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Maths change of sign coursework

Extracts from this document...

Introduction

                Osaamah Mohammed

Maths A2 Coursework

Change of Sign Method:-

3x3 – 6x + 1 = 0

Sketch: -

image15.pngimage25.pngimage33.pngimage31.pngimage25.pngimage25.pngimage09.pngimage14.pngimage00.pngimage01.png

Looking at the graph we find that the roots lie between these intervals:
[-2,-1]  [0,1]  [1,2]

Finding root interval between [0,1] using a Decimal Search

image03.pngimage32.pngimage04.pngimage02.pngimage34.pngimage06.pngimage05.pngimage06.png

Root [0 , 1]

X1 = 0

...read more.

Middle

image07.png

0.17

-0.005261

Root [0.16 , 0.17]

X

f(X)

0.16

0.052298

0.161

0.046520

0.162

0.040755

0.163

0.034992

0.164

0.029233

0.165

0.023476

0.166

0.017723

0.167

0.011972

0.168

0.0062249

0.169

0.00048043image08.pngimage07.png

0.170

-0.005261

Root [0.169 , 0.170]

X

f(X)

0.1690

0.00048043image07.pngimage08.png

0.1691

-0.0000923852

Root [0.1690 , 0.1691]

X

f(X)

0.1690

0.00048043

0.16901

0.00042299

0.16902

0.00036557

0.16903

0.00030814

0.16904

0.00025071

0.16905

0.00019328

0.16906

0.00013586

0.16907

0.00078427

0.16908

0.00021image08.pngimage07.png

0.16909

-0.000036426

From this method we find that the root lies between[0.16908 , 0.16909]

Root = 0.169085 ± 0.000005

testf (0.16908) = 0.00021

        f (0.16909) = -0.000036426

There is a change

...read more.

Conclusion

Start X1 = 0        Use Xn+1 = 2xn3 + 1
                                      
5

X2 = 2(0)3 + 1 = 0.2      

                  5

X3 = 2(0.2)3 + 1 = 0.216      

                  5

X4 = 2(0.216)3 + 1 = 0.22016      

                  5

X5 = 0.22134

X6= 0.22169

X7 = 0.22179

X8 = 0.22182

X9 = 0.22183                        root to 5 s.f (took 9 iterations)image30.png

X10 = 0.22183

Failure: -

Solve 2x3 – 5x + 1 = 0

Re-arrange this so that “X=….”

  • 2x3 – 5x + 1 = 0

      2x3 +1 = 5x

      2x3 = 5x – 1

      x3 = 5x – 1

                 2

      x = (5x – 1)

                 2

      x = g(x)

Sketch : -

...read more.

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