# The position of an object moving along a line is given by #p(t) = 3t - sin(( pi )/6t) #. What is the speed of the object at #t = 2 #?

The speed is

The equation indicates the object's position.

The derivative of position is the speed.

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To find the speed of the object at ( t = 2 ), we need to find the derivative of the position function ( p(t) ) with respect to time ( t ), and then evaluate it at ( t = 2 ).

The derivative of ( p(t) ) with respect to ( t ) is ( p'(t) ).

( p'(t) = 3 - \frac{\pi}{6} \cos\left(\frac{\pi}{6}t\right) )

Now, evaluate ( p'(2) ):

( p'(2) = 3 - \frac{\pi}{6} \cos\left(\frac{\pi}{6} \times 2\right) )

( p'(2) = 3 - \frac{\pi}{6} \cos\left(\frac{\pi}{3}\right) )

( p'(2) = 3 - \frac{\pi}{6} \times \frac{1}{2} )

( p'(2) = 3 - \frac{\pi}{12} )

Therefore, the speed of the object at ( t = 2 ) is ( 3 - \frac{\pi}{12} ) units per time.

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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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