Description of task. 7. This is a method which involves using an iterative formula. You must rearrange f(x) = 0 to the form x= g(x) to get the Iterative Formula Xr+1= g(xr). 8. You can rearrange this as x = g(x) in several ways. One of these iterative formulae may be used to find a root of the equation f(x) = 0. 9. If it diverges, then you need to choose a different iterative formula. 10. You must show the iterative process on your graph, using your chosen software which produces a staircase (if gradient of g(x) is positive at the intersection) or a cobweb (if gradient of g(x) is negative at the intersection). You must label x. to x3 on your graph. 11. State the equation you are trying to find a root of, which is f(x) = 0, where you define what your f(x) is. (There should be a different f(x) used for each method.) 12. Show how you have rearranged it to the form x = g(x). 13. State the iterative formula that you are therefore using. This should be of the form Xr+1= g(x₂). 14. Show a graph with your y = f(x), y = g(x) and y = x. 15. Show your calculations, clearly stating the integer near the root that you have chosen to be x and a zoomed-in graph showing the staircase or cobweb with the first four x values labelled (x. to x3). 16. Describe how the iterations are performed and the x, values converge to the root, referring to both your calculations and your zoomed-in graph illustration. 17. Perform a change of sign check by looking at f(x) for x values 0.000005 above and below your root. 18. State that your root is x = ..... with a maximum error of ±0.000005 19. Confirm that the value of g'(x) near the root is between -1 and 1 and say that the method only works when this is the case. You can do this in several ways, such as one of the following: i. draw a tangent to g(x) near the root and find the gradient of this tangent. ii. use the gradient function, y = g'(x) to find the gradient of g(x) near the root. iii. compare the steepness of the curve y = g(x) near the root with the steepness of the line y = x (or y = -x) at this point. iv. evaluate the gradient of g(x) near the root by differentiating g(x). 20. You must repeat the steps above for a case where this method fails to find a root you are looking for, using the same equation f(x)=0 with the same f(x). It can be a different rearrangement x = g(x), but it doesn't have to be. Is this required? Yes or No Completed? Date?

Description of task. 7. This is a method which involves using an iterative formula. You must rearrange f(x) = 0 to the form x= g(x) to get the Iterative Formula Xr+1= g(xr). 8. You can rearrange this as x = g(x) in several ways. One of these iterative formulae may be used to find a root of the equation f(x) = 0. 9. If it diverges, then you need to choose a different iterative formula. 10. You must show the iterative process on your graph, using your chosen software which produces a staircase (if gradient of g(x) is positive at the intersection) or a cobweb (if gradient of g(x) is negative at the intersection). You must label x. to x3 on your graph. 11. State the equation you are trying to find a root of, which is f(x) = 0, where you define what your f(x) is. (There should be a different f(x) used for each method.) 12. Show how you have rearranged it to the form x = g(x). 13. State the iterative formula that you are therefore using. This should be of the form Xr+1= g(x₂). 14. Show a graph with your y = f(x), y = g(x) and y = x. 15. Show your calculations, clearly stating the integer near the root that you have chosen to be x and a zoomed-in graph showing the staircase or cobweb with the first four x values labelled (x. to x3). 16. Describe how the iterations are performed and the x, values converge to the root, referring to both your calculations and your zoomed-in graph illustration. 17. Perform a change of sign check by looking at f(x) for x values 0.000005 above and below your root. 18. State that your root is x = ..... with a maximum error of ±0.000005 19. Confirm that the value of g'(x) near the root is between -1 and 1 and say that the method only works when this is the case. You can do this in several ways, such as one of the following: i. draw a tangent to g(x) near the root and find the gradient of this tangent. ii. use the gradient function, y = g'(x) to find the gradient of g(x) near the root. iii. compare the steepness of the curve y = g(x) near the root with the steepness of the line y = x (or y = -x) at this point. iv. evaluate the gradient of g(x) near the root by differentiating g(x). 20. You must repeat the steps above for a case where this method fails to find a root you are looking for, using the same equation f(x)=0 with the same f(x). It can be a different rearrangement x = g(x), but it doesn't have to be. Is this required? Yes or No Completed? Date?

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter4: Polynomials

Section4.2: Adding And Subtracting Polynomials

Problem 33WE

See similar textbooks