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AravindG

  • 3 years ago

Please help: can u explain detailed what is a tangent to a curve .I know its a line just touching the curve at only one point but i need to understand it more and also would like to know how to draw the tangent if i am given any graph .Is THERE ONLY ONE tangent possible at a point?

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  1. UnkleRhaukus
    • 3 years ago
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    |dw:1346418663569:dw|

  2. AravindG
    • 3 years ago
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    @Callisto , @.Sam. , @eliassaab , @lgbasallote , @satellite73 , @mukushla

  3. Mikael
    • 3 years ago
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    In the general case the slope is computed by derivative. But for circles - the straight line is PERPENDICULAR to the radius at the point of tangent

  4. UnkleRhaukus
    • 3 years ago
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    |dw:1346418733121:dw|

  5. AravindG
    • 3 years ago
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    i knw about tangent in circles ..my prob is i dont get idea on tangents of curves

  6. AravindG
    • 3 years ago
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    i cant draw tangent at a point if i am given a figure

  7. UnkleRhaukus
    • 3 years ago
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    |dw:1346418814483:dw|

  8. AravindG
    • 3 years ago
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    |dw:1346418865060:dw|

  9. UnkleRhaukus
    • 3 years ago
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    |dw:1346418835881:dw|

  10. AravindG
    • 3 years ago
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    perpendicular to what?

  11. AravindG
    • 3 years ago
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    |dw:1346418935365:dw| what is tangent at P?

  12. Mikael
    • 3 years ago
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    Aravind - where the curve itself is flat (straight) the tangent coincides with it.

  13. AravindG
    • 3 years ago
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    @Mikael WHY???

  14. Mikael
    • 3 years ago
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    |dw:1346418660977:dw|

  15. AravindG
    • 3 years ago
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    if it is coincidnt it will touch at more than one point!!

  16. Mikael
    • 3 years ago
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    Because the tangent to a straight i this straight line !

  17. AravindG
    • 3 years ago
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    but tangent is defined to touch only at a point...err i am confused

  18. Mikael
    • 3 years ago
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    Yes it will "touch" at infinitely many points

  19. amishra
    • 3 years ago
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    A tangent is a line outside the circle which touches only one point of the circle. Through one given point, infinite tangents can be drawn. A tangent at a given point is perpendicular to the radius of the circle.

  20. UnkleRhaukus
    • 3 years ago
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    yes , @Mikael the tangent is the flat line approximation of the curve

  21. AravindG
    • 3 years ago
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    Through one given point in a curve, infinite tangents can be drawn. "is that true?

  22. Mikael
    • 3 years ago
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    No, no no

  23. AravindG
    • 3 years ago
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    only one ryt?

  24. Mikael
    • 3 years ago
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    Almost - two

  25. AravindG
    • 3 years ago
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    ????

  26. Mikael
    • 3 years ago
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    |dw:1346418836231:dw|

  27. AravindG
    • 3 years ago
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    @Mikael read the qn agian Through one given point in a curve, infinite tangents can be drawn. "is that true?

  28. AravindG
    • 3 years ago
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    Through ONE given point in a curve, infinite tangents can be drawn. "is that true?

  29. Mikael
    • 3 years ago
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    Read what @UnkleRhaukus said: this is the full definition: flat approximation

  30. AravindG
    • 3 years ago
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    i think the answer is 1

  31. Mikael
    • 3 years ago
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    No - just one single well-defined tangent

  32. AravindG
    • 3 years ago
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    yep :) only one tangent is allowed at point ryt?

  33. UnkleRhaukus
    • 3 years ago
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    i can not think of any example where a curve could have more than one tangent

  34. Mikael
    • 3 years ago
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    At point ON the curve YES "there can be only one" As Duncan McCloud says

  35. AravindG
    • 3 years ago
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    @UnkleRhaukus i liked ur definition the tangent is the flat line approximation of the curve

  36. Mikael
    • 3 years ago
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    It is not his (he would be lucky to be Isaac Newton, but he is unfortunately NOT)

  37. AravindG
    • 3 years ago
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    those are newtons words??

  38. UnkleRhaukus
    • 3 years ago
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    it's not mine you say?

  39. AravindG
    • 3 years ago
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    @UnkleRhaukus ?

  40. Mikael
    • 3 years ago
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    I say that "tangent is a flat approximation to the curve" is ORIGINALLY Newton's definition.

