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The function \(\small{s(t)}\) describes the position of a particle moving along a coordinate line, where \(s\) is in feet and \(t\) is in seconds.

\(\small{s(t) = t^{3}-10t^{2}+25t+9, \qquad t \ge 0}\)

 

(a) Find the velocity and acceleration functions.

\(v(t)\)3t^2 - 20t + 25

\(a(t)\)6t - 20

 

(b) Over what interval(s) is the particle moving in the positive direction? Use inf to represent \(\small{\infty}\), and U for the union of sets.

Interval: 

 

(c) Over what interval(s) is the particle moving in the negative direction? Use inf to represent \(\small{\infty}\), and U for the union of sets.

Interval: 

 

(d) Over what interval(s) does the particle have positive acceleration? Use inf to represent \(\small{\infty}\), and U for the union of sets.

Interval: 

 

(e) Over what interval(s) does the particle have negative acceleration? Use inf to represent \(\small{\infty}\), and U for the union of sets.

Interval: 

 

(f) Over what interval is the particle speeding up? Slowing down? Use inf to represent \(\small{\infty}\), and U for the union of sets.

Speeding up: 

Slowing down: 

 Mar 30, 2022
 #1
avatar+36434 
+1

a) correct

b)  solve    the velocity funtion > 0      3t^2 -20t+25 > 0

c)  Solve   3t^2 - 20t +20 < 0

d)  Similar to the velocity function  :   6t -20 >0

e)    6t-20 <0

f) Positive acceleration = speeding up

   slowing down   acceleration < 0

 Mar 30, 2022
 #2
avatar+36434 
+1

   Here is a graphical representation of the first one ( you can use the  quadratic fromula to find the roots...then test for > 0 in each of the three intervals)

 

 

https://www.desmos.com/calculator/5advoaevql

ElectricPavlov  Mar 30, 2022

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