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So, I'm taking Calculus BC in school as a 11th grader currently. My teacher tends to put extremely hard problems on the test. Most of these questions require a lot of work or thinking out of the box. They aren't simple calculus questions like homework questions. I want a book that has multiple extremely hard math problems just to practice for her test. It doesn't have to be tailored around the AP (Advanced Placement) exam but it just needs to have the content of what the AP exam covers. It should contain derivatives, integrals, limits, multivariable/polar... etc. Currently we are on applications of derivatives and it is a test that multiple people fail and get 50s on an average so I would really appreciate any help. Our teacher gives us practice tests but they aren't enough to cover the vast questions that she could put. She has multiple-choice questions and free-response questions and I'm hoping to try and get any book to assist.

I have already looked at Tom Apostol's book, Demidovich's book, and Spivak's and they focus a lot into the concept rather than just problems. Also their problems are fairly easy than what my teacher would put.

I'm willing to either get the book online as a PDF/download/link or even buy it if available so that I can practice hundreds of problems.

Below, I will share two questions that were on one of the prior tests for a difficulty range.

(Please don't edit the post from the link to the image. I would prefer keeping it as a link as the image takes up a bit too much space on the post itself.)

Here is the link to the image for the two questions for the applications of derivatives test: [1]: https://i.sstatic.net/TMdeo6iJ.png

Also, I want to point out that I have solved these questions. These questions are from prior year tests that our teacher has provided to us and I am not seeking for answers to these questions. I provided these two questions because I would like to have a calculus book to practice for the class.

VLH
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    What are mcq and frq, please? – Gerry Myerson Sep 30 '24 at 05:39
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    "MCQ" = "Multiple Choice Question"; "FRQ" = "Free Response Question" – PrincessEev Sep 30 '24 at 05:39
  • How does https://math.unl.edu/faculty/Rammaha/Challenge-Problems.pdf look? Here's a previous question like yours, with links to other questions like yours: https://math.stackexchange.com/questions/78906/challenging-problems-in-calculus – also, https://math.stackexchange.com/questions/1927440/on-finding-sites-with-difficult-practice-problems and https://math.stackexchange.com/questions/102514/multivariable-calculus-hard-problems-with-solutions – Gerry Myerson Sep 30 '24 at 05:43
  • https://www.amazon.com.au/Challenging-Calculus-Problems-Fully-Solved/dp/1941691269 might be worth a look. Also, https://www.ncssm.edu/stories/ncssm-offers-collection-of-challenging-calculus-problems – Gerry Myerson Sep 30 '24 at 05:49
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    The “calculus” in these two example problems is not particularly difficult. These problems may be difficult for you because they are difficult for you to set up. How did you find world problems in your earlier classes? – user317176 Sep 30 '24 at 08:35
  • My teacher posts previous year tests. I have solved these two questions and I'm able to do them fairly well but a lot of students in my class struggle with these types of questions. I would like to find similar types of these questions if available on a book/pdf/website for extra practice. – VLH Sep 30 '24 at 08:50
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    Ah thanks for the websites and links to the books Gerry. I have checked out some of the problems and they are interesting and something that my teacher would put. Thank you – VLH Sep 30 '24 at 08:51
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    Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. – Henrik supports the community Sep 30 '24 at 10:23
  • I’m voting to close this question because it looks like an attempt to answer two exam questions without understanding the material. – John Douma Sep 30 '24 at 13:24
  • Sorry for that Henrik, I had it as a link prior but someone had suggested to make an edit to the post and have the image instead of the link itself. I will change that back – VLH Oct 01 '24 at 00:28
  • My intended purpose was to get a couple calculus books based on the difficulty of the problems provided. I had never intended to get the answer or solution through this post. I will edit the post to change it to say previous year questions that our teacher provided for us and that I do not want the answers or methods to solve the questions itself. – VLH Oct 01 '24 at 00:31
  • @JohnDouma I have changed the post. If there is anything wrong with it please inform me and I will make the changes. I am a newbie to posting on stack exchange and didn't know if I needed to specify that or not. Also, we have finished the concept fully and today was a review day for the upcoming exam. – VLH Oct 01 '24 at 00:44
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    @VLH I have retracted my close vote. Thank you for the clarification. – John Douma Oct 01 '24 at 02:16
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    Ok, I know that you have to cope with this kind of thing "in school", but, if you are actually interested in mathematics (as opposed to the school-subject version of it), please do believe that these are fake questions. They are not at all "applied"... any more than having one person taking a train south from NYC to Philadelphia, with a leaky conical bucket, while his sister is taking a train from... Yes, there is a test of fighting through the nonsense. Happily, in real life, the obstacles to understanding the mathematics of real situations is not as ridiculous as the contrived [contd] – paul garrett Oct 01 '24 at 05:13
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    ... in "popular" calculus texts. The pretense is to give "real" applications comprehensible to students who do not know any science, finance, statistics... That's really ridiculous, because calculus was/is a wonderful solution exactly to issues in those fields. This disconnect has always disappointed me, and more so as the years go by... Calculus is mostly used as gate-keeping, in which case it apparently doesn't matter how ridiculous it may be. – paul garrett Oct 01 '24 at 05:17
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    Maybe just ask your teacher what her source for questions is, and she might give you an answer. – Cornman Oct 01 '24 at 08:58
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    If you want more questions like those in your example, you might take a look at a calculus-based physics text. Regarding scoring 50% on a test: some teachers (maybe yours?) have the opinion that a test is as much of a teaching tool as a measuring tool. If many of the students are scoring 100%, then those students have no room to stretch themselves. – spuck Oct 01 '24 at 19:35
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    The first question has been taken directly from Stewart's Calculus (section Related Rates). This is an average problem there, so you can find many problems of similar difficulty in that book. The book also contains harder problems at the end of each chapter ("Problem +"), some of them being quite hard. – Taladris Oct 02 '24 at 07:41
  • For what you're looking at, maybe start by working through Calculus in the Shaum's Outlines series. – A rural reader Oct 03 '24 at 22:07

