For **revision** and **self-testing** of your pre-knowledge of algebra, pre-calculus & calculus, use the following two links from Glencoe Mathematics *Self-Check Algebra Quizzes * and *Self-Check PreCalculus & Introductory Calculus Quizzes *. Tests are sorted by chapter & section, standardized or mixed tests, with test results that can be e-mailed to any person.

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**Sketchpad 4 files**

Download the Antiderivative Plotter for *Sketchpad* from: Follow Tangent

Download the Slope Field Plotter for *Sketchpad* from: Slope Fields

Download the Euler Method sketch for *Sketchpad* from: Euler Method Graph & Table

Download the Predator-Prey Solutions (unzip after downloading) for Problems 11-19 on pp. 357-358 of the notes.

Visit the **Sketchpad User Group** site, Sketchpad SketchExchange, to download free Sketchpad sketches for various topics at school ranging from primary to high school and college. You can also join the site if you want to contribute, discuss or ask questions from other teachers and users in the Discussion Forum.

If you have an iPad, you can download for free the *Sketchpad Explorer* application. With this app you can manipulate and use all the above sketches. However, note that with this app you cannot make your own new sketches and constructions - for that you require the full version of *Sketchpad* - see below.

**Note**: Obtain your own **LEGAL** copy of * Sketchpad 5* (the latest version) by ordering from Prof Michael de Villiers at
Send Mail.

**Animation and Interactive applets: Online or Downloads**

Dynamically explore translations, reflections, rotations and dilations of graphs at:
*Dynamic Transformations of Graphs*

See the animation applet for *Torricelli's Law: Water leaking from a tin* at: Torricelli Animation. Then complete the **optional** Exercise Activity for the theoretical derivation from high school physics of the differential equation, modeling the situation at: Torricelli Exercise.

Go to the free online, ordinary differential equation solver of *WolframAlpha* at: WolframAlpha Ordinary Differential Equation Solver and in the window at the top, input or edit different equations to be solved. You can also download *WolframAlpha* to your iPhone or iPad at: Download WolframAlpha for iPhone or iPad.

See the Euler Method in Pics with interactive applets at: Euler Method in Pics

If you have a Java Enabled browser go to the following link to use the *Graph Applet* (general function *y* = *f*(*x*) plotter): Graph Applet

Or download the Graph Applet to your own computer to use offline from: Graph Applet

*IDEA: Internet Differential Equations Activities* with interactive applets for interesting applications in several different contexts such as ecological models, wolf hunting, bungee jumping, salmon migration, etc. at Internet Differential Equations Activities

Download Microsoft Mathematics 4.0. It provides a graphing calculator that plots in 2D and 3D, step-by-step equation solving, calculus (differentiation, integration), linear algebra (matrices), and many other useful tools for mathematics and science.

**Mathematica Demonstrations**

The four *Mathematica* Notebooks below, can all be viewed with a computer with *Mathematica* installed. If your computer does not have *Mathematica*, download the free *Mathematica Player* from: Free Mathematica Player to view the notebooks.

Download the Slope Fields Demonstration for *Mathematica* from: Slope Fields Demo

Download the Numerical Methods for Differential Equations Demonstration for *Mathematica* from: Numerical Methods for Differential Equations Demo

Download the "Predator-Prey Dynamics with Type-Two Functional Response" interactive notebook for *Mathematica* from: Predator-Prey Dynamics

Download the "Predator-Prey Ecosystem: A Real-Time Agent-Based Simulation" interactive notebook for *Mathematica* from: Predator-Prey Ecosystem Simulation

**Numerical Methods**

Download the following notes on the Bisection method and Fixed point iteration on solving equations numerically. The whole text from which these excerpts have been taken can be downloaded from Numerical Methods: African Virtual University.

Read about Hero's method (or the Babylonian method) for finding square roots at Methods of Computing Square Roots or Hero of Alexandria.

View directions on how to use *Sketchpad* to create an iteration using Hero's method to find the square root of any number at Hero's method with Sketchpad.

