Note: I edited this and added some text shown in green below to reflect new findings. If reading for the first time, just ignore the color differences.
Iâ€™ve been looking at potential calculus textbooks to be used mostly for individual study by an eleventh grader. A particular career path has not been chosen, but early inclinations are the natural sciences or social sciences. A bit of mathematical rigor is desired, but more for a flavor and some experience rather than preparation for a career as a mathematician. The student will have used Foersterâ€™s Precalculus, a moderately rigorous text (IMO well below AoPS, very slightly above Brown, above Larson or Sullivan, and well above Lial) and would like to take the AP Calculus BC test after the course. There are some calculus texts listed in the pinned High School Math thread and the College Board website lists quasi-approved texts as well as detailed AP class descriptions. Of course, time was limited and these are only my opinions, but I thought some other folks might be interested. Please let me know if you have comments or questions. A one paragraph review for each book follows:
Foerster Calculus, 2nd edition, 2010 The look and feel, as well as the wording, of the Foerster text seems more like a high school text (mostly a plus for Foerster in this context) and written with the AP test in mind. I couldnâ€™t find any coverage of the rigorous epsilon-delta proofs in Foerster, although it might be there (at least the definition of limit is given in epsilon-delta form on p. 34). Foerster doesnâ€™t seem to prove even some of the simplest theorems that many textbooks prove, including Larson. I consider that a significant negative. While proofs may not be covered on the AP calculus tests, I think that some understanding that the results of calculus have been proved with rigor, and at least a couple of examples, is pretty important. For us, that means that Foerster would only be considered if some kind of supplementation were done with another text. Throughout the book, Foerster, as many other texts published in the last 20 years, seem to cover four ways of seeing the math: verbal, algebraic, visual, and numerical points of view. The problem sets seemed of average interest and application, mostly in a physics and engineering context -- the most important evaluation is when you go to do them and how tightly they fit with the text, and I can't really assess that. Any experience with the problem sets of Foerster or other texts? In sum, I'm not as impressed with Foerster's Calculus as his Precalculus and would only use it together with another text. But the student reviews of the book on Amazon are pretty positive, so it would seem reasonable to use it together with another book. Note that the new AP Calculus BC exam, starting about 2016, has somewhat more emphasis on linking definitions with results and more emphasis on series, so Foerster's minimalist calculus approach might need to be supplemented slightly even if your only goal is to get to the minimum needed for the new AP test.
Anton, Bivens, and Davis, Calculus (Early Transcendentals), 7th edition, 2002, looks and feels like a college calculus text. I really like how Anton separates out rigorous (epsilon-delta) proofs so that they can be covered (or not) along the way; weâ€™d probably do a few early on and then skip them later, unless DS really gets into it. I couldnâ€™t see that Anton had better problems than Foerster. They both seemed average at first glance, mostly in a physics and engineering context. I will say that some of the Anton problems and text had what I'd consider excellent practical advice for using calculus in science, exceeding the Hughes-Hallett and Larson texts reviewed below. This edition of Anton starts with an 100-page introductory review chapter on functions, whereas Foerster starts at about the same place as Chapter 2 of Anton. If you feel like your precalculus was strong, that seems like extra text to wade through; if you feel like your precalculus was weaker or partially forgotten, that chapter might be a plus. Most texts have a brief introductory precalculus or functions chapter, some tightly integrated into the text and some more optional. The last five chapters in Anton go beyond topics covered in Foerster in this version, perhaps only multivariable calculus topics. One possible approach for us would be to use Foerster most of the way, but to get some experience with rigorous proofs in some other book, like Antonâ€™s Chapter 2 (Limits and Continuity), Chapter 3 (The Derivative), and maybe a few other places if interest exists. After reviewing the text myself, I later read the reviews on Amazon: they weren't good; a number of reviewers said that Anton did not explain concepts well for a first-time student. This has definitely given me some pause. On the other hand, a comment below says that her (extremely talented) DS found it a useful supplement to AoPS calculus.