4 Answers4

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Posted as an answer because this is way too long to be a comment, but isn't really a meaningful/full/thorough answer to the question posed in my opinion since I focus on a couple of peripheral issues more than anything.

Honestly, those problems look like fairly basic questions from my university's calculus textbook of choice:

  • Name: University Calculus: Early Transcendentals
  • Amazon: 4th Edition Link (though honestly you can probably find a PDF online of an outdated edition)
  • Authors: Joel Hass / Christopher Heil / Maurice Weir / Przemyslaw Bogacki / George Thomas
  • Publisher: Pearson
  • ISBN-10: 0134995546 (for the 4th edition)
  • ISBN-13: 978-0134995540 (for the 4th edition)

That is to say, probably any standard undergraduate calculus textbook has what you're looking for in spades. Try to focus on textbooks geared towards Calculus I-III students.[1] I think a big problem your search resulted in was textbooks like Apostol & Spivak, which from what I recall are more geared towards a real analysis class.[2]

Aside[1]: I say to focus on textbooks for Calculus I-III students because, from what I recall from teaching those courses at my university and what I recall of the AP Calculus AB/BC curricula, AP Calculus BC is mostly a hasty blend of Calculus I/II topics.

Aside[2]: "Real analysis" is the field calculus lives in, but in university classes it usually corresponds to courses in which you go through calculus in full rigor: $\varepsilon$-$\delta$ definitions, Darboux integrals, Fubini's theorem and when it can apply, etc., whereas most basic calculus classes will handwave those details. In short then, I think you were looking at textbooks way more advanced than what you need.

