# Whitman Calculus

David Guichard, Whitman College

Copyright Year: 2010

Publisher: David Guichard

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

The current open textbook under review is Whitman Calculus (single variable) in pdf format. The text covers appropriately all areas and ideas of standard calculus 1 and calculus 2 courses taught at US universities and colleges, although the... read more

The current open textbook under review is Whitman Calculus (single variable) in pdf format. The text covers appropriately all areas and ideas of standard calculus 1 and calculus 2 courses taught at US universities and colleges, although the ordering of the contents might be a little bit different from other popular calculus texts such as Stewart Calculus or Thomas’ Calculus. There are eleven chapters in the text including: chapter 1 analytic geometry, chapter 2 instantaneous rate of change: the derivative, chapter 3 rules of finding derivatives, chapter 4 transcendental functions, chapter 5 curve sketching, chapter 6 applications of the derivative, chapter 7 integration, chapter 8 techniques of integration, chapter 9 applications of integration, chapter 10 polar coordinates, parametric equations and chapter 11 sequences and series. The Whitman Calculus provides an effective index and glossary with linked page numbers for easy and quick referencing purposes.

The contents of Whitman Calculus are accurate, error-free and unbiased at the same level as other popular calculus texts normally used by US institutions. Unavoidably there might be a few typos or minor changes, but the author maintains an active change log at http://www.whitman.edu/mathematics/multivariable/changelog.txt.

As in other popular calculus textbooks, the contents in Whitman Calculus are up-to-date and will not make the text obsolete within a short period of time. The text is written in a logical manner and the content is arranged in chapters and sections, easy for future updates. In addition the author maintains an active change log in case there might be a few typos found or there might be minor changes needed in future.

Whitman Calculus is written in a clear and straightforward manner. The technical terms are explained with enough context elaborating the terms for a better understanding, and are sometimes described in a conversational style which is friendly to readers.

The text is consistent in terms of the terminology used in standard calculus. The text is written in traditional math textbook format logically with chapters, sections and exercises after each section, selected answers, useful formulas and the index.

Whitman Calculus is easily and readily divisible into short sections that can be assigned section-wise within the course. The author even provides two additional versions of the text: 2-up version (two book pages per printed page) and 4-up version (four book pages per printed page). I think this is really a great feature for college professors and students using the text in classroom. The text can easily be reorganized and realigned without much disruption to the reader.

The topics in Whitman Calculus are presented in a logical and clear manner. The text is organized in chapters and sections with a logical flow of the materials of calculus, covering chapter 1 analytic geometry, chapter 2 instantaneous rate of change: the derivative, chapter 3 rules of finding derivatives, chapter 4 transcendental functions, chapter 5 curve sketching, chapter 6 applications of the derivative, chapter 7 integration, chapter 8 techniques of integration, chapter 9 applications of integration, chapter 10 polar coordinates, parametric equations and chapter 11 sequences and series.

The navigation of the text is simple and easy. The images/charts used in the text are nice and clean to reflect the related contents without any confusion to the reader. The text would be even better if the author could add more images/graphics to the text.

The text generally contains no grammatical errors.

The text is not culturally intensive or offensive in any way. The examples/exercises used in the text are appropriate in terms of races, ethnicities and backgrounds.

I think that Whitman Calculus is a wonderful open source calculus textbook overall, and would like to recommend Whitman Calculus to math professors and college students for classroom use. One area in which the text could be improved is the volume of the exercises. The text could be enhanced if the author would add more exercises to the text.

The book covers standard first semester calculus topics. Topics in the second semester calculus tend to vary a little more from program to program. For the second semester course some instructors might find this text missing some topics, such... read more

<p> The book covers standard first semester calculus topics. Topics in the second semester calculus tend to vary a little more from program to program. For the second semester course some instructors might find this text missing some topics, such as first order differential equations. Overall, the text is structured, organized, clear, and free of major errors. It is definitely worth considering for those who are considering adopting an open textbook. Instructors will, however, likely need to provide more in-text examples and post-section exercises as the book does not provide as many as some instructors may like to have. Compared to a standard calculus text, this book has limited figures. For example, Section 4.8: Implicit Differentiation has no graphs presented. Students often find it difficult to understand and visualize what implicit differentiation is and a few graphs in this section would greatly benefit their understanding of the material. With that said, I also would like to point out that some of the currently present figures (marked with AP) allow user interactions, which promotes further understanding of the material with more ease.</p>

