Search Menu

Contents

Problems for "Average Value and Second Fundamental Theorem"

Problems for "Average Value and Second Fundamental Theorem"

Problems for "Average Value and Second Fundamental Theorem"

Problems for "Average Value and Second Fundamental Theorem"

Problems for "Average Value and Second Fundamental Theorem"

Problems for "Average Value and Second Fundamental Theorem"

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  
    = 1-3-(-1-3)  
    = 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

sin(x) = 0 at x = 0 .

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      

x 2+9x - (18 + 18)
     
F(x) = x 2 +9x - 36      
F'(x) = 9x + 9      

This verifies the result given by the second fundamental theorem.