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Question 6

Page history last edited by PBworks 17 years ago

Question 6


You MAY NOT use a calculator to solve this question.


Let on the closed interval [0, 4π].


(a) Approximate F(2π) using four inscribed rectangles.


(b) Find F'(2π).


(c) Find the average value of F'(x) on the interval [0, 4π].




(a) This question asks us to approximate the value of F(x) from 0 to 2π. To do this, we may use the methods we learned in class. The easiest approximation we can use (note that we can't use a calculator) that suits our need is the midpoint sum. Here's a graph of what we want to do:





(No one edited my work, so I did... is that allowed Mr. K?)


(b) Finding F'(2π).


Note: S = The indefinite integral sign. 0 S 2π reads "integrate from zero to two pi".


F(x)            = (0 S x) [cos(t/2) + (3/2)] dt

F(x)(dt/dx) = (0 S x) [cos(t/2) + (3/2)] (dt/dx)           The "0 S x" and (dt/dx) cancel each other out.


F'(x)           = cos(x/2) + (3/2)                                  Write what's left.

F'(2π)         = cos(2π/2) + (3/2)                                Plug in "2π" where x is.   


                  = cos(π) + (3/2)                                     Solve.

                  = -1 + (3/2)  

                  = (-2/2) + (3/2)

                  = 1/2


(c) Finding the average value of F'(x) on the interval [0, 4π]


Average value =    1      (a S b) F'(x) dx



                        =     1    (0 S 4π) cos(x/2) + (3/2)


                        =    1   [sin(x/2)(1/2) + (3/2)x](from zero to four)


                        =    1   [(0+6π) - (0)]  


                        =    1   (6π)




                        =  3









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