<p> No issues discovered.</p>

<p> Given the characteristics of the subject matter, I do not see the need to change any of the topics in the near or distant future. Overtime, in a very slow rate, some applications could be modified or added. The relevancy and longevity of the book is not an issue.</p>

<p> Clarity is not an issue.</p>

<p> The text is internally consistent in terms of the structure, style, terminology, and framework. In some places, the format is not entirely consistent. For example, on Page 69, Exercise 12 has a fraction written as $\frac{169}{x}$ and the fraction in Exercise 22 is written as $1/x$. On Page 45, Exercise 6 has a period at the end of the question whereas all the other similar questions do not. These are very minor issues and they do not affect the effectiveness of the textbook.</p>

<p> The division of chapters and sections of this text is very reasonable and is similar to many other calculus texts.</p>

<p> Overall, the material is organized in a logical and standard order. The transition from section to section and from chapter to chapter is natural and smooth. Theorems, definitions, and examples, etc., are properly labeled and organized. For some examples and exercises, it would be better to arrange them differently as typically we put similar examples and exercises together and order them by increasing level of difficulty. In some cases, some examples and exercises appear to belong more appropriately in another section. For example, in Section 3.5: Chain Rule on Page 69, Exercises 3, 9, 17, 18, 19, 20, 21, and 23 are of the same type and Exercises 1, 2, and 22 are of the same type. Further, Exercises 1, 2, and 22 are not in the right section since it does not require the Chain Rule to solve these problems. Instead, they belong in Section 3.2. For Example 3.5.6. on Page 69, I believe most students' first instinct would be to use the Quotient Rule directly instead of rewriting the expression and using the Product Rule and the Chain Rule. Since directly applying the Quotient Rule is a more natural and straightforward method, I think a different example would work better here. Also, in terms of the difficulty level, this example is easier than the proceeding examples (Examples 3.5.4 and 3.5.5). Thus, it would be better to put 3.5.6. before 3.5.4.</p>

<p> The book contains many hyperlinks which users can click on and conveniently go to the suggested definitions, theorems, figures, etc. In some cases, the link does not perfectly bring the users to the referred object. For example, on Page 43, if you click on the blue print of Theorem 2.3.6, it will bring you to the content that is right below the suggested theorem. On Page 91, if you click on the blue print of Example 4.8.3. in Exercise 10, it will bring you to the page that starts with the very end of Example 4.8.3. This also happens to many hyperlinks of figures that I have clicked on. However, these imperfect links do not cause much trouble in practice. For each of the exercise questions, there is a blue arrow following the question if the answer to the question is available. By clicking on the blue arrow, users can directly go to the corresponding answer. This is a wonderful feature that brings users lots of convenience. If there were links that can bring the users back from the answers to the questions it would be even more convenient.</p>

<p> I did not discover any grammatical errors. On Page 21, in the first paragraph under the graphs, "((see figure 1.3.1)" has an extra left parenthesis. On Page 69, in Exercise 10, the parentheses in the numerator and the denominator should both be removed.</p>

<p> No cultural insensitivity discovered.</p>

<p> Overall it is a good text to consider if one is looking to move to an open textbook. Although some additional editing may be necessary, it provides a solid, main body for a first semester calculus text. I appreciate the author making this text open!</p>

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... read more

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

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.

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

It is internally consistent.

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.

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.

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.

I found no grammatical errors.

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.

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.

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... read more

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).

No major inaccuracies were discovered.

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.

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

No major issues found.

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

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.

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.

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.

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.

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.

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... read more

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.

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].

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.

The writing of this text is exemplary.

The text uses standard mathematical terminology throughout.

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

The organization and flow of this text is exemplary.

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.

I did not notice any grammatical errors in the text.

The text is culturally neutral.

It has been a real pleasure reading this book.

