# What is the instantaneous velocity of an object moving in accordance to # f(t)= (t^2,tcos(3t-pi/4)) # at # t=(pi)/8 #?

Instantaneous velocity of the object is

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To find the instantaneous velocity of the object at ( t = \frac{\pi}{8} ) in accordance with the function ( f(t) = (t^2, t\cos(3t - \frac{\pi}{4})) ), we need to differentiate the function with respect to time ( t ) and then evaluate it at ( t = \frac{\pi}{8} ).

Differentiating each component of the function ( f(t) ) with respect to time gives:

( \frac{df}{dt} = \left(\frac{d(t^2)}{dt}, \frac{d(t\cos(3t - \frac{\pi}{4}))}{dt}\right) )

( \frac{df}{dt} = (2t, \cos(3t - \frac{\pi}{4}) - 3t\sin(3t - \frac{\pi}{4})) )

Now, plug in ( t = \frac{\pi}{8} ) into each component:

( \frac{df}{dt} \bigg|_{t = \frac{\pi}{8}} = \left(2\left(\frac{\pi}{8}\right), \cos\left(3\left(\frac{\pi}{8}\right) - \frac{\pi}{4}\right) - 3\left(\frac{\pi}{8}\right)\sin\left(3\left(\frac{\pi}{8}\right) - \frac{\pi}{4}\right)\right) )

Finally, compute the values:

( \frac{df}{dt} \bigg|_{t = \frac{\pi}{8}} = \left(\frac{\pi}{4}, \cos(\frac{3\pi}{8} - \frac{\pi}{4}) - \frac{3\pi}{8}\sin(\frac{3\pi}{8} - \frac{\pi}{4})\right) )

This gives the instantaneous velocity of the object at ( t = \frac{\pi}{8} ).

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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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