AravindG
  • 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?
Mathematics
schrodinger
  • schrodinger
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UnkleRhaukus
  • UnkleRhaukus
|dw:1346418663569:dw|
AravindG
  • AravindG
anonymous
  • 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

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UnkleRhaukus
  • UnkleRhaukus
|dw:1346418733121:dw|
AravindG
  • AravindG
i knw about tangent in circles ..my prob is i dont get idea on tangents of curves
AravindG
  • AravindG
i cant draw tangent at a point if i am given a figure
UnkleRhaukus
  • UnkleRhaukus
|dw:1346418814483:dw|
AravindG
  • AravindG
|dw:1346418865060:dw|
UnkleRhaukus
  • UnkleRhaukus
|dw:1346418835881:dw|
AravindG
  • AravindG
perpendicular to what?
AravindG
  • AravindG
|dw:1346418935365:dw| what is tangent at P?
anonymous
  • anonymous
Aravind - where the curve itself is flat (straight) the tangent coincides with it.
AravindG
  • AravindG
@Mikael WHY???
anonymous
  • anonymous
|dw:1346418660977:dw|
AravindG
  • AravindG
if it is coincidnt it will touch at more than one point!!
anonymous
  • anonymous
Because the tangent to a straight i this straight line !
AravindG
  • AravindG
but tangent is defined to touch only at a point...err i am confused
anonymous
  • anonymous
Yes it will "touch" at infinitely many points
anonymous
  • 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.
UnkleRhaukus
  • UnkleRhaukus
yes , @Mikael the tangent is the flat line approximation of the curve
AravindG
  • AravindG
Through one given point in a curve, infinite tangents can be drawn. "is that true?
anonymous
  • anonymous
No, no no
AravindG
  • AravindG
only one ryt?
anonymous
  • anonymous
Almost - two
AravindG
  • AravindG
????
anonymous
  • anonymous
|dw:1346418836231:dw|
AravindG
  • AravindG
@Mikael read the qn agian Through one given point in a curve, infinite tangents can be drawn. "is that true?
AravindG
  • AravindG
Through ONE given point in a curve, infinite tangents can be drawn. "is that true?
anonymous
  • anonymous
Read what @UnkleRhaukus said: this is the full definition: flat approximation
AravindG
  • AravindG
i think the answer is 1
anonymous
  • anonymous
No - just one single well-defined tangent
AravindG
  • AravindG
yep :) only one tangent is allowed at point ryt?
UnkleRhaukus
  • UnkleRhaukus
i can not think of any example where a curve could have more than one tangent
anonymous
  • anonymous
At point ON the curve YES "there can be only one" As Duncan McCloud says
AravindG
  • AravindG
@UnkleRhaukus i liked ur definition the tangent is the flat line approximation of the curve
anonymous
  • anonymous
It is not his (he would be lucky to be Isaac Newton, but he is unfortunately NOT)
AravindG
  • AravindG
those are newtons words??
UnkleRhaukus
  • UnkleRhaukus
it's not mine you say?
AravindG
  • AravindG
anonymous
  • anonymous
I say that "tangent is a flat approximation to the curve" is ORIGINALLY Newton's definition.
UnkleRhaukus
  • UnkleRhaukus
wow, where can i see that?
AravindG
  • AravindG
"GREAT MINDS THINK ALIKE "
anonymous
  • anonymous
"Principia Mathematicae" (National Library, also the Library of the British Academy of Sciences
anonymous
  • anonymous
Sorry - The Royal Academy od Sciences
UnkleRhaukus
  • UnkleRhaukus
we i havent read that one, ( i think its in latin)
anonymous
  • anonymous
I can get you a recent English Translation by a nobel prize winner (countrymanof Aravind originally...)
AravindG
  • AravindG
HMM..CAN u guys give me some real life situations where calculation of tangents is necessary?
anonymous
  • anonymous
@UnkleRhaukus for a small fee naturally >;]
anonymous
  • anonymous
Yeah , Let\[f(x) = 33x^2 + 18x - 15\] Find the tangent to this curve whose slope equals 99
AravindG
  • AravindG
@Mikael i mean not that way
UnkleRhaukus
  • 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|
AravindG
  • AravindG
well i need real life example where calculation of tangent is necessary
AravindG
  • AravindG
for eg.space vehicle trajectory after it leaves earth
anonymous
  • anonymous
Yes this is very much related to original problems Newton had to solve - instantaneous velocity
AravindG
  • AravindG
more examples?
anonymous
  • anonymous
1 Attachment
UnkleRhaukus
  • UnkleRhaukus
|dw:1346419864709:dw|
anonymous
  • 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
anonymous
  • anonymous
This will be solved by a tangent to exponential function graph
AravindG
  • AravindG
@UnkleRhaukus nic pic :P
AravindG
  • AravindG
@Mikael do u have more?
UnkleRhaukus
  • UnkleRhaukus
does it look like some athlete throwing a hammer ?
