The function s(t) describes the motion of a particle along a line. s(t) = t3 − 13t2 + 35t − 220

Question

The function s(t)

 describes the motion of a particle along a line.

s(t) = t³ − 13t² + 35t − 220

(a) Find the velocity function v(t)

 of the particle at any time t ≥ 0.

v(t) = 

 

(b) Identify the time interval(s) on which the particle is moving in a positive direction. (Enter your answer using interval notation.)

 

(c) Identify the time interval(s) on which the particle is moving in a negative direction. (Enter your answer using interval notation.)

 

(d) Identify the time(s) at which the particle changes direction. (Enter your answers as a comma-separated list.)

t = 

 

Answer 1

s(t) = t^3 -13t^2 + 35t -200

take the derivative

v(t) = s'(t) = 3t^2 -26t + 35 = velocity at time t

s'(t) = 3t^2 -26t + 35 = 0

(3t -5)(t-7) = 0

t = 5/3 and 7 seconds

the velocity >0 when s'(t) > 0, when t>7 or t<5/3 seconds

velocity <0 when 5/3 < t < 7 seconds

b) moving in a positive direction in the intervals

(-infinity, 5/3) U (7, infinity)

c) moving in a negative direction in the interval (5/3, 7)

d) times when direction changes are 5/3 and 7 seconds