## mahmit2012 Group Title 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 2 years ago 2 years ago

1. mahmit2012 Group Title

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2. mahmit2012 Group Title

It is not easy to find out.!

3. experimentX Group Title
4. mahmit2012 Group Title

5. experimentX Group Title

the left hand derivative must be equal to right hand derivative at x=1

6. experimentX Group Title

for b, the function must be continuous ... so equate left hand limit to right hand limit at x=1

7. niravshah08 Group Title

the right hand derivative is just "a" and left hand derivative can be found using quotient rule

8. mahmit2012 Group Title

I know but how can you reach to -1/6 ?

9. experimentX Group Title

wolfram is pretty handy ... :D

10. mahmit2012 Group Title

yeap,but I didn't find any ways in there.

11. niravshah08 Group Title

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12. experimentX Group Title

$a = \left ( \frac{d}{dx} \left ( \frac{\cos^{-1}(\sqrt{x})}{\sqrt{1-x}} \right ) \right )_{x=1}$

13. mahmit2012 Group Title

Ok find it out?it gets so complexity.Doesn't it?

14. experimentX Group Title

$\lim_{x->1}ax+b = \lim_{x->1}\frac{\cos^{-1}(\sqrt{x})}{\sqrt{1-x}}$

15. experimentX Group Title

i prefer to find out by using wolfram first .. then use hand to verity it.

16. experimentX Group Title
17. mahmit2012 Group Title

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 Group Title

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 Group Title

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

20. mahmit2012 Group Title

please show me -1/6 . I didn't find it in wolfram...!Did you see any?

21. asnaseer Group Title

The link I pasted above shows how to find the derivative. Then just plug x=1 into this to get your answer.

22. asnaseer Group Title

sorry - I meant take the limit as x tends to 1

23. mahmit2012 Group Title

Ok copy past the answer and find it out for x=1.

24. asnaseer Group Title

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 Group Title

That was wrong site you posted.

26. experimentX Group Title

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 Group Title

experimentX: thats interesting - looks like wolfram is automagically using the limit as x tends to 1 to calculate this :)

28. mahmit2012 Group Title

I have never found it yet.so could you please copy past the way (lim x->1)

29. mahmit2012 Group Title

only i looked the answer -1/6 and derivation of function.

30. experimentX Group Title

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 Group Title

there is no calculation i chose step but there is only answer.

32. mahmit2012 Group Title

I accepted the conclusion but show me how?

33. experimentX Group Title

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 Group Title

$\frac{d}{dx} \left ( \frac{\cos^{-1}(\sqrt{x})}{\sqrt{1-x}} \right )$ I mean find this first.

35. mahmit2012 Group Title

ok find out this limit|dw:1337458049743:dw|

36. experimentX Group Title

yeah, that would be -1/6

37. experimentX Group Title

using L'hospital's rule

38. mahmit2012 Group Title

ok show me.i really appreciate

39. experimentX Group Title

are you having trouble with limits?

40. experimentX Group Title

which part .. first or last??

41. mahmit2012 Group Title

no but this is not easy to find out.check it out.

42. experimentX Group Title

Ah sorry ... we need only right hand limit

43. mahmit2012 Group Title

both of them to each other you should take limit,because each goes to infinity !

44. experimentX Group Title

ah .. yes ... looks like i'm nuts

45. experimentX Group Title

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 Group Title

im waiting yet.

47. experimentX Group Title

$\frac{\sqrt x \cos^{-1}(\sqrt x) - \sqrt{1-x} }{(1-x)^{3/2} \sqrt x }$ 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 Group Title

yes it is going to 0/0 so what is the rest?

49. mahmit2012 Group Title

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50. mahmit2012 Group Title

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51. mahmit2012 Group Title

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52. experimentX Group Title

Ahh ... looks like you got it!! good night!!

53. mahmit2012 Group Title

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