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Self Learning Calculus From the Beginning

  1. Feb 16, 2014 #1
    Math Topics to Self Study

    Any recommendations for a 9th grader?
    I have taken algebra 1 and 2 and am currently taking geometry.

    I find that math class is sometimes very slow. I understand that some students need extra help, but for me it's a bit slow.

    So to take initiative, I want to learn some extra math during my free time. Any recommendations on what math topics to study are appreciated.

    I won't mind buying some books, but some good internet sources would be appreciated.

    Last edited: Feb 16, 2014
  2. jcsd
  3. Feb 16, 2014 #2
    Calculus is quite a bit away. I suggest you learn Algebra 3/Precalculus through Art of Problem Solving. I have used it and it's a great resource for making you actually understand the material rather then memorizing it. It has many hard problems and is ideal for learning problem solving skills and participating in math contests such as AMC. A set of books I used were from http://www.ncert.nic.in/ncerts/textbook/textbook.htm Those books are available for free online and are the most-used books in the Indian curricula. They are quite nice, from what I have used. You would probably need only 11-12 grades of those books (they cover calculus as well). Another book you can try is "Basic Mathematics" by Serge Lang.
  4. Feb 16, 2014 #3
    Yes, you do need to learn pre-calculus, but if you want to learn the bare minimum, I feel like the trigonometric identities and the unit circle would suffice. Many of the key concepts of calculus (e.g., derivatives) require some advanced thinking, but not necessarily advanced knowledge.
  5. Feb 16, 2014 #4

    Thanks, I will definetly look into those.

    I want to change my question to what math topics I can study during my free time, because sometimes in school the math seems very slow. The teacher reviews the same thing for days, and I understand that some students need more help, but it is a bit slow ofr me.
  6. Feb 16, 2014 #5
    You can just use the Edit button in your first post it you want to change it. I agree, much of math and science taught in regular schools (even in the gifted programs where I am) is quite slow and frankly boring as there is a lot of memorization. In that case, I highly recommend AoPS as there you are informed of understanding the topics and learn about topics beyond the regular curriculum. There are lots of hard problems (many of which you won't get correct the first time).

    As for topics specifically, there are several:
    Discrete maths (counting/combinatorics, number theory, probability)
    Geometry (beyond what you studied in school)
    And then Calculus.
  7. Feb 16, 2014 #6
    Cohen: pre calculus a problem solving approach (under 10 I think barnes n noble)

    Lang (forget the name but does algwbra/trig)

    Understand those then get a copy of swokoski caluclus and thomas calculus.

    Swokoski gives excellent examples as to what is happening. The book is also easy to read. The problem with swokoski is that the exercisea are rather easy. Thomas has great exwrcises or even a copy of stewart calculus. (Find cheaper edition). Do not skip learning algebra and trigonometry.
  8. Feb 17, 2014 #7


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    Art of Problem Solving books were created for those who love math and are good at it. Just get them and go through them. Do all the problems, especially the challenge problems. When you're done with the calculus book, then it's time to go on to other things.

    Personally I would suggest the Number Theory and Counting and Probability books to get a bit outside the usual curriculum. Richard Rusczyk's The Calculus Trap was written for you; read it.
  9. Feb 17, 2014 #8
    :D So I read the article and smiled throughout. Definetly gave me a new perspective.

    Do you recommend I get all, or skip some (e.g algebra). If you recommend skipping some, which should I skip and which should I get.
  10. Feb 17, 2014 #9


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    Rusczyk's a good guy.

    I recommend that you ask these questions on their forums. The people there will have specific and useful advice. Some people like the classes. Some like the books alone. You are the one who will be best able to judge what will work for you.

    See if you can find a math teacher at your school who will advocate for you. Suggest that you ditch the regular school math classes and replace them with something more useful from AoPS. Many people have gotten that sort of thing to work. One nice thing about that approach is that they'll likely cover the cost of books and/or classes.

    As to specifically which books make sense, take a look at their diagnostic tests (pre-test and post-test) and let them help you decide. Me, I'd try going through all the challenge problems in the books covering the subjects I think I know. There might be some interesting surprises there.

    Remember, if you want people to give you the good stuff you have to ask. Then when they say no you have to keep looking until you find the people who say yes. It's not easy, but it can be done.
  11. Feb 17, 2014 #10
    What do you mean by this? How will the teacher pay for the books? By classes, do you mean I actually pay for one of those online classes from AoPS?
  12. Feb 17, 2014 #11


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    I mean that there's always money to pay for "official" activities, so if you can find a way to be officially sanctioned then you can get support. Perhaps there are means to be officially designated as independent study. Anyway, I don't know anything about your specific schooling situation. I'm just letting you know that if you figure out what you want and ask around you might find that all sorts of things are possible.