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... read more

<h3>The BCcupms Core Calculus Report (revised 2013):</h3> <p>In 2002 the British Columbia Committee on the Undergraduate Program in Mathematics and Statistics (BCcupms) accepted the <a href="http://bccat.ca/pubs/calculus.pdf">Core Calculus Report</a>. It was reviewed in 2007 and revised in <a href= "http://bccupms.ca/Documents/Project_Documents/CoreCalc.pdf">2013</a>. 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. </p> <h3>Core topics:</h3> <ol> <li>Limits, continuity, intermediate value theorem. <ul> <li> 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. </li> <li> 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 <em>not</em> 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 $x=1$). <div align="center" id="Discontinuities"> image not available <h4>Figure 2.3.a in the Text, Discontinuities.</h4> </div> There is no discussion of removable or jump discontinuities. </li> <li> 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. </li> </ul> </li> <li> Differentiation <ul> <li> First and second derivatives with geometric and physical interpretations. The following are covered: <ul> <li>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.</li> <li> 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. </li> <li> 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). </li> <li> Interpretations of the second derivative: concavity and acceleration. </li> </ul> </li> <li> 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. </li> <li> 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. </li> <li>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. </li> <li> 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 <em>last</em> 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. </li> <li> 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. </li> <li> Optimization --- local and absolute extrema with applications Optimization is well covered with a large number of exercises. </li> </ul> </li> <li> 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. </li> <li> 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. </li> <li>Integration <ul> <li> 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. </li> <li> Areas of plane regions Covered. </li> <li> 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. </li> <li> Fundamental Theorem of Calculus Both forms of the Fundamental Theorem are covered. </li> <li> Integration techniques: substitution (including trig substitution), parts, partial fractions. The following integration techniques are covered: <ul> <li> $u$-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. <!-- I prefer to have them look for patterns. --> </li> <li> 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. </li> <li> trigonometric substitutions are covered, but not systematically. </li> <li> 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. </li> <li> 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 $x^2+bx+c$. </li> </ul> </li> <li> Applications of integration. Applications of integration are in Chapter 9. <ul> <li> Areas between curves. </li> <li> Distance, velocity, acceleration. </li> <li> Volumes by slicing, circular laminae, cylindrical shells. </li> <li> Average value of a function. </li> <li> Work. </li> <li> Centre of mass. The formula for $x$-centroid \[\overline{x}=\frac{\int_a^bxy\,dx}{\int_a^by\,dx}\] is given, but they rely on finding the inverse function to get $\overline y$ where in practice it is easier to have a second formula: \[\overline y=\frac{\frac{1}{2}\int_a^by^2\,dx}{\int_a^by\,dx}\] </li> <li> Kinetic energy. </li> <li> Probabliity. </li> <li> Arc length. </li> <li> Surface area. </li> </ul> </li> </ul> </li> <li> Improper integrals: evaluation and convergence estimates Improper integrals including the Cauchy Principal Value are covered tangentially in Section 9.7. There are no convergence estimates. </li> <li> Separable differential equations There are no differential equations in the text. </li> </ol> <h3>Additional Topics</h3> <ol> <li> Sequences and series. For example, the following tests: integral, comparison, alternating series, root, and limit ratio. Sequences and series are covered in Chapter 11. The following tests are covered: <ul> <li> divergence test ($\lim_{x\rightarrow\infty}a_n\neq0\Rightarrow \sum_{i=1}^\infty a_i$ diverges). </li> <li> the integral test, $p$-series, truncation error. </li> <li> the alternating series test, truncation error. </li> <li> the direct comparison test. The limit comparison test is missing. </li> <li> absolute, conditional convergence. </li> <li> the ratio and root tests. </li> </ul> </li> <li> Additional applications of integration. See the core topics for the list of applications of integration. </li> <li> Additional differential equations topics There are no differential equations. </li> <li> Complex numbers There are no complex numbers. </li> <li> Continuous probability density functions Continuous probability density functions are covered in Section 9.8. </li> <li> Polar coordinates and parametric equations (with calculus applications) Polar coordinates and parametric equations are covered in Chaper 10. The calculus applications discussed are <ul> <li> polar coordinates: slopes and areas. </li> <li> parametric curves: area and arc length. </li> </ul> </li> <li> Additional numerical methods (eg. Simpson's Rule) and error bounds The Trapezoid Rule and Simpson's Rule together with their error bounds are covered in Section 8.6. </li> <li> Related rates Related rates are covered in Section 6.2. </li> <li> L'Hôpital's Rule L'Hôpital's Rule is covered in Section 4.2, but only for the indeterminate the forms $\left[\frac{0}{0}\right]$, $\left[\frac{\infty}{\infty}\right]$, and $\left[0\cdot\infty\right]$. The more difficult ones, viz $\left[\infty-\infty\right]$, $\left[0^0\right]$, $\left[\infty^0\right]$, and $\left[1^\infty\right]$, are not discussed. </li> </ol> <h3>Exercises:</h3> There is an adequate selection of exercises at the end of each section. Most are routine, although the exercise sets usually end with a few which are more challenging. <h3>The Index:</h3> The index is quite good, although the following terms are not in it: arc length, area, average value, curve sketching, decreasing, increasing, indefinite integral, Newton's Method (the Newton, a unit of mass, is there), second derivative, substitutions, surface area, volume. <h3>Conclusion:</h3> <ul> <li> The following topics are not in the text: complex numbers, differential equations, limit comparison test, the derivative of $\text{arcsec }x$ and the integration techniques which depend on it, and logarithmic differentiation. </li> <li> The following are inadequately covered: partial fraction decomposition, integrals of powers of trigonometric functions, trigonometric substitutions, and improper integrals. </li> </ul>