AravindG
  • AravindG
i thought someone was spinning thee bucket :P
anonymous
  • anonymous
Yes - the typical (boring) school/colledge example would be:
UnkleRhaukus
  • UnkleRhaukus
instantaneous velocity or speed is the tangent of distance/time
anonymous
  • 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
anonymous
  • anonymous
Btw @UnkleRhaukus did u open the picture I have attached above ?
AravindG
  • 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.
UnkleRhaukus
  • UnkleRhaukus
looks like 150$ i dont have @Mikael
AravindG
  • AravindG
help please
anonymous
  • 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
AravindG
  • AravindG
\[y=\frac{(x-7)}{(x-2)(x-3)}\]
anonymous
  • anonymous
1. Find the point (in ur case the intersection with the axis)
AravindG
  • AravindG
@Mikael i knw dat i done so many problems
anonymous
  • anonymous
2 Derivative at that point
AravindG
  • AravindG
i cannot differentiate this coorectly
AravindG
  • AravindG
i tred logarithmic differentiation
AravindG
  • AravindG
nt getting
anonymous
  • anonymous
Now you have THREE ( 3 ) data: (x, f(x)) and the slope=derivative
AravindG
  • AravindG
help please
anonymous
  • anonymous
Hey what the fuss it is simple Ratio function
AravindG
  • AravindG
i did like this
AravindG
  • AravindG
|dw:1346420490258:dw|
anonymous
  • anonymous
\[(\frac{ f }{ g }) = \frac{ f'*g - f*g'}{ g^2}\]
AravindG
  • AravindG
now when i put in x=7 i am stuck
anonymous
  • anonymous
You did fine (beginning at least)
anonymous
  • anonymous
Then do the formula above
AravindG
  • AravindG
i think using quotient rule complicates the answer
AravindG
  • AravindG
what is wrong with my working
anonymous
  • anonymous
Yes , but it GETS YOU THERE
AravindG
  • AravindG
@UnkleRhaukus sharre ur view
AravindG
  • AravindG
my text says another method :
anonymous
  • anonymous
Which is ...?
AravindG
  • AravindG
|dw:1346420702829:dw|
AravindG
  • AravindG
i they havent shown how they gt it :(
AravindG
  • AravindG
@Mimi can u help?
anonymous
  • anonymous
WHAAT the f. ??? How did y get there - in the denumer ?
AravindG
  • AravindG
ya i am also puzzled
AravindG
  • AravindG
anonymous
  • anonymous
Why not use simple and direct Ratio deriv. formula ??!
AravindG
  • AravindG
i dont knw can u fig out what the text has used ?
AravindG
  • AravindG
@ash2326 , @myininaya PLS HELP
anonymous
  • anonymous
Yes - Now i understand
AravindG
  • AravindG
HOW?
anonymous
  • anonymous
Stop - you will awake all the evil spirits. It is a simple trick
AravindG
  • AravindG
?
anonymous
  • anonymous
Lets start:
AravindG
  • AravindG
go on
anonymous
  • anonymous
When one applies the ratio formula one gets
UnkleRhaukus
  • UnkleRhaukus
.
anonymous
  • anonymous
\[\frac{ Polynom }{ ((x-2)(x-3))^2}\]
AravindG
  • AravindG
so?
anonymous
  • anonymous
Now as u notice in ur text's form the denomin is WITHOUT the square
AravindG
  • AravindG
yep how?
anonymous
  • anonymous
They "transfered" one "sqrt of the denominator" as original y TO THE DENUMERATOR
anonymous
  • anonymous
In fact\[\frac{ 1*(x-2)(x-3) - (x-7)*(Monomial ) }{(x-2)^2(x-3)^2} =\]
AravindG
  • AravindG
i see
AravindG
  • AravindG
then how did 2x-5 come?
AravindG
  • AravindG
wait i gt it !! 2x-5 is differential of x^2-5x+6 !!
anonymous
  • anonymous
\[=\frac{ 1 - (x-7)*(Monomial) *((x-2)(x-3))^{-1}}{ (x-2)(x-3)} \]
anonymous
  • anonymous
@AravindG Interact with this one : http://demonstrations.wolfram.com/TangentToACurve/
anonymous
  • anonymous
\[= \frac{ 1 - Monomial*{Original Function} }{ (x-2)(x-3)} =\]
anonymous
  • anonymous
\[= \frac{ 1 - y*Monomial }{ (x-2)(x-3) }\]
AravindG
  • AravindG
gt it !!! thx a lot!!!
anonymous
  • anonymous
I have given you a complete calculation
anonymous
  • anonymous
And thx is due
anonymous
  • anonymous
Close this question afterwards
AravindG
  • AravindG
i knw d rest
AravindG
  • AravindG
my only doubt is why i didnt get it by logarithmic differentiation
anonymous
  • anonymous
What does it mean ?
AravindG
  • AravindG
show u sas new qn
AravindG
  • AravindG
closing thisone as page lagging

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