Overview
You need to prepare a report based on the tasks that you can find below. You need to include appropriate images of your work done using GeoGebra to attach as proof of your answers. Pictures of written work will not be marked. You need to use the Word equation editor and the GeoGebra software.
This assessment consists of 66 marks and will comprise 10% of your overall mark for the module.
Knowledge and Understanding
2) identify whether a real-life situation is best solved exactly, numerically, or probabilistically;
3) evaluate the strengths and weaknesses of alternative models and consequently of any conclusions drawn;
Subject-specific Skills
5) choose and apply appropriate techniques in algebra, co-ordinate geometry, trigonometry and calculus in finding a solution to a problem;
7) employ a range of numeric techniques to find approximate solutions to problems;
Key and Employability Skills
8) use a range of IT packages, including calculators, spreadsheets, word processors, and skills in the analysis and solution of problems;
Please note: You are reminded that plagiarism is the uncited use of other’s work – this includes graphs, diagrams, images and written text. It is therefore important to reference all source material used in your essay. If you are found to have plagiarised work you are guilty of Academic Impropriety and are likely to have your grade for that component of the assessment reduced to zero.
Evidence for Assessment
• Word document containing your report
• Images exported from GeoGebra
Academic Impropriety:
You are reminded that plagiarism is the unreferenced use of other people’s work or your own previous work. This could include visual images, sound recordings, diagrams, as well as written text. You can however use other people’s work as examples, supporting evidence or inspiration as long as it is referenced appropriately.
Academic Impropriety also includes copying or using other people’s work and presenting this as your own. This could include work produced by family members, friends or unknown people on the Internet.
If you are guilty of Academic Impropriety you are likely to have your grade for the assignment reduced to zero.
Task 1 – Cubics (27 marks) |
Learning Outcome |
Part A
A monic cubic π(π₯) is such that · it has two turning points, and · the maximum difference between two of its roots is 2π. The vertical distance between the turning points of π(π₯) is π . Use GeoGebra to investigate cubics of this type to find an expression for the minimum and maximum values of π and the relationship between all three roots in each case.
Part B Another monic cubic π(π₯) is such that it has · two turning points, and · rotational symmetry of 180° about the origin. Let π be a straight-line intersecting π(π₯) at three distinct points. Use GeoGebra to investigate cubics of this type to find any relationship between the π₯ coordinates of the intersection points between π and π(π₯). |
2) identify whether a real-life situation is best solved exactly, numerically, or probabilistically;
5) choose and apply appropriate techniques in algebra, co-ordinate geometry, trigonometry and calculus in finding a solution to a problem;
8) use a range of IT packages, including calculators, spreadsheets, word processors, and skills in the analysis and solution of problems; |
Task 2 – Geometry (12 marks) |
Learning Outcome |
A garden is in the shape of a rectangle with a semicircle entirely along one edge. The length of the fence around the garden is π metres. The corresponding π is the radius of the semicircle that maximises the area of the garden.
a. Use GeoGebra to find the corresponding π for a range of different perimeters, π.
b. Use GeoGebra to show that the points (π, π) corresponding to your values all lie, subject to rounding errors, on a straight line and find the equation of this line.
c. Use your equation to find the lengths of the sides of the garden which maximises its area when the perimeter is 130 metres. Use GeoGebra to show your result in this case. |
2) identify whether a real-life situation is best solved exactly, numerically, or probabilistically;
5) choose and apply appropriate techniques in algebra, co-ordinate geometry, trigonometry and calculus in finding a solution to a problem;
7) employ a range of numeric techniques to find approximate solutions to problems;
8) use a range of IT packages, including calculators, spreadsheets, word processors, and skills in the analysis and solution of problems; |
Task 3 - Calculus (12 marks) |
Learning Outcome |
You are given that any reasonable function π(π₯) can be approximated by a Taylor Series about π₯ = 0 of appropriate order, with the formulas:-
Order 1 (linear) π(π₯) ≈ π(0) + π′(0)π₯ Order 2 (quadratic) π′′(0) π(π₯) ≈ π(0) + π′(0)π₯ + π₯2 2! Order 3 (cubic) π′′(0) π′′′(0) π(π₯) ≈ π(0) + π′(0)π₯ + π₯2 + π₯3 2! 3! and similarly for higher orders. 2π₯ a. For the function π(π₯) = π , use GeoGebra to find the value at √π₯2+1 π₯ = 0 of π(π₯) and up to the 5th derivative for the function
b. Use the values found in part a. to write down the equations of the linear, cubic and quintic approximations to the function.
c. Use GeoGebra to plot the functions found in part b and the Integral function in GeoGebra to find the approximate area under each of the approximations in part b. between π₯ = 0 and π₯ = 1.
d. Also use the Integral function in GeoGebra to find the area under the original curve over the same domain and comment on the accuracy of the approximations. |
2) identify whether a real-life situation is best solved exactly, numerically, or probabilistically;
3) evaluate the strengths and weaknesses of alternative models and consequently of any conclusions drawn;
5) choose and apply appropriate techniques in algebra, co-ordinate geometry, trigonometry and calculus in finding a solution to a problem;
8) use a range of IT packages, including calculators, spreadsheets, word processors, and skills in the analysis and solution of problems; |
Task 4 – Numerical Integration (15 marks) |
Learning Outcome |
The integral π ∫ π(π₯) ππ₯ π can be approximated by modelling π(π₯) by a piecewise linear or quadratic function respectively. You are asked to investigate the area under the curve with definition π(π₯) = 25π₯2π−3π₯ between π₯ = 1 and π₯ = 2. Part A Assume first that you are using a linear model and so using the Trapezium rule to find an estimate of the area. a. Plot the function π¦ = π(π₯) explain whether you expect the estimate to be an overestimate or underestimate.
b. Use an appropriate spreadsheet to find the minimum number of ordinates necessary to find an estimate to the area correct to 1 decimal place.
Part B Assume now that you are using a quadratic model and so using Simpson’s rule to find an estimate of the area.
c. Use an appropriate spreadsheet to find the minimum number of ordinates necessary to find an estimate to the area correct to 1 decimal place.
Part C d. Compare the two models, discussing the advantages and disadvantages of both. |
2) identify whether a real-life situation is best solved exactly, numerically, or probabilistically;
3) evaluate the strengths and weaknesses of alternative models and consequently of any conclusions drawn;
5) choose and apply appropriate techniques in algebra, co-ordinate geometry, trigonometry and calculus in finding a solution to a problem;
7) employ a range of numeric techniques to find approximate solutions to problems;
8) use a range of IT packages, including calculators, spreadsheets, word processors, and skills in the analysis and solution of problems; |
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