Calculus

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A particle moves along a horizontal line so that at any time t its position is given by x(t)=cost-t. Time is measured in seconds and x is measured in meters.
a.) Find the velocity as a function t. Use your answer to determine the velocity of the particle when t=pi/6 seconds. Indicate units of measure.
b.) Find the acceleration as a function of the time t. Use your answer to determine the velocity of the particle when t=pi/6 seconds. Indicate units of measure.
c.) What are the values of t,0≤t≤2pi, for which the particle is at rest?
d.) What are the values of t, 0≤t≤2pi, for which the particle moves to the right?
e.) At t=pi/6 seconds, is the particle speeding up or slowing down? Justify your answer.

• Calculus -

x = cos t - t

dx/dt = v = -sin t - 1
sin pi/6 = sin 30 = 1/2
so
v = -1.5

a = dv/dt = -cos t
I think you want the acceleration a = -sqrt3/2

when is v = 0 ?
when sin t = -1 which is t = 3 pi/2

when is v >0? when sin t <-1 which is never

when t = pi/6 , a is negative so v is getting less negative so slowing down

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