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IB Math HL Portfolio Type 1 (Ratio)

Extracts from this document...

Introduction

  1. image03.pngimage00.pngimage01.png

image02.png

Area A formed in the function y=x2 from y=0 to y=1 and the y-axis can be found by integrating the function with respect to y.

Area A:

 y=x2

 x=image04.pngimage04.png

 x=image81.pngimage81.png

image151.pngimage151.png dy

=[image171.pngimage171.pngimage05.pngimage05.pngimage11.pngimage11.png]image14.pngimage14.png

=[image27.pngimage27.png]image14.pngimage14.png

=(image41.pngimage41.png

=image50.pngimage50.png units2

Area B formed in the function y=x2 from x=0 to x=1 and the x-axis can be found by integrating the function with respect to x.

Area B:

y=x2

image59.pngimage59.png dx

=image66.pngimage66.png

=image72.pngimage72.png

=image79.pngimage79.png

=image86.pngimage86.png units2

Therefore, the ratio of the areas A and B for the function y=x2 is:

A : B = image50.pngimage50.png : image86.pngimage86.png

           = 2 : 1

Now try the same with the function y=xn when nimage105.pngimage105.pngZ+ and between x=0 and x=1.

Area A formed in the function from y=0 and y=1n and the y-axis can be found by integrating the function in respect for y.

Area A:

y=xn

 x=image51.pngimage51.png

image117.pngimage117.png dy

=[image53.pngimage53.pngimage05.pngimage05.pngimage54.pngimage54.png]image14.pngimage14.png

=[image157.pngimage157.png]image14.pngimage14.png

=(image164.pngimage164.png

=image165.pngimage165.png units2

Area B formed in the function from x=0 and x=1 and the x-axis can be found by integrating the function in respect of x.

Area B:

y=xn

image166.pngimage166.png dx

=image167.pngimage167.png

=image168.pngimage168.png

=image169.pngimage169.png

=image170.pngimage170.png units2

...read more.

Middle

 : image35.pngimage35.png

          = 47 : 1

In y=x47, the value of n is 47, and therefore the conjecture n : 1 works when n=47.

When n=113,

Area A:

y=x113

 x=image36.pngimage36.png

image37.pngimage37.png dy

=[image38.pngimage38.pngimage05.pngimage05.pngimage39.pngimage39.png]image14.pngimage14.png

=[image40.pngimage40.png]image14.pngimage14.png

=(image42.pngimage42.png

=image43.pngimage43.png units2

Area B:

y=x113

image44.pngimage44.png dx

=image45.pngimage45.png

=image46.pngimage46.png

=image47.pngimage47.png

=image48.pngimage48.png units2

Therefore, the ratio of the areas A and B in the function y=x113 is:

A : B = image43.pngimage43.png : image48.pngimage48.png

          = 113 : 1

In y=x113, the value of n is 113, and therefore the conjecture n : 1 works when n=113.

image49.pngimage49.png It can be assumed that the conjecture n : 1 works in any positive real numbers.

  1. To determine whether the conjecture only holds for areas between x=0 and x=1, now try the same thing but changing the values of x.

When x is from 0 to 2,

Area A:

y=xn

x=image51.pngimage51.png

image52.pngimage52.png dy

=[image53.pngimage53.pngimage05.pngimage05.pngimage54.pngimage54.png]image55.pngimage55.png

=[image56.pngimage56.png]image55.pngimage55.png

=(image57.pngimage57.png

=image58.pngimage58.png units2

Area B:

y=xn

image60.pngimage60.png dx

=image61.pngimage61.png

=image62.pngimage62.png

=image63.pngimage63.png

=image64.pngimage64.png units2

Therefore, the ratio of the areas A and B when x is between 0 and 2 is:

A : B = image58.pngimage58.png : image64.pngimage64.png

          = n : 1 ………. Conjecture works

When x is from 1 to 2,

Area A:

y=xn

x=image51.pngimage51.png

image65.pngimage65.png dy

=[image53.pngimage53.pngimage05.pngimage05.pngimage54.pngimage54.png]image67.pngimage67.png

=[image56.pngimage56.png]image67.pngimage67.png

=(image68.pngimage68.png

=image58.pngimage58.pngimage69.pngimage69.png

=image70.pngimage70.png units2

Area B:

y=xn

image71.pngimage71.png dx

=image73.pngimage73.png

=image74.pngimage74.png

=image75.pngimage75.png

=image76.pngimage76.png

=image77.pngimage77.png units2

...read more.

Conclusion

A : B = image122.pngimage122.png   : image127.pngimage127.png

          = n : 1 ………. Conjecture works

image128.pngimage128.pngs.

  1. When y=xn, x is between a and b and y is between an and bn,

Area A:

y=xn

x=image51.pngimage51.png

image129.pngimage129.png dy

=image130.pngimage130.png

=image131.pngimage131.png

=(image132.pngimage132.png

=image133.pngimage133.png

=n(image134.pngimage134.png units2

Area B:

y=xn

image135.pngimage135.png dx

=image136.pngimage136.png

=(image134.pngimage134.png units2

Therefore, the ratio of the areas A and B is,

A : B = n(image134.pngimage134.png : (image134.pngimage134.png

          = n : 1 ………. Conjecture works

image49.pngimage49.pngThe conjecture n : 1 is true for the general case y=xn from x=a to x=b where a<b and the area A is defined as y=xn, y=an, y=bn and the y-axis and the area B is defined as y=xn, x=a, x=b and the x-axis.

  1. (a) We need to find the volume B in order to find out the volume A so we start off with volume B.

Volume B:

y=xn

image137.pngimage137.png dx

=image138.pngimage138.png

=image139.pngimage139.png

=image140.pngimage140.png units3

And then volume A.

Volume A:

image141.png

=image142.pngimage142.png

=image143.pngimage143.png

=image144.pngimage144.png

=image145.pngimage145.png

=image146.pngimage146.png

=image147.pngimage147.png units3

Therefore, the ratio of the volumes A and B is:

A : B = image147.pngimage147.png : image140.pngimage140.png

          =2n : 1

image148.png

(b)Volume A:

y=xn

x=image51.pngimage51.png

image149.pngimage149.png dy

=image150.pngimage150.png

=image152.pngimage152.png

=image153.pngimage153.png

=image154.pngimage154.png units3

Volume B:

image155.png

=image156.pngimage156.png

=image158.pngimage158.png

=image159.pngimage159.png

=image160.pngimage160.png

=image161.pngimage161.png

=image162.pngimage162.png units3

Therefore, the ratio of the volumes A and B is:

A : B = image154.pngimage154.png : image162.pngimage162.png

          = n : 2

image163.png

...read more.

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