OCR MEI C3 Coursework - Numerical Methods

C3 Coursework Change of sign (decimal search) Finding a root of an equation with graphical illustration f(x)=x3-7x2+2x+1 This is graph of y=x3-7x2+2x+1 The graph shows that roots of f(x)=0 exist in the intervals [-1,0]; [0,1]; [6,7] We shall test for a root in the interval [6,7] x f(x) 6.0 -23 6.1 -20.289 6.2 -17.352 6.3 -14.183 6.4 -10.776 6.5 -7.125 6.6 -3.224 6.7 0.933 Change of sign indicates root exists in interval [6.6,6.7] This means that x=6.65±0.05 x=7 (0d.p.) x f(x) 6.60 -3.224 6.61 -2.81992 6.62 -2.41327 6.63 -2.00405 6.64 -1.59226 6.65 -1.17787 6.66 -0.7609 6.67 -0.34134 6.68 0.080832 Change of sign indicates root in interval [6.67,6.68] This means that x=6.675±0.005 x=6.7 (1d.p.) x f(x) 6.670 -0.34134 6.671 -0.29924 6.672 -0.25711 6.673 -0.21496 6.674 -0.17278 6.675 -0.13058 6.676 -0.08835 6.677 -0.04609 6.678 -0.00381 6.679 0.038498 Change of sign indicates root in interval [6.678,6.679] This means that x=6.6785±0.0005 x=6.68 (2d.p.) X f(x) 6.6780 -0.00381 6.6781 0.000419 Change of sign indicates root in interval [6.6780,6.6781] This means that x=6.67805±0.00005 x = 6.678 (3d.p.) Failure of the decimal search Two of the roots of f(x)=0 where f(x)=5x3-20x2+2x-0.05 could not be found with this method. This is because they are so close together that there is no sign change

  • Word count: 1678
  • Level: AS and A Level
  • Subject: Maths
Access this essay

This is an investigation to identify some compounds containing oxygen AimThis experiment is to identify the chemical by use of their functional group in simple reactions

This is an investigation to identify some compounds containing oxygen Aim This experiment is to identify the chemical by use of their functional group in simple reactions. With the information that all the compound contain 3 carbon atoms, 6 or 8 hydrogen atoms, 1 or 2 oxygen atoms and no other elements simple tests can be performed which will identify the functional group of the compound and the name of the compound can be deduced from this information. I have chosen to use: * Propanoic acid * Propanal * Propanone * Propan-1-ol * Propan-2-ol * Methyl ethanoate I will identify the compound following the flowchart below Tollens reagent (ammoniacal silver nitrate) reduces the to a carboxylic acid, the complex silver nitrate is reduced to metallic silver which adheres to the surface of the test tube forming a mirror effect. 2,4-dinitrophenylhydrazine is a yellow solid which when dissolved in methanol and acidified with sulphuric acid it is able to react with the keytone to form a condensation product. The precipitate can be filtered off and the melting point determined. CH3COCH3 + C6H3(NO2)2N2H3 C6H3(NO2)2NHN=C(CH3)2 +H2O Warm acidified dichromate oxidises alcohols, primary alcohols are oxidised to an aldehyde then to a carboxylic acid if there is enough dichromate to oxidise the aldehyde, so if the Tollens reagent is added the aldehyde is

  • Word count: 758
  • Level: AS and A Level
  • Subject: Maths
Access this essay

My hypothesis is: 'Girls obtain better grades than boys'.

Sampling Data I am choosing a pupil to represent our school by their grades; I need to find out which sex obtains better grades so I can make my choice. My hypothesis is: 'Girls obtain better grades than boys' I have been given some data of some student's CATs and SATs results and by taking different types of samples of the data I will find out if girls obtain better grades than boys. Sampling There are four different types of sampling that can be used. These are: Systematic Sampling This is where there is a systematic way of choosing the sample, e.g. every 10th item will be sampled. Attribute Sampling This is where a sample is taken by choosing the sample by using unrelated attribute, e.g. to sample every person who's birthday is in the first two months. Stratified Sampling This is where a sample is taken according to what the population consists of, e.g. if there is 200 girls and 100 boys in the population then in your sample take 20 girls and 10 boys. Random sampling This is when a sample is taken at random, this means that every data item is likely to be chosen. I have chosen to do random sampling; this is because it was the most appropriate. I tried systematic sampling and here were my results: Males Females 81 82 91 84 94 88 96 92 98 98 01 00 03 14 04 16 06 18 07 13 18 Males Females Mean = 102 Mean = 101 Median = 102 Median

  • Word count: 1171
  • Level: AS and A Level
  • Subject: Maths
Access this essay

There are three snails; slippery, slimy and slidey. They enter a ten-metre race for food. Each snail runs according to the following rules. Slippery : d= 4.4 + 0.55t Slimy : d= 0.3t(t-7) Slidey : d= 0.3t(t-3.4)(t-9)