I notice you mention the grades being $50$ or so on average. Honestly ... that's about where some of my averages were for some units of Calculus I & II when I taught them at my university. I don't think this is a problem of question choice or test design. (In fact, I would argue your instructor is going easy on you by having multiple-choice questions. I never use those.) Rather, I think these are just artifacts of the difficulty of the material and the speed of the course, along with lack of preparedness (but rarely does anyone come into calculus truly "prepared").

Calculus exposes you to perhaps a new way of thinking for the first time. Conceptually, it's difficult to deal with these infinite and (loosely!) infinitesimal quantities. Blind algorithms and symbol manipulation can't save you like they might have in algebra classes. It also is where you begin to broach on a mathematics far more general than what you learned over the past ten years, which makes whole new types of problems accessible to you, provided you are able to model and visualize very well.

But the grade school system in the US (where you presumably live since you mention AP) is just very poorly fit for those sorts of things, so most students going through Calculus I have to struggle and adapt a lot. Calculus II & III especially are often considered weed-out courses for STEM majors.

But I'm not here to soapbox. I guess the point I'm trying to make is that $50$ as an average for this class is frankly not that unusual. It's hard material. The key things you'll need to survive are the intuitions for each of the objects you're studying and the ability to translate real-world phenomena into basic mathematical equations (e.g. "rate of change" means a derivative).

Just especially keep in mind your focus should be on how to solve problems and where those solutions come from - and not trying to handle specific problem types that while superficially different are actually testing the same concepts. That's definitely a trap I've seen students fall into before.

PrincessEev
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    Honestly, those problems look like fairly basic questions --- That's what I was thinking, more specifically these are straightforward related rates problems. @VLH: Probably any difficulty in them will be translating to appropriate equations for differentiation with respect to time. Some nice problems can be found in William Burton Ford's A First Course in the Differential and Integral Calculus (1937 revised edition) -- some related rates problems are on pp. 31-32 and a lot of max/min problems are on pp. 73-79. – Dave L. Renfro Sep 30 '24 at 12:28
  • Thank you Dave. Also thank you PrincessEev. I really appreciate both books. I will check them out. – VLH Oct 01 '24 at 00:32
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    You seem to relate the average score on the exams to the difficulty of the material. These two concepts are completely unrelated. For a given class and a given set of students I could design a final exam with an average score anywhere in the interval say 25% to 90%. All of them would test exclusively the material tought in the course. Of course students should be aware of what the teacher envisions and different approaches are for different teaching goals but you can't conclude anything about the course from 50% average without further context. – quarague Oct 01 '24 at 06:48
  • @VLH: I really appreciate both books. --- Ford's book is old, so it's not going to drill on a lot of things the AP tests focus on (definite integrals as accumulation functions -- at least not the terminology, use of calculators, attempt to avoid physics and engineering applications in favor of social science applications), but the applied problems -- especially the max/min problems -- in Ford's book range from straightforward to challenging, and pretty much all of them have their origins in actual problems people needed to solve (sometimes as a "toy model" for a more complicated situation). – Dave L. Renfro Oct 01 '24 at 11:59
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Based on the two example questions provided, I would suggest (though I could be incorrect) that it may be that you find such questions difficult because they are applied questions. That is, they are questions where you are given information concerning the set-up, and from this it is up to you to put the information and context together to find where, and how, to apply the calculus content. It isn’t necessarily that such questions are extremely tricky calculus questions, but rather they are tricky because they are real-world application questions.

If you feel that this is an area which you find tricky, there are a few resources which may help.

  1. Everything for Applied Calculus Webpage by Stefan Waner and Steven R. Costenoble. I suggest this because it is full of calculus concepts applied to real world examples. There are visualisations and explanations, but also many, many exercises to try. Some exercises involve applying calculus to radioactive decay, and even COVID data. It’s a really good resource to practice these sort of questions.
  2. Schaum’s Outline 3000 Solved Problems in Calculus. This book contains tonnes of questions, some of which are applied. They are of varying difficulty, so search for some which you find tricky.
  3. Introduction to Calculus by Courant and John. This textbook covers all of the essential topics, and it designed with the aim for application of calculus, so you should find the exercises useful.
  4. Calculus with Applications by Lax and Terrell. Lax is a well known mathematician, and this book is full of useful information and tricky exercises.