Explore iteration with these excellent Step by Step activities for calculator and spreadsheet from Mason, Graham & Johnston Wilder (2005), *Developing Thinking in Algebra*.

Iteration is not restricted to algebra, as these two interactive Java sketches show iteration examples in geometry. Also included is a link to a 2014 paper in *Learning & Teaching Mathematics*: Over and over: two geometric iterations with triangles.

**iTunes University lectures & other clips**

Download the *mp4* movie lecture clip from MIT on the *Euler's Numerical Method* for solving differential equations from (**note** it is a large file of 120.1 MB): Euler's Numerical Method. This mp4 clip will play on any MP4 player such as QuickTime, Windows Media Player, etc.

Download other *iTunes U* movie lecture clips from MIT on *Differential Equations* at: iTunes U. Then click on button *Go to iTunes U in the iTunes Store* with right arrow in blue on lower right hand side of screen. You will need to already have iTunes (free) installed on your computer BEFORE you go to to iTunes U. Then look under Categories on the left for the Mathematics button and then search for the MIT lectures on Differential Equations - the 'Euler Numerical Method' clip above is Lecture 2.

*Interesting calculus problem*: Modelling Activities with YouTube video clip showing graph of height of water in tilted container against time: Tilted container being filled. From mathematical principles, explain WHY the graph has this shape. More precisely, determine what type of function the height is of time.

**Assignment**

The 2014 Assignment is now uploaded. Download/access it at: Solving Differential Equations with Technology

**Some articles**

The following papers of mine may be useful background reading, not only about this course, but also generally about the teaching and learning of mathematics:

* The role of technology in mathematical modelling (1994)

* Place Kicking Locus in Rugby (1999)

* Mathematical Applications, Modeling & Technology (2007)

Also read this older paper by esteemed mathematician Peter Hilton already written in 1981 and published in 1989: Mathematics in 2001: Implications for today's undergraduate teaching.

**Exercise Answers**

1) From course textbook ** CALCULUS** by Paul Foerster: Chapter 7 - 3.5 MB - Go to: Differential Equations Exercise Answers to odd numbered questions.

2) Download the Chapter Review Test, 406 KB, for Self Study. Check your answers in the worked solutions below.

3) Download and study: Worked Solutions to Review Problems 7.7 and Chapter Self-test, 172 KB.

**Warning**: Please do NOT look at the solutions BEFORE you have attempted to solve the problems yourself; if you do, you'll NOT learn anything!

**Online Differential Equations Videos**

Watch or download the first 10 of the free Differential Equations Videos at Learners TV.

**Calculus PDF Books for Sale**

The range of PDF books at the World Wide Center of Mathematics Textbooks are recommended additions to consider for your professional library and/or that of your prospective school. The site also has several free instructional videos - scroll down for the free videos on *differential equations*. The PDF books are available at $10 each and can be downloaded directly from the internet after payment.

**Free e-Books**

*Real Functions in One Variable: Calculus* 1a - 9 MB - (complex numbers, elementary functions, differentiation, integration, simple differential equations, approximation of functions). Go to: Calculus 1a

*Real Functions in One Variable: Simple Differential Equations* - 6.9 MB - (theorems, separation of variables, linear differential equations of first order, existence & uniqueness, Bernoulli differential equations, setting up model equations, etc.). Go to: Simple Differential Equations

**Math 420: Geometry link**

**UKZN Library link**

Visit UKZN Library or Education Library Guide for further resources such as e-books, e-journals and many other useful links.

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Read (or subscribe to) the latest: *Math e-Newsletter* with information about new math books, websites, conferences, quotes, humour, and challenging explorations for students.

The July 2014 issue of the *Learning & Teaching Mathematics* journal of AMESA (Association for Mathematics Education for South Africa) is available at:

(** click on pic below**)

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For an AMESA membership form, click *here*. For more information about AMESA go to *more info* and/or join AMESA on Facebook at *Facebook*.

*Click on any of the above pictures for more information about these books and/or to purchase.*