(Speedy Snails) INTRODUCTION Mathematics can be used to solve the problem happened in our life such as finding distance or time. Now, we've got a problem here. We want to know following questions below. Let us solve the problem by using mathematical method. There are three snails; slippery, slimy and slidey. They enter a ten-metre race for food. Each snail runs according to the following rules. Slippery : d= 4.4 + 0.55t Slimy : d= 0.3t(t-7) Slidey : d= 0.3t(t-3.4)(t-9) The snails race from a designated starting point toward a designated finish line. The distance, d, is measured in metres, and the time, t, is measured in minutes. QUESTIONS . Find the distance of each snail from the start after: (An appropriate window and graph) There are two ways to solve the question 1. Firstly, we can substitute the time into each formula. (a) 0 minutes Slippery : d = 4.4 + 0.55t = 4.4 + 0.55(0) = 4.4 + 0 = 4.4(m) Slimy : d = 0.3t(t-7) = 0.3(0)(0-7) = 0(-7) = 0(m) Slidey : d = 0.3t(t-3.4)(t-9) = 0.3(0)(0-3.4)(0-9) = 0(-3.4)(-9) = 0(m) Name of snails Slippery Slimy Slidey Distance (m) 4.4 0 0 (A distance of each snail when the time is 0 min.) (b) 2minutes Slippery : d = 4.4 + 0.55t = 4.4 + 0.55(2) = 4.4 + 1.1 = 5.5(m) Slimy : d = 0.3t(t-7) = 0.3(2)(2-7) = 0.3(-10) = -3(m) (This means Slimy is going backwards) Slidey : d =

  • Word count: 2026
  • Level: AS and A Level
  • Subject: Maths
Access this essay

An investigation into the density of 'mock blood'

Chris Ellison An investigation into the density of 'mock blood' Sample A - blood taken from a normal healthy adult male who lives at sea level. Sample B - blood taken from the same male after six months of aerobic exercise. Sample C - blood taken from the same male after training for three months at altitude. Results Time taken for a drop of the sample to fall through 100cm3 of Copper (II) Sulphate Solution. Sample A (Seconds) Sample B (Seconds) Sample C (Seconds) 1.5 2.0 0.0 1.5 3.0 8.0 4.5 2.0 0.0 4.0 2.0 7.5 2.0 3.5 0.0 2.5 4.0 1.0 4.0 6.0 7.0 2.5 2.0 8.5 6.5 4.5 9.5 2.0 1.5 7.5 Mean 13.10 Mean 13.05 Mean 8.5 Now I am going to carry out some statistics on my results to find if they occurred by chance or not. I am going to use the t-test because I need to compare two sets of results that I collected. T-test tables Sample A x x-x (x-x)2 1.5 .60 2.56 1.5 .60 2.56 4.5 .40 .96 4.0 0.90 0.81 2.0 .10 .21 2.5 0.60 0.36 4.0 0.90 0.81 2.5 0.60 0.36 6.5 3.40 1.56 2.0 .10 .21 x = 13.10 ? = 23.4 Sample B x x-x (x-x)2 2.0 .05 .10 3.0 0.05 0.30 2.0 .05 .10 2.0 .05 .10 3.5 0.45 0.20 4.0 .05 .10 6.0 2.95 8.70 2.0 .05 .10 4.5 .45 2.10 1.5 .45 2.10 x = 13.05 ? = 18.63 Sample C x x-x (x-x)2 0.0 .10 .21 8.0 0.90 3.61 0.0 .10 .21 7.5

  • Word count: 1037
  • Level: AS and A Level
  • Subject: Maths
Access this essay

Which three factors affect the price of a second hand car.

Used Car Prices: Coursework Aim The aim of this project is to decide which three factors affect the price of a second hand car. I will choose the three factors, which I think, will influence the price the most out of several lines of enquiry. Introduction I plan to investigate what factors influence the price of a second hand car. To begin with I had to look at a database of 100 cars. This database gave me a lot of information about each car (e.g. age, air con, mileage etc). All these factors were my lines of enquiry. To choose my three factors I had to decide which three of the eighteen factors would affect the price the most. Lines Of Enquiry There are a few lines of enquiries, which could be possible factors in affecting a second hand cars price. To choose my three factors I put all the lines of enquiry into a list from most important (1) to least important (18) . Mileage - This is my top factor because it tells you exactly how long the car has left in it. If the mileage is high it means the car has traveled far and may be in need of repair. 2. Age - This tells you how long the car has been running, if it's been running a long time its parts may be worn from the rust and may need a repair. 3. Engine Size - Many people want a large engine, this may be because they want to go fast or maybe they live up a hill and want to make sure that it can go up without much

  • Word count: 2170
  • Level: AS and A Level
  • Subject: Maths
Access this essay

Investigate the three different numerical methods used to solve equations.

