## AravindG 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? one year ago one year ago

1. UnkleRhaukus

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2. AravindG

@Callisto , @.Sam. , @eliassaab , @lgbasallote , @satellite73 , @mukushla

3. Mikael

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

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5. AravindG

i knw about tangent in circles ..my prob is i dont get idea on tangents of curves

6. AravindG

i cant draw tangent at a point if i am given a figure

7. UnkleRhaukus

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8. AravindG

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9. UnkleRhaukus

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10. AravindG

perpendicular to what?

11. AravindG

|dw:1346418935365:dw| what is tangent at P?

12. Mikael

Aravind - where the curve itself is flat (straight) the tangent coincides with it.

13. AravindG

@Mikael WHY???

14. Mikael

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15. AravindG

if it is coincidnt it will touch at more than one point!!

16. Mikael

Because the tangent to a straight i this straight line !

17. AravindG

but tangent is defined to touch only at a point...err i am confused

18. Mikael

Yes it will "touch" at infinitely many points

19. amishra

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

yes , @Mikael the tangent is the flat line approximation of the curve

21. AravindG

Through one given point in a curve, infinite tangents can be drawn. "is that true?

22. Mikael

No, no no

23. AravindG

only one ryt?

24. Mikael

Almost - two

25. AravindG

????

26. Mikael

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27. AravindG

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

28. AravindG

Through ONE given point in a curve, infinite tangents can be drawn. "is that true?

29. Mikael

Read what @UnkleRhaukus said: this is the full definition: flat approximation

30. AravindG

i think the answer is 1

31. Mikael

No - just one single well-defined tangent

32. AravindG

yep :) only one tangent is allowed at point ryt?

33. UnkleRhaukus

i can not think of any example where a curve could have more than one tangent

34. Mikael

At point ON the curve YES "there can be only one" As Duncan McCloud says

35. AravindG

@UnkleRhaukus i liked ur definition the tangent is the flat line approximation of the curve

36. Mikael

It is not his (he would be lucky to be Isaac Newton, but he is unfortunately NOT)

37. AravindG

those are newtons words??

38. UnkleRhaukus

it's not mine you say?

39. AravindG

@UnkleRhaukus ?

40. Mikael

I say that "tangent is a flat approximation to the curve" is ORIGINALLY Newton's definition.

41. UnkleRhaukus

wow, where can i see that?

42. AravindG

"GREAT MINDS THINK ALIKE "

43. Mikael

"Principia Mathematicae" (National Library, also the Library of the British Academy of Sciences

44. Mikael

Sorry - The Royal Academy od Sciences

45. UnkleRhaukus

we i havent read that one, ( i think its in latin)

46. Mikael

I can get you a recent English Translation by a nobel prize winner (countrymanof Aravind originally...)

47. AravindG

HMM..CAN u guys give me some real life situations where calculation of tangents is necessary?

48. Mikael

@UnkleRhaukus for a small fee naturally >;]

49. Mikael

Yeah , Let$f(x) = 33x^2 + 18x - 15$ Find the tangent to this curve whose slope equals 99

50. AravindG

@Mikael i mean not that way

51. UnkleRhaukus

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

well i need real life example where calculation of tangent is necessary

53. AravindG

for eg.space vehicle trajectory after it leaves earth

54. Mikael

Yes this is very much related to original problems Newton had to solve - instantaneous velocity

55. AravindG

more examples?

56. Mikael

57. UnkleRhaukus

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58. Mikael

Well - suppose Mr. Buffet has 23*10^9 $in a hedge fund. Assume that his interest accumulates at 1% per second, draw the linear graph (straight line) approximating his wealth grouth after 27 seconds 59. Mikael This will be solved by a tangent to exponential function graph 60. AravindG @UnkleRhaukus nic pic :P 61. AravindG @Mikael do u have more? 62. UnkleRhaukus does it look like some athlete throwing a hammer ? 63. AravindG i thought someone was spinning thee bucket :P 64. Mikael Yes - the typical (boring) school/colledge example would be: 65. UnkleRhaukus instantaneous velocity or speed is the tangent of distance/time 66. Mikael Let f(x) = (45x^2 -33x + 7) sin (x-5) . Find the approximate value at 5.01 using the linear approximation with its slope = derivative at x=5 67. Mikael Btw @UnkleRhaukus did u open the picture I have attached above ? 68. AravindG One last question which i gt stuck today : Find the equation of the tangent to the curve y=x-7/((x-2)(x-3)) at the point where it cuts the x axis. 69. UnkleRhaukus looks like 150$ i dont have @Mikael