The diagrams are very good. I managed to spot six errors: <ul> <li> page 53, figure 2.3.a: It is not clear that the function is defined at the discontinuities (This has been discussed under Comprehensiveness). </li> <li> 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 $a$." </li> <li> page 83: ``figure p. 4.3.'' (``p.'' should not be there.) </li> <li> page 111: ``vertical asymptote where the <em>derivative</em> is zero'' (should be ``where the <em>denominator</em> is zero''). </li> <li> Page 163: $\int x^3\sqrt{1-x^2}$ should be $\int x^3\sqrt{1-x^2}\,dx$ (the ``$dx$'' is missing). </li> <li> Page 200, Exercise 9.3.1: ``$dy$'' missing at the end of the second integral which should read $\int_1^4\left(1+\sqrt y\right)^2-\left(y-1\right)^2dy$. </li> </ul>

There are no problems.

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: <div align="center" id="Discontinuities"> image not available <h4>Visualizations for the Product and Chain Rules.</h4> </div> 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. </li>

Sometimes the authors use terminology before it is discussed.

There are no problems

I have a few minor concerns: <ul> <li> 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). </li> <li> 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. </li> <li> 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! </li> <li> Page 91: $\mathbb R$ is used without any explanation. </li> <li> Page 99: needs to be preceded by a discussion of even and odd functions. which are not defined until page 112. </li> <li> Page 102: <em>injective</em> is used without definition. </li> <li> Page 112, Exercise 5.5: students are asked to find intercepts even though they are not discussed in the text. </li> </ul>

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.

I didn't notice any.

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

## Table of Contents

- 1 Analytic Geometry
- 2 The Derivative
- 3 Rules for Finding Derivatives
- 4 Transcendental Functions
- 5 Curve Sketching
- 6 Applications of the Derivative
- 7 Integration
- 8 Techniques of Integration
- 9 Applications of Integration
- 10 Polar Coordinates, Parametric Equations
- 11 Sequences and Series

## Ancillary Material

## About the Book

An introductory level single variable calculus book, covering standard topics in differential and integral calculus, and infinite series. Late transcendentals and multivariable versions are also available.

**This textbook has been used in classes at: **Boise State University, Claremont McKenna College, University of Minnesota, University of Puget Sound, Western Connecticut State University, Whitman College.

## About the Contributors

### Author

**David Guichard** is a Professor of Mathematics at Whitman College in Walla Walla, Washington. He received his Ph.D. from the University of Wisconsin, and his research interests include Graph Theory.