• 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)

    The gradient function can find out any function of any curve of the y = axn. "A loose end" Some parts of this investigation require that I do a straight line graph for y = axn. I will only binomially expand this, and compare the result to the gradient function - y= x (x+h)-x = h = 1 =x0.

  2. Mathematical Investigation

    the value of c tells how much each wave moves 3. The sign of "c" of the zero of (bx+c) dictates the direction of translation of the waves; positive "c" tells the wave to move to the right (in reference to y=sin(x).); negative "c" tells the wave to move to the left (in reference to y=sin(x)).

  1. Sequences and series investigation

    � 3 = 5a + b = 4Y If I use the equation above 6a + b = 6. I can take my latest equation and subtract it from it to find 'a'. So, _6a + b = 6 5a + b = 4Y a = 15 Now that we

  2. Numerical Method (Maths Investigation)

    equation as it leads to the failure of basis iteration. FAILURE OF REARRANGEMENT METHOD The failure of basis iteration to find the required root is also considered as the failure of the rearrangement method. Now, let me show it graphically here.

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

    24-sided: The triangle ACD in 24-sided is one twelfth of the . Known: Segment BC=1 Angle ABC=90 degree Angle ACB=7.5 degree Angle ACD=15 degree Segment AB = Segment BD So Segment BC=the radius of circle So Then we use the area formula of triangle: As we known, the area of triangle ACD is one twenty-fourth of area of equilateral polygon.

  2. By the software Equation Grapher, we can find the smallest positive root of y=0 ...

    Therefore, by the Factor Theorem, the equation in (ii) has roots at x = -2, and 2. Hence, the smallest positive root is . 1.(b) From the graph, it seems that the smallest positive root is f () =()

  1. Change of sign method.

    I can use a spreadsheets and close ups of my graph to find the roots to a given number of decimal places. I have chosen to look closely at the root in the interval [0, 1]. Shown below are the formulae that I entered into the spreadsheet to obtain my required results.

  2. The Gradient Fraction

    graphs firstly using the Triangle Method and then comprise other methods such as the Increment Method. I will begin by drawing the table of values, and then finding the results of the graphs. 'y=x2' solved by the 'Triangle Method' x -4 -3 -2 -1 0 1 2 3 4 x2

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