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Question 8

Page history last edited by Rence 15 years ago

A Calculator MAY NOT be used to help answer 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].


Quick Find Code For This Problem.


Legend: [QUES8LGND]

Solution For Part a.): [QUES8PA]

Solution For Part b.): [QUES8PB]

Solution For Part c.): [QUES8PC]


How To Use Quick Find.


-To Use Quick Find, copy (Ctrl+C/Command+C) the code (All the stuff INSIDE the square brackets in the code region above) for which ever section you'd like to go to, then, hit in this order;






or if your on a Mac You'd hit






Then, your webbrowser should bring you directly to the beginning of the solution indicated by the code, making navigation through longish solutions much easier. Enjoy :]







Red Stuff for Hot Stuff. Aka, read this.

Green Means Go! Follow these instructions to solve the problem.

Blue Is for Chill. Basically, this is any other kind of interesing note or idea that may be of interest.





A) The main pattern that helps you figure out what method to use is given when the question asks for an approximation with the use of four rectangular intervals. It also says that these rectangles are inscribed into the function. The function is decreasing as we go to the right, and the rectangles are inscribed. Therefore, we are looking for a Right Hand Riemann Sum.


The inner function that is integrated helps define what the function in terms of x is, but the inner function is in terms of t.


Area of a rectangle is length x width. This is needed to find the area under the curve which are represented with four subintervals.

[will put picture here soon]


Between 0 and 2pi, the four subintervals occur at 0 to pi/2, pi/2 to pi, pi to 3pi/2, 3pi/2 to 2pi.


RHS = pi/2 [F(pi/2) + F(pi) + F(3pi/2) + F(2pi)]

[antidifferentiation picture here... will be posted tomorrow because i am FAMISHED.(grr.....)]

RHS = 7.8539 units.




B) The second fundamental theorem of calculus:


If F is a continuous function on some interval [a,b] and an accumulation function A is defined by


 then A'(x) = f(x)


This just means that the derivative of an accumulation function is the underlying function. 



In this part of the problem all I have to do is just plug the numbers in and find the answer.

Since F(x) equal to the antiderative of cos(t/3)+(3/2) therefore F'(x) is cos(t/3)+(3/2). 




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

Average value formula:





In simpler words, this formula is saying that somewhere on the interval, the function will equal its average value.  It's the same thinking as finding the average to any set of numbers.  We would add those numbers up and divide by how many of them there is.  That's exactly what's going on in this formula.  The integral sign means we are adding up the area of an infinte number of rectangles under the function and dividing it by the size of the interval.


Then plug the equations and numbers into the formula.

Since F(x) equal to the anitderative of cos(t/3)+(3/2) therefore F'(x) is cos(t/3)+(3/2).




intergate it from 0 to 4π




then work out to find the answer.


Therefore the average value of F'(x) on the interval [0, 4π] is 1.4602 

Comments (2)

Darren Kuropatwa said

at 12:28 am on Apr 19, 2009

Yinan, look at the top right corner of the page. There's a link there that says: "Upload Files"

yinan said

at 10:26 am on Apr 19, 2009

Thank you Mr.K I got it

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