An intriguing text upon first review is Hughes-Hallett, Gleason, McCallum, et al, Calculus (Single Variable), 3rd ed. It strikes me as concise, elegant, appropriate for high school or college, and has some nice problems across disciplines. While it has the standard background chapter on functions to start out, itâ€™s efficient, 54 pages vs. longer in some books, and leads nicely into the next chapter on limits and derivatives; for example, chapter one provides motivation for limits and derivatives along the way, so it doesnâ€™t seem like a waste, even though it starts out at a pretty basic level. The text does use the rigorous epsilon-delta form in the definition of the limit as part of the main text and does prove some easier results. The text seems very careful with the mathematics, and has â€œapplicationsâ€ across several disciplines. Unlike Anton, however, the text seems highly disjointed from real data, real science, and probabilistic phenomena. Anton gives a solid introductory section on modeling (Section 1.7), including deterministic and probabilistic, and gives enlightening scientific points along the way in the text; on the other hand, I couldnâ€™t find evidence in the Hughes-Hallett text that the authors were even aware of the points in the Anton text let alone clearly conveying them. In addition to not finding these modeling and scientific points covered, I found the initial example of a function unnecessarily confusing and begging for a probabilistic explanation. Hughes-Hallett does have a clean, concise style which would seem pleasant for both high school and college students, so we have some trade-offs here. Even though itâ€™s one of the shorter texts, the presentation of key concepts like limits and the derivative seemed to go slowly, which is a big plus. I was not able to evaluate how easy or hard the problems were nor how well they complemented the presentation, but they donâ€™t seem like just routine calculation. For comparison in length, Hughes-Hallett is 558 pages â€“ it has quite a bit of white space, so itâ€™s on the shorter side; Foerster is 658 pages with some white space; Anton 788 pp. before the vectors chapter starts, and itâ€™s pretty dense; Larson 760 pp. before vectors and itâ€™s pretty dense too. You get more depth with Anton, but it is longer. (For comparison, AoPS has about 304 pp of fairly dense text.) ETA: While I personally still like the Hughes-Hallett text for the reasons described above, the reviews on Amazon are pretty negative, more so than for most competitor texts. The main complaints seem to be that: (1) the text doesn't explain things well to the student (hard for me to tell myself -- looks reasonable to me, but I've already learned the material); and (2) the problems are too big a jump from the text. I might still use it, but it would be with a text like Larson or Thomas readily available if DS doesn't understand it. I'd also carefully preview the problems for appropriateness.
Larson, Hostetler, and Edwards Calculus, 8th ed., seems like a basic, standard text, with a feel of either a high school or college book; some problems like those marked â€œWriting About Conceptsâ€ seems somewhat geared to the AP exam. Many of the theorems have proofs in the text. The proofs given didnâ€™t seem particularly hard; there were, however, a few Putnam Exam challenge problems, clearly marked as such, in the first chapter, so thereâ€™s at least some exposure there (if you get those, you probably need a different book!). The formal epsilon-delta limit definition is given (p. 52, and used on pp. 53-54), but I didnâ€™t check the later proofs for rigor. There was a short discussion of Fitting Models to Data on pp. 31-33, but it was poor; not only wasnâ€™t it very informative, but it even gave bad advice by saying â€œThe closer |r| is to 1, the better the model fits the data.â€ (For example, a perfectly described relationship where there is error in the measured response variable can be far less than 1 and lower than the correlation coefficient of a poorly modeled phenomenon.) Some of the problems arenâ€™t well stated either (e.g., p. 89, #60 which doesnâ€™t start with something like â€œAccording to ___ , â€¦â€ and the reader may confuse this approximation with a physical law. It may sound picky, but if weâ€™re trying to teach good problem solving skills, it would be good to be careful. On the plus side, there are a lot of problems ranging from simple exercises to somewhat real world problems in a wide range of disciplines, color figures, and there are a number of helpful hints where students have had problems; it appears to me that theyâ€™ve probably had some helpful comments over the years with this widely used and modified text. Overall, the text seems OK but not great. We might use it as a supplement for an alternate explanation if something is unclear to DS from the main text. Later note: upon reading the reviews on Amazon, which were very positive, and the positive comment by Lisa in VA is in IT, and AK_Mom4's comment that Larson's explanations are clear, and both WTMers comments about the availability of good supplemental materials, I'm reconsidering Larson's text; while I definitely have reservations with the book, it does have a lot of fans in its ability to explain concepts to first-time calculus students, and that's indeed an important use for a textbook! Speaking of supplemental materials for Larson, there have been a number of positive comments on the Chalkdust videos: Lisa in VA is in IT (linked above), Brenda in MA, Jane Elliot, plansrme and a whole general thread on Chalkdust math videos (not cheap new, however).
Stewart Calculus, 5th ed.: seems a bit more wordy. For the epsilon-delta rigorous proofs, there is a separate section (2.4), which redefines limits from an earlier section. It appears that the rigorous definition is an optional section (per Preface, p. viii, 3rd paragraph from end) and is not used in later proofs. There are also separate sections on applications in different fields, which may be useful to use in a college setting where you have people from different majors taking different classes, but for use with a high schooler with an unknown major, it just seems like more text to wade through. To me, the text seems to have a college text flavor; while not a deal breaker, by any means, I donâ€™t consider a college flavor an advantage for a high schooler taking calculus, especially since the rigor seems less. We might use it to supplement a troublesome section here and there, but we wonâ€™t use it as our base text; Iâ€™d probably use Larson or another text as a backup reference text before Stewart, which may have been my least favorite. I prefer Hughes-Hallett, Anton, Larson or Thomas. Some reviewers on Amazon and elsewhere seemed to like Stewart but some strongly disliked it.
Life of Fred Calculus looks to be one of the least mathematically rigorous. The rigorous epsilon-delta definition of a limit is left to an appendix; even Stewart has it as an optional section in the main text. If your kids love Fred and just want to get a feel for calculus, it may be a good choice. AMA has a brief review of Fredâ€™s calculus. Weâ€™ll go with a different text.
