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

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Answer 1 · 2017-11-22T07:44:48

The instantaneous velocity is $= (- 0.0418, 1) m s^{- 1}$ $=(-0.0418,1)ms^-1$

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 $t=4$

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

$=(-0.0418,1)$

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