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posted by  Del Piero on 11/4/2009 11:19:12 PM  |  status: Closed  |  Earned Karma: 25

please help q2 ,Will give lifesaver rating

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Calculus N/A N/A N/A 11/5/2009 at 5:00:00 PM
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Thanks a lot bro...... ;)
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Oracle
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posted by anonymousxyz on 11/4/2009 11:44:22 PM  |  status: Live
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x  =  sin t

dx / dt  =  cos t


y  =  sin 2t

dy / dt  =  2 cos 2t

dy / dx  =  ( dy / dt )  /  ( dx / dt )  =  ( 2 cos 2t ) / ( cos t )


The tangent line is horizontal when dy / dx equals zero:


dy / dx  =  0  =>

( 2 cos 2t ) / ( cos t )  =  0  =>

2 cos 2t  =  0  =>

cos 2t  =  0  =>

2t  =  kπ / 2, where k is an odd integer  =>

t  =  kπ / 4


0  <=  t  <=  2π  =>

t  =  π / 4,  3π / 4,  5π / 4,  7π / 4


x( π / 4 )  =  sin ( π / 4 )  =  √2 / 2

y( π / 4 )  =  sin ( π / 2 )  =  1


x( 3π / 4 )  =  sin ( 3π / 4 )  =  √2 / 2

y( 3π / 4 )  =  sin ( 3π / 2 )  =  -1


x( 5π / 4 )  =  sin ( 5π / 4 )  =  -√2 / 2

y( 5π / 4 )  =  sin ( 5π / 2 )  =  1


x( 7π / 4 )  =  sin ( 7π / 4 )  =  -√2 / 2

y( 7π / 4 )  =  sin ( 7π / 2 )  =  -1
Sage
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posted by DaveD on 11/4/2009 11:46:53 PM  |  status: Live
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, so , giving  for any integer k. Note that the denominator is  for every possible value of t, so the derivative is defined at each t. Thus, the answer is  for any integer k.
Novice
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posted by CMackey on 11/5/2009 12:35:43 AM  |  status: Live
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Response Details:
The easiest way to do this is to just take derivatives. ( and on a side note the graph looks like a bowtie kinda)

You need the derivative of x=sin(t) and y=sin(2t) to be be equal to zero for the same value of t. To do this, first take a derivative.


dx/dt = cos(t)

dy/dt = 2cos(2t)

now for the tangent line to be horizontal we set the slope to zero so we plug that in for our derivative and we get :

0 = cos(t)

and

0 = 2cos(2t)

which is 2cos(2t) - cos(t)= 0


Now we think of all the values which satisfy this.

The easiest way at this point is to plug 2cos(2t)-cos(t) into your calculator and find all points where
y=0.

you should find four points when you are done.


The other way is to solve for t in your original two equations and plug those in to their counterparts.

e.g   (      arcsin(x)=t   y=2sin(2(arcsin(x)))  )

then take the derivative and set to 0

Hope that helps!
Charles M.
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