U.S. focused e.g. U.S. geographical locations, baseball, U.S. income tax data, etc. Imperial (rather than metric) measurement units are frequently used e.g. feet, miles, pounds etc.

Comments

The text is straightforward in appearance, e.g. no sidebars, boxed material, or special highlighting. No special attention is made, therefore, on highlighting key material and core ideas. On the other hand, students can have free reign of the highlighter pen and annotate the text to their hearts content without any fear of reducing the resale prize on the second-hand textbook market! The text is also free of the little historical vignettes or anecdotes that are often found in the major Calculus textbooks. The material is too the point and keeps the book to a reasonable length. There are less figures and diagrams than is standard in the major textbooks. More graphs (and in some cases coloured lines on existing graphs) may improve explanations for students. Calculus students may find themselves wanting more worked examples, although presumably these would be provided in class lectures. On a similar note, the question sets are small, instructors may find themselves needing to set problems outside of those provided. This would also be important to avoid too much repetition with multiple offerings of the course year in, year out. Students themselves may like to try further exercises than the textbook currently supplies. A supplementary worked examples and problem set may need to be provided in addition to the textbook.

This review originated in the BC Open Textbook Collection and is licensed under CC BY-ND.

Reviewed by James L. Bailey, Chair, British Columbia Committee on the Undergraduate Program in Mathematics and Statistics, College of the Rockies on 7/15/13

Comprehensiveness rating: 4 see less

The BCcupms Core Calculus Report (revised 2013):

In 2002 the British Columbia Committee on the Undergraduate Program in Mathematics and Statistics (BCcupms) accepted the Core Calculus Report. It was reviewed in 2007 and revised in 2013. This document has a list of core topics which all first year (two semester) Science Calculus courses must include and a list of additional topics, at least four of which must be chosen. Any text which is adopted for a first year Science Calculus course must be consistent with this report.

Core topics:

Limits, continuity, intermediate value theorem.


Limits are introduced in Section 2.3 where Definition 2.3 is the $\epsilon, \delta$-definition of a limit. The definition is used to show that \[\lim_{x\rightarrow2}x^2=4\] (Example 2.5, page 40). Properties of limits are stated in Theorem 2.7 (page 42). One sided limits are defined, together with an example, in Section 2.3.

Continuity is covered in Section 2.5. There is a problem with Figure 2.3(a) (the left half of the figure below is my attempt at reproducing it). The author states that ``a function $f$ is continuous if it is continuous at every point in its domain'' (page 53). It is claimed that Figure 2.3(a) is the graph of a discontinuous function, but it is not clear that the function is defined at the discontinuities, viz. $x=-1,0,\text{ and }2$. In fact, the function would be continuous if it were not defined at these values. Something like the right half of the figure would have made it clearer what function values were intended at the discontinuities ($x=-1$ and $x=1$).

image not available

Figure 2.3.a in the Text, Discontinuities.

There is no discussion of removable or jump discontinuities.

The Intermediate Value Theorem is found in Section 2.5 together with an application, using a binary search to approximate a zero of a function.



Differentiation


First and second derivatives with geometric and physical interpretations. The following are covered:


The derivative of a function is introduced at the bottom of page 32, as a summary of the procedure used to find the slope of the tangent to $\sqrt{625-x^2}$ at any point. Section 2.4 introduces the main notations, $y'=f'(x)$ and $\frac{dy}{dx}$, and has a discussion of places where a function does not have a derivative (corners and cusps). The dot notation, $\dot x$, is introduced on page 128.

Interpretations of the derivative: slope of tangent line; velocity, acceleration (velocity and acceleration are also discussed in Section 9.2 when discussing integration); rate of change in general.

The second derivative does not have its own section. It is first introduced with the second derivative test for extrema (Section 5.3) and concavity (Section 5.4).

Interpretations of the second derivative: concavity and acceleration.



Mean Value Theorem The Mean Value Theorem is treated in Section 6.5. The authors first prove Rolle's Theorem and then use that to prove the Mean Value Theorem.

Derivatives of the exponential and logarithm functions, exponential growth and decay. The derivatives of the exponential and logarithm functions are covered. On page 85 where the authors find the derivative of $\log_ax$ they show that $\log_ae=\frac{1}{\ln a}$. For no more work they could have derived the change of base formula, $\log_ax=\frac{\ln x}{\ln a}$ and then found the derivative of $\log_ax$ more economically. Exponential growth and decay is not covered, presumably because there are no differential equations.

