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  1. #1 Calculus 
    Forum Ph.D. Heinsbergrelatz's Avatar
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    what is the difference between Calculus (1) and Calculus (2)? i know one is one step ahead of another, but in terms of what topics differ? and also the age level??

    thank you


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  3. #2  
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    most likely, it will be calculus 1 will cover differentiation, rates of change, related rates, and the like, and calculus 2 should cover integration, Riemman sums, and things of that nature.


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  4. #3 Re: Calculus 
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    Quote Originally Posted by Heinsbergrelatz
    what is the difference between Calculus (1) and Calculus (2)? i know one is one step ahead of another, but in terms of what topics differ? and also the age level??

    thank you
    That depends entirely on how your school chooses to break up the calculus course. Calculus 1 and Calculus 2 are not standard terms and can vary between schools.

    Since the courses are usually taken one after the other, there should be no expectation as to age difference.

    It might help if you described what Calculus 1 and Calculus 2 mean to you -- perhaps in terms of the text being used or the topics listed in the syllabus (or table of contents).
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  5. #4  
    Forum Sophomore An inconvenient lie's Avatar
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    Hey im in the 8th grade gifted so algebra 2

    However though i want to conquer both calculus 1 and 2 over the summer before high school does any one know a good web site that will teach me for FREE?
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  6. #5  
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    Quote Originally Posted by An inconvenient lie
    Hey im in the 8th grade gifted so algebra 2

    However though i want to conquer both calculus 1 and 2 over the summer before high school does any one know a good web site that will teach me for FREE?
    You are being overoptimistic. It is extremely doubtful that you would learn calculus quite that quickly, particularly from a web site.

    Try reading a book and working with someone who konws calculus well.

    Generally you need to have a firm handle on trigonometry and what we used to call high school "advanced mathematics" (aka colloege algebra) before you tackle calculus. That is usally the class that follows algebra 2.
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  7. #6  
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    still any thing that could help :-D

    lol i just found this site and it is absolutely wonderful!
    Im a quick learner though and im trying hard to get prepared for the international baccalaureate program
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  8. #7  
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    Quote Originally Posted by An inconvenient lie
    still any thing that could help :-D

    lol i just found this site and it is absolutely wonderful!
    Im a quick learner though and im trying hard to get prepared for the international baccalaureate program
    Be careful.

    There are lots of texts and classes that teach students manipulation of symbols for differentiation and integration. The result is students who can evaluate expressions proficiently, but who don't really understand what derivatives and integrals really are. This is pretty common with high school calculus classes, unless the teacher is exceptional.

    It is the reason why I am very leery of high school calculus classes. In many cases it is better for the student to learn calculus at a university, from someone with a deep knowledge of the subject.

    Have you considered applying for any of the many summer camps that are held at various universities that have participation from professional mathematicians ?
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  9. #8  
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    Yes but they cost to much money.

    So for the most part i learn on my own time but i try to stay a few grade levels above the average American lol i think i might try that this year if i can save up some money from mowing lawns
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  10. #9  
    Forum Sophomore An inconvenient lie's Avatar
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    lmao i need to learn calulus so i know what the heck you guy's are talking about when the equations appear on this forum! Im just staring at them going hmm that sounds nice but in reality i have no idea what the equations represent!
    http://whatreallyhappened.com/WRHART...alwarming.html
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  11. #10  
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    You can probably find a calculus textbook at your local library. I do think it is possible to learn beginning calculus from a reasonably good textbook (and really, aren't all calculus textbooks interchangeable, with the exception of Spivak's book?).

    Be sure, however, not to rush the learning process. You can't learn all of calculus in one summer--at least not well. Make sure you can do the most difficult exercises at the end of each section before you move on to the next section. This will take some time, but it will be well worth the effort.
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  12. #11  
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    Quote Originally Posted by An inconvenient lie
    still any thing that could help :-D

    lol i just found this site and it is absolutely wonderful!
    Im a quick learner though and im trying hard to get prepared for the international baccalaureate program
    A friend of mine was in the International Baccalaureate program and I was requested by a teacher to join. I'd caution you to be certain you would like to join before actually going through with it: the work is exceptionally excessive and after the end of your four years you must take an IB test to recieve your diploma.

    About your actual question regarding calculus. You must first study trigonometry and logarithms (including natural logarithms). Your trigonometry course should cover the six fundamental trigonometric functions (that is, the sine, cosine, tangent, cosecant, secant, and cotangent) and their inverses (the arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent). Essentially, it seems to me what you're mising is a course in preliminary calculus. The IB program offers this course to students their freshman year--if you join the program, take it, unless you would like to study it over the summer, therefore preparing you for calculus your freshman year if you're a fast learner.

