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Math Help - Gradient of an Unknown Function!!!

  1. #1
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    Arrow Gradient of an Unknown Function!!!

    Hello, The latest research of mine concerning the derivative of a function without using diffrentiation and even without having the knowledge of what the equation of the funtion is, is attached to the post, please send me your comments on this latest document of mine.
    With regards.
    Mubashir
    Attached Files Attached Files
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  2. #2
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    Quote Originally Posted by mubashirmansoor
    Hello, The latest research of mine concerning the derivative of a function without using diffrentiation and even without having the knowledge of what the equation of the funtion is, is attached to the post, please send me your comments on this latest document of mine.
    With regards.
    Mubashir
    Over complicated compared to classical numerical methods.

    RonL
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  3. #3
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    Arrow Complicated But....

    Well Captain, It is complicated compared with diffrentiation But the good thing about it is that we dont need to know the function with which the terms are related, The main Uses of this formula is to find the accleration of the heavenly bodies etc...
    Now what did you really mean by numerical methods? Do we needto know the polynomial for finding the gradient????

    Thanks for Your Attention.
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  4. #4
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    Quote Originally Posted by mubashirmansoor
    Well Captain, It is complicated compared with diffrentiation But the good thing about it is that we dont need to know the function with which the terms are related, The main Uses of this formula is to find the accleration of the heavenly bodies etc...
    Now what did you really mean by numerical methods? Do we needto know the polynomial for finding the gradient????

    Thanks for Your Attention.
    No complicated compared to

    <br />
f'(x)\approx \frac{f(x+\Delta)-f(x-\Delta)}{2\Delta}<br />

    RonL
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  5. #5
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    Editted

    Well yes but this formula is applicable to all curves while there are diffrent types of the diffrentiated formula you are talking about for diffrent graphs...
    For clarification and more accuracy I've done some changes written in the attached file.
    The ultimate formula is;

    dy/dx =10^10*tan (½(tan-1((f(x+1)–f(x))/10^10)+tan-1((f(x)–f(x-1))/10^10)))


    Regards
    Mubashir
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  6. #6
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    Quote Originally Posted by mubashirmansoor
    Well yes but this formula is applicable to all curves while there are diffrent types of the diffrentiated formula you are talking about for diffrent graphs...
    For clarification and more accuracy I've done some changes written in the attached file.
    The ultimate formula is;

    dy/dx =10^10*tan (½(tan-1((f(x+1)–f(x))/10^10)+tan-1((f(x)–f(x-1))/10^10)))


    Regards
    Mubashir
    For your example, what is the numerical value that your method gives for
    the estimate of the derivativeof x^2 at x=2?

    RonL
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  7. #7
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    Talking Exact....

    The answer is "4",
    It's answers are abseloutly the same as diffrentiation technique,
    I've tried it in all types of graphs & works for all.
    Do you still call it Useless???
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  8. #8
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    Quote Originally Posted by mubashirmansoor
    The answer is "4",
    It's answers are abseloutly the same as diffrentiation technique,
    I've tried it in all types of graphs & works for all.
    Do you still call it Useless???
    No it is exact for 2-nd degree curves only. Also I beleive it is identical in
    practice with what I gave above.

    Try it for x^3 at x=2.

    Asking for what numerical value your method gives for your example is
    to indicate to you that the calculation is not simple. For instance how
    many transendental function evaluations are involved? (transendental
    functions include the trig functions and their inverses, as well as logs
    and exponential functions).

    RonL
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  9. #9
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    What's going on???

    Sorry for the delay in my response,

    You have written that the answers are not accurate in the formula provided, There is something happening which I don’t understand, (I’m a student of O’levels so I’d be glad if you help me figure out the problem) if we use the formula with small gaps between the terms, it works very accurately & there is just .00000001 difference in the answers compared to differentiation. I mean the formula given below;

    dy/dx = tan ( 0.5 ( tan-1 (( f(x+0.1) – f(x) ) / 0.01) + tan-1 (( f(x) – f(x-0.01) / 0.01)))

    This equation is even applicable for x^20 or higher degree equations… To get rid of inserting these extremely nearby points, I thought to decrease the size of the curve (not the dimensions) by 10^10, in this way 1 unit would turn to 10 billionth of a unit and even the points with huge gaps between each other would become extremely near. So when the calculation was taking place the curve was small enough to get good approximations, and after the evaluation the curve was brought to its original size hence multiplied by 10^10.
    I don’t feel that the technique has a problem but its answers are absolutely the same as differentiation for degree 2 equations but as the degrees increase the answers become larger and larger by about 1 to 3 units for the degrees 3 and 13/15 respectively.

    We know that differentiation is itself not accurate and that we say “delta x” approaches zero which is impossible, and the real answers should be a little bit more than that of differentiation. What’s the problem with the last part of my technique???

    I’ll be really thankful if you help me figure out what’s going on…
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  10. #10
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    Quote Originally Posted by mubashirmansoor

    We know that differentiation is itself not accurate and that we say “delta x” approaches zero which is impossible, and the real answers should be a little bit more than that of differentiation. What’s the problem with the last part of my technique???
    Differentiation is exact, it gives the exact value of the slope at a point on a curve.

    You will have to learn some calculus before you will understand what
    is going on in detail.

    A good (cheap-ish) book to learn calculus from is "Calculus: An Intuitive and
    Physical Approach" by Morris Kline (see here )

    Then a good book to learn some numerical analysis from is "Numerical
    Methods that Work" by Forman Acton (see here )

    RonL
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    If diffrentiation is exact then why do we say dy/dx is aproximately equal to delta y/delta x???
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  12. #12
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    Quote Originally Posted by mubashirmansoor
    If diffrentiation is exact then why do we say dy/dx is aproximately equal to delta y/delta x???
    Because \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}. They are not the same.

    -Dan
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  13. #13
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    Quote Originally Posted by mubashirmansoor
    If diffrentiation is exact then why do we say dy/dx is aproximately equal to delta y/delta x???
    \Delta y / \Delta x is used as a stage in the development of the derivative.

    The derivative is the slope of a curve at a point, we approximate this by
    calculating the slope of a secant of the curve near the point of interest.
    Then by letting the points at which the secant meets the curve near the
    point both approach the point of interest in the limit obtain the exact slope
    of the curve at the point if this should happen to exist. The slope of the
    secant is \Delta y / \Delta x .

    If you consult a reference on calculus as I have suggested earlier you will
    learn what is going on better than you will form a discussion here.

    RonL
    Last edited by CaptainBlack; July 30th 2006 at 07:56 AM.
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  14. #14
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    Arrow

    The link provided is to be used for having a look at the detailed discusion done on this topic;

    http://www.physicsforums.com/showthr...65#post1044865
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  15. #15
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    Quote Originally Posted by mubashirmansoor
    The link provided is to be used for having a look at the detailed discusion done on this topic;

    http://www.physicsforums.com/showthr...65#post1044865
    I'm not sure what your point is. They seem to be having the same issues with your programme as we are...

    -Dan
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