Use GeoGebra to investigate cubics of this type to find an expression for the minimum and maximum values of 𝑑 and the relationship

Task 1 – Cubics (27 marks)

Learning Outcome

 Part A

A monic cubic 𝑃(𝑥) is such that

·         it has two turning points, and

·         the maximum difference between two of its roots is 2𝑘. The vertical distance between the turning points of 𝑃(𝑥) is 𝒅.

Use GeoGebra to investigate cubics of this type to find an expression for the minimum and maximum values of 𝑑 and the relationship

between all three roots in each case.

Part B

Another monic cubic 𝑄(𝑥) is such that it has

·         two turning points, and

·         rotational symmetry of 180° about the origin.

Let 𝑙 be a straight-line intersecting 𝑄(𝑥) at three distinct points.

Use GeoGebra to investigate cubics of this type to find any relationship between the 𝑥 coordinates of the intersection points between 𝑙 and

𝑄(𝑥).

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 𝑃 metres. The corresponding 𝑟 is the radius of the semicircle that maximises the area of the garden.

a.    Use GeoGebra to find the corresponding 𝑟 for a range of different perimeters, 𝑃.

b.    Use GeoGebra to show that the points (𝑃, 𝑟) 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 𝑓(𝑥) can be approximated by a Taylor Series about 𝑥 = 0 of appropriate order, with the formulas:-

Order 1 (linear)

𝑓(𝑥) ≈ 𝑓(0) + 𝑓′(0)𝑥

Order 2 (quadratic)

𝑓′′(0)

𝑓(𝑥) ≈ 𝑓(0) + 𝑓′(0)𝑥 +           𝑥2

2!

Order 3 (cubic)

𝑓′′(0)         𝑓′′′(0)

𝑓(𝑥) ≈ 𝑓(0) + 𝑓′(0)𝑥 +            𝑥2 +              𝑥3

2!                 3!

and similarly for higher orders.

2𝑥

a.    For the function 𝑓(𝑥) =   𝑒        , use GeoGebra to find the value at

√𝑥2+1

𝑥 = 0 of 𝑓(𝑥) 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 𝑥 = 0 and

𝑥 = 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

𝑏

∫ 𝑓(𝑥) 𝑑𝑥

𝑎

can be approximated by modelling 𝑓(𝑥) by a piecewise linear or quadratic function respectively.

You are asked to investigate the area under the curve with definition

𝑓(𝑥) = 25𝑥2𝑒−3𝑥

between 𝑥 = 1 and 𝑥 = 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 𝑦 = 𝑓(𝑥) 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;