Why aren’t books like these used as texts? After all, user-friendly, and user-fun, can do a lot of good, and should do no harm. Well, I now know why not. At least in the case of this one.

It’s way full of sloppiness, inaccuracies, misleading statements, omissions, and downright errors (and therefore *can* do harm). I found them at the averate rate of one every third or fourth page. Relying on this book is like making the mistake that many students make — namely, having your brother or best friend tutor you, when that bro or friend isn’t anywhere near capable of doing such tutoring.

It was annoying almost to the point of deliciousness to keep finding less-than-ideal statements and explanations, page after page. Listing them all would require a book in itself.

Here are some examples of sheer sloppiness:

- On page 14 he writes, “A line in standard form looks like this: Ax + By = C. ... to officially be in standard form, the coefficients must be integers...” Well, what about equations like x + √2y = 1 ? (That kind of thing comes up in calculus a lot.)
- On page 20 he gives a problem for readers to do which is quite different and much harder than the example given. (And he misses the opportunity to tell his readers that, in doing what I call “take-out” factoring, one can “take out” algebraic quantities like x, as well as numbers like 2.)
- And on page 21: “To begin, set your quadratic equation equal to 0”, and in general he often says “equation” when he means “expression” (also, equation when he means graph of an equation).

And so it goes; moving right along,

- on page 87, he uses his “Date vs. Weight” graph to illustrate the Intermediate Value Theorem: “From the graph, we can see that I weighed 180 pounds on December 1 and porked up to 191 by the time December 30 ‘rolled around’.. According to the theorem, I can choose any value between 180 and 191 (for example, 183), and I am guaranted that at some time between December l and December 30, I actually weighted that much.” This is true, but, the way all this is stated, a student might wonder on which day that happened. (whereas it’s really a matter of which time of which day). So that isn’t a good example, not the way it’s stated.
- Page 102: “Even if you get a numerical value when calculating a derivative [from the limit of difference quotients definition], it’s possible that the answer is invalid, because there actually is no derivative!” It might be helpful to students to add that the reason that one might get a numerical value is that one hasn’t taken enough values of Δx.
- Page 112: “You must use the Chain Rule to differentiate any functon that contains something other than just x.” Paraphrases of that statement appear throughout that chapter, and although I know what he means, I found it extremely confusing. I assume that many students would, too. After all, EVERY function, besides x, contains “something other than just x”.
- As our last example, page 262: “If a series contains alternating negative and positive terms, you have to apply the Alternating Series Test...” As though that’s a guarantee. (And he does that kind of thing throughout the chapters on series, giving the wrong impression (although he eventually states otherwise) that there’s an algorithm for deciding about convergence for all infinite series.

Probably most math books contain a fair amount of omissions, or what I call “missed opportunities”, but this one was chock-full of them. On pages 21-22, in going over “reverse FOIL”, he declines to give the quick explanation of why it works (namely, the product of two quantities — even algebraic quantities — is zero if and only if at least one of the factors is zero). On page 26, he neglects to give an example of a relation that is not a function. On page 27, he doesn’t emphasize the idea of the composition of functions, giving only the example (and analagous problem): If f(x) = √x and g(x) = x + 6, evaluate g(f(25)).

And so it goes on. For example, on page 151 we find the usual “box problem” as an example of an application of “the first derivative test”, but no figure illustrating the 3D box made from the sheet of paper. And on pages 159-160, while estimating the area under a curve between two points, he skims over the actual calculation, in particular the calculation of the function evaluated at the various division points.

And then there are the downright mistakes.

- Page v, the first page of his “Contents at a Glance” (and there’s also a printing error; the page number is omitted, as are several other page numbers in the small Roman numeral section of the book): Part 2, Section 7: “Continuity — ensuring a smooth ride for the rest of the course”. Actually, “smooth” is misleading (although I see the joke); it should be “uninterrupted” — “Smooth” is about differentiability, not continuity — and he could probably make some joke out of “uninterrupted”, too...)
- Page 25: “When is an equation a function?” (Answer: Never, not unless we’re talking about much more advanced abstract math)
- Page 26: “Here’s the most basic definition of a relation... just a list of ordered pairs: s:{ (-1, 5), (1, 6), (2, 4) }. This relation, called s, gives a list of inputs and outputs. In essence, you’re asking s, ‘What will you give me if I give you -1 ?’ ...” Well, this particular s gives us 5 and only 5, but other s’s that aren’t functions give two or more numbers; that’s the whole point.
- And a little further along: Page 84: “... if no general limit exists, you have jump discontinuity..”
- And perhaps his most serous error, on page 94: “This limit is called the difference quotient”. (No, it’s the
*limit* of difference quotients, and it’s called the derivative (of f(x) at x = c. )

But it’s far from his final error; on page 99, very top, he writes the Product Rule using differentials on the right side, but not on the left, and on page 146, about the Mean Value Theorem, describing Figure 13.1: “Here, the secant line is drawn connecting the end points of the closed interval [a, b].” Wrong! The two points that get connected by that secant are on the plane, and they’re (a, f(a)) and (b, f(b)). And page 161: “Calculating midpoint sums is very similar to right and left sums. The only difference is… how you define the heights of the rectangles.” — WHAT?!! The difference is which rectangles you take the heights of. (Also, there’s no figure illustrating the Midpoint Rule.)

