# What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(sqrtt),1/t+2) # at # t=1 #?

Please read below.

We have:

We manipulate the equations a bit.

We now have:

Power rule:

Chain rule:

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To find the instantaneous velocity of the object at ( t = 1 ), we differentiate the function ( f(t) ) with respect to time ( t ) and then evaluate it at ( t = 1 ).

Given ( f(t) = (e^{\sqrt{t}}, \frac{1}{t} + 2) ), the velocity vector is ( \frac{df}{dt} = \left(\frac{d}{dt} e^{\sqrt{t}}, \frac{d}{dt} \left(\frac{1}{t} + 2\right)\right) ).

Differentiating each component separately, we get:

( \frac{d}{dt} e^{\sqrt{t}} = \frac{1}{2\sqrt{t}}e^{\sqrt{t}} )

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

At ( t = 1 ), we have:

( \frac{d}{dt} e^{\sqrt{1}} = \frac{1}{2}e )

( \frac{d}{dt} \left(\frac{1}{1} + 2\right) = -1 )

So, the instantaneous velocity of the object at ( t = 1 ) is ( \left(\frac{1}{2}e, -1\right) ).

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The instantaneous velocity of an object moving according to the parametric function ( f(t) = (e^{\sqrt{t}}, \frac{1}{t} + 2) ) at ( t = 1 ) can be found by taking the derivative of each component of the function with respect to time ( t ), and then evaluating them at ( t = 1 ).

The derivative of ( e^{\sqrt{t}} ) with respect to ( t ) is ( \frac{d}{dt}(e^{\sqrt{t}}) = \frac{1}{2\sqrt{t}} e^{\sqrt{t}} ).

The derivative of ( \frac{1}{t} + 2 ) with respect to ( t ) is ( \frac{d}{dt}(\frac{1}{t} + 2) = -\frac{1}{t^2} ).

Evaluating these derivatives at ( t = 1 ), we get:

[ \frac{1}{2\sqrt{1}} e^{\sqrt{1}} = \frac{1}{2} e ]

[ -\frac{1}{1^2} = -1 ]

Therefore, the instantaneous velocity of the object at ( t = 1 ) is ( (\frac{1}{2} e, -1) ).

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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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