Famous Equations

Do you have a favourite equation? Peter Alfeld of The University of Utah has a collection he thinks are important or intriguing. My personal favourite, Euler’s identity, is his first on the list.

                    e^iπ = -1

For more on beautiful equations try the 17 equations that changed the world on the World Economic Forum where for each equation we have the following information:

• What does it mean?
• History
• Importance
• Modern use

Or from BBC Earth, have a look at “What is the most beautiful equation?”

For UK school age students note the famous equation poster contest from the excellent Maths Careers site. This competition closes on Friday 8th June; winners in each age group will win an Android tablet. There will also be five ‘highly commended’ certificates awarded in each group.

 

 

 

Differential Equations

You can check solutions to Differential Equations using WolframAlpha. The slides here illustrate the syntax for first and second order differential equations. Examples like this and more are available from WolframAlpha: Examples for Differential Equations.

Differential Equations on Slideshare.

Notes and examples on Differential Equations.

• CIMT Section 18.5 on First Order Differential Equations
• Units 15.1 to 15.10 from AJ Hobson’s Just the Maths (individual pdf files hosted by UEA) (or a complete pdf from the Math Centre:   AJ Hobson’s ‘Just the Maths’) are all on Differential Equations.
• mathcentre: Differential Equations Resources
  mathcentre: First Order Differential Equations
  mathcentre Second Order Differential Equations
•  A series of notes from the HELM Project.
  
  
  

  Helm Notes – First order ODEs 

 

 

 

Underground Mathematics

A reminder that the excellent Underground Mathematics Resources are all still available, now hosted by Nrich.

Have you looked at the many Review Questions? Choose your question type and level by topic, all with very full solutions.

Simultaneous Equations

MathisFun Easter Puzzle

An Easter puzzle from MathisFun – as an excuse for solving simultaneous equations. We could of course use algebra. Using the notation, h, q and t for the egg Horace wants, the egg with the small square pattern and the egg with the stripey pattern respectively.
We have:
h + 2q      =550 (1)
h + q   +t =600 (2)
—-2q  +t =500 (3)
Subtracting equation (3) from equation (2) gives
h−q=100 (4)

(1)−(4) gives 3q=450 so q=150 and h must be 250 ($2.50).

We could check our solution on WolframAlpha of course:

Select image for WolframAlpha query

Did you know that you can easily invert matrices and solve simultaneous equations using Excel?

Select image for Excel file

Select the image or this link for the Excel file. Excel simultaneous equations

To enter the MINVERSE function in the example above, select cells C7:E9, enter the MINVERSE function as shown then press CTRL + SHIFT + ENTER; similarly to enter the MMULT function in this example, select cells H7:H9, enter the MMULT function as shown and then press CTRL + SHIFT + ENTER. (Using formulae like these demonstrates a neat Excel technique, to learn more see MrExcel on Array Formulae).

Plymouth University workbook

For notes / examples / tutorials on Simultaneous Equations try the mathcentre resources or this workbook from Plymouth University. For more on solving three equations in three unknowns for older students see Simultaneous Linear Equations from AJ Hobson’s ‘Just the Maths’

Since it’s Easter an updated version of the best Easter Eggs of all – from WolframAlpha.

WolframAlpha – a little fun!  

If you want some more puzzles, MathisFun has plenty more, or try some of the puzzles here.

Linear Programming

For Linear Programming there are some excellent ways of using technology to help your learning. This post will look at online utilities from zweigmedia and the use of the Excel Solver.

This example from MEI (MEI Decision 1 Legacy Specimen Paper) will be used to illustrate the use of these tools.

A vet is treating a farm animal. He must provide minimum daily requirements of an antibiotic, a vitamin and a nutrient. He has two types of medicine available, tablets and liquid. The table summarises what the medicines contain and the requirements.

The tablets cost £0.38 each and liquid medicine costs £1 per dose.
The vet wants to find the cheapest way to treat the animal.
We have x as the number of tablets which the vet prescribes per day, and y is the number of doses of liquid medicine, giving us the linear programming problem:
Minimise    0.38x + y
Subject to
3x +   2y   ≥ 18
2x +   4y   ≥ 16
10x + 50y ≥ 100

Firstly we will look at zweigmedia’s Linear Programming Grapher. This is intuitive to use. Note the option to see examples. Fraction mode is selected here.
Selecting Solve gives a clear solution illustrated graphically.

Linear Programming Grapher – zweigmedia

We see that the optimal solution, if x and y can take any values, is a cost of £3.20,
with x = 20/3 and y = 2/3.

We could also solve the problem using the Simplex Tool. As well as a solution it is possible to see the simplex tableaus. The Simplex Tool is needed for any problems with more than two variables.

If we wish to restrict x and y to integers, we can turn to Excel and use Solver. This allows a choice of making x and y integer or not.

Using the same example, we can set up the problem in Excel.
Vet Problem with Excel Solver is the Excel spreadsheet with annotation to explain the answer report and for full instructions on setting up the spreadsheet, use this file:
Solver Excel to solve LP problems – Colleen Young

Set up the problem in Excel

and use Solver..

