MAT136H1 Lecture Notes - Antiderivative

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4 Feb 2013
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5.4 Integration & Anti-derivatives
Indefinite Integrals
Question #4 (Medium): Working With Acceleration, Velocity, Displacement, and Distance
Strategy
Acceleration is the derivative of velocity. Velocity is the derivative of displacement.
Thus, 
, and 
.
Displacement is the net distance travelled, so   
.
But Distance is the total distance travelled, so   
.
Sample Question
The given acceleration (in and the initial velocity (in describe a particle moving along a line.
  ,  ,     
1) Find the velocity at time .
2) Find the total distance travelled during the given time interval.
Solution
1) Velocity is the anti-derivative of acceleration:
Thus, 
  
   
    
a. Since  , then      . So the constant factor
 .
So the velocity function at time is:     .
2) Note that the distance is not the net but total distance travelled:
  
, then 

   

Factoring          , the -intercepts are  
.
So   over the intervals 
and ; and   over the interval
Split the interval according to these  intervals:
    

   
   
   

Taking the anti-derivative, then plugging in endpoints of each subinterval:
 
 
 
 
 


 
  
  

  
       
  
  

     
  
     

      
  
  
  
      

  
 
     
Therefore, the total distance travelled during the time interval     is .
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