Derivatives of trigonometric functions and their inverses. The derivatives of $\sin x$, $\cos x$, $\tan x$, and $\sec x$ are covered; $\cot x$ and $\csc x$ are left as exercises. On pages 75--76, in giving the usual geometric argument that \[\lim_{x\rightarrow0}\frac{\sin x}{x}=1,\] the authors argue that, with a little algebra, \[\frac{\cos x \sin x}{2} \leq \frac{x}{2}\Rightarrow \frac{\sin x}{x}\leq \frac{1}{\cos x}.\] They do not point out that we need $0\lt x\lt\frac{\pi}{2}$ in order to keep the various quantities positive and avoid problems with the inequalities. Because the argument is essentially geometric, and this is the restriction which is implied by the diagram, they may feel that it is unnecessary to point this out. The derivatives of the inverse trigonometric functions: the derivative of $\arcsin x$ is done, but the derivatives of $\arccos x$, $\arctan x$, and $\text{arccot}\, x$ are left as exercises. The derivative of $\text{arcsec}\, x$ is not discussed.

Differentiation rules (including chain rule, implicit differentiation) The authors start by deriving the power rule $\frac{d}{dx}x^n=nx^{n-1}$ for integer $n$ using an ad hoc argument which gives the first two terms of the Binomial Theorem (page 56); rational exponents are handled after they have covered implicit differentiation. The other rules (constant multiple, sum, product, quotient, and chain) are presented in order. Finding derivatives by implicit differentiation is covered, but finding the second derivative of an function defined implicitly is discussed only in the section on polar coordinates. Logarithmic differentiation is not covered. On page 67, where they show how to differentiate \[f(x)=\frac{x^2-1}{x\sqrt{x^2+1}}\] they say: "The last operation here is division, so to get started we need to use the quotient rule first," but there is no indication why this is important. It may be better to state the implied rule, that the differentiation rules are applied in the reverse order to that which is used when doing a calculation.

Linear approximations and Newton's Method Newton's Method is well covered but the section on Linear Approximations is a little thin. In particular, I would have liked to see problems such as "Use a linear approximation to estimate $\sqrt{10}$," and some problems which do not have a unique answer because the student has to make choices.

Optimization --- local and absolute extrema with applications Optimization is well covered with a large number of exercises.



Taylor polynomials and special Taylor series $\left(\sin x,\,\cos x,\,e^x,\, \frac{1}{1-x}\right)$, plus enough sequences and series to understand the radius of convergence; in particular, the concept of series and convergence, the ratio test, and how to find the radius of convergence. These are all covered. In addition, differentiation and integration of power series are covered and there is a proof of the Lagrange form of the remainder.

Curve Sketching. Chapter 5 covers curve sketching. Intercepts are not discussed. Horizontal and vertical asymptotes are discussed but the authors say that slant asymptotes "are somewhat more difficult to identify and we will ignore them." Even and odd symmetry is mentioned.

Integration


Definition of the definite integral and approximate integration. Both are covered. There is an example of using the limit of a Riemann sum to calculate an area, although the term ``Riemann sum'' is not used.

Areas of plane regions Covered.

Average value of a function. Covered by example although the general formula, \[f_{\text{avg}}=\frac{1}{b-a}\int_a^bf(x)\,dx\] is not given.

Fundamental Theorem of Calculus Both forms of the Fundamental Theorem are covered.

Integration techniques: substitution (including trig substitution), parts, partial fractions. The following integration techniques are covered:


$u$-substitutions. I have a problem with the authors' approach: the authors allow both $x$ and $u$ in the same integral. In the example on page 163 they have \begin{eqnarray*} \int x^3\sqrt{1-x^2}dx&=&\int x^3\sqrt u\frac{-2x}{-2x}dx\qquad u=1-x^2,\,du=-2x\,dx \\ &=&\int \frac{x^2}{-2}\sqrt u\frac{du}{dx}dx\\ &=&\int \frac{x^2}{-2}\sqrt u\, du\qquad x^2=1-u\\ &=&\int -\frac{1}{2}(1-u)\sqrt u\, du\\ \end{eqnarray*} The authors advise that it is necessary to "translate the given function so that it is written entirely in terms of u, with no x remaining in the expression" but I have found that students often miss this nicety.

powers of $\sin x$ and $\cos x$ are covered in Section 8.3, using examples only; it is not explicitly stated that for $\int\sin^nx\cos^mx\,dx$ use a $u$-substitution if one of $n$, $m$ is odd and the double angle formula if both are even. Powers of $\tan x$ and $\sec x$ are not covered. The use of reduction formulae is not discussed.

trigonometric substitutions are covered, but not systematically.

there is a section on integration by parts and tabular integration. There are no rules of thumb to help students decide when to use integration by parts. Reduction formulae are not discussed.

rational functions are covered, but only the easy cases: when the denominator is of the form $(ax+b)^n$, $(x-r)(x-s)$, or an irreducible quadratic $x^2+bx+c$.



