Problem :
Find the average value of
f
for
f (x) = 3x
^{2}  6x
on
[ 1, 1]
.
f _{avg}  = f (x)dx  
= x ^{3}3x ^{2} _{1} ^{1}  
= 13(13)  
= 1 
Problem :
Find the average value of
f
for
f (x) = sin(x)
on
[ Π, Π]
.
f _{avg}  = f (x)dx  
= cos(x) _{Π } ^{ Π }  
= 1((1))  
= 0 
Problem :
For the function above, find the
c
on
[ Π, Π]
such that
f (c) = f
_{avg}
Problem : Use the second fundamental theorem of calculus to find F' .
F(x) = (9t+9)dt 
F'(x) = 9x + 9
Problem : Now integrate the above expression for F and then take the derivative to find F'(x) and verify the result from the previous question.
F(x) = (9t+9)dt 
9t+9 dt = t ^{2}+9t _{2} ^{x}  


F(x) = x ^{2} +9x  36  
F'(x) = 9x + 9 
This verifies the result given by the second fundamental theorem.