4500 words - 18 pages

Change of Sign Method - Mathematical Essay

In order to find the roots of an equation that cannot be solved

algebraically, I can use numerical methods to do this instead. One of

these methods is the change of sign method. From looking at a graph of

my equation I can find two integers that my root lies between, then

from there, using spreadsheets, I can use the change of sign method to

discover where the root lies to five decimal places.

I have chosen to try to solve the equation: 5x3-7x+1=0

First, I drew the graph of y=5x3-7x+1 in autograph to find where the

roots roughly lie by looking where the graph cuts the x-axis.

Y=5x3-7x+1

If it

is negative it replaces f(a) as the lower bound of the interval.

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If this number is negative, it is the lowest bound for the interval.

1. These numbers are the integers that I have discovered my roots to

be between.

From this spread sheet I can find the upper and lower bounds of the

intervals so that I can narrow down the accuracy of the root of the

equation. Shown above are only the first few rows of the spreadsheet,

to find the roots to a high enough accuracy, the formulas are filled

down for more rows. If I look at the numbers that are present in the

spreadsheet once the equation has been entered, we can find the root

to a give number of decimal places.

Change of sign method

y=5x^3-7x+1

a

b

(a+b)/2

f(a)

f(b)

f((a+b)/2)

1.0000000000

2.0000000000

1.5000000000

-1.0000000000

27.0000000000

7.3750000000

1.0000000000

1.5000000000

1.2500000000

-1.0000000000

7.3750000000

2.0156250000

1.0000000000

1.2500000000

1.1250000000

-1.0000000000

2.0156250000

0.2441406250

1.0000000000

1.1250000000

1.0625000000

-1.0000000000

0.2441406250

-0.4401855469

1.0625000000

1.1250000000

1.0937500000

-0.4401855469

0.2441406250

-0.1140441895

1.0937500000

1.1250000000

1.1093750000

-0.1140441895

0.2441406250

0.0609855652

1.0937500000

1.1093750000

1.1015625000

-0.1140441895

0.0609855652

-0.0275378227

1.1015625000

1.1093750000

1.1054687500

-0.0275378227

0.0609855652

0.0164708495

1.1015625000

1.1054687500

1.1035156250

-0.0275378227

0.0164708495

-0.0055966303

1.1035156250

1.1054687500

1.1044921875

-0.0055966303

0.0164708495

0.0054213097

1.1035156250

1.1044921875

1.1040039063

-0.0055966303

0.0054213097

-0.0000916085

1.1040039063

1.1044921875

1.1042480469

-0.0000916085

0.0054213097

0.0026638633

1.1040039063

1.1042480469

1.1041259766

-0.0000916085

0.0026638633

0.0012858806

1.1040039063

1.1041259766

1.1040649414

-0.0000916085

0.0012858806

0.0005970744

1.1040039063

1.1040649414

1.1040344238

-0.0000916085

0.0005970744

0.0002527175

1.1040039063

1.1040344238

1.1040191650

-0.0000916085

0.0002527175

0.0000805507

1.1040039063

1.1040191650

1.1040115356

-0.0000916085

0.0000805507

-0.0000055299

1.1040115356

1.1040191650

1.1040153503

-0.0000055299

0.0000805507

0.0000375101

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