Solving Equations by Numerical Methods Introduction In this coursework I am going to investigate the three different numerical methods used to solve equations. These include the change of sign method, Newton Raphson method and the Rearranging method. To carry out the investigation I will explain how each method works, using an example of a working equation in each case. I will also show when each of the methods will not work with other equations. I will then compare all three of the methods. Change of sign method Decimal search and Interval bisection are both ways of finding an interval where there is a change of sign. The change of sign can be found on a graph when the line crosses the x-axis. Wherever there is a change of sign there will be a root. These two methods find the interval where the root lies. Decimal Search To find the roots using decimal search, the y-values must be found, using the values of X from 0.1, 0.2, 0.3, all the way up to 1. When a change of sign occurs, this means there is a root lying here. Once the root interval to 1 decimal place has been found, it must then be found to 2 and 3 decimal places and so on to the required number of places. Y=x3-5x+1 can be solved using decimal search. The table shows that the routes can be found. X Y X Y X Y 0 2 -1 -2 3 0.1 0.501 2.1 -0.239 -2.1 2.239 0.2 0.008 2.2 0.648 -2.2 .352 0.3

  • Word count: 2555
  • Level: AS and A Level
  • Subject: Maths
Access this essay

In this coursework I will be looking at equations that cannot be solved algebraically

Core 3 Coursework In this coursework I will be looking at equations that cannot be solved algebraically. Instead I will look at three numerical methods for solving them: the change of sign method; the Newton-Rhapson method; and the rearranging method. Change of Sign Method - Decimal Search I am going to solve the equation using the decimal search (change of sign) method. I will find the y-values and look for the point at which there is a change of sign, because if there is a change from positive y to negative y then the curve must pass through the x-axis so there will be a root. Here is a graph of the function There is only one root; it lies between the integer bounds [-1,0] I have calculated the corresponding y-value for each x-value within this range, 1 decimal place apart. x y -1.0 -5 -0.9 -3.635 -0.8 -2.52 -0.7 -1.625 -0.6 -0.92 -0.5 -0.375 -0.4 0.04 -0.3 0.355 -0.2 0.6 -0.1 0.805 x y -0.50 -0.375 -0.49 -0.3281 -0.48 -0.2826 -0.47 -0.2382 -0.46 -0.1951 -0.45 -0.1531 -0.44 -0.1123 -0.43 -0.07263 -0.42 -0.03404 -0.41 0.003495 -0.40 0.04 Again the change of sign is highlighted; the root must lie between [-0.42,-0.41], so I will investigate these values. x y -0.420 -0.03404 -0.419 -0.03024 -0.418 -0.02645 -0.417 -0.02267 -0.416 -0.0189 -0.415 -0.01514 -0.414 -0.01139 -0.413 -0.007656 -0.412 -0.003929

  • Word count: 2692
  • Level: AS and A Level
  • Subject: Maths
Access this essay

Guessing the length of a line

Introduction Most people appear to think that the older you are the smarter you are. I am going to investigate this by looking gat data from year 7 and comparing it to data from year 11. Both year groups were asked to complete the same task, to guess the size Hypothesis . Pupils in year 11 are better at guessing the length of a line than pupils in year 7. 2. People who are good at estimating lengths are good at estimating angles. For both of my hypothesis I am going to be using secondary data given to me by my school, as this is less time consuming than collecting data by myself. My sampling frame will be the results from all the girls in School from year 7 to year 11 in 2002 that completed the task. I could take a census survey meaning that I could use every piece of data, which would be more accurate, but would also be immensely time consuming so I am going to take a sample of data. I could also do cluster sampling but this would be too time consuming and inappropriate for my data. It wouldn't give me a large enough sample and therefore wouldn't be in proportion to my data. Once I have collected my sample I am going to use mean, mode, median nd standard deviation to analyse the spread of data. For my first hypothesis I am going to use stem and leaf diagrams and histograms to analyse my data whereas, for my second hypothesis I am going to use scatter graphs to

  • Word count: 1482
  • Level: AS and A Level
  • Subject: Maths
Access this essay

India PopulationPyramid for 1995

India Population Pyramid for 1995 Age and sex distribution for the year 1995: India Population Pyramid for 2000 Age and sex distribution for the year 2000: India Population Pyramid for 2003 Age and sex distribution for the year 2003: India Population Pyramid for 2005 Predicted age and sex distribution for the year 2005: India Population Pyramid for 2010 Predicted age and sex distribution for the year 2010: India Population Pyramid for 2020 Predicted age and sex distribution for the year 2020: India Population Pyramid for 2050 Predicted age and sex distribution for the year

  • Word count: 95
  • Level: AS and A Level
  • Subject: Maths
Access this essay