upper lower rectangles (1 Viewer)

[fullonoob]

Use upper and lower rectangles to prove that


how do you do upper lower rectangles o-o

 

[Aquawhite]

What are we proving?

Edit: I see it now, and I do not know what you even mean by upper and lower rectangles. Is this some kind of colloquial lingo that you kids are using these days... >_>

 

[Pwnage101]

Think about what you are trying to prove, now you have an integral - draw what this means. The integral you have supplied is the area between the curve 1/x and teh x-axis bounded by x=2^n and x=2^n+1

So your 'lower' bound for the integral will be the area of the rectangle with width (2^n+1-2^n) and height of the lower of f(2^n) and f(2^n+1) where f(x)=1/x. Clearly the larger x is (x>0), the smaller 1/x=f(x) is, so your lower limit is the rectangle with height 1/(2^n+1) and the upepr limit will have height 1/(2^n).

A diagram will be very useful. So:

area of smaller rectangle < supplied integral < area of larger rectangle

area of smaller rectangle = height x width = 1/(2^n+1) x (2^n+1-2^n) = 1/(2^n+1) x (2-1)2^n = 1/(2^n+1) x 2^n = 2^n/2^n+1 = 1/2

area of larger rectangle = height x width = 1/2^n x 2^n (same width as above) = 1

therefore 2^-1 < supplied integral < 1

Q.E.D.

 

[Mature Lamb]

Do we need to know upper and lower rectangles? I thought that wasn't part of the syllabus anymore D=

 

[shaon0]

Do we need to know upper and lower rectangles? I thought that wasn't part of the syllabus anymore D=

I've seen it in textbooks before. Wouldn't be harmful to learn the method if you're doing 4unit though.

 

[jet]

Yeah I've seen it in recent HSC papers. It's definitely something you should know.

 

[fullonoob]

how do you know the height and width of the rectangles?

 

[fullonoob]

Sketch the region cut off in the first quadrant by y = 2x-3 / (x^2 - 3x -4) and find its area.
weird question o-o
you get ln {-} which you cant do, both limits end up with with - ln
but the answer you get is just positive ln of the two O_O