Statistical information is a familiar aspect of modern life, which features routinely in, for example, news reports, sports commentaries and advertising. Scientists who have collected large amounts of data by either counting or measuring quantities also rely on statistical techniques to help them make sense of these data. Suppose you had information collected from, say, three thousand patients, all with the same medical condition but undergoing a variety of treatments. First you would need te

The calculator is small and slips conveniently into a bag or pocket. You will be able to carry it around with you and use it unobtrusively as and when you want â€“ perhaps in a shop, on a train or in a restaurant.

Most aspects of the calculator are straightforward to use. Calculations are entered on the screen in the same order as you would write them down. More complicated mathematical functions and features are also reasonably intuitive, and there are â€˜escapeâ€™ mechanisms, so that you can explore without worrying about how you will get back to where you were.

In Section 1 we discuss intuitive ideas of symmetry for a two-dimensional figure, and define the set of *symmetries* of such a figure. We then view these symmetries as functions that combine under composition, and show that the resulting structure has properties known as *closure*, *identity*, *inverses* and *associativity*. We use these properties to define a *group* in Author(s):

3.4 Drawing and interpreting graphs

A graph shows the relationship between two quantities. These quantities may be very different: for instance, the price of coffee in relation to different years, or the braking distance of a car in relation to different speeds, or the height of a child at different ages. Because the quantities are different, there is no need to have equal scales on the graph, and it is often impractical to do so. However, it is essential that the scales are shown on the axes: they should indicate exactly what

This unit lays the foundations of the subject of mechanics. Mechanics is concerned with how and why objects stay put, and how and why they move. In particular, this unit â€“ *Modelling static problems* â€“ considers why objects stay put.

**Please note that this unit assumes you have a good working knowledge of vectors**.

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

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.

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 *

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

*In the following subsections, we apply the vector ideas
introduced so far to displacements and velocities. The
examples will feature directions referred to points of the compass,
known as bearings.*

*The direction of Leeds relative to Bristol can be described
as â€˜15Â° to the East of due Northâ€™,
or NÂ 15Â°Â E. This is an instance of a
bearing. Directions on the ground are typically given like
this, in terms of the directions NorthÂ (N),
SouthÂ (S), EastÂ (E)*

*1.3: Summing vectors given in geometric form *

*The following activity illustrates how the conversion
processes outlined in the preceding sections may come in useful. If
two vectors are given in geometric form, and their sum is sought in
the same form, one approach is to convert each of the vectors into
component form, add their corresponding components, and then convert
the sum back to geometric form.*

*Author(s):*

*1.2: Converting to geometric form *

*You have seen how any vector given in geometric form, in
terms of magnitude and direction, can be written in component form.
You will now see how conversion in the opposite sense may be
achieved, starting from component form. In other words, given a
vector
aÂ =Â a
_{1}
iÂ +Â a
_{2}
j,
what are its magnitude |a| and direction Î¸?*

*The first part of this question is dealt with using
Pythagorasâ€™ Theorem: the magnitude of a v*

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

*The material acknowledged below is Proprietary and used under licence, see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence*

Grateful acknowledgement is made to the following:

## Figures

Figur

*The polar bear has become an international climate change icon. But how much is known about this bear, its habitat and life? This unit will talk about the role of language, but by way of introduction how about the name of this bear? To me it is the polar bear; to a German it is an EisbÃ¤r (ice bear) and to a French person it is an ours blanc (white bear). In these three examples the bear is referred to as polar, white, or an ice bear â€“ eminently sensible. The Latin name for th*

*The scientific theory of plate tectonics suggests that at least some of these Arctic lands were once tropical. Since then the continents have moved and ice has changed the landscape. This unit will concentrate on evidence from the last 800,000 years using information collected from ice cores from Greenland and Antarctica, and will use this evidence to discuss current and possible future climate. The cores show that there have been nine periods in the recent past when large areas of the Earth *

*6.3 The role of active citizens and communities *

*Few people agree that individuals should take the main responsibility for tackling environmental issues. For example, in a 2007 poll of over 2000 UK citizens, 70% agreed that the government should take a lead in combating climate change, even if it means using the law to change people's behaviour. However, over 60% disagreed that there was nothing they could do to avert climate change and over half agreed that they would do more if others did more too, although 40% thought that recycling was *

We must abandon the conceit that individual, isolated, private actions are the answer. They can and do help. But they will not take us far enough without collective action.

(Al Gore, 2007)

There are some things that we can do as individuals: making this an energy-efficient house and making smart transport choices. Then there a

*5.3 Moving towards a sustainable carbon footprint *

*So far, you've been considering reductions in average individual or household carbon footprints by 20% to 30% or more.*

*But it is becoming increasingly clear that this will not be enough. As I mentioned in Section 4, developed countries, like Britain, Germany and America, will have to reduce their CO _{2}e emissions by 60% to 80% or more by 2050 to prevent climate change running out of control, while at the same time allowing the growing populations of Africa, India and China to r*

*2.2 Records of the Earth's temperature *

*To put the temperature records reported by the IPCC in context, we start with a longer-term geological perspective on the Earth's GMST.*

*
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