    Regardless, you're taking geometry so get the geometry book and go through that at the same time. I'm sure you'll find some of the problems quite challenging.
  13. Feb 17, 2014 #12
    Hi ecoo!

    Grab a textbook and solutions manuel and start learning! Art of problem solving is an excellent
    website with good math forums. Try math competitions to increase your skill in geometry.
    If you want to take Calculus, learn precalculus. I recommend watching youtube videos, and reading any textbook. Many textbooks are published free online by generous authors.
    Just don't procrastinate--only working will help you succeed!!

    Good luck in your future endeavors!

    Credentials: Took AP calculus BC (last year) in 9th grade. Self learned physics honors, trig, precalculus, algebra II. :)
  14. Feb 18, 2014 #13
    I strongly suggest you spend the next year supplementing your math courses with a self-study of trigonometry. Buy this book and work through it until it is coldly understood. After that start tackling calculus.
  15. Feb 19, 2014 #14
    Just my opinion, but I have gone through parts of the book mentioned by Quadratic and found it rather childish and "dumbed-down". But then again, preview the book and if it suits you then great.
  16. Feb 20, 2014 #15
    Anything specific? I haven't used the book in 10 years but I remember enjoying it and it was recommended by the most respected professor in my math department.
  17. Feb 20, 2014 #16
    The book introduces trigonometry from the beginning? I have no knowledge in trigonometry except the very basic like sin, cosin, etc.
  18. Feb 21, 2014 #17
    Yes, it is very much an introductory trig text.
  19. Feb 21, 2014 #18
    I don't remember much as of now, except I disliked it quite a bit. I will try to get hold of a copy though...

    AoPS books "Intro to Geometry", "Intermediate Algebra" and "Precalculus" all cover trigonometry from more or less the very basics.
  20. Feb 22, 2014 #19
    What do you think of the Physics Classroom?


    I'm getting the trigonometry book recommended by Quadratic so let's hope things work out :D
  21. Feb 23, 2014 #20
    Gwt the cohen precalculus book better. It actually has rigor..
  22. Oct 11, 2014 #21
    Any Precalculus textbook will have what you need. Your teacher's probably would not mind giving you suggestions, unless they are stuck up. I would get an older edition of a textbook and start doing problems from there. Geometry is fine and all, but the truth is that you will see the concepts some of the time should you decide to go into higher level maths. Much of geometry that is needed are your basic area and volume formulas. Some proofs in calculus use geometric proofs to prove a result such as the limit as h-->0 of Sin(h)/h=1.
  23. Oct 11, 2014 #22
    I would recommend Lang's Basic Mathematics for a more thorough treatment of precalculus topics, perhaps supplemented by KhanAcademy videos if you find them helpful.

    After you have a better understanding of precalculus you can then try a book on calculus. Personally, I watched KhanAcademy videos on the subject and then went straight for Calculus by Apostol which provided a much more rigorous introduction. However, it would probably be easier for you to get a more relaxed book as your first. Regardless, if you do find that you continue to enjoy the topics, I'd recommend you to study the theory as well as the methods.
  24. Oct 11, 2014 #23


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    Last edited by a moderator: May 7, 2017
  25. Oct 11, 2014 #24
    First of all, good for you for taking the initiative. I wish I had done that at your age. As it is now, I am 31 and going back to school (since I DIDNT do what you are doing and instead let my boredom drive me away from school). A year ago I was in Pre-Calc 2 and now I am in multivariate calculus. So, I have a tad bit of insight as to what you are going through now and what you will go through as you complete your education.

    You will need to start with pre-calculus topics (trig, functions, etc). Even if your "normal" math is slow it is really important not to skip any topics in the learning process as math builds on itself the whole way through. Check out the textbook section on Amazon, you should be able to find some textbooks on there that could help. You could also visit a local college or community college and check out their student store to see what texts they use for those classes. That would be a good first step. Sadly, my college pre-calc text was garbage... even the teachers didn't use it. So I do not have a recommendation for you regarding that.

    Keep reaching out, though, man! It is so awesome to hear about young, motivated people like yourself. Best of luck to you!
  26. Oct 11, 2014 #25


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    For motivated high school math students,
    I understand your situation. First off, I suggest you derive formulas in integration table.

    Another to deal with the situation is to participate in Olympiad math competition or other competition like Putnam. You can learn a lot even if you lose. Some questions are challenging regardless of whether you have knowledge or not and they can be solved even if you do not have the knowledge.

    If you want to learn more about calculus, you can read Thomas's calculus.

    Pros: Easy to read. Good for obtaining knowledge.
    Cons: Problems seem difficult if you do not have knowledge, but become easy after you have the knowledge. The best way to deal with this to to solve Olympiad calculus problems.
    Last edited by a moderator: May 7, 2017
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