What is the instantaneous velocity of an object moving in accordance to # f(t)= (ln(t^2),3t-2) # at # t=2 #?

Answer 1

Instantaneous velocity at #t=2# is #1/3#

When an object moves in accordance to #f(t)=(ln(t^2),3t-2)#
its instantaneous velocity is given by #(dy)/(dx)=((dy)/(dt))/((dx)/(dt))#
As in parametric equation #y=ln(t^2)# and #x=3t-2#
#(dy)/(dt)=2txx1/t^2=2/t# and #(dx)/(dt)=3# or
#(dy)/(dx)=(2/t)/3=2/(3t)#
Hence, instantaneous velocity at #t=2# is #2/(3xx2)=1/3#
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Answer 2

To find the instantaneous velocity of an object moving according to the function ( f(t) = (\ln(t^2), 3t - 2) ) at ( t = 2 ), we first differentiate the function with respect to time to get the velocity function, and then evaluate it at ( t = 2 ).

The velocity function, ( v(t) ), is given by the derivative of the position function, ( f(t) ):

[ v(t) = \frac{d}{dt} f(t) ]

Differentiating each component of ( f(t) ) separately:

[ v(t) = \left( \frac{d}{dt} \ln(t^2), \frac{d}{dt} (3t - 2) \right) ]

[ v(t) = \left( \frac{2}{t}, 3 \right) ]

Now, to find the instantaneous velocity at ( t = 2 ), we substitute ( t = 2 ) into the velocity function:

[ v(2) = \left( \frac{2}{2}, 3 \right) ]

[ v(2) = (1, 3) ]

Therefore, the instantaneous velocity of the object at ( t = 2 ) is ( (1, 3) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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