• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Finding the root of an equation

Extracts from this document...

Introduction

Finding the root

Change of sign method 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. 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

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. 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. 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. 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.

Conclusion  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.  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. 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.

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

# Related AS and A Level Core & Pure Mathematics essays

1.  5 star(s)

+ an(n-1) (xn-2)h� + ... + ahn - axn) 2 h I can now multiply this through by h to get - anxn-1 + an(n-1) (xn-2)h + ... ahn-1. 2 Like all previous binomial expansion, every single value contains an h term, and continually limits to 0. The only term left over is naxn-1 My investigation is now at end, and perhaps the coursework is now complete.

2. ## Sequences and series investigation

Sequence 5 : N = 5 _(5�) - 2(52) + 2Y(5) - 1 = 166Y - 50 + 135 - 1 = 129 The formula on this sequence seems to be successful. I will now apply it to another sequence to be 100% correct: N = 4 _(4�) - 2(42)

1. ## 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

2. Results x Gradient 2 7.82 3 12.41 4 16.8 -2.5 -9.23 -4 -17.77 Values at point: 2: x= 1.15, y=9 [9/1.15] 3: x= 1.45, y= 18 [18/1.45] 4: x= 1.9, y= 32 [32/1.9] -2.5: x= 1.3, y = -12 [-12/1.3] -4: x= 1.8, y=-32 [-32/1.8] Above are the differences in the 'x' and 'y' scales.

1. ## Mathematical Investigation

The exact value of a stretched period of one complete cycle (two wave peaks or 2pi radians) can be obtained by the formula 2pi/b. Part III Investigate the family of curves y=sin(x+c) Figure#5: Graphs of different functions of the family curve y=sin(x+c): y=sin(x); y=sin[x+ (pi/4)] and y=sin[x-(pi/4)].

2. ## 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)

1. ## 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

2. ## Change of sign method - Finding a root by using change of sign method

Because this is the middle point between the interval. Fail example by using Exel It is not guaranteed to use this method, because there still has some problems in it. See the graph below: As we can see the curve touches the x axis. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 