Looking for a good Calc I/II workbook.

August 16, 2008 11:28 PM

Planning on teaching myself Calculus I and II in order to take the AP Calculus BC exam this May. If you've taught or taken either class, at a high school, university, or independently, read on.

To preface: I'm a very self-motivated person. I can stick with something to completion. My math background is very strong, I have all the requisite skills for Calculus.

This year in school (I'm a senior in high school) I'm taking an AP Calculus AB course. This is more or less equivalent to a Calc I course at the university level, just taught over the course of a year instead of a semester. We're two weeks into the class.

This past summer, however, I started studying calc independently. I found a good set of recorded lectures (Thinkwell's, if anyone cares) and watched a good bit, did some practice questions, and moved on. I went through basic limits and such, through taking derivatives of functions both with the definition of the derivative and with the power, product, quotient and chain rules. I started implicit differentiation around the end of the summer.

Now, I just *know* that my Calc AB class will move way too slow for me. We're approaching week three of the class, and we've been on "Exploring Limits Graphically" for like four days. This is unacceptable.

So, my plan is to cover everything we cover in our AB class (Calc I) and the material that will be on the BC exam (Calc II) by the time May rolls around, in order to take the exam. I'd love to get a 4 or 5 on the BC exam without ever taking a formal Calc II course. This is made a bit more doable by the fact that our textbook was designed for covering both exams, the class just never gets more than halfway through the book.

I've already started working ahead; we're on section 1.2 in class, I'm on section 1.5 on my own. The problem I'm finding is, I really need more practice problems. My textbook has a decent bit, but it tries to make everything a word problem. This is fine, I agree that knowing how to translate English into math is important, but sometimes I just feel the need to practice the calc more than the application.

So, I'm looking for a workbook that covers calc I and/or II. One for each or one for both is fine. Preferably, it would have questions more like "given f(x) = 4x^2, what is f'(2)?" than "A car accelerates at 4 m/s^2. how fast is the car traveling after two seconds?"

Book should come with answers definitely. Worked out solutions are not necessary, but nice. My calc teacher or my physics teacher from last year would be glad to help me out if I ever actually got stuck on something. If you took Calc I or II and worked out of a workbook in addition to your textbook, I'd love to know. I really don't need more instruction, just more problem sets.

I'm looking at This book and this book right now. Any thoughts from people who've used either would be appreciated.

Basically, my current plan is work through what our class will get through this year by Christmas, and cover the rest of the book by May. I bought two one-subject notebooks the other day, and I'm using one for straight notes from lectures/books, and the other for working out problems. I figure the notes will be useful in college and whenever I need a quick refresher.

Further, if anyone has any advice/tips/handy sites/books/whatever to share, please do!

Any more tips would be great!

posted by Precision to education (28 comments total)

Couldn't you just go with one of the study guides for the AP test? It's been a while, but I remember using something like this when I was in high school. I'd be surprised if the types of problems on the test change much from year to year, so these types of study guides are probably pretty good indications of what you'll see on the test.
posted by SAC at 11:41 PM on August 16, 2008

...and you could probably just get older editions in the library. I think that's where mine went.
posted by SAC at 11:48 PM on August 16, 2008

1. Did I miss something here? It seems like you've written about five pages just to ask, "Can anyone recommend a good book to use to prepare for the Calc BC AP exam?" Is that it, or is there something else I've missed?

2. What is the rush to get through so much math so quickly? Calculus forms the basis of many mathematical and math-related fields, so it's really important to learn it carefully and learn it well. If that means taking part of it in college, wouldn't it be better to do so and really understand it and be able to use it, rather than to rush through on your own during your senior year (when there will be 8000 other things going on to distract you from your goals of teaching yourself calculus) and have a less-solid foundation to build on? I'm not trying to throw cold water on your plan; but, as a physicist, calculus has been probably the single most important course I ever took, and I'm glad I took my time with it.