70. AravindG

71. Mikael

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

$y=\frac{(x-7)}{(x-2)(x-3)}$

73. Mikael

1. Find the point (in ur case the intersection with the axis)

74. AravindG

@Mikael i knw dat i done so many problems

75. Mikael

2 Derivative at that point

76. AravindG

i cannot differentiate this coorectly

77. AravindG

i tred logarithmic differentiation

78. AravindG

nt getting

79. Mikael

Now you have THREE ( 3 ) data: (x, f(x)) and the slope=derivative

80. AravindG

81. Mikael

Hey what the fuss it is simple Ratio function

82. AravindG

i did like this

83. AravindG

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84. Mikael

$(\frac{ f }{ g }) = \frac{ f'*g - f*g'}{ g^2}$

85. AravindG

now when i put in x=7 i am stuck

86. Mikael

You did fine (beginning at least)

87. Mikael

Then do the formula above

88. AravindG

i think using quotient rule complicates the answer

89. AravindG

what is wrong with my working

90. Mikael

Yes , but it GETS YOU THERE

91. AravindG

@UnkleRhaukus sharre ur view

92. AravindG

my text says another method :

93. Mikael

Which is ...?

94. AravindG

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95. AravindG

i they havent shown how they gt it :(

96. AravindG

@Mimi can u help?

97. Mikael

WHAAT the f. ??? How did y get there - in the denumer ?

98. AravindG

ya i am also puzzled

99. AravindG

@Mimi_x3

100. Mikael

Why not use simple and direct Ratio deriv. formula ??!

101. AravindG

i dont knw can u fig out what the text has used ?

102. AravindG

@ash2326 , @myininaya PLS HELP

103. Mikael

Yes - Now i understand

104. AravindG

HOW?

105. Mikael

Stop - you will awake all the evil spirits. It is a simple trick

106. AravindG

?

107. Mikael

Lets start:

108. AravindG

go on

109. Mikael

When one applies the ratio formula one gets

110. UnkleRhaukus

.

111. Mikael

$\frac{ Polynom }{ ((x-2)(x-3))^2}$

112. AravindG

so?

113. Mikael

Now as u notice in ur text's form the denomin is WITHOUT the square

114. AravindG

yep how?

115. Mikael

They "transfered" one "sqrt of the denominator" as original y TO THE DENUMERATOR

116. Mikael

In fact$\frac{ 1*(x-2)(x-3) - (x-7)*(Monomial ) }{(x-2)^2(x-3)^2} =$

117. AravindG

i see

118. AravindG

then how did 2x-5 come?

119. AravindG

wait i gt it !! 2x-5 is differential of x^2-5x+6 !!

120. Mikael

$=\frac{ 1 - (x-7)*(Monomial) *((x-2)(x-3))^{-1}}{ (x-2)(x-3)}$

121. Rohangrr

@AravindG Interact with this one : http://demonstrations.wolfram.com/TangentToACurve/

122. Mikael

$= \frac{ 1 - Monomial*{Original Function} }{ (x-2)(x-3)} =$

123. Mikael

$= \frac{ 1 - y*Monomial }{ (x-2)(x-3) }$

124. AravindG

gt it !!! thx a lot!!!

125. Mikael

I have given you a complete calculation

126. Mikael

And thx is due

127. Mikael

Close this question afterwards

128. AravindG

i knw d rest

129. AravindG

my only doubt is why i didnt get it by logarithmic differentiation

130. Mikael

What does it mean ?

131. AravindG

show u sas new qn

132. AravindG

closing thisone as page lagging