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mahmit2012

  • 3 years ago

Q3 If f(x) is a differentiable at x=1, find a,b. f(x)=ArcCos(sqrt(x))/sqrt(1-x) for 0<=x<1 ax+b for x=>1

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  1. mahmit2012
    • 3 years ago
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    |dw:1337454011771:dw|

  2. mahmit2012
    • 3 years ago
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    It is not easy to find out.!

  3. experimentX
    • 3 years ago
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    a = http://www.wolframalpha.com/input/?i=d%2Fdx+Cos^-1%28sqrt%28x%29%29%2Fsqrt%281-x%29+%2C+x%3D1

  4. mahmit2012
    • 3 years ago
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    experimentX can you give me a way to your answer?

  5. experimentX
    • 3 years ago
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    the left hand derivative must be equal to right hand derivative at x=1

  6. experimentX
    • 3 years ago
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    for b, the function must be continuous ... so equate left hand limit to right hand limit at x=1

  7. niravshah08
    • 3 years ago
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    the right hand derivative is just "a" and left hand derivative can be found using quotient rule

  8. mahmit2012
    • 3 years ago
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    I know but how can you reach to -1/6 ?

  9. experimentX
    • 3 years ago
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    wolfram is pretty handy ... :D

  10. mahmit2012
    • 3 years ago
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    yeap,but I didn't find any ways in there.

  11. niravshah08
    • 3 years ago
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    |dw:1337455871084:dw|

  12. experimentX
    • 3 years ago
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    \[ a = \left ( \frac{d}{dx} \left ( \frac{\cos^{-1}(\sqrt{x})}{\sqrt{1-x}} \right ) \right )_{x=1} \]

  13. mahmit2012
    • 3 years ago
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    Ok find it out?it gets so complexity.Doesn't it?

  14. experimentX
    • 3 years ago
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    \[ \lim_{x->1}ax+b = \lim_{x->1}\frac{\cos^{-1}(\sqrt{x})}{\sqrt{1-x}}\]

  15. experimentX
    • 3 years ago
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    i prefer to find out by using wolfram first .. then use hand to verity it.

  16. experimentX
    • 3 years ago
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    http://www.wolframalpha.com/input/?i=lim+x-%3E1+Cos^-1%28sqrt%28x%29%29%2Fsqrt%281-x%29

  17. mahmit2012
    • 3 years ago
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    in right hand side the limit is going to be 1 .then a+b=1 but give me a proof for a=-1/6

  18. asnaseer
    • 3 years ago
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    you can find the proof using wolfram - click show steps to see how it is done: http://www.wolframalpha.com/input/?i=d%2Fdx+Cos^-1%28sqrt%28x%29%29%2Fsqrt%281-x%29+

  19. experimentX
    • 3 years ago
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    \[ \left ( \frac{d}{dx}(ax+b) \right )_{x=1} = a = \left ( \frac{d}{dx} \left ( \frac{\cos^{-1}(\sqrt{x})}{\sqrt{1-x}} \right ) \right )_{x=1}\] http://www.wolframalpha.com/input/?i=d%28Cos^-1%28sqrt%28x%29%29%2Fsqrt%281-x%29%29%2Fdx+%2C+x%3D1

  20. mahmit2012
    • 3 years ago
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    please show me -1/6 . I didn't find it in wolfram...!Did you see any?

  21. asnaseer
    • 3 years ago
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    The link I pasted above shows how to find the derivative. Then just plug x=1 into this to get your answer.

  22. asnaseer
    • 3 years ago
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    sorry - I meant take the limit as x tends to 1

  23. mahmit2012
    • 3 years ago
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    Ok copy past the answer and find it out for x=1.

  24. asnaseer
    • 3 years ago
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    you can find the derivation of this limit here: http://www.wolframalpha.com/input/?i=limit+%281%2F%28%28-1+%2B+x%29+Sqrt [x]%29+%2B+ArcCos[Sqrt[x]]%2F%281+-+x%29^%283%2F2%29%29%2F2+as+x+tends+to+1 again, just click show steps to see it

  25. mahmit2012
    • 3 years ago
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    That was wrong site you posted.