Applications of integration. Applications of integration are in Chapter 9.


Areas between curves.

Distance, velocity, acceleration.

Content Accuracy rating: 5

The diagrams are very good. I managed to spot six errors:

page 53, figure 2.3.a: It is not clear that the function is defined at the discontinuities (This has been discussed under Comprehensiveness).

Page 83: ``the limit $\displaystyle\left[\lim_{\Delta x\rightarrow0}\frac{a^{\Delta x}}{\Delta x}\right]$ varies directly with the value of a''. I think that ``varies directly'' usually means ``is directly proportional to.'' This is a quibbling point, and I don't know how it could have been worded better, certainly not "is a monotone increasing function of $a$."

page 83: ``figure p. 4.3.'' (``p.'' should not be there.)

page 111: ``vertical asymptote where the derivative is zero'' (should be ``where the denominator is zero'').

Page 163: $\int x^3\sqrt{1-x^2}$ should be $\int x^3\sqrt{1-x^2}\,dx$ (the ``$dx$'' is missing).

Page 200, Exercise 9.3.1: ``$dy$'' missing at the end of the second integral which should read $\int_1^4\left(1+\sqrt y\right)^2-\left(y-1\right)^2dy$.

Relevance/Longevity rating: 5

There are no problems.

Clarity rating: 5

The style of writing is clear, informal, almost chatty. The authors keep jargon to a minimum, perhaps to a fault. For example, the term ``Riemann sum'' is not used even though there is an example of calculating $\int_0^x3t\,dt$ using $n$ rectangles of equal width using the left endpoint approximation and letting $n\rightarrow\infty$. The style of the book is to work from the concrete to the abstract, from the particular to the general. For example, to introduce the idea of the derivative and to motivate the idea of a limit they have a long discussion about the slope of the tangent to the semicircle $y=\sqrt{625-x^2}$ at the point $(7,24)$. First working numerically, they calculate the slope of the secant lines between $x=7$ and $x=7.1$. Next they find the general formula \[\frac{\sqrt{625-(7+\Delta x)^2}-24}{\Delta x}\] and substitute $\Delta x=0.01$ to get a better approximation for the slope of the tangent. There is a link in Figure 2.1 to a \href{http://www.whitman.edu/mathematics/calculus/live/jsxgraph/secant_lines.html}{Sage worksheet} of the function $y=2x(1-x)$ in which one end of a secant line is fixed at $x=0.15$ but the other can be moved, so it is possible to watch the secant line approach the tangent line. Next they rationalize the previous expression to get \[\frac{-14-\Delta x}{\sqrt{625-(7-\Delta x)^2}+24}\] and argue that as $\Delta x\rightarrow0$ the slope of the secant line approaches the slope of the tangent line, $-\frac{7}{24}$. They point out that we are able check this answer because the line from the centre of a circle is perpendicular to the tangent, so their slopes must be negative reciprocals. They continue by doing the same calculation but with 7 replaced by $x$ to get the slope at an arbitrary point on the circle and then set $x=7$ to see that the new formula gives the earlier answer. They then replace $\sqrt{625-x^2}$ by $f(x)$ get the derivative of an arbitrary function as the limit of the difference quotient. The three approaches used in this example (numerical, graphical, and algebraic) together with the rather leisurely pace help the student understand this difficult concept. Unfortunately the authors are not consistent in the use of this three-pronged attack. For example, their approaches to the product rule and the chain rule are purely algebraic; their argument is rigorous but doesn't give the student any insight into what is happening. It would have been helpful to have some visual representation of these rules, perhaps something like the following figure:

image not available

Visualizations for the Product and Chain Rules.

For the most part the authors do not state general results, but rather expect the student either to imitate the process in the examples or to invent their own formulae. This has the advantage of discouraging students from memorizing a formula and ``plug and chug'' in exercises.

Consistency rating: 5

Sometimes the authors use terminology before it is discussed.

Modularity rating: 5

There are no problems

Organization/Structure/Flow rating: 4

I have a few minor concerns:

Page 54, Exercise 2.5.4: assumes that the natural logarithm $\ln x$ is known (the logarithm function, $\log x$, is defined on page 80 and the natural logartithm function, $\ln x$ on page 83).

Page 62: the authors assume that the meaning of $\sum$ is known when they introduce the notation for a product, $\Pi_{k=1}^{n}f_k$. They do not define sigma notation until page 150.

