## AravindG 4 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?

1. UnkleRhaukus

|dw:1346418663569:dw|

2. AravindG

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

3. anonymous

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

|dw:1346418733121:dw|

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

|dw:1346418814483:dw|

8. AravindG

|dw:1346418865060:dw|

9. UnkleRhaukus

|dw:1346418835881:dw|

10. AravindG

perpendicular to what?

11. AravindG

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

12. anonymous

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

13. AravindG

@Mikael WHY???

14. anonymous

|dw:1346418660977:dw|

15. AravindG

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

16. anonymous

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. anonymous

Yes it will "touch" at infinitely many points

19. anonymous

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. anonymous

No, no no

23. AravindG

only one ryt?

24. anonymous

Almost - two

25. AravindG

????

26. anonymous

|dw:1346418836231:dw|

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. anonymous

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

30. AravindG

i think the answer is 1

31. anonymous

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. anonymous

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. anonymous

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. anonymous

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. anonymous

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

44. anonymous

Sorry - The Royal Academy od Sciences

45. UnkleRhaukus

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

46. anonymous

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. anonymous

@UnkleRhaukus for a small fee naturally >;]

49. anonymous

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. anonymous

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

55. AravindG

more examples?

56. anonymous

57. UnkleRhaukus

|dw:1346419864709:dw|

58. anonymous

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

70. AravindG

71. anonymous

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. anonymous

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

74. AravindG

@Mikael i knw dat i done so many problems

75. anonymous

2 Derivative at that point

76. AravindG

i cannot differentiate this coorectly

77. AravindG

i tred logarithmic differentiation

78. AravindG

nt getting

79. anonymous

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

80. AravindG

81. anonymous

Hey what the fuss it is simple Ratio function

82. AravindG

i did like this

83. AravindG

|dw:1346420490258:dw|

84. anonymous

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

85. AravindG

now when i put in x=7 i am stuck

86. anonymous

You did fine (beginning at least)

87. anonymous

Then do the formula above

88. AravindG

i think using quotient rule complicates the answer

89. AravindG

what is wrong with my working

90. anonymous

Yes , but it GETS YOU THERE

91. AravindG

@UnkleRhaukus sharre ur view

92. AravindG

my text says another method :

93. anonymous

Which is ...?

94. AravindG

|dw:1346420702829:dw|

95. AravindG

i they havent shown how they gt it :(

96. AravindG

@Mimi can u help?

97. anonymous

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

98. AravindG

ya i am also puzzled

99. AravindG

@Mimi_x3

100. anonymous

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. anonymous

Yes - Now i understand

104. AravindG

HOW?

105. anonymous

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

106. AravindG

?

107. anonymous

Lets start:

108. AravindG

go on

109. anonymous

When one applies the ratio formula one gets

110. UnkleRhaukus

.

111. anonymous

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

112. AravindG

so?

113. anonymous

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

114. AravindG

yep how?

115. anonymous

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

116. anonymous

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. anonymous

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

121. anonymous

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

122. anonymous

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

123. anonymous

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

124. AravindG

gt it !!! thx a lot!!!

125. anonymous

I have given you a complete calculation

126. anonymous

And thx is due

127. anonymous

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. anonymous

What does it mean ?

131. AravindG

show u sas new qn

132. AravindG

closing thisone as page lagging