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Calculus Exploration – Grade 11 Extended Maths Class January 2022

发布时间：2022-01-25

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Calculus Exploration – Grade 11 Extended Maths Class January 2022

This is a task for criteria C and D of the MYP Extended Mathematics Curriculum. It is a summative task and the success criteria are given t the end of this document. The tsk id divided into two sections: Section 1 contains a problem to do with calculus that you are asked to solve, with a certain amount of structure; Section 2 asks you to come up with your own idea of a practical application of calculus, for which you will create and then solve a problem.

Your answers should be in the form of a short report, maximum 10 pages (excluding contents and Bibliography pages) and you should use a variety of means of mathematical communication (graphs, charts, tables, equations using an equation editor etc.) to ensure that your readers understand what you are doing clearly.

Section 1 – The Sliding Ladder

Bingxian has a ladder placed against a wall. The ladder is 5 metres long and its top is 4 metres from the base of the wall, whilst its bottom is 3metres away from the base of the wall (see diagram 1).

Diagram 1: Initial Position of Ladder

Task 1 – Find the coordinated of the midpoint of the ladder.

The ladder now begins to slip down the wall, so that the new coordinate of its bottom is at (3 + t , 0).

Task 2 – Find the new height of the ladder in terms of t. Find also the new midpoint of the ladder in terms of t.

Task 3 – Using substitution, eliminate t and hence find an equation for the midpoint of the ladder as it slides down the wall.

Task 4 – Use a suitable method to sketch the graph of this equation and comment on what it tells you geometrically about the position of the midpoint as the ladder slides down the wall.

As the ladder slips down the wall, Bingxian plots its position at a number of stages. When these are all plotted on the same graph, Bingxian ends up with the following diagram (see diagram 2).

Diagram 2 – Positions of the Ladder

The name for the shape create by the slipping ladder is an asteroid.

Task 5 – Explain why diagram 2 shows only a quarter of the asteroid.

The equation for the asteroid created by Bingxian’s ladder is

Task 6 – Find in terms of x only.

Task 7 – Find the gradient of the asteroid when y = x, assuming x > 0, y > 0.

Task 8 – Explain the significance of your answer to Task 7.

Section 2 – Your idea.

In this section, you will need to find a practical application of calculus (differentiation, integration, or both). You need to come up with a problem that requires you to use calculus to solve. The problem does not have to be very complex, but it should have a reasonable level of difficulty.

For your problem, you need to describe what it is, then explain how calculus can be used to solve it. Remember that you should try to use a variety of methods of communication, so drawing graphs and diagrams could be useful, as could tables of data if you think they help to communicate what you are doing.

The deadline for completing the report, which should be no more than 10 pages long in total, is Monday 17^th January 2022 at 9am. You will have some time in lessons at the start, and then all of the homework tasks until the deadline date. ON the next page are the success criteria for the report.

Criterion C - Communication		Level	Criterion D – Using Mathematics in a Practical Context
Your work involves academic dishonesty, is completed late or does not meet any of the criteria below.	0		Your work involves academic dishonesty, is completed late or does not meet any of the criteria below.
Your report has a basic structure; explanations are incomplete.	1		Section 1 – Tasks 1 and 2 are completed correctly.
Your report contains some structure – section headings, a contents page.	2		Section 1 – Tasks 3 and 4 are completed correctly.
Your report contains some use of algebra or formulae.	3		Section 1 – Tasks 5 to 8 are attempted.
Your report contains a variety of forms of communication ,including graphs, charts or diagrams.	4		Section 1- at least three of tasks 5 to 8 are completed correctly, with only minor errors.
An equation editor has been used.	5		A suitable problem has been chosen for Section 2.
Variables are clearly defined, algebra and notation are correct.	6		The problem in Section 2 has been explained and a sensible attempt has been made to solve it.
Methods are clearly explained for a grade 11 audience.	7		The problem in Section 2 has been completed satisfactorily.
The report is complete, concise and flows well when read.	8		The problem in Section 2 is sufficiently sophisticated to challenge the student.