Consider the function f(t) = 6sqrt(x) + 9

Question

Consider the function f(t) = 6sqrt(x) + 9

on the inverval [1,10]. 

Find the average or mean slope of the function on this interval 

 

By the Mean Value Theorem we know there exists a c in the open interval (1,10) such that f'(c) is equal to this mean slope . For this problem there is only one c that works. Find it. 

Answer 1

The average slope on (1,10)  is   the same as on the closed interval [1,10] so

 

   m_avg =    (1/9) ( f(10) - f(1)) =   (2/3) ( √10  -1 )

 

To find c  set  f'(c) =  m_avg

 

  (2/3)( √10 - 1) =   3 / √c

 

     This equation can be solved for x  

   c =  (81/4) (√10 - 1)^-2    ~ 4.33