And there’s plenty more where all that came from. May I offer the perhaps-sly comment that it’s quite possible to be colloquial and “friendly” without being inaccurate?

There are also a number of pedagogical bones that I would pick with him.

His treatment of Partial Fractions is way too sketchy. He barely gives examples. Ditto his treatment of improper integrals; he doesn’t give even an example of what I call “infinitely long integrals”, meaning integrals with plus or minus infinity as one or both or the limits of integration. In fact, the more advanced the topics become, the sketchier the book gets — as though its author were getting as tired and saturated as his readers.

And every once in a while he’s guilty of writing down, with too much emphasis, the way to not do a problem. The very *first* teaching advice that I got, before I began teaching, was: *never* write anything wrong on the the board. Indeed, a sure-fire way to fix an undesirable misconception in a student’s mind is to make it possible for that student to see it. That student is likely to remember the misconception and forget the “not”.

On page 11 he says, “If a formula, rule, or theorem has a proof that I deem unimportant to you mastering the topic in question, I will omit it, and you’ll just have to trust me that it’s for the best.” I don’t think that I’ve ever, in all my teaching, come across a theorem for which there isn’t at least some sketch of a proof — perhaps only a sentence, or a picture — something that can make at least partial-believers out of at least some students. That, I feel, is one of the purposes that a book like this should serve — helping students to believe, without taxing brains too much.

At least in my own teaching experience, what makes students at *all* levels and abilities feel anxous is not being given any reason at all to believe the definitions and theorems. Students might not like full-blown proofs, but they almost always would like to get at least some idea as to why. In fact, in my experience, they *ask* why. (Yes, even engineers; yes, even “bad” students. In fact, the tendency to ask why does not seem to correlate with a student’s grade.)

And alas, his four-step plan for calculus success is, as probably any such plan (no matter how many steps...), easier said than done.

- “Make sure to understand what the major vocabulary words mean.” Yes, and?
- “Sift through the complicated wording of the important calculus theorems and strip away the difficult language.” Again, how?
- “Develp a mathematical instinct.” Hmmm.
- And finally, “Sometimes you just have to memorize.”

Actually, this book stresses memorization far more than the syllabi in the courses that I’ve taught, in universities which vary as to difficulty, and far more than I believe it should. Most calculus courses do not require students to memorize all of the integral formulas, in particular the trig integrals. What’s required is understanding and knowing what to look up, and when. (Reality then takes over; for example, if a student goes on to take a more advanced course, she then will need to, not necesarily memorize, but remember — otherwise she’ll spend so much time looking things up that she won’t have time to do anything else. Moreover, she will remember *because she’ll have used the formulas often enough*.)

Naturally, as a user-friendly calculus teacher, Kelley has a different style from mine; that I would expect, since part of the point of user-friendly is that the individual personalities of the teachers are allowed to come out.

Here’s one difference between him and me: Occasionally his passages seem, perhaps subtly, to impart negativity towards math. The most blatant example of this is on page 187: “Let’s be honest with each other for a moment. Integration sort of stinks. In fact, integration really stinks.” Perhaps it’s uptight or nitpicky of me to recoil at that, but the way I would (and do) put it is: ”I need to warn you: Integration is harder than differentiation. There are no formulas which work for anti-diff-ing all functions; there’s nothing like the Product Rule or Chain Rule, like for diff-ing. In fact, some functions can’t be integrated at all — or not to get as ‘the answer’ any recognizable (“elementary”) function — no matter what techniques we use. I’m saying this because I don’t want people thinking that there’s something wrong with *them* because they find integrating harder than diff-ing.”

And I prefer to then continue on to the positive idea, that this means that there are “new” functions, and how exciting that is. There is a fine line between joking, skepticism, etc., and negativity. Also, there is a fine line between allowing the students their negativity and conveying negativity oneself; I prefer to convey the message that they’re allowed to feel negative towards math but that I don’t — indeed, that it’s possible to love math, and that I do.