Page 87: Exercise 4.7.22 asks the reader to use implicit differentiation to find the derivative of $y=\log_ax$ but implicit differentiation is the topic of the next section. In fact, using implicit differentiation to find the derivative of $y=\ln x$ starts of the next section!

Page 91: $\mathbb R$ is used without any explanation.

Page 99: needs to be preceded by a discussion of even and odd functions. which are not defined until page 112.

Page 102: injective is used without definition.

Page 112, Exercise 5.5: students are asked to find intercepts even though they are not discussed in the text.

Interface rating: 5

The text is free of significant interface issues, including navigation problems, distortion of images/charts, and any other display features that may distract or confuse the reader. I have only a minor concern: some of the figures are marked with "(AP)" which points to Sage worksheet or an interactive applet. I found that not all the Sage worksheets opened correctly with Internet Explorer (for example Figure 2.1) although they work fine in Firefox and Google Chrome.

Grammatical Errors rating: 5

I didn't notice any.

Cultural Relevance rating: 5

The examples are similar to those found in any Calculus text.

Comments

This review originated in the BC Open Textbook Collection and is licensed under CC BY-ND.

Reviewed by Michael Nyenhuis, Instructor, Kwantlen Polytechnic University on 7/2/13

Comprehensiveness rating: 4 see less

The text covers the standard topics in first-year calculus, but I think a regular student would have trouble using it. It covers the contents at a more mathematically sophisticated level than students are usually used to. For example, limits are introduced using the epsilon-delta definition. That may be appropriate in an honours section of calculus, but not for a regular section. Graphs or diagrams that would help in understanding the material are missing. For example, the text states that "rightward" is the positive x direction and upward is the positive y direction, but includes no diagram. Exercises are often skimpy, and emphasize only one view of calculus. For example, the section on the quotient rule has only 4 algebraic exercises, none of which combine other rules. I would have liked to have seen graphical questions, such as asking students to sketch the derivative of f(x) given the graph of f(x). I think students would find the text more difficult to read than Stewart, say, and instructors will need to supplement the exercises. The index is good.

Content Accuracy rating: 4

Content is mathematically correct, but students might prefer a less mathematically rigourous approach.

Relevance/Longevity rating: 5

Calculus does not change much, but how it is taught has changed over the years. The text seems to focus on the algebraic approach. Instructors who want to emphasize the graphical approach may not be happy with it. Because the source TeX files are publicly available, it is, in principle, easy to edit the text. Some experience in TeX would be needed, however.

Clarity rating: 4

The text is written in fairly standard math. Definitions are provided.

Consistency rating: 5

It is internally consistent.

Modularity rating: 3

Sections are usually fairly brief and most cover a single topic, so they could be assigned as reading. I think, however, that most students will find the text hard to understand, making assigned reading kind-of moot. Many sections begin with a reference to previous sections, but these could be edited out. Math texts usually assume material is covered in the same order as it is presented in the text, and this text is no exception. However, instructors who like to cover topics "out of order" are already familiar with this problem.

Organization/Structure/Flow rating: 4

For a mathematician, the topics are presented in a clear, logical fashion. Students may wonder what the author is getting at, but math instructors should have no difficulty in doing so.

Interface rating: 4

The text is a pdf document, so navigation is kind-of primitive. My one beef is that exercises are followed by an arrow leading to the answers, but no arrow follows answers leading back to the questions! Images and charts are clear.

Grammatical Errors rating: 5

I found no grammatical errors.

Cultural Relevance rating: 4

I did not find the text culturally insensitive or offensive. The author works in Washington State, so some exercises and examples refer to locations in Washington.

Comments

The text strikes me as OK from a Canadian mensuration perspective. Most questions involving miles do not require a conversion to feet. Some acceleration questions are in feet. The work section has several examples in feet and pounds, which will obviously cause problems. How many Canadian students are aware that the unit of mass in the Imperial System is the slug? Additional observations: I checked the internet for other open-source calculus texts. Most are either lecture notes, or multimedia (video-based), or non-traditional (emphasizing infinitessimals, for example). The only other complete, standard text was a scan (sometimes of low image quality) of Strang's 1991 Calculus text. It is dated (there's a reference to "A Thousand Points of Light") and cannot be edited. I think Guichard's book may be a good choice for an honours calculus class, but I would hesitate recommending it for any other. However, it is also the only candidate. If an open-source text must be chosen, I think Guichard's text is the only choice.

This review originated in the BC Open Textbook Collection and is licensed under CC BY-ND.