# Calc

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A particle moves along the x-axis in such a way that it's position in time t for t is greator or equal to 0 is given by x= 1/3t^3 - 3t^2 +8

A) show that at time t= 0 the particle is moving to the right.
B) find all values of t for which the particle is moving to the left.
C) what is the position of the particle at time t=3
D) when t=3, what is the total distance the particle has traveled

• Calc -

x = 1/3t^3 - 3t^2 + 8

dx/dt = t^2 - 6t

a)
when t=0 ,
dx/dt = 0-0 = 0

showing that x is not moving at all at that time.

b) to be moving to the left
t^2 - 6t < 0
t(t-6) < 0
which is true for all
0 < t < 6

c) when t=3
x = (1/3)(27) - 3(9) + 8 = -10

d) when t= 0 , x = 8
when t=3 , x = -10
the particle has traveled 18 units

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