Some word choices and some of the jokes bothered me:

- Page 131: “The concavity of a curve describes the way the curve bends. Notice that the concave up curve in Figure 11.7 would catch water poured into it from above, whereas the concave down curve would dump the water onto the floor, causing your mother to get angry.” (As a mother, I have always felt upset by remarks like that, and when confronted with them, I’ve quipped “What about Daddy?” Indeed, it would be very easy for him to replace “mother” with “parents”. )
- And page 145: “Man has struggled for centuries to define life...” I always say “humankind”; it’s quick and easy.
- And perhaps the following example, on page 158, in the introduction to “Approximating Area”. is subtle, and doesn’t bother everybody, but it bothers me: On the previous page he invokes the movie “Speed” to describe how his readers must feel after all they’ve been through, mentioning the “romance” in the movie, then he concludes that introduction: “For now, take a deep breath, and enjoy a much slower pace for a few chapters as we talk about something different. And if you see Sandra Bullock, tell her I said hi.” How might a women student in his class feel? How might Sandra Bullock feel? Things like this convey a subtle attittude that I for one don’t like, and that concluding remark isn’t funny enough to make it worth it.

Many of his jokes are really cute. Page 91: “Honey, I shrunk the Δx”. Page 146, about the Mean Value Theorem: “It has no twin called the Kind Value Theorem”. Page 197, in the list, “In This Chapter”: “Teaching improper integrals some manners”.

His longer, more “extended” jokes (possibly “personal” jokes from his own classrooms), such as the one on page 254, about the Comparison Test: “If you have siblings, you know what it’s like to be compared to them. ‘Why can’t you get grades like your sister?’ and “I wish you could rebuild a transmission with the clarity and sense of purpose that your brother Hank can’ probably sound familiar to you. Don’t look now — you’re turned into your parents. ‘Why can’t you be a well-behaved geometric series like that one?’ “

It’s jokes like those — the ones that relate specifically to the math involved — that I prefer, as opposed to jokes which do *not* relate to the math involved. Of course, I’m not saying that such jokes should be outlawed; every joke (and everything friendly) is helpful to students, in an emotional and/or psychological way. But I personally aim for jokes that relate directly to and enhance the math — and that therefore, as an added bonus, can help students remember and understand the concepts. And there are jokes in this book (such as the Sandra Bullock comment) about which I would advise, “When in doubt, leave it out”.

There are some smaller matters which I’d like to mention, things which I believe all calculus books that I’ve seen are guilty of.

First, a “device” that I use when teaching calculus is to say, for each topic, whether or not calculus (in the sense of derivatives or integrals) is directly used for that topic. For example, the Midpoint Rule doesn’t use calculus techniques. (That’s one of the beauties of definite integration theory, that calculus makes it possible, in many cases, to find the exact definite integral, and in a way that’s often easier than using Midpoint-like rules.) Nor does average rate of change. If not the students, this helps *me* to keep things in perspective, and to appreciate calculus. I often wonder why calculus texts neglect to do this.

Also, two of my pet peeves about all textbooks, especially math textbooks, are: (1) they don’t underline or italicize enough. (I’ve talked about this in another review for *MAA Online*.) I often come across ideas which are beautifully explained, but which I feel would come across even better if the appropriate words were underlined or italicized. Teachers of course can, in the classroom , emphasize those words but students merely reading them won’t necessarily get their emphasis. (2) The answers to the examples aren’t circled. It isn’t always clear just what the answer to an example actually is, and circling that answer would help immensely. This book is no exception in those failings.

There are also matters about which I feel positive. I like that he includes the solutions to “You’ve Got Problems” (which are the problems in the book, after most of the examples). And there are many passages which I imagine are very clarifying to students. For example, page 59, about one-sided limits: “To keep from confusing right- and left-hand limits, remember the key word: *from*.” Also, “a left-hand limit is the height toward which you’re heading as you approach the given x-value *from* the left, not as you go *toward* the left on the graph.” (And, yes, here he commendably italicizes that key word — but it might be even better if he also italicized “not”.) I also like his treatment of related rates on pages 148-150. And page 163, on the Trapezoid Rule: “You may not be used to seeing trapezoids tipped on their side like this — in geometry, the bases are usually horizontal, not vertical.” He also occasionally offers encouragement, advice, or disguised advice, to students as to how to learn: For example, page 252: “Have you ever noticed that it’s easy to choose between things if only given a few options?”

Indeed, this book succeeds in many places. But it fails in more. And the world probably still needs a *good* user-friendly calculus book.

Marion D. Cohen has a poetry book in press, forthcoming from Plain View Press (http://www.plainviewpress.net ), about the experience of mathematics. The title of the book is “Crossing the Equal Sign”. She would love to receive emails at: [email protected]