integral of 1/x

This is a discussion on integral of 1/x within the A Brief History of Cprogramming.com forums, part of the Community Boards category; trig identities go as such, remember that f' = [f(x+h) - f(x)]/h as h ->0 sin(x+h) = sin x cos ...

  1. #31
    'AlHamdulillah
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    trig identities go as such,

    remember that f' = [f(x+h) - f(x)]/h as h ->0
    sin(x+h) = sin x cos h + cos x sin h

    therefore:

    f' = lim (sin x cos h + cos x sin h - sin x )/h as h ->0

    using the properties of limits:

    f' = lim (sin x cos h - sin x)/h + lim cos x sin h/h as h->0

    factor sin x:

    lim sin x(cos h - 1)/h + lim cos x sin h/h as h->0

    based on the fact that sin x and cos x have nothing to do with h(i.e. they are not involved in the limit):

    sin x lim (cos h-1)/h + cos x lim sin h/h as h->0

    computing the limit of (cos h - 1)/h and sin h/h as h -> 0 we have

    sin x * 0 + cos x * 1 = cos x

    EDIT: I know you can find out for yourself, just wanted to save you some trouble :P

    and about chain rule, you are right, it is pretty intuitive:

    if y = f(x)
    and z = g(x)
    and h(x) = f(g(x))

    dy/dx = dy/dz * dz/dt

    It is just that doing it this way tends to make people think of leibnix notation as a fraction, which in some ways it is, but most of the time it is better to think of it as just a notation.
    Last edited by EvBladeRunnervE; 03-17-2004 at 08:02 PM.

  2. #32
    Toaster Zach L.'s Avatar
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    Sandwich rule?

  3. #33
    'AlHamdulillah
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    Sandwich rule
    Sandwiching is the following, say if I have 3 functions:

    f(g(h(x)))

    the derivative of this is : f'(g(h(x))) * g'(h(x)) * h'(x)

  4. #34
    Toaster Zach L.'s Avatar
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    Right... I can't say I had ever heard that term before.

  5. #35
    Registered User axon's Avatar
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    lmao...I've never heard of it either...made me hungry though

    some entropy with that sink? entropysink.com

    there are two cardinal sins from which all others spring: Impatience and Laziness. - franz kafka

  6. #36
    l'Anziano DavidP's Avatar
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    Sandwiching is the following, say if I have 3 functions:

    f(g(h(x)))

    the derivative of this is : f'(g(h(x))) * g'(h(x)) * h'(x)
    ive never heard that called the Sandwich rule...i usually here it called the Chain Rule.
    My Website

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  7. #37
    'AlHamdulillah
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    I know, but I was just making a guess at it, as I honestly dont know what it is either....

  8. #38
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    sandwich rule == sandwich theorem

    Maybe you've heard of it differently but it's where:

    sin(h) / h == 1
    lim h -> 0
    Last edited by Silvercord; 03-18-2004 at 09:45 AM.

  9. #39
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    Originally posted by Silvercord
    sandwich rule == sandwich theorem

    Maybe you've heard of it differently but it's where:

    sin(h) / h == 1
    lim h -> 0
    Is that what you call sandwich rule??

    I think i took it in Calculus but i ve never heard the word sandwich when i studied it

    to be frank man the law you ve stated is derived from L'Hopital rule because if you substitute both sin(h) and h with zeros you will get 0/0
    so in Hopital rule u differentiate both numerator and denominator by h so it will be cos(h)/1 where cos(0)=1
    thus sin(h)/h =1

    so it's L'hopital rule not sandwich one

  10. #40
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    no i re-read it in my book im right

  11. #41
    ¡Amo fútbol!
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    Originally posted by Silvercord
    no i re-read it in my book im right
    No you aren't. What book are you reading? The only "Sandwich Rule" I've heard of is used to describe limits of a function by "sandwiching" it between two functions.

    EDIT: The Squeezing (Sandwich) Theorem
    The Actual Sandwich Theorem

  12. #42
    Software Developer jverkoey's Avatar
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    i don't know too much about calc, but i found a site on this theorem, if it helps any...:

    http://www.sosmath.com/calculus/limc.../limcon03.html

  13. #43
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    No you aren't. What book are you reading? The only "Sandwich Rule" I've heard of is used to describe limits of a function by "sandwiching" it between two functions.
    well maybe it's not the only example of the sandwich/pinching theorem but i'm on page 127 of "Calculus, Graphical, Numerical, Algebraic" by "Finney, Thomas, Demana, Waits" and it uses sin(h) / h, and that's the only way I had ever seen it done until you posted your links.
    Last edited by Silvercord; 03-19-2004 at 08:43 AM.

  14. #44
    'AlHamdulillah
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    thanks jverkoey, that stuff is actually in my calculus book. wonder why I never realized it.

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