    Another course that the IB program offers is IB higher level math studies; this course will cover vector geometry, augmented matrices and quadratic transformations, mathematical logic (including truth tables), proof by mathematical induction, and some calculus material not covered in the traditional AP calculus course (such as the Taylor series and asymptotic continuity of hyperbolic curves). i would recommend again if you join the IB program to take this course your senior year.
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  13. #12  
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    Quote Originally Posted by salsaonline
    You can probably find a calculus textbook at your local library. I do think it is possible to learn beginning calculus from a reasonably good textbook (and really, aren't all calculus textbooks interchangeable, with the exception of Spivak's book?).
    Yes, and all are pretty much terrible, with the exception of Spivak's book.
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  14. #13  
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    This is all great advice!

    Now im going to make sure to keep this all in mind and maybe go check out this spivack's book immediately from the library!
    http://whatreallyhappened.com/WRHART...alwarming.html
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  15. #14  
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    This is what i think of Calculus(1)

    -Differentiation----------
    -Product Rule
    -Chain rule
    -Quotient Rule
    tangent to the curve-normal to the curve.
    Second derivative
    Trigonometry Differentiation- Sinx, cox, tanx, sec^2x, cosecx, sec, cotx etc....
    Logarithmic Differentiation- e^x, e^ax+b, ln(x),ln(ax+b)
    Rates of change

    -Integration---------

    Normal integration with simple function. (x)^{n}, and

    definite integrals
    integration-to find the area under a curve and two intersecting curves.
    Trigonometric Integration, logarithmic integration etc...


    Calculus(2)-----

    Differentiation
    -Going deeper in to the idea of Limits and functions.
    -reviewing the basic concepts of differentiation in Cal(1) with proofs.
    -concepts of the usage of differentiation in economic models- marginal cost, profit max. etc...
    -differentiating complex variables
    -Studying some curve properties
    -Rational functions
    -Motion in a straight line
    -Kinematics-Acceleration, velocity, the laws of motion
    -Implicit Differentiation
    -proving the trig. functions and their derivatives as well as the logarithmic functions.
    -Maxima and minima with Trig.
    -Derivative of the inverse trig function- arcsinx arccosx arctanx

    Integration-
    -all the stuff in Cal(1) +proving definite integrals on finding the area under a curve.
    -distance from velocity functions
    -Basic circular function integration
    -Volumes of revolution on both y-x axis.
    -integration by parts
    -integration with unusual substitution.
    -separable differentiation equations.
    -further integrals of the inverse Trigonometry functions.
    -functions which cannot be integrated.


    my year level at school just finished calculus(1), and apparently most students are finding it difficult. im not boasting, but i found Cal(1) easy, so around about 7 months ago i started my Big brother's Calculus book (2)- he is 2 years older than me. He is doing IB, and does pure mathematics higher. im working on his books now, as the books at my current level does not seem to enlighten me.

    so does the Cal(1) and Cal(2) i have stated above seem pretty close to the courses you took?
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  16. #15  
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    Quote Originally Posted by Heinsbergrelatz
    Th

    so does the Cal(1) and Cal(2) i have stated above seem pretty close to the courses you took?
    The two combined are about what I had as first-year calculus of one variable (but see the next paragraph). Standard stuff. This is what I would expect to find any standard calculus textbook. About the only difference among such courses is the emphasis on the underlying theory. I cannot get a sense for that from the list of topics alone, and it is as much a function of a particular instructor as of anything else.

    Missing from the list are sequences anb series which are usually tied to the understanding of limits. This is a serious omission if indeed it is not taught with significant emphasis. In fact, without a discussion of sequences and limits of sequences I would think that the treatment of limits in general is rather perfunctory and that is the key to all of calculus. This makes me wonder if the emphasis of the course is calculation and symbol manipulation without much real understanding. Is this omission an actual gap in the courses, or just a failure to include it on the list ?
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  17. #16  
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    The two combined are about what I had as first-year calculus of one variable (but see the next paragraph). Standard stuff. This is what I would expect to find any standard calculus textbook. About the only difference among such courses is the emphasis on the underlying theory. I cannot get a sense for that from the list of topics alone, and it is as much a function of a particular instructor as of anything else.