3. Are you sure that doing well on the BC exam will buy you anything more than doing well on the AB exam? Where I went to college, the C (Sequences & Series) portion of the exam was attached to another course that pretty much everybody had to take. So a good score on BC (like I had) just meant you were very slightly better prepared to take the same class that everybody who got a good score on the AB version of the exam (like most of my equally well-prepared classmates had). So before you go ahead with this, you might want to inquire at schools you might attend about exactly what AP math courses are worth.
posted by dseaton at 11:53 PM on August 16, 2008

I took Calc 1 in college, never had to take Calc 2 though. From what my buddies used to say there's a good jump from 1-2, they were pretty good at math and had a lot of trouble with it. Who knows though, you sound like you're good at math and you're definetly motivated (study ahead? who does that?!!) so I'm sure a good AP prep book would do you good, Kaplan is always a good brand to look at.
posted by BrnP84 at 11:54 PM on August 16, 2008

Dseaton gives some pretty good advice too, if you're majoring in a math related field you don't want to test out of your entry courses and get thrown into upper levels in your first semester. If you're majoring in like English or something than hell yea, test out of all that shit you can, saves you tons of time and money. I was a bio major and I chose not to test out of my english courses because I got really bad advice from my sister, she told me to not test out of anything because it'll be easy A's your first year. Horrible advice, I could've tested out of my freshman english but instead I didn't and ended up getting a C. English wasn't integral to my degree and I should've tested out of it. On the other hand I could've tested out of two entry bio courses but chose not to. I'm extremely glad I didn't because it helped solidify my knowledge base and helped me off to a good start for the rest of my college career. So I guess it just depends on what you want to do, if you don't really need math but they require calc than hell yea get rid of that shit now.
posted by BrnP84 at 12:02 AM on August 17, 2008

Don't let anybody discourage you from learning. It's great if you're motivated to work ahead.

A couple of years ago, I decided to go back to school for mechanical engineering. I took many math classes almost ten years ago (including calculus, differential equations, etc), but could remember almost nothing. I asked my cousin (who is working towards his PhD in mathematics) what I should do. He recommended buying a Schaum's manual for calculus.

Over the next few weeks, I busted my bum to relearn that stuff in the Schaum's manual. It worked, and worked well. There were numerous examples to work out with the answers. Each section was brief enough to cover quickly, but helpful so I actually relearned it (and keep in mind, I remembered almost NOTHING from calculus).

However, be careful in working ahead. If you concentrate on learning ALL of it too fast, you'll forget it. Then you're screwed on both the Calculus AB and the BC exams.

I took the Calc AB exam (I believe I got a 4 if I remember correctly). There are a lot of distractions if you're taking the exam at the end of your senior year (which I'm assuming you are). Mine took place on the last day of school. While we were inside testing, the bells rang and all the other seniors ran outside and partied. How's that for a distraction?

Another tip: My first degrees were in English and Finance. You don't need calculus for either, but I took a bunch of math classes and physics anyway. Since I took all those classes then, I don't have to take them now--7 years after I first graduated college. That saves time in the long run.
posted by rybreadmed at 12:21 AM on August 17, 2008 [1 favorite has favorites]

As I recall, Calc A AP was mostly differential and Calc BC AP was almost entirely integral. Unless you are dealing with differential equations, these two things don't have much overlap. If you expect to do well in your Calc BC class you are going to have to teach yourself about integrals from top to bottom (not incredibly hard, but certainly more challenging than the concept of a derivative).

The concept of the derivative is introduced very early in most lines of math education. The fundamental basis is already there when they introduce the limit definition of the derivative. In most cases, the concept of the integral is not introduced long before the application of it (usually in a pre-calc or "calc 1" type of course). Since that is usually the case, it leaves you a lot of knowledge to develop on your own. There are plenty of materials, but there are new concepts that you have probably not yet touched or heard of.
posted by milqman at 12:25 AM on August 17, 2008

What textbook is your class using? I would recommend James Stewart's book. George Thomas's book is also highly regarded, but in my experience Stewart is more straightforward.

I would simply get the first two volumes of Stewart's three-volume set (the third volume repeats a couple of chapters from the second and then adds Calc III, which is the calculus of multivariate functions, along with some basic second-order diffeq material), and work through many of the problems. The solutions manual is readily available from many BitTorrent sites.
If you want to contact me, I can provide you with the assignment sheets used in the Calculus BC course at my high school; those might give you an idea of some problems to work. I can also give you the sheets used as lesson material. Shoot me a message if you're interested in that. Good luck!
posted by likedoomsday at 1:00 AM on August 17, 2008

