What is the instantaneous velocity of an object moving in accordance to # f(t)= (e^(t-t^2),e^t/t^2) # at # t=2 #?
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To find the instantaneous velocity of an object moving according to the function ( f(t) = (e^{t - t^2}, \frac{e^t}{t^2}) ) at ( t = 2 ), we need to differentiate each component of the function with respect to time ( t ) and then evaluate them at ( t = 2 ).
The first component of the function is ( e^{t - t^2} ), so its derivative with respect to ( t ) is ( \frac{d}{dt} e^{t - t^2} = e^{t - t^2} (1 - 2t) ).
The second component of the function is ( \frac{e^t}{t^2} ), so its derivative with respect to ( t ) is ( \frac{d}{dt} \left( \frac{e^t}{t^2} \right) = \frac{e^t (t^2 \cdot 1 - 2t \cdot e^t)}{t^4} = \frac{e^t (t - 2t^2)}{t^4} ).
Now, we can evaluate these derivatives at ( t = 2 ):
For the first component: ( e^{2 - 2^2} (1 - 2 \cdot 2) = e^{-2} (-3) ).
For the second component: ( \frac{e^2 (2 - 2 \cdot 2^2)}{2^4} = \frac{e^2 (-6)}{16} ).
So, the instantaneous velocity of the object at ( t = 2 ) is ( \left( e^{-2} (-3), \frac{e^2 (-6)}{16} \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.
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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