Ascentis Mathematics

Ascentis Mathematics

Carrying Out a Practical Investigation

INVESTIGATION D

The Apprentice’s Vase

A cabinet maker’s apprentice, like all young people should, wishes to please his parents with a gift of an elaborately turnedwooden vase.  The vase shape is shown in Diagram B.  It comprises a curved inner and outer profile with a flat base of width 1 unit.

Diagram A defines the shape of the curves and the shows how they relate to the flat base.

The outer curve is given by the equation for

The inner curve is given by the equation   for

The flat bottom of the vase is defined by the lines  and

1. Estimate , the root of  and  , the root of using numerical approximation techniques.  In doing this you must demonstrate the use of at least two different numerical approximation methods.

2. Estimate the volume of wood in the vase, using techniques of numeric integration. You must demonstrate the use of two differentnumeric integration techniques for estimating this volume.

3. What is the level of accuracy of your final volume calculation, given the accuracy of your initial estimates of the roots α and β?

Planning

You must teach yourself about numerical methods, using standard A-level textbooks and the Internet as back up.  The following is a suggested plan of action:

1. Examine the Ascentis L3 specification for Unit 6 Numeric Methods. Similarly, examine the specifications for AS, A2 and Further Pure mathematics from any A-level examining body (e.g. MEI, Edexcel or AQA).

This will get you familiar with the standard terms used to refer to numerical methods techniques.  It will also indicate the appropriate order in which to study them – avoid trying to short-circuit the learning process by picking up later techniques without understanding the underlying ideas.

[2 hours]

2. Get a suitable textbook from the library and study each sub-topic, making sure you try some of the graded exercises as the textbook develops the topic. Learn how to teach yourself from a textbook – you will have to do this on a weekly basis at university!

During this phase, your teachers are available to help you understand the concepts, but you must be pro-active in seeking help.

[16 hours]

3. Once you feel you have mastered the concepts and techniques, start on the task and keep going until you have completed it to the best of your ability.

You are very much on your own in this phase.  Do not ask your teacher to complete the task for you.  Complete the assignment yourself!

[5 hours]

4. Write up your project neatly (handwritten work is fine, so long as it is legible). We are not looking for a long-winded essay.  Mathematics is a precise and concise language, so take care to use correct mathematical notation at all times.

Your final submission should include:

• A short introduction explaining the purpose of the document. (note: this document will be lodged on NCG Online, so you can copy the diagrams above for inclusion in your write-up).
• A clear mathematical development leading to an estimation of volumes and a discussion on accuracy.
• A critical review of your work with observations on any improvements you could have made in its execution.
• A bibliography.

[5 hours]

Keywords

Absolute and relative errors

Numerical methods, Numerical analysis, Numerical approximation

Approximate solution to equations:

• Interval bisection
• linear interpolation
• Iteration (x_n+1=g(x_n))
• NewtonRaphson method

Numeric Integration

• Simpson’s Rule
• Trapezium Rule

Volume of revolution