• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14

Finding the root of an equation

Extracts from this document...

Introduction

Finding the root

Change of sign method

image00.png

This is the graph of the equation y=3x3+4x2-3x-1. The change of sign method will be used to find out the value of the root in this graph which lies between -1 and 0. The change of sign method requires finding out where a change of sign occurs in the y-co-ordinate and establish where, within the gap found out, the next change of sign will occur. This will result in a narrowing down of the group of values the root could lie between which will result in an answer of a satisfactory degree of accuracy.

The first part of the investigation involves finding out where, between -1 and 3, a change of sign occurs.

X

Y

-1

3

-0.9

2.753

-0.8

2.424

-0.7

2.031

-0.6

1.592

-0.5

1.125

-0.4

0.648

-0.3

0.179

-0.2

-0.264

-0.1

-0.663

0

-1

        There is a change of sign and therefore a root in the region between -0.3 and -0.2.

image01.png

        The next part of the investigation is to find out where a change of sign occurs between -0.3 and -0.2.

X

Y

-0.3

0.179

-0.29

0.133233

-0.28

0.087744

-0.27

0.042551

-0.26

-0.00233

-0.25

-0.04688

-0.24

-0.09107

-0.23

-0.1349

-0.22

-0.17834

-0.21

-0.22138

-0.2

-0.264

...read more.

Middle

0.002145

-0.26

-0.00233

        There is a change of sign and therefore a root in the region between -0.261 and -0.260.

image13.png

        The next part is to establish where, between -0.261 and -0.26, a change of sign occurs.

X

Y

-0.261

0.0021453

-0.2609

0.0016978

-0.2608

0.0012503

-0.2607

0.0008029

-0.2606

0.0003556

-0.2605

-0.0000918

-0.2604

-0.0005391

-0.2603

-0.0009864

-0.2602

-0.0014336

-0.2601

-0.0018808

-0.26

-0.0023280

        There is a change of sign and therefore a root in the region between -0.2606 and -0.2605.

image14.png

        The next part of the investigation is establishing where, between -0.2606 and -0.2605, a change of sign occurs.

X

Y

0.2606

0.000355557

-0.26059

0.000310821

-0.26058

0.000266086

-0.26057

0.000221351

-0.26056

0.000176616

-0.26055

0.000131882

-0.26054

0.000087148

-0.26053

0.000042414

-0.26052

-0.000002319558

-0.26051

-0.000047053

-0.2605

-0.000091785

        There is a change of sign and therefore a root between -0.26053 and -0.26052.

image15.png

        As shown visibly, and as can be proved mathematically, the root of the equation which lies between -1 and 0 is closer to -0.26052 than it is to -0.26053. Therefore, the root of the equation which lies between -1 and 0 is -0.26052 to 5 significant figures.

        Because of the nature of the answer when given to five significant figures, the amount of error is +0.000005. This makes the percentage error of the answer is 0.

...read more.

Conclusion

image07.pngimage08.png

        This shows the process of Newton-Rapshon iterations aiming to find out the value of the root between -1 and 0 which lies closer to zero. The starting value chosen in this case in 0. This has resulted in a value of -0.22708 to 5.s.f. being obtained for the root.

        The other root lies between 1 and 2.

image09.pngimage10.png

        The starting value for this root was 2. This root has been found to be 1.7368 to 5.s.f.

        However, there are occasions where finding the root using the Newton-Rapshon method doesn’t always work.

image11.png

        This is the graph y=2x3+2x2-2x-1. If the root between -2 and -1 was under investigation and the starting value of -1 was used, the gradient of the tangent at this point is zero. Therefore, no tangent can be drawn from this point to intercept the x-axis because the tangent at this point is horizontal.

        y=2x3+2x2-2x-1; When y=-1, x=1

        dy/dx=6x2+4x-2; When x=-1, dy/dx=6-4-2=0

        y=mx+c; y=-1, m=0, x=1, c=?

        y-mx=c; -1-0=-1 Therefore, equation of tangent is y=-1

        When y=0, y cannot =0. Therefore the Newton-Rapshon method cannot be applied here.

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    The Gradient Function

    5 star(s)

    This fits in with a pattern i.e. 1(nx n-1) = gradient function, where a = 1 2(nx n-1) = gradient function, where a =2. 3(nxn-1) = gradient function, where a =3. It is clear from these results that the overall equation for curves of ax^n is nax^n-1.

  2. Investigation of circumference ratio - finding the value of pi.

    48-sided: The triangle ACD in 24-sided is one twelfth of the . Known: Segment BC=1 Angle ABC=90 degree Angle ACB=3.75 degree Angle ACD=7.5 degree Segment AB = Segment BD So Segment BC=the radius of circle So =0.065 Then we use the area formula of triangle: As we known, the area

  1. Three ways of reading The Bloody Chamber.

    Umbeto Eco characterises the post-modern the following way: "I think of a post-modern attitude as that of a man who loves a very cultivated woman and knows he cannot say to her, 'I love you madly', because he knows that she knows (and that she knows that he knows)

  2. Mathematical Investigation

    Therefore based upon the observations the following points can be conjectured: 1. varying values of "c" only changes the position of the waves in a horizontal translation along the x-axis while the amplitude and period remain the same with the original function 2.

  1. The Gradient Fraction

    All the points at the 'x' scale have gradients of 2. I begin to comprehend the relationship between the equation and the gradient. The equation of this graph is 'y=2x'. The 2 in front of the 'x' shows the gradient in the straight line.

  2. Sequences and series investigation

    2(n -1) (n - 1) + 2n - 1 2) 2(n2 - 2n + 1) + 2n - 1 3) 2n2 - 4n + 2 + 2n - 1 4) 2n2 - 2n + 1 Therefore my final equation is: 2n2 - 2n + 1 Proving My Equation and Using

  1. Triminoes Investigation

    = a x 2� + b x 2� + c x 2 + d = 8a + 4b + 2c + d = 10 - equation 2 f (3) = a x 3� + b x 3� + c x 3 + d = 27a + 9b + 3c + d = 20 - equation 3 f (4)

  2. Examining, analysing and comparing three different ways in which to find the roots to ...

    4.19 4.191 4.192 4.193 4.194 4.195 4.196 4.197 4.198 4.199 4.2 -0.29 -0.25 -0.21 -0.18 -0.14 -0.10 -0.06 -0.03 0.01 0.05 0.09 Here the interval was between [4.197, 4.198] but in order to decide on a root with a 3 decimal place accuracy, I must go to 4 decimals and

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work