  26. experimentX
    • 3 years ago
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    Jeez what am i getting instead of indeterminate http://www.wolframalpha.com/input/?i=1%2F2+%281%2F%28%28-1%2Bx%29+sqrt%28x%29%29%2B%28cos^%28-1%29%28sqrt%28x%29%29%29%2F%281-x%29^%283%2F2%29%29+where+x%3D1

  27. asnaseer
    • 3 years ago
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    experimentX: thats interesting - looks like wolfram is automagically using the limit as x tends to 1 to calculate this :)

  28. mahmit2012
    • 3 years ago
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    I have never found it yet.so could you please copy past the way (lim x->1)

  29. mahmit2012
    • 3 years ago
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    only i looked the answer -1/6 and derivation of function.

  30. experimentX
    • 3 years ago
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    it seems the case http://www.wolframalpha.com/input/?i=lim+x-%3E1++1%2F2%281%2F%28%28-1%2Bx%29+sqrt%28x%29%29%2B%28cos^%28-1%29%28sqrt%28x%29%29%29%2F%281-x%29^%283%2F2%29%29 but limit of derivative ... sounds fisht @mahmit2012 click Show Steps .. below "=" icon

  31. mahmit2012
    • 3 years ago
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    there is no calculation i chose step but there is only answer.

  32. mahmit2012
    • 3 years ago
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    I accepted the conclusion but show me how?

  33. experimentX
    • 3 years ago
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    Oh ... this is a bit fishy ... the slope of f(x) will be a at x=1 (left hand slope as well as slope) While on the other side, the right hand slope tends to this value 'a' ... i think limiting the value of the derivative will just do fine. You can find the derivative of arccos... right??

  34. experimentX
    • 3 years ago
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    \[ \frac{d}{dx} \left ( \frac{\cos^{-1}(\sqrt{x})}{\sqrt{1-x}} \right ) \] I mean find this first.

  35. mahmit2012
    • 3 years ago
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    ok find out this limit|dw:1337458049743:dw|

  36. experimentX
    • 3 years ago
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    yeah, that would be -1/6

  37. experimentX
    • 3 years ago
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    using L'hospital's rule

  38. mahmit2012
    • 3 years ago
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    ok show me.i really appreciate

  39. experimentX
    • 3 years ago
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    are you having trouble with limits?

  40. experimentX
    • 3 years ago
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    which part .. first or last??

  41. mahmit2012
    • 3 years ago
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    no but this is not easy to find out.check it out.

  42. experimentX
    • 3 years ago
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    Ah sorry ... we need only right hand limit

  43. mahmit2012
    • 3 years ago
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    both of them to each other you should take limit,because each goes to infinity !

  44. experimentX
    • 3 years ago
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    ah .. yes ... looks like i'm nuts

  45. experimentX
    • 3 years ago
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    It seems to be the case of \( \infty - \infty \) so we can apply L"hospital's rule \[ \frac{1}{(-1+x) \sqrt x} = -\frac{1}{(1-x) \sqrt x}\] make them a fraction ... and use L'hospital's rule

  46. mahmit2012
    • 3 years ago
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    im waiting yet.

  47. experimentX
    • 3 years ago
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    \[ \frac{\sqrt x \cos^{-1}(\sqrt x) - \sqrt{1-x} }{(1-x)^{3/2} \sqrt x }\] Use L'hospital rule ... Since now both numerators are denominators are zero ... you can use L'Hospital rule http://www.wolframalpha.com/input/?i=lim+x-%3E1++%28sqrt%281-x%29++-+sqrt%28x%29+cos^-1%28sqrt%28x%29%29%29

  48. mahmit2012
    • 3 years ago
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    yes it is going to 0/0 so what is the rest?

  49. mahmit2012
    • 3 years ago
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    |dw:1337460455981:dw|

  50. mahmit2012
    • 3 years ago
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    |dw:1337460652950:dw|

  51. mahmit2012
    • 3 years ago
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    |dw:1337460769484:dw|

  52. experimentX
    • 3 years ago
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    Ahh ... looks like you got it!! good night!!

  53. mahmit2012
    • 3 years ago
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    |dw:1337460890186:dw|

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