Perhaps this is too easy, but is there a BC-level class at your high school? Teachers who might moonlight as a BC-level teacher at a community college? You could just ask them for some supplemental material. I doubt they'd let you have a set of tests and quizzes, but you might convince them to let you take them at or near the time the BC kids were taking them, if not before. Of course, I imagine that if such a class existed, you'd be in it, but still- asking your teacher to teach you to the BC standard would not be hard, and that way you could use class time to work at your own pace, monitored by your teacher. I took AB-level calc last year as a senior, and there was a sophomore in my class who had started taking our class, then transitioned to BC work when he decided to aim higher. We actually had a few BC classes at my school, but he was hampered by scheduling related to having to bus back to the 9-10 school for the rest of his classes. If you really have a good relationship with your teacher, I should think they'd love to help you with your goal.
posted by MadamM at 1:13 AM on August 17, 2008

I did this with a friend under basically identical circumstances; we finished the BC part well ahead of our class's AB class. I ended up getting a 5 on the AB portion and a 3 on the BC portion.

You can just go to your local used bookstore (try ones near a local college for best results) and see if you can find a copy of something that covers the BC material. These things don't change much over the years so you can probably find several books on the cheap and just take problems from both of them; most college books will have answers in the back for at least half the problems (and it's easy to check when you're in the store).

I would highly recommend finding someone else to do this with - the true test of understanding is whether or not you can explain to another person, and it's a good way to stay motivated and get some extra insight if there's a concept you have problems with.
posted by 0xFCAF at 1:33 AM on August 17, 2008

ryebreadmed: Don't let anybody discourage you from learning.

Just to be clear here, in case this comment was referring to my advice, I'm not trying to discourage Precision from learning. Rather, I am suggesting that the best definition of learning, in this case, is not necessarily covering more material at shallower depth. I can say from experience (I have an undergraduate degree in math and a Ph.D. in physics -- I have taken a LOT of math) that when it comes to learning math, quality always trumps quantity.

Extra studying is good, and could help you do better, but not if it means you're sacrificing real understanding just to get through more material. Especially if your eventual goal is to pursue a math-related field.
posted by dseaton at 3:43 AM on August 17, 2008

Following up on the comments of ryebreadmed and dseaton, here's a question you might want to consider: What do you think you will study in college? If it is math or science, it might be in your best interest to retake differential/integral calculus in college. If you want to go into the humanities, by all means take and pass the AP exam and dispense with the calculus formalities.

I teach math at the college level, which has included an uncountable (math joke) number of calculus courses. Too often, students who have taken the AP calculus exams (both AB and BC) are really good at taking calculus exams, but not really good at calculus. You seem like a highly motivated student, so this might not hold for you, but I actually prefer it when my students have had no calculus background.

To answer your actual question though, I have found the Schaum's series helpful as well. I've also used Stewart's Calculus text and found it to be pretty good. If you want more problems with worked solutions, you can google my name (in my profile) to the first hit and follow the Math 212 link to a bunch of calc. II resources (wait a few weeks for the course to develop).
posted by El_Marto at 4:55 AM on August 17, 2008

The best books in the world for teaching yourself mathematics are Engineering Mathematics and K.A. Stroud and the sequel Advanced Engineering Mathematics. I cannot recommend them highly enough. Unlike the textbooks for most classes, they are specifically designed for self-teaching and follow a self-checking workbook format.

They are not, however, large repositories of practice problems. But it sounds like you've got that aspect covered with the books you mentioned.
posted by LastOfHisKind at 5:25 AM on August 17, 2008

If you're serious about learning the material, calc 1 at MIT covers almost exactly the same material as BC. If you search 18.01 on the Open Courseware page, you'll find links to a bunch of different versions of the class with lecture notes, homework and answers and tests with answers as well as textbook recommendations.

If your primary goal is to do well on the calc BC exam, definitely get a copy of the Princeton Review book. The questions on the exams are pretty formulaic so it's not hard to just figure out how to answer every type of question that you'll see.
posted by martinX's bellbottoms at 5:35 AM on August 17, 2008

So, I'm looking for a workbook that covers calc I and/or II. One for each or one for both is fine. Preferably, it would have questions more like "given f(x) = 4x^2, what is f'(2)?" than "A car accelerates at 4 m/s^2. how fast is the car traveling after two seconds?"

I should also add that any calculus exam worth its shit will be full of word problems. You can solve your first example question with any graphing calculator. In order to solve the second you have to understand what a derivative is and how it relates to acceleration and velocity.

