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Introduction

In unit MSXR209_1 you saw how some of the stages of a mathematical modelling process can be applied in the context of modelling pollution in the Great Lakes. In this unit you are asked to relate the stages of the mathematical modelling process to another practical example, this time modelling the skid marks caused by vehicle tyres. By considering the example you should be able to draw out and clarify your ideas of mathematical modelling.

This unit, the second in a series of five, builds
Author(s): The Open University

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Introduction

This unit explores a real-world system – the Great Lakes – where mathematical modelling has been used to understand what is happening and to predict what will happen if changes are made. The system concerned is extremely complex but, by keeping things as simple as possible, sufficient information will be extracted to allow a mathematical model of the system to be obtained.

This unit is an adapted extract from the course Author(s): The Open University

5.3 Powers

Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent).

Definition

If a > 0, m Author(s): The Open University

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5.1 Arithmetic with real numbers

At the end of Section 1, we discussed the decimals and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least Upper Bound Property of Author(s): The Open University

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6.2 Getting the feel of big and small numbers

Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 1099 are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big?

First express 1 000 000 000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109
Author(s): The Open University

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3.1.1 Try some yourself

1 Express each of the following numbers in scientific notation.

  • (a) Light travels 9460 700 000 000 km in a year.

  • (b) The average distance from the centre of the Earth to the centre o
    Author(s): The Open University

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2.1 The impact of a power

Here is a tale based on an ancient Eastern legend, which gives an idea of the impact of raising a number to a power.

Example 6

A long time ago there lived a very rich king whose son's life was saved by a poor old beggar woman. The king was naturally very grateful to the woman, so he offered to
Author(s): The Open University

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1.4 Cubes

To find the cube of a number, multiply three copies of it together. For example:

You can use your calculator to find cubes. 23 is ‘two cubed’ or ‘two to the power three’. Just as ‘square root’ is the opposite process to squaring, so 'cube root' is the o
Author(s): The Open University

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Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

All materials included in this unit are derived from content originated at the Open University.


Author(s): The Open University

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1 Using vectors to model

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 1 MB).

Learning outcomes

After studying this unit you should:

  • know some basic definitions and terminology associated with scalars and vectors and how to represent vectors in two dimensions;

  • understand how vectors can be represented in three (or more) dimensions and know both plane polar and Cartesian representations;

  • know ways to operate on and combine vectors.


Author(s): The Open University

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Introduction

This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.

This is an adapted extract from the Open University course Mathematical methods and models (MST209)
Author(s): The Open University

Learning outcomes

After studying this unit you should:

  • be able to solve homogeneous second-order equations;

  • know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;

  • know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.


Author(s): The Open University

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Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

All materials included in this unit are derived from content originated at the Open University.


Author(s): The Open University

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Learning outcomes

After studying this unit you should be able to:

  • understand how the wave and diffusion partial differential equations can be used to model certain systems;

  • determine appropriate simple boundary and initial conditions for such models;

  • find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the meth
    Author(s): The Open University

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Introduction

This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations.

This unit is an adapted extract from the course Mathematical methods and models (MST209
Author(s): The Open University

1 First-order differential equations

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 1.6 MB).

Learning outcomes

During this unit you will:

  • learn some basic definitions and terminology associated with differential equations and their solutions;

  • be able to visualize the direction field associated with a first-order differential equation and be able to use a numerical method of solution known as Euler's method;

  • be able to use analytical methods of solution by direct integration; separation of variables; and the integrating factor method.


Author(s): The Open University

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First-order differential equations

This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration.

This unit is an adapted extract from the course Mathematical methods and
Author(s): The Open University

Acknowledgements

All materials included in this unit are derived from content originated at the Open University.


Author(s): The Open University

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