  41. UnkleRhaukus
    • 3 years ago
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    wow, where can i see that?

  42. AravindG
    • 3 years ago
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    "GREAT MINDS THINK ALIKE "

  43. Mikael
    • 3 years ago
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    "Principia Mathematicae" (National Library, also the Library of the British Academy of Sciences

  44. Mikael
    • 3 years ago
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    Sorry - The Royal Academy od Sciences

  45. UnkleRhaukus
    • 3 years ago
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    we i havent read that one, ( i think its in latin)

  46. Mikael
    • 3 years ago
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    I can get you a recent English Translation by a nobel prize winner (countrymanof Aravind originally...)

  47. AravindG
    • 3 years ago
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    HMM..CAN u guys give me some real life situations where calculation of tangents is necessary?

  48. Mikael
    • 3 years ago
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    @UnkleRhaukus for a small fee naturally >;]

  49. Mikael
    • 3 years ago
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    Yeah , Let\[f(x) = 33x^2 + 18x - 15\] Find the tangent to this curve whose slope equals 99

  50. AravindG
    • 3 years ago
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    @Mikael i mean not that way

  51. UnkleRhaukus
    • 3 years ago
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    The only example i can think of where you could have multiple tangents is if they had som scale associated with them |dw:1346419654145:dw|

  52. AravindG
    • 3 years ago
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    well i need real life example where calculation of tangent is necessary

  53. AravindG
    • 3 years ago
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    for eg.space vehicle trajectory after it leaves earth

  54. Mikael
    • 3 years ago
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    Yes this is very much related to original problems Newton had to solve - instantaneous velocity

  55. AravindG
    • 3 years ago
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    more examples?

  56. Mikael
    • 3 years ago
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    1 Attachment
  57. UnkleRhaukus
    • 3 years ago
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    |dw:1346419864709:dw|

  58. Mikael
    • 3 years ago
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    Well - suppose Mr. Buffet has 23*10^9 $ in a hedge fund. Assume that his interest accumulates at 1% per second, draw the linear graph (straight line) approximating his wealth grouth after 27 seconds

  59. Mikael
    • 3 years ago
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    This will be solved by a tangent to exponential function graph

  60. AravindG
    • 3 years ago
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    @UnkleRhaukus nic pic :P

  61. AravindG
    • 3 years ago
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    @Mikael do u have more?

  62. UnkleRhaukus
    • 3 years ago
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    does it look like some athlete throwing a hammer ?

  63. AravindG
    • 3 years ago
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    i thought someone was spinning thee bucket :P

  64. Mikael
    • 3 years ago
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    Yes - the typical (boring) school/colledge example would be:

  65. UnkleRhaukus
    • 3 years ago
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    instantaneous velocity or speed is the tangent of distance/time

  66. Mikael
    • 3 years ago
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    Let f(x) = (45x^2 -33x + 7) sin (x-5) . Find the approximate value at 5.01 using the linear approximation with its slope = derivative at x=5

  67. Mikael
    • 3 years ago
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    Btw @UnkleRhaukus did u open the picture I have attached above ?

  68. AravindG
    • 3 years ago
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    One last question which i gt stuck today : Find the equation of the tangent to the curve y=x-7/((x-2)(x-3)) at the point where it cuts the x axis.

  69. UnkleRhaukus
    • 3 years ago
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    looks like 150$ i dont have @Mikael

  70. AravindG
    • 3 years ago
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    help please

  71. Mikael
    • 3 years ago
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    Let's show you the general idea to Find the equation of the tangent to the curve y=f(x) at the point x1

  72. AravindG
    • 3 years ago
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    \[y=\frac{(x-7)}{(x-2)(x-3)}\]

  73. Mikael
    • 3 years ago
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    1. Find the point (in ur case the intersection with the axis)

  74. AravindG
    • 3 years ago
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    @Mikael i knw dat i done so many problems

  75. Mikael
    • 3 years ago
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    2 Derivative at that point

  76. AravindG
    • 3 years ago
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    i cannot differentiate this coorectly

  77. AravindG
    • 3 years ago
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    i tred logarithmic differentiation

  78. AravindG
    • 3 years ago
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    nt getting

  79. Mikael
    • 3 years ago
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    Now you have THREE ( 3 ) data: (x, f(x)) and the slope=derivative

  80. AravindG
    • 3 years ago
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    help please

  81. Mikael
    • 3 years ago
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    Hey what the fuss it is simple Ratio function

