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

Instantaneous velocity is

or in vector form

or

I assume that f(t) is a position, Standard convention for a position in space would be

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To find the instantaneous velocity of an object moving according to the function ( f(t) = (t - e^t, t e^{2t}) ) at ( t = 3 ), we need to find the derivative of the function with respect to time, which gives us the velocity vector. Then, we evaluate this velocity vector at ( t = 3 ) to find the instantaneous velocity at that specific time.

Taking the derivative of ( f(t) ) with respect to ( t ) gives us the velocity function:

[ f'(t) = \left( 1 - e^t, e^{2t} + 2t e^{2t} \right) ]

Evaluating this velocity function at ( t = 3 ) yields:

[ f'(3) = \left( 1 - e^3, e^6 + 6e^6 \right) ]

So, the instantaneous velocity of the object at ( t = 3 ) is ( \left( 1 - e^3, e^6 + 6e^6 \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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