Also, if you haven't already, buy or borrow a TI-89. Especially if you're going to be teaching yourself, it makes it easy to experiment with a bunch of similar problems and the people writing the AP exams assume you have one.
posted by martinX's bellbottoms at 5:40 AM on August 17, 2008

you want practice problems: schaums...
posted by geos at 5:51 AM on August 17, 2008

I took AB at high school, and BC at the local community college over the summer. That allowed me to concentrate on one at a time. I nailed the AB exam, and I never had to take the BC exam because the course transferred. We used Stewart for BC, I think.

The first semester of AB in high school, we learned material. The second semester, we did practice AB tests. Seriously, we just did practice AB tests the whole semester. Something like 90% of the people in the class got a 5.

You might look into taking BC at the community college your second semester. I'm sure your high school would accept it as a substitute for second semester AB. And I'm sure the community college offers classes at night.

If that's not an option for you, I recommend using this book. It is a good first-year college book. In order to cover AB and BC in one semester each, you must average 3 sections per week. For example, 2.1, 2.2, and 2.3 in one week. That is what they do in the first-year college course. protip: buy the older edition, it's cheaper and pretty much the same.

If you're really just looking for more problems, go to Borders and buy one of the Calc AB/BC test-prep books. They have tons of problems exactly like the AP.

One last idea. How about this: take AB, forget about BC completely. But, study AB at a deeper level, on your own time. You know, here how it works: the engineers and math majors take the same classes the first two years, then the third year the math majors have to take that same material again, at a deeper level. Calculus and "Analysis" are the same thing. In fact, some hard core kids go straight to analysis. They use Rudin's Principles of Analysis as their first year college text. I'm not suggesting you do that. That would ruin your senior year of high school for sure. But perhaps you could read Lang's book alongside your watered-down AB class. I think that would be a great experience. Possibly more valuable than jamming the BC material in your brain, only to forget it over the summer.
posted by metastability at 6:25 AM on August 17, 2008

sorry, that second link should be this
posted by metastability at 6:36 AM on August 17, 2008

It's been 12 (good grief) years since I took calculus in high school, and as I remember it, the BC exam was significantly more difficult than the AB. If you are doing this for your own edification, more power to you. But if you are wanting to apply your AP exam grade to place out of college courses, you should double check the requirements at your prospective schools. Different schools and even different majors/departments within schools can have very different approaches to providing AP credit. Some will accept lower scores than others,

For instance, I could have placed out of two semesters of chemistry, but ONLY if I started with organic chem my first semester of freshman year. As full of myself as I was, I knew that probably wasn't a good idea, so I chose to place out of only one semester. Likewise, I had the option of placing out of Calc I and II if I passed an additional exam provided by the school. However, the school of engineering strongly encouraged the students to take Calc II at the college level - and besides, I figured that it would be an easy grade since I had seen most of the material before.

At any rate, good luck.

(Side note, the Schaum's editions are good for lots of topics - I used them for stats and mech engineering courses and still have them on my shelf for reference today).
posted by mbd1mbd1 at 8:09 AM on August 17, 2008

I'd beg to differ with likedoomsday - stay away from Stewart. I used Paul Dawkin's online notes to get through Calc I-III (they mostly follow Stewart's sequence if you do end up with that monstrosity).
posted by djb at 8:13 AM on August 17, 2008

I took a Calc AB course my senior year in high school, took the Calc BC exam and got a 4 on it, which is good enough for credit almost anywhere. I spent maybe a week beforehand cramming the extra stuff (its basically just Taylor series & a few misc topics). If you understand the Calc AB material, you'll have no trouble.
posted by devilsbrigade at 8:53 AM on August 17, 2008

When I was in a similar situation years ago, instead of dealing with teaching myself to the AP exam, I took Calc 2/3 at a local college. This ended up being worth WAY more credit when I went to university, and got me exposed to better and more in-depth math than my peers in BC. If class A isn't moving quickly, do more problems or do work for some other classes. Or spend the time looking for scholarships. Or program, or get a hobby or go out on dates or something.

It will look much better to get ahold of a college calc book and do the portions which correspond to AB material, get a 5, and do calc 2/3 somewhere else than to get a 4 and a 3. This is be a much more solid introduction than teaching yourself.

