C3 Coursework - different methods of solving equations.

C3 Coursework - Numerical Solutions Decimal Search There a numerous ways to solve a problem and in finding the unknown. Some methods give you the exact and precise answer but usually are harder and more complex. The Decimal search method enables you to get a very close approximate to the real solution but more easily. The way this method works is by looking between two numerical values (for example 1 and 2) and then As a demonstration in applying this method, I will be attempting to solve this equation using the Decimal Search method and going through the method step by step: Below is what this function looks when plotted on a graph: We know that the solution for F(x) = 0 is the point on the X axis where the sign changes from a positive to a negative. So if we zoom in a little bit further, from this graph we can tell where the solution lies, somewhere between 0 and 10 Now that we know the solution is roughly between these two values, I will use excel to solve the problem with firstly taking increments in x, the size of 1. So when I substitute the incremented values of x between -10 and 0 into the equation, I get the following results: x F(x) -1 20 -2 9 -3 0 -4 -13 -5 -56 -6 -125 -7 -226 -8 -365 -9 -548 You can tell that the sign changed between -3 and -4. So I set these as my initial values. The fact that the solution lies between -3 and -4 can

  • Ranking:
  • Word count: 3460
  • Level: AS and A Level
  • Subject: Maths
Access this essay

Estimate a consumption function for the UK economy explaining the economic theory and statistical techniques you have used.

Estimate a consumption function for the UK economy explaining the economic theory and statistical techniques you have used. Consumption has been considered as the most important single element in aggregate demand, accounting for almost 66% of GDP in 1989. Therefore, it is essential that the level of consumption be predicted accurately, for even a small percentage error may lead to a large absolute error. Another reason why consumption such important is that the marginal propensity of consume is one of the factor which is used to determine the size of the multiplier. This will influence the changes of investment and government spending. Moreover, as saving ratio, another factor that used to predict the behavior of consumers, has fluctuated rapidly during these years, so the consumption become more important. Thus, many economists attempt to develop many theories and equations to predict consumers' expenditure. Therefore, in this project, there are two main theories and their equations will be given. Some data from Economic Trends Annual Supplement and graphs will be used to estimate the performance of the model on the basis of some econometric tests. Keynes, John Maynard introduced consumption function in 1936. In The General Theory of Employment Interest and Money, Keynes pointed out that " We should therefore define what shall call the propensity to consume as the

  • Ranking:
  • Word count: 4377
  • Level: AS and A Level
  • Subject: Maths
Access this essay

Numerical solutions of equations

A2 Mathematics Coursework C3 Year 12 Numerical solutions of equations Solving 0 = x5+x-5 using the "Change Of Sign" Method The method I will use to solve 0 = x5+x-5 is the Change of Sign Method involving the Decimal Search method. I have drawn this graph using the Autograph Software, and the print screen of this is below: From my graph above, I can see that the root of this equation is between x =1 and x = 1.5. The table of x values and f(x) values is shown below. I can work out the f(x) values by substituting the x-values into the equation. x .1 .2 .3 .4 .5 f(x) -3 -2.28949 -1.31168 0.01293 .77824 4.09375 From my table of values above, it is clear that the change of sign from negative to positive occurs between x = 1.2 and x = 1.3. So, I can narrow these values down further to find another change of sign. x f(x) .21 -1.19626 .22 -1.07729 .23 -0.95469 .24 -0.82837 .25 -0.69824 .26 -0.56420 .27 -0.42616 .28 -0.28403 .29 -0.13769 .30 0.01293 I can see that the change of sign is between x = 1.29 and x = 1.30. x f(x) .291 -0.12283 .292 -0.10792 .293 -0.09296 .294 -0.07797 .295 -0.06293 .296 -0.04784 .297 -0.03271 .298 -0.01754 .299 -0.00233 .300 0.01293 The change of sign is in the interval [1.299, 1.300] x f(x) .2991 -0.000805 .2992 0.000720 .2993 0.002244 The root of this equation lies in the interval

  • Ranking:
  • Word count: 2743
  • Level: AS and A Level
  • Subject: Maths
Access this essay

The Gradient Function

The Gradient Function Aim: To find the gradient function of curves of the form y=axn. To begin with, I should investigate how the gradient changes, in relation to the value of x. Following this, I plan to expand my investigation to see how the gradient changes, and as a result how a changes in relation to this. Method: At the very start of the investigation, I shall investigate the gradient at the values of y=xn. To start with, I shall put the results in a table, but later on, as I attempt to find the gradient through advanced methods, a table may be unnecessary. As I plot the values of y=x2, this should allow me to plot a line of best fit and analyze, and otherwise evaluate, the relationship between the gradient and x in this equation. I have begun with n=2. After analyzing this, I shall carry on using a constant value of "a" until further on in the investigation, and keep on increasing n by 1 each time. I shall plot on the graphs the relative x values and determine a gradient between n and the gradients. Perhaps further on in the investigation, I shall modify the value of a, and perhaps make n a fractional or negative power. Method to find the gradient: These methods would perhaps be better if I demonstrated them using an example, so I will illustrate this using y=x2. This is the graph of y=x2. I will find out the gradient of this curve, by using the three methods -

  • Ranking:
  • Word count: 6489
  • Level: AS and A Level
  • Subject: Maths
Access this essay