# The position of an object moving along a line is given by #p(t) = sin(3t- pi /4) +2 #. What is the speed of the object at #t = (3pi) /4 #?

Velocity of an object is the time derivative of it's position coordinate(s). If the position is given as a function of time, first we must find the time derivative to find the velocity function.

Distinguishing the term,

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To find the speed of the object at ( t = \frac{3\pi}{4} ), we need to differentiate the position function ( p(t) ) with respect to time ( t ), and then evaluate the derivative at ( t = \frac{3\pi}{4} ).

The derivative of ( p(t) ) is ( p'(t) = 3\cos(3t - \frac{\pi}{4}) ).

Now, plug in ( t = \frac{3\pi}{4} ) into the derivative: ( p'(\frac{3\pi}{4}) = 3\cos(\frac{3\cdot3\pi}{4} - \frac{\pi}{4}) = 3\cos(\frac{9\pi}{4} - \frac{\pi}{4}) = 3\cos(2\pi) = 3\cos(0) = 3 ).

Therefore, the speed of the object at ( t = \frac{3\pi}{4} ) is ( 3 ).

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