Calculus-Sec-6-5-Solution


   6.5   Average Value of a Function                 by SGLee - HSKim, YJLim

                                                                                                              http://youtu.be/BVahd-DJoe8

 

1-7. Find the average value of the function on the given interval.

 

1. 

    http://matrix.skku.ac.kr/cal-lab/cal-6-5-1.html 

 

 

    .




2. 

    http://matrix.skku.ac.kr/cal-lab/cal-6-5-2.html 

 

 

    




3. 

    http://matrix.skku.ac.kr/cal-lab/cal-6-5-3.html 

 

 

    




4. 

 

     .

      To calculate this integral, use a change of variable , then . So,

      




5. 

    http://matrix.skku.ac.kr/cal-lab/cal-6-5-5.html 

 

 

     .




6. 

 

     .




7. 

    http://matrix.skku.ac.kr/cal-lab/cal-6-5-7.html 

 

 

     .

      Use a change of variable , then . So

      




8-10. (a) Find the average value of  on the given interval.

         (b) Find  such that .

         (c) Sketch the graph of  and a rectangle whose area is the same as the area under the graph of .

 

8. 

    http://matrix.skku.ac.kr/cal-lab/cal-6-5-8.html 

 

(a)

    

          

          .

(b)

     




9. 

 

      (a)

         

     (b)

        




10. 

 

 

      (a)   .

            Use change of variable to , then . So

            

          Also change of variable to ,

          then .

          .

                                

(b)http://matrix.skku.ac.kr/cal-lab/cal-6-5-10.html 

   




(c)  http://matrix.skku.ac.kr/cal-lab/cal-6-5-10-c.html 




11. If  is continuous and , show that  takes on the value  at least once on the interval .

 

 

    Since  is continuous and , there exists a constant  in  such that

      .




12. Find the numbers  such that the average value of  on the interval  is equal to .

 

 

         .

      That is   .




13. In a certain city the temperature (in  hours after  A.M. is modeled by the function  .

      Find the average temperature during the period from  A.M. to  P.M.

      http://matrix.skku.ac.kr/cal-lab/cal-6-5-13.html 

 

 

 .  




14. If a cup of coffee has temperature  in a room where the temperature is  then, according to Newton's Law of Cooling,

      the temperature of the coffee after  minutes is .

     What is the average temperature of the coffee during the first half hour?

     http://matrix.skku.ac.kr/cal-lab/cal-6-5-14.html 

 

 

     

              




15. The linear density in a rod  long is , where  is measured in meters from one end of the rod.

      Find the average density of the rod.

 

      Let . Then

      . Change of variable to , then . So

       




16. If  denotes the average value of  on the interval  and ,

     show that   

 

 

     

                                                     




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