What is the instantaneous velocity of an object moving in accordance to # f(t)= (2sin(2t+pi),tcos2t) # at # t=pi/3 #?

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Answer 1 · 2018-03-07T20:10:16

#f'(pi/3)=(3 + 2 sqrt(3) π)/(12 - 6 π)#

We have the path of an object as a function of time given by #f(t)=(2sin(2t+pi),tcos2t)#.

This is the parametric form of the Cartesian equation with

#x=2sin(2t+pi)#
#y=tcos2t#

We want to find the instantaneous velocity of the object at #t=pi"/"3#. This is equivalent to evaluating the first derivative of #x# at #t=pi"/"3#.

The way to find the first derivative of #x# using these parameters is given by the following formula

#dy/dx=(dy"/"dt)/(dx"/"dt)#

#dx/dt=4cos(2t+pi)#

#dy/dt=cos2t-2tsin2t#

#rArr dy/dx= (cos2t-2tsin2t)/(4cos(2t+pi))#

We now plug in #t=pi/3# to get

#f'(pi/3)=(3 + 2 sqrt(3) π)/(12 - 6 π)#