Suppose that a particle moves according to the law of motion s = t^2 - 11 t + 33. Use interval notation to indicate when the particle is moving in the positive

Question

Calculus I, Derivative

t>=0

Answer 1

Hi Bickey.

 

A particle that is moving to the right is moving in a positive direction.

A partical that is moving to the left is moving in a negative direction.

 

If s is the position, then ds/dt is the velocity of the particle.  

 

What we need to do is find (ds/dt) and determine when it is positive, that is the time interval that it will be moving in the positive direction.

 

ds/dt = 2t-11

 

We need to find the critical number, the value of t where the derivative is 0.  This is the value of t where the particle stops (velocity = 0) and changes direction.  Then we just need to figure out where (ds/dt) is positive.

 

We set 2t-11=0  and find that the velocity is 0 (the particle stops to change direction) at  t=5.5   Now we just have to figure out which direction it was going before t=5.5 and after t=5.5

 

Use any value for t that is less than  5.5, let's choose t=1.  2(1)-11 = -9  

Since (ds/dt) is negative before t=5.5, it means that the particle must have been moving left before t=5.5

 

At t=5.5 it stops (velocity = 0) and changes to moving right - verify this by testing a value of "t" that is >5.5

 

If we test t=10, we get (ds/dt)=2(10)-11 = 9, this is positive, so we know that after t=5.5, the particle is moving in the positive direction, or moving right.

 

In interval notation, the particle moves in a positive direction from t=5.5 until the end of time (infinity) but at t=5.5 it is not positive nor negative, it is 0.  So with interval notation, both endpoints will be curved because the end point t=5.5 is not included nor is the end point of infinity - inifinity is never included because "t"  never actually gets there.

 

Answer:  (5.5,+∞)