What is the instantaneous velocity of an object moving in accordance to # f(t)= (t/(e^t+2t),t) # at # t=4 #?

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Answer 1 · 2017-11-22T07:44:48

The instantaneous velocity is #=(-0.0418,1)ms^-1#

We need

#(u/v)'=(u'v-uv')/(v^2)#

Start by calculating the derivative of #(t/(e^t+2t))# wrt #t#

#x=t/(e^t+2t)#

#u=t#, #=>#, #u'=1#

#v=e^t+2t#, #=>#, #v'=e^t+2#

So,

#dx/dt=(1*(e^t+2t)-t(e^t+2))/(e^t+2t)^2#

#=(e^t+2t-te^t-2t)/((e^t+2t)^2)#

#=(e^t-te^t)/((e^t+2t)^2)#

#=(e^t(1-t))/(e^t+2t)^2#

Also,

#y=t#

#dy/dt=1#

Therefore,

#v(t)=f'(t)= ((e^t(1-t))/(e^t+2t)^2,1)#

So, when #t=4#

#v(4)=f'(4)= ((e^4(1-4))/(e^4+2*4)^2,1)#

#=(-0.0418,1)#