I hope that you find some of the suggestions on this list useful for your studies.

Topology Titan
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    Ah thank you a lot. Yeah, they aren't hard but because our teacher has stressed about this test being hard, I wanted to do as much practice as possible. – VLH Oct 01 '24 at 00:35
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Spivak focuses on real analysis, which is different from the algebra/numerical orientation that Calculus BC has. Instead, pick the hardest problems from college calculus books:

  • Thomas is the easiest textbook covering a comprehensive engineering curriculum. (It's my favourite).
  • Stewart is more difficult than Thomas.
  • Edwards/Penny is the hardest of the three

Doing college level worksheets for a Calculus II course will be useful (you need access to those notes).

Online links to Notes/Textbooks:

  • Paul's notes includes both theory and practice questions with solutions
  • Aziz's Notes has good number of questions with solutions.

College competitions are not organized by topic, but they have solutions:

Russian books have a different feel (and some difficult questions):

Starlight
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  • Yup, we use the Stewart and Thomas books in school. The other links are extremely useful in prep as they are similar questions as for my teacher to put on the test. – VLH Oct 01 '24 at 00:34
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I personally like the CLP textbooks which are used at the University of British Columbia

https://personal.math.ubc.ca/~CLP/CLP1/

They have exercises of varying difficulties and solutions to those exercises and are completely free.

Severin Schraven
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  • I looked at the CLP books, and they are really good. Do you know any resources like this for mathematical statistics? – Starlight Sep 30 '24 at 17:39
  • @Starlight Unfortunately, I know nothing about mathematical statistics. Maybe you could ask a question and people more proficient than me could advise you. – Severin Schraven Sep 30 '24 at 21:19
  • Thank you for the link, these are extremely nice problems and I'm interested in these. Thank you – VLH Oct 01 '24 at 00:32
  • @VLH You are very welcome. I used to teach Calculus at UBC using that material. I find the textbook to be very clear and even entertaining at times (one encounters occassionaly funny footnotes). – Severin Schraven Oct 01 '24 at 10:14
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    one encounters occassionaly funny footnotes --- This led me to look at some footnotes. I only looked at about $10$ to $12$ footnotes, but among those I did find one that qualifies: Footnote 33 on p. 247. This reminds me of a 1947 College Algebra text by Moses Richardson (my interest in this book is discussed in a comment to this Mathematics Educators SE answer). See the footnote on p. 111 of Richardson's book, which I also mention in this 10 Sep 1999 sci.math post. – Dave L. Renfro Oct 01 '24 at 12:22
  • @DaveL.Renfro Thanks for sharing Richardson's footnote! I really like footnotes being used to smuggle some humorous comment into a math textbook. – Severin Schraven Oct 01 '24 at 13:41
  • If anyone wonders why I began that 10 Sep 1999 sci.math post with I don't know if this has anything to do with "Math in the Usual Words"--oops, I mean "Common Tongue", but I'm pretty sure it doesn't matter., see the beginning post in that thread. That particular person had been advocating for his "better math terminology for children" back then. I suspect I had (back then) recently come across (again; I've known about it since the early 1970s) that footnote by Richardson and wanted an excuse to post it for the amusement of others, and that sci.math thread seemed about as appropriate as any. – Dave L. Renfro Oct 01 '24 at 13:59
  • That particular person had been advocating for his "better math terminology for children" back then. --- It seems this went on long after 1999: SpeakEasy Math: Part I & SpeakEasy Math: Part II & SpeakEasy Math: Part III & SpeakEasy Math: Part IV & SpeakEasy Math: Experimental Word List. – Dave L. Renfro Oct 01 '24 at 15:15