The Group Application Task will be marked using the Group Task rubric. It accounts for 15% of the overall mark for the subject.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Applications of Calculus Task

Information for students

• Working in groups (maximum of 3 members), students will be allowed one week to collaborate in solving the problems in the task and communicating their findings.

• Each group will need to establish a viable method of communication. Some Zoom time will be provided.

• Students should decide how they will approach the task in their groups. All group members must be prepared to make a fair and reasonable contribution to the group effort.

• Every member of a group must access the assignment from LMS and work individually on the questions until they are ready to share their ideas with the rest of the group. The answers recorded by each student will be the ones that the group members have accepted as the best. Each student will complete an Answer Summary by typing the group’s consensus answers into their own Summary Sheet.

• An Answer Summary from every student must be submitted onto Moodle in accordance with the set deadline. Instructions for doing this will be made available.

• Students will be assessed on the criteria set out in the assessment rubric.

• Students may refer to their notes, textbooks and online resources, but discussion of the assignment with others should be restricted to members of their group.

• At any time, students may seek clarification from a teacher regarding the wording of information or questions, but teachers may not help with mathematical calculations or with explanations of mathematical terminology.

Assessment criteria

• The Group Application Task will be marked using the Group Task rubric. It accounts for 15% of the overall mark for the subject.

• Students are expected to complete a peer review for the task on LMS by the deadline set by the teacher. Failure to do so on time may affect the collaboration score.

Task 1

An outdoor sports complex is located in a field in the shape of an ellipse given by the equation

𝑥2 + 𝑦2 = 1. The field is surrounded by a fence in the

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shape of a parallelogram as shown in the diagram. The fence touches the edge of the field in four places

a) Find 𝑑𝑦 for the ellipse 𝑥2 + 𝑦2 = 1.

b) The sides of the fence are made up of four straight lines. Find these equations given that the fence touches the edge of the field at 𝑥 = ±4.

c) Explain why the sides of the fence must be a rhombus.

d) Find the exact area of the region enclosed by the four fences.

Task 2

Part of the field will be used to for the sport javelin throw. The

javelin area is defined by A1. The area is enclosed by the curve

𝑓(𝑥) = −𝑥3 + 4𝑥2 − 𝑥 − 5 and the ellipse 𝑥2 + 𝑦2 = 1. The area

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A1 will be filled with grass for the javelin throw.

a) Determine, the exact 𝑥-value within the field where the tangent line of 𝑓(𝑥) = −𝑥3 + 4𝑥2 − 𝑥 − 5 is parallel to the line 𝑔(𝑥) = 2𝑥 − 7.

The javelin area needs to be filled with grass.

Pre-cut grass can only be laid if the curve 𝑓(𝑥) = −𝑥3 + 4𝑥2 − 𝑥 − 5 is concave up within the field. If the curve is concave down within the field, seeds will need to be planted.

b) Find the inflection point(s) and the intervals of concavity up and down of the javelin area within the domain 𝑥 ∈ [0,5].

c) Describe how the javelin area can be filled with grass within the domain 𝑥 ∈ [0,5].

d) Express the equation of the ellipse as 𝑦 = ℎ(𝑥).

e) Using your calculator or otherwise, find the coordinates of the intersection points of the graphs 𝑓(𝑥)

with the edges of the field correct to 1 decimal place. Label these on the given diagram.

f) Write down the integral expression for the area to be filled with grass to cover A1. Use values from part e).

g) Hence, using your calculator determine the area A1 correct to the nearest unit.

Attachments:

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