If you are interested and want to see the real version and the whys of what you're learning in calc, get a copy of Foundations of Mathematical Analysis or another intro book on real analysis and follow along.
posted by a robot made out of meat at 8:58 AM on August 17, 2008

FWIW, from Wikipedia (and this is my recollection as well):

AP Calculus BC includes all of the topics covered in AP Calculus AB, as well as convergence tests for series, Taylor and/or Maclaurin series, the use of parametric equations, polar functions, including arc length in polar coordinates, calculating curve length in parametric and function (y = f(x)) equations, L'Hôpital's rule, integration by parts, improper integrals, Euler's method, differential equations for logistic growth, and using partial fractions to integrate rational functions.

You should go through L'Hopitals rule, integration by parts, partial fractions, and the other basic stuff in any calc class. Convergence tests are quick. Taylor/Maclaurin series had the most extra questions in my recollection. Parametric eqns & polar functions are easy to work through. Arc length & curve length take some getting used to, but again, they're pretty easy to get used to. I don't remember any improper integrals or Euler's method being on the exam. Differential equations for logistic growth are dead easy, and I don't remember ever having to use anything more than knowing what logistic growth is.

As 0xFCAF did, I did it with two other people in the AB class, which was very helpful. Our calc teacher had old AB and BC exams, & we worked through one or two BC exams together as prep.
posted by devilsbrigade at 9:01 AM on August 17, 2008

First of all, I'm *not* interested in learning the material for the sole purpose of taking the exam and testing out of the college course(s). I want to truly learn the theory and application behind calculus, and I want to learn it quicker than second semester of my freshman year in college. My AP physics class last year showed me how beautiful and useful calculus is--I can't believe I took a physics class without knowing calculus!

I'm very interested in learning the material at great depth--which is why I'm looking for practice problems. I actually *enjoy* math. I'm not doing this just to "get ahead" or anything like that. Feynman taught himself calculus in high school--why can't I? I'm not planning on "racing through" the material or anything of the such. I think that if I work weekends and over Christmas/Spring holidays, I should be able to cover the ABC material in a fairly good manner. Another good thing is that this year I'm only taking 4 classes (of course, they are all either AP or college courses), so I'm out of school by 11:10 every morning. I have plenty of time to devote to actually learning calculus. I want to learn the subject, not learn to how take the test. Sorry if that's how I came off.

I think you're wrong in assuming that I'm trying to "get through so much math". Maybe this would be true for most people asking this question, but I genuinely want to learn the calc involved. I'm planning on majoring in Computer Science, possibly with a math minor. If after this year I feel that I need to retake Calc II at the university level, even with the credit from the exam, I will.

And, yes, you missed something. I was looking for more than just "recommend a book to prep for the BC exam." I'm looking for a book to help learn calculus I and II.
posted by Precision at 10:10 AM on August 17, 2008

First off, as a very strong math student who just completed Calculus AB during her senior year of high school, I recommend waiting before dismissing your class as too slow. I have found that many math classes -- calculus included -- often begin with review, and feel slow and repetitive. That said, it's great that you're interested in preparing for the BC exam.

I missed about a month of school due to medical reasons and was able to keep up (and score a 5 on the exam) by watching video lectures that came with my textbook's teacher edition and do practice problems with the answers in the back of the book. Ask your calculus teacher if the textbooks came with videos. I also recommend buying a test prep book or two (such as Barrons or SparkNotes) so that you can be sure you're covering all of the neccessary material for the exam.

Good luck!
posted by i_am_a_fiesta at 11:36 AM on August 17, 2008

First off, as a very strong math student who just completed Calculus AB during her senior year of high school, I recommend waiting before dismissing your class as too slow. I have found that many math classes -- calculus included -- often begin with review, and feel slow and repetitive.

This would be fine and all, but I've independently covered nearly everything we're going to cover in the first semester of our class. I was friends with a lot of seniors last year that took the course, and looking back, they moved slow, even throughout both semesters.
posted by Precision at 12:27 PM on August 17, 2008

The Thinkwell materials are designed to work in place of a book, so you should just use those, work the problems on their website, take all their exams, and you should be well prepared to take the BC exam.
posted by yellowcandy at 8:42 PM on August 17, 2008

Precison, if you really want to learn the theory then ask for an Introduction to Real Analysis or Foundations of Mathematical analysis type book. You'll want to talk to a prof to get something that matches your level of background, since few in HS have much in formal theory. These books will teach you why the formulas work.

Also, Feynman was an ET sent to teach us about physics (and bongos). It's a high standard to hold yourself to.
posted by a robot made out of meat at 5:48 AM on August 18, 2008