# What is the instantaneous velocity of an object moving in accordance to # f(t)= (t/(t-5),3t-2) # at # t=-2 #?

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To find the instantaneous velocity of the object at ( t = -2 ), you need to find the derivative of the function ( f(t) ) with respect to ( t ), which represents the velocity vector. Then, evaluate this derivative function at ( t = -2 ).

Given ( f(t) = \left(\frac{t}{t - 5}, 3t - 2\right) ), the derivative of ( f(t) ) with respect to ( t ) is:

( f'(t) = \left(\frac{d}{dt}\left(\frac{t}{t - 5}\right), \frac{d}{dt}(3t - 2)\right) )

To compute the derivatives:

( \frac{d}{dt}\left(\frac{t}{t - 5}\right) = \frac{(t - 5) - t(1)}{(t - 5)^2} )

( \frac{d}{dt}(3t - 2) = 3 )

Now, you have the derivative function:

( f'(t) = \left(\frac{-5}{(t - 5)^2}, 3\right) )

To find the instantaneous velocity at ( t = -2 ), substitute ( t = -2 ) into ( f'(t) ):

( f'(-2) = \left(\frac{-5}{(-2 - 5)^2}, 3\right) )

( f'(-2) = \left(\frac{-5}{(-7)^2}, 3\right) )

( f'(-2) = \left(\frac{-5}{49}, 3\right) )

So, the instantaneous velocity of the object at ( t = -2 ) is ( \left(\frac{-5}{49}, 3\right) ).

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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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