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?
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UnkleRhaukus
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AravindG
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@Callisto , @.Sam. , @eliassaab , @lgbasallote , @satellite73 , @mukushla
Mikael
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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
UnkleRhaukus
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AravindG
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i knw about tangent in circles ..my prob is i dont get idea on tangents of curves
AravindG
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i cant draw tangent at a point if i am given a figure
UnkleRhaukus
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AravindG
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UnkleRhaukus
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AravindG
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perpendicular to what?
AravindG
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what is tangent at P?
Mikael
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Aravind - where the curve itself is flat (straight) the tangent coincides with it.
AravindG
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@Mikael WHY???
Mikael
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AravindG
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if it is coincidnt it will touch at more than one point!!
Mikael
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Because the tangent to a straight i this straight line !
AravindG
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but tangent is defined to touch only at a point...err i am confused
Mikael
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Yes it will "touch" at infinitely many points
amishra
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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.
UnkleRhaukus
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yes , @Mikael
the tangent is the flat line approximation of the curve
AravindG
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Through one given point in a curve, infinite tangents can be drawn. "is that true?
Mikael
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No, no no
AravindG
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only one ryt?
Mikael
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Almost - two
AravindG
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????
Mikael
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AravindG
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@Mikael read the qn agian
Through one given point in a curve, infinite tangents can be drawn. "is that true?
AravindG
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Through ONE given point in a curve, infinite tangents can be drawn. "is that true?
Mikael
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Read what @UnkleRhaukus said: this is the full definition: flat approximation
AravindG
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i think the answer is 1
Mikael
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No - just one single well-defined tangent
AravindG
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yep :) only one tangent is allowed at point ryt?
UnkleRhaukus
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i can not think of any example where a curve could have more than one tangent
Mikael
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At point ON the curve YES "there can be only one" As Duncan McCloud says
AravindG
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@UnkleRhaukus i liked ur definition the tangent is the flat line approximation of the curve
Mikael
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It is not his (he would be lucky to be Isaac Newton, but he is unfortunately NOT)
AravindG
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those are newtons words??
UnkleRhaukus
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it's not mine you say?
AravindG
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@UnkleRhaukus ?
Mikael
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I say that "tangent is a flat approximation to the curve" is ORIGINALLY Newton's definition.
UnkleRhaukus
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wow, where can i see that?
AravindG
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"GREAT MINDS THINK ALIKE "
Mikael
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"Principia Mathematicae" (National Library, also the Library of the British Academy of Sciences
Mikael
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Sorry - The Royal Academy od Sciences
UnkleRhaukus
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we i havent read that one, ( i think its in latin)
Mikael
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I can get you a recent English Translation by a nobel prize winner (countrymanof Aravind originally...)
AravindG
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HMM..CAN u guys give me some real life situations where calculation of tangents is necessary?
Mikael
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@UnkleRhaukus for a small fee naturally >;]
Mikael
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Yeah , Let\[f(x) = 33x^2 + 18x - 15\]
Find the tangent to this curve whose slope equals 99
AravindG
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@Mikael i mean not that way
UnkleRhaukus
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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
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AravindG
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well i need real life example where calculation of tangent is necessary
AravindG
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for eg.space vehicle trajectory after it leaves earth
Mikael
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Yes this is very much related to original problems Newton had to solve - instantaneous velocity
AravindG
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more examples?
Mikael
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UnkleRhaukus
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Mikael
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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
Mikael
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This will be solved by a tangent to exponential function graph
AravindG
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@UnkleRhaukus nic pic :P
AravindG
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@Mikael do u have more?
UnkleRhaukus
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does it look like some athlete throwing a hammer ?
AravindG
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i thought someone was spinning thee bucket :P
Mikael
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Yes - the typical (boring) school/colledge example would be:
UnkleRhaukus
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instantaneous velocity or speed is the tangent of distance/time
Mikael
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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
Mikael
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Btw @UnkleRhaukus did u open the picture I have attached above ?
AravindG
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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.
UnkleRhaukus
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looks like 150$ i dont have @Mikael
AravindG
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help please
Mikael
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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
AravindG
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\[y=\frac{(x-7)}{(x-2)(x-3)}\]
Mikael
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1. Find the point (in ur case the intersection with the axis)
AravindG
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@Mikael i knw dat i done so many problems
Mikael
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2 Derivative at that point
AravindG
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i cannot differentiate this coorectly
AravindG
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i tred logarithmic differentiation
AravindG
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nt getting
Mikael
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Now you have THREE ( 3 ) data:
(x, f(x)) and the slope=derivative
AravindG
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help please
Mikael
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Hey what the fuss it is simple Ratio function
AravindG
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i did like this
AravindG
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Mikael
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\[(\frac{ f }{ g }) = \frac{ f'*g - f*g'}{ g^2}\]
AravindG
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now when i put in x=7 i am stuck
Mikael
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You did fine (beginning at least)
Mikael
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Then do the formula above
AravindG
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i think using quotient rule complicates the answer
AravindG
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what is wrong with my working
Mikael
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Yes , but it GETS YOU THERE
AravindG
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@UnkleRhaukus sharre ur view
AravindG
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my text says another method :
Mikael
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Which is ...?
AravindG
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AravindG
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i they havent shown how they gt it :(
AravindG
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@Mimi can u help?
Mikael
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WHAAT the f. ??? How did y get there - in the denumer ?
AravindG
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ya i am also puzzled
AravindG
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@Mimi_x3
Mikael
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Why not use simple and direct Ratio deriv. formula ??!
AravindG
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i dont knw can u fig out what the text has used ?
AravindG
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@ash2326 , @myininaya PLS HELP
Mikael
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Yes - Now i understand
AravindG
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HOW?
Mikael
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Stop - you will awake all the evil spirits. It is a simple trick
AravindG
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?
Mikael
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Lets start:
AravindG
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go on
Mikael
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When one applies the ratio formula one gets
UnkleRhaukus
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.
Mikael
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\[\frac{ Polynom }{ ((x-2)(x-3))^2}\]
AravindG
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so?
Mikael
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Now as u notice in ur text's form the denomin is WITHOUT the square
AravindG
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yep how?
Mikael
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They "transfered" one "sqrt of the denominator" as original y TO THE DENUMERATOR
Mikael
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In fact\[\frac{ 1*(x-2)(x-3) - (x-7)*(Monomial ) }{(x-2)^2(x-3)^2} =\]
AravindG
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i see
AravindG
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then how did 2x-5 come?
AravindG
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wait i gt it !! 2x-5 is differential of x^2-5x+6 !!
Mikael
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\[=\frac{ 1 - (x-7)*(Monomial) *((x-2)(x-3))^{-1}}{ (x-2)(x-3)} \]
Mikael
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\[= \frac{ 1 - Monomial*{Original Function} }{ (x-2)(x-3)} =\]
Mikael
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\[= \frac{ 1 - y*Monomial }{ (x-2)(x-3) }\]
AravindG
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gt it !!! thx a lot!!!
Mikael
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I have given you a complete calculation
Mikael
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And thx is due
Mikael
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Close this question afterwards
AravindG
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i knw d rest
AravindG
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my only doubt is why i didnt get it by logarithmic differentiation
Mikael
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What does it mean ?
AravindG
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show u sas new qn
AravindG
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closing thisone as page lagging