# Use Maple to plot graphs of such that only one of the stationary points is clearly visible and identifiable on each graph (by using the correct Maple syntax).

Coursework: YOU SHOULD ANSWER ALL PARTS TO THIS COURSWORK. THERE ARE 3 SEPARATE PARTS

This coursework with worth 30% of the module mark in total.

Part 1: Statistics (5% of the module mark)

First you have to find some data. You can find you own data or use a sample data files within the Minitab computer package.

It is recommended you use one of these, and if you do, you MUST do all of your analysis on a 90% sample of this data. This is how to take a 90% sample of the data.

Step 1: Find an appropriate data set (minimum 1 column, 100 rows of data).

Step 2: Click Calc … Random Data … Sample from columns (see below)

This comes up with the following dialog box:

Then click ‘OK’

Once you have done this, you need to find out the summary statistics and draw at least 2 different charts USING THIS 90% SAMPLE. Use the techniques we covered in Workshop 1 and Workshop 2.

Ensure your tutor can access the 90% sample when you hand in your coursework.

Write a brief report: you need to explain what the summary statistics mean. Identify the ‘best’ measure of central tendency; justify you answers.

Describe your charts, again, explaining what they are demonstrating. Is your data normally distributed? Again you need to justify your answers using the features you have spotted in your charts.

Part 2: Differentiation (10% of the module mark)

Choose a function for which you need to use the Quotient Rule in order to successfully differentiate it. You can use a tutorial sheet question or a question from one of the recommended texts for this module (there are lots of copies in the Learning Centre).

Credit will be given for choosing a difficult function to differentiate which demonstrates a variety of different methods needed to calculate the derivatives by hand. Note your function must contain at least 1 stationary point. IF YOU NEED HELP WITH THIS PLEASE ASK A TUTOR IN THE COMPUTER WORKSHOPS

• Calculate the first and second derivatives of your chosen function ‘by hand’. Find and classify at least 1 stationary point

• Define as a Maple function (i.e. a functional operator) and then use it in the rest of this work. DO NOT define it as a Maple expression.

• Plot the graph of, ensuring that the main features of the function are clear (by using the correct Maple syntax).

• Use Maple to calculate the first and second derivatives of . Check you get the same answer as part 1.

• Use Maple to find and classify the stationary points of . Give both actual values and decimal values correct to 3 decimal places.

• Use Maple to plot graphs of such that only one of the stationary points is clearly visible and identifiable on each graph (by using the correct Maple syntax).

• Use Maple to determine the area enclosed by the graph of , the x axis, and the lines and . You should first choose appropriate values for a and b and then plot the appropriate section of the graph (ensuring the required area is visible).

Part 3: Integration (15% of the module mark)

For this piece of coursework you must use the methods for numerical integration to produce approximations to an area which cannot be calculated exactly using other techniques covered in the module 4ET005.

Credit will be given for good knowledge of the methods themselves and the way that they behave in relation to error. By using a spreadsheet program you will be able to produce very accurate approximations to your chosen area and be able to judge how reliable your approximations are.

IMPORTANT:

Your chosen problem should not have an analytical solution and you need to demonstrate this using all the integration techniques you have covered in this module. You should explain why each technique does not work, rather than stating ‘this integral is really hard’.

This means that you shouldn’t choose an integral that can be solved using the techniques you have learnt in this module. For example, you will know how to evaluate the following integrals, so you shouldn’t pick anything similar for this coursework.

For example would an appropriate integral for this coursework, but is not.

What you need to do:

Phase 1 – Problem Specification

Make sure that you tell the reader exactly what you are attempting to do in your coursework. When you are trying to approximate an integral make sure that you tell the reader between which values you are integrating – show this on a diagram. You should briefly explain why your problem cannot be solved using normal analytical methods. See above. This phase should be fairly short.

Phase 2 – Strategy

Say what you are going to do to solve the problem. State which Numerical Methods you will use and the formulae involved in them. Where appropriate, you may wish to include a diagram to explain the formulae, but do not derive any of the standard formulae – you will earn no marks for deriving standard results.

Phase 3 – Formula Application

Do this on a spreadsheet. Try to set it out as clearly as you possibly can, so that the reader can see how all of the cells relate to one another. Remember to include the version of the spreadsheet where formulae are displayed somewhere in your report (perhaps in an appendix or maybe next to the spreadsheet itself). Label cells where appropriate. Try to stick to a standard notation.

Phase 4 – Use of technology

You will use Excel to devise your spreadsheets. You should say what its limitations are (e.g how many decimal places can it handle?). You should say how many decimal places your calculations are being made to.

Phase 5 – Error Analysis

This is a very important section. You should try to give an estimate of how much error there is in your solution to the problem. You may wish to analyse differences between successive approximations and then the ratio of differences, in order to see how fast your series of approximations in tending to the true value; you can use this to give an estimate of the error involved (see notes below). You could also use Maple to check the true error values.

Phase 6 – Interpretation

Relate the solution you have obtained back to the original problem. Maybe you will want to discuss how you could improve your estimate and how long this would take in terms the number of extra iterations required. If you think your results are as good as the calculating power of the spreadsheet will allow, you should say why you think this.

Phase 7 – Oral Communication

You should be completely familiar with your work and be able to discuss it with your tutor during the Semester 2 workshop sessions identified.

IF THIS ASPECT OF THE COURSEWORK IS NOT COMPLETED THE MAXIMUM GRADE YOU CAN ACHIEVE IS 40% FOR THIS ASSESSMENT.

SUBMISSION OF COURSEWORK:

Use the assessment upload tools in the 4ET005 WOLF Topic. ENSURE YOU UPLOAD THE CORRECT FILE TO THE CORRECT FOLDER.

For part 1, a copy of your statistics report with a copy of the data you used.

For part 2, a copy of your Maple project plus a scan of your handwritten differentiation.

ENSURE YOU ATTEMPT ALL PARTS TO YOUR COURSEWORK.

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