Saxon only goes through AB and probably too mundane for DS.
Finney, et al., Calculus: Graphical, Numerical, Algebraic (2003). This is clearly a standard high school textbook, which seems to provide the most basic of AP Calculus BC training. While the limit is defined in epsilon-delta form, it doesn't seem to be used except for some exercises in an appendix, and I didn't find any proofs in the book. The material, including definitions, does seem carefully presented, but it's a very elementary text, and there are quite a few exercises rather than challenging problems. There is a very, very heavy use of graphing calculators to give a graphical view of functions, limits, etc. -- not necessarily bad, but I'm trying to give an idea of the type of text. While the text has a strong school text feel, it isn't plagued by excessive, distracting sidebars, at least in the edition I reviewed. The text is too basic for us, but I could see the text's usefulness to explain concepts which are not grasped via the primary text you're using. The order of most of the texts seems pretty standard, except for some including transcendental functions earlier vs. some later, so some cross-use of texts isn't out of the question.
MIT Open Courseware looks like a good possibility, but DS prefers a book rather than lecture focus. We might try it with an early topic and see how it works.
Strang also has a calculus text available for free download on the MIT Open Courseware site. I have not had a chance to review it yet, but his Linear Algebra and Its Applications text was my favorite math text as an undergraduate. My prof. (admittedly) was a terrible teacher, but I had no problem learning the material from the text, and I've heard that his calculus text is clear and direct too. At the same MIT OCW site, Dr. Strang has a series of videos "to show ways calculus is important in our lives."
AoPS (Patrick) Calculus, 2nd ed. (2012) would merit serious consideration for folks whoâ€™ve successfully used AoPSâ€™s courses in the past. While the AoPS text doesnâ€™t prove as many results as Spivak or Apostol, it is a rigorous course, and seems comparable in rigor to Hughes-Hallett or Anton, all being solid books. In tone and style, AoPS strikes me as distinctly high school like, even appropriate for a prepared early high school student. If your DC thrives on AoPS, it seems like a great choice. It has that math competition preparation flavor, but I didnâ€™t see a lot of real-world application in AoPS except in a highly mathy sort of way; Hughes-Hallett has problems using the context of a variety of disciplines, which would be more appealing to my more science-oriented DS, although they are still pretty artificial IMO; Anton goes beyond that and really enhances scientific understanding of the uses of calculus in experimental sciences (the actual problems seem mostly physics and engineering). I think Anton provides the best combination of rigor and promotion of use of calculus in non-math settings, but it is a bit longer and has more of a college feel. Larson has a number of proofs and a number of real-world examples, across many disciplines, but Iâ€™m not sure theyâ€™re as tightly integrated into the presentation as in Anton nor promote the same level of understanding.
A lot of text choice has to do with how clear the explanations are in practice, and I donâ€™t have experience on that with these 21st Century texts. Your experiences with using calculus texts to explain concepts would be appreciated!
ETA: some of the links and the Finney text review. Some later notes shown in green and will be discussed in a later post. The review of Thomas' Calculus below was added in response to a question by Mike:
Thomas' Calculus: Early Transcendentals 12th edition (2010) based on the original work by George Thomas as revised by Weir and Hass. There are many editions of Thomas' Calculus and this one is intended as a college course either for those with high school calculus or without. (For example, the text University Calculus by the same authors is a streamlined version meant for those who took calculus in high school, so it's outside the scope of this review. The next text to be reviewed is an older, explicitly high school text by George Thomas.) Thomas' Calculus seems to be at about the same level as Anton; there's perhaps more theory in Thomas' text but there's a good amount of rigor in Anton, and probably more than Thomas, if you select it, and Anton seems to have a significantly greater depth of practical problems. Thomas' Calculus 12 ed. appears rigorous, although it does not prove every result -- none of the texts except perhaps Spivak and Apostol do, but that's not what we or most high school students want anyhow. Thomas' has a lot of exercises, both simple and medium in difficulty. Thomas' seems like a good book, although I like Anton better for deeper applied problems and seemingly more flexibility with the theory. The most important thing, however, is how well the text explains concepts to the new learner, and, unfortunately, I can't say for sure. I'll just say it's worth considering.
Thomas Elements of Calculus and Analytic Geometry 2nd edition (1976) is an adaptation of George Thomas' calculus for high school students, by George Thomas himself. It looks like a 1970s high school textbook. Transcendental functions are covered late in the book, which I'm not a fan of since it leaves less time for working with transcendental functions during the course -- this is hotly debated, which is why most textbooks nowadays have versions with late transcendentals and early transcendentals. The presentation in this relatively simple, short book does strike me as straightforward and probably fairly easy to follow. There are a few simple theorems proved. Most of the problems are simple exercises with little sense for the range of applications of calculus and not many challenging problems. In summary, I don't think this is the best choice as the core text, but it seems to have value in having around as a way of describing a problematic topic if a student is struggling.