    Missing from the list are sequences anb series which are usually tied to the understanding of limits. This is a serious omission if indeed it is not taught with significant emphasis. In fact, without a discussion of sequences and limits of sequences I would think that the treatment of limits in general is rather perfunctory and that is the key to all of calculus. This makes me wonder if the emphasis of the course is calculation and symbol manipulation without much real understanding. Is this omission an actual gap in the courses, or just a failure to include it on the list ?
    the proofs and series like Taylor series, Riemann Sums etc... is in the book itself. and the whole sequence and series topic is just before the calculus topic. i believe, multi-variable will not collide within my age level, though there is nothing wrong with going ahead if you are capable of course.
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  18. #17  
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    Quote Originally Posted by Heinsbergrelatz
    [
    the proofs and series like Taylor series, Riemann Sums etc... is in the book itself. and the whole sequence and series topic is just before the calculus topic. i believe, multi-variable will not collide within my age level, though there is nothing wrong with going ahead if you are capable of course.
    From the way that you say this I infer that little emphasis is given to the underlying theory behind calculus. You give the impression that proofs, limits, series, etc. are a topic distinct from calculus. In truth they are the heart of the subject. If this is the case, then you are learning to do calculations, essentially by rote and at some point if you really need to understand the subject you will have to take another class that emphasizes the fundamental and the theory.
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  19. #18  
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    we never learned or actually required to learn the proofs according to our teachers at our current level of understanding.
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  20. #19  
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    Quote Originally Posted by Heinsbergrelatz
    we never learned or actually required to learn the proofs according to our teachers at our current level of understanding.
    In that case your classes are a bit different from those of my experience. We generally presented proofs of the major theorems to the students without expecting them to be able to reproduce them. But we did expect the students to understand clearly the statements of the theorems and have some idea why they are true. That would include being able to produce examples in which both the hypothesis and conclusions of the theorems did not hold.

    Based on what you have described, I would anticipate that there is a subsequent class that has somewhat greater emphasis on the theorems than what you have described here. That would, I think, result in a solid pedagogical approach to calculus. You learn it in steps and deeper understanding comes with each step. Repetitiion at that level can be beneficial.

    Unfortunately to really do the proofs required for calculus, you need the intermediate value theorem and that is dependent on the least upper bound property of the real numbers. It is fairly standard for the least upper bound property to not be treated in depth until after one has learned basic calculus, but if you can accept the intermediate value theorem "on faith" then one is in a position to understand the remainder of the proofs of the basic theorems of calculus.

    You would see calculus developed rigorously, from first priinciples in an introductory class with a title something like "introduction to real analysis'. [One needs to be a bit careful because the title "real analysis" is used to describe classes that range from the development of the real numbers and simple calculus starting with the Peano axioms, to advanced treatments of measure and integration and introductory functional analysis (say rigorous treatment of Fourier series and Fourier transforms on the real line).]
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  21. #20  
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    do you know any websites that would help me govern the fundamental proofs of calculus? of course i dont expect to understand the whole matter through some website..
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  22. #21  
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    Quote Originally Posted by Heinsbergrelatz
    do you know any websites that would help me govern the fundamental proofs of calculus? of course i dont expect to understand the whole matter through some website..
    I don't know of any such websites, nor have I looked.

    Why not look in a calculus text ? Spivak's calculus book would certainly do the trick and is probably the best one for this purpose.
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  23. #22  
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    oo yes i heard his name alot,

    BTW when were you introduced to the fundamental concepts of integration, where you prove how to get the area under a curve, not plainl;y solving it by calculation?

    i myself have thought about it, and i thought solving the questions without comprehending the meaning to the entire body of the integral seemed absurd. So i decided to study the proofs, and yes now i understand the concepts and how to proving the Integrals, but of course not entirely. I realized there were alot of dividing in to little fractions of geometrical shapes and polygons before approaching the entire thing.
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  24. #23  
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    Quote Originally Posted by Heinsbergrelatz
    oo yes i heard his name alot,

    BTW when were you introduced to the fundamental concepts of integration, where you prove how to get the area under a curve, not plainl;y solving it by calculation?

    i myself have thought about it, and i thought solving the questions without comprehending the meaning to the entire body of the integral seemed absurd. So i decided to study the proofs, and yes now i understand the concepts and how to proving the Integrals, but of course not entirely. I realized there were alot of dividing in to little fractions of geometrical shapes and polygons before approaching the entire thing.
    I saw the basic theory of Riemann integration in my first calculus course.
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  25. #24  
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    Ah, i see, well thank you for your help on the subject matter.
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