  82. AravindG
    • 3 years ago
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    i did like this

  83. AravindG
    • 3 years ago
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    |dw:1346420490258:dw|

  84. Mikael
    • 3 years ago
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    \[(\frac{ f }{ g }) = \frac{ f'*g - f*g'}{ g^2}\]

  85. AravindG
    • 3 years ago
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    now when i put in x=7 i am stuck

  86. Mikael
    • 3 years ago
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    You did fine (beginning at least)

  87. Mikael
    • 3 years ago
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    Then do the formula above

  88. AravindG
    • 3 years ago
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    i think using quotient rule complicates the answer

  89. AravindG
    • 3 years ago
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    what is wrong with my working

  90. Mikael
    • 3 years ago
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    Yes , but it GETS YOU THERE

  91. AravindG
    • 3 years ago
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    @UnkleRhaukus sharre ur view

  92. AravindG
    • 3 years ago
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    my text says another method :

  93. Mikael
    • 3 years ago
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    Which is ...?

  94. AravindG
    • 3 years ago
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    |dw:1346420702829:dw|

  95. AravindG
    • 3 years ago
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    i they havent shown how they gt it :(

  96. AravindG
    • 3 years ago
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    @Mimi can u help?

  97. Mikael
    • 3 years ago
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    WHAAT the f. ??? How did y get there - in the denumer ?

  98. AravindG
    • 3 years ago
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    ya i am also puzzled

  99. AravindG
    • 3 years ago
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    @Mimi_x3

  100. Mikael
    • 3 years ago
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    Why not use simple and direct Ratio deriv. formula ??!

  101. AravindG
    • 3 years ago
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    i dont knw can u fig out what the text has used ?

  102. AravindG
    • 3 years ago
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    @ash2326 , @myininaya PLS HELP

  103. Mikael
    • 3 years ago
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    Yes - Now i understand

  104. AravindG
    • 3 years ago
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    HOW?

  105. Mikael
    • 3 years ago
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    Stop - you will awake all the evil spirits. It is a simple trick

  106. AravindG
    • 3 years ago
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    ?

  107. Mikael
    • 3 years ago
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    Lets start:

  108. AravindG
    • 3 years ago
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    go on

  109. Mikael
    • 3 years ago
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    When one applies the ratio formula one gets

  110. UnkleRhaukus
    • 3 years ago
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    .

  111. Mikael
    • 3 years ago
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    \[\frac{ Polynom }{ ((x-2)(x-3))^2}\]

  112. AravindG
    • 3 years ago
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    so?

  113. Mikael
    • 3 years ago
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    Now as u notice in ur text's form the denomin is WITHOUT the square

  114. AravindG
    • 3 years ago
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    yep how?

  115. Mikael
    • 3 years ago
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    They "transfered" one "sqrt of the denominator" as original y TO THE DENUMERATOR

  116. Mikael
    • 3 years ago
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    In fact\[\frac{ 1*(x-2)(x-3) - (x-7)*(Monomial ) }{(x-2)^2(x-3)^2} =\]

  117. AravindG
    • 3 years ago
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    i see

  118. AravindG
    • 3 years ago
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    then how did 2x-5 come?

  119. AravindG
    • 3 years ago
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    wait i gt it !! 2x-5 is differential of x^2-5x+6 !!

  120. Mikael
    • 3 years ago
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    \[=\frac{ 1 - (x-7)*(Monomial) *((x-2)(x-3))^{-1}}{ (x-2)(x-3)} \]

  121. Rohangrr
    • 3 years ago
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    @AravindG Interact with this one : http://demonstrations.wolfram.com/TangentToACurve/

  122. Mikael
    • 3 years ago
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    \[= \frac{ 1 - Monomial*{Original Function} }{ (x-2)(x-3)} =\]

  123. Mikael
    • 3 years ago
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    \[= \frac{ 1 - y*Monomial }{ (x-2)(x-3) }\]

  124. AravindG
    • 3 years ago
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    gt it !!! thx a lot!!!

  125. Mikael
    • 3 years ago
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    I have given you a complete calculation

  126. Mikael
    • 3 years ago
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    And thx is due

  127. Mikael
    • 3 years ago
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    Close this question afterwards

  128. AravindG
    • 3 years ago
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    i knw d rest

  129. AravindG
    • 3 years ago
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    my only doubt is why i didnt get it by logarithmic differentiation

  130. Mikael
    • 3 years ago
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    What does it mean ?

  131. AravindG
    • 3 years ago
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    show u sas new qn

  132. AravindG
    • 3 years ago
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    closing thisone as page lagging

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