Excel Solver – Answer Report

We could choose to restrict x and y to integers with Solver.

giving us the solution:

For a further example, this time a maximisation problem from Edexcel have a look at this Excel file, Badges LP Edexcel. Each sheet in the workbook is fully annotated to explain the method.

 

 

 

 

 

Two problems …

…to think about, from Transum Mathematics.

1. Algebraic Product
2. Find the Radius

If you like these problems you will find more of these Advanced Starters on Transum Mathematics.

Transum Mathematics

Casio FX991EX-ClassWiz

Are you getting used to a new ClassWiz calculator?
There are guides, manuals and videos available to help.

On the Casio website, you will find quick guides for Casio Calculators including the ClassWiz.

Casio ClassWiz quick guide

You can find an excellent calculator guide for the Casio FX991EX-ClassWiz on Dr Frost’s site.

For the complete manual from Casio see this page, note the calculator is listed under fx-570EX / fx-911EX on the Casio manuals page.

From The Calculator Guide on YouTube see this comprehensive playlist on the Casio Classwiz fx-99EX. For example see this introductory video on using Statistics mode to find the mean, variance and other statistical summary data.

Other examples include further statistical videos such as Normal Distribution calculations, integration and differential calculations and using Complex mode.

Matrix Calculations.

Correlation and Regression

Some useful resources to help you study Correlation and Regression.

This GeoGebra applet allows you to move points and watch the effect on the line of best fit.

Why not try to plot the points, draw a lines of best fit and then compare your line with the computer. This works very well on a phone, use this link.

Correlation coefficients and lines of best fit can also be studies with this PhET simulation on Least Squares Regression.

Choose from a range of examples or choose Custom to add your own points and guess then check the correlation coefficient. You can also draw your own line of best fit and compare it to the theoretical line of best fit. Note the option to include residuals for both your own attempt and the line of best fit.

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We can check Regression Calculations using this Linear Regression calculator from Social Science Statistics.

On the subject of correlation coefficients, we can play a game to see how well we can guess the correlation coefficient! Guess the Correlation Coefficient.

From Cambridge PhD student, Omar Wagih ‘Guess the Correlation‘, a rather addictive game with a purpose – Omar Wagih is collecting the data on the guesses collected and using it to analyse how we perceive correlations in scatter plots. Select About to read the rules and further details.

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For ranked data you must be able to calculate Spearman’s rank correlation coefficient from raw data or summary statistics. Again, Social Science Statistics, offers us a calculator which is useful for checking work.

Calculation details provide a useful check on work.
Note Social Science Statistics also has a calculator for calculating the Pearson correlation coefficient.

 

Integration

Firstly for some useful notes and examples on Integration:

University of Plymouth Notes

• Plymouth University – Indefinite Integration
• Plymouth University – Definite Integration
• Units 12-1 – 13.16 from AJ Hobson’s ‘Just the Maths’ offers very comprehensive coverage starting with the basics and going well beyond school Mathematics
• Choose Integration from the MathCentre topics
• Helm Workbooks – section 13 

(These sources of Notes can all be found on the Notes page.)

You can use Desmos and WolframAlpha to check your work and see some excellent visual representations.

As an example, we can find the total area bounded by f(x) = x⁴−3x³−4x²+12x, the x-axis, the line x=−1 and the line x=3.

We could use WolframAlpha for a quick check. The visual representation shows clearly that we are dealing with areas above and below the x-axis.

Scrolling down the page we see that this query also returns the indefinite integral.

For the total shaded area, we can change the limits of the query to evaluate each section.
See Calculus & Analysis for more examples of WolframAlpha queries.

Or we could turn to the excellent Desmos where we can very simply change the limits.

If you are unfamiliar with Integration with Desmos, turn to Learn Desmos: Integrals.
Note you can explore the graph shown in the video.

We could also look at this introduction to Integration on GeoGebra.

 

University Preparation

An updated version of an earlier post ….

STEP (Sixth Term Examination Paper) Mathematics is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics. For students studying for STEP papers try this excellent portal from stepmaths.co.uk which has (all free) access to STEP questions and solutions. Create an account, login and you have access to a complete library of resources.

The resources are very clearly presented. For each question you have access to a pdf with the question, Examiners’ Report and both an Official and thanks to Peter Mitchell a fully worked handwritten solution.

Note that you can also download a copy of  Stephen Siklos’ Advanced Problems in Mathematics and Core Mathematics. Advanced Problems in Mathematics is excellent preparation for ANY undergraduate Mathematics course.

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 43 problems. Each problem has been given a title and a rough indication of the mathematical content which means you can pick out questions by topic.

Stephen Siklos Advanced Problems in Mathematics and Core Mathematics

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Underground Mathematics has STEP questions within their Review Questions. Each question comes with a full worked solution.

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See also, from Cambridge University, their STEP Support Programme.

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And from Nrich, Prepare for University.

See also, a related post 16+ Challenge Questions. Also, Multiple Choice Questions which includes Oxford Admissions test questions.