Introduction to the calculus of variations
This free course concerns the calculus of variations. Section 1 introduces some key ingredients by solving a seemingly simple problem – finding the shortest distance between two points in a plane. The section also introduces the notions of a functional and of a stationary path. Section 2 describes basic problems that can be formulated in terms of functionals. Section 3 looks at partial and total derivatives. Section 4 contains a derivation of the Euler-Lagrange equation. In Section 5 the Euler
Author(s): Creator not set

Now that you have rules for addition and subtraction of negative numbers, think about multiplication and division.

Example 27

Describe each of the following in terms of the number line and the value of Thomas's piggy bank:

• (a) the mul
Author(s): The Open University

Now what about subtraction? You can think of subtraction as undoing addition: adding 3 to 8 gets you 11, and so subtracting 3 from the answer, 11, gets you back to 8. Therefore, in terms of the number line, subtracting 3 from 11 means starting at 11 and moving 3 units to the left.

Author(s): The Open University

Activity 30

Carry out the following calculations, without your calculator.

• (a) 3 × (60 + 70).

• (b) (3 × 60) + 70.

• (c) (70 − 60) ÷ 5.

• Author(s): The Open University

When using a calculator many people have ‘blind
Author(s): The Open University

Question 1

Which of these triangles are similar?

There is another kind of symmetry which is often used in designs. It can be seen, for instance, in a car wheel trim.

Look at the trim on the left. It does not have line symmetry but
Author(s): The Open University

Mediating Change: Culture and Climate Change
Every generation faces challenges that previous generations could scarcely imagine. Twenty years ago, few people were talking about climate change, now it's one of the most hotly-contested areas in politics. How do artists, writers, musicians and broadcasters respond when a new subject appears that is as large and significant as this? What kind of novels, plays, paintings, sculptures, movies and music begin to emerge? ‘Mediating Change’ is a four-part series, chaired by BBC Radio 4’s Que
Author(s): The OpenLearn team

Studying mammals: Food for thought
Who were our ancestors? How are apes and humans related? And where does the extinct Homo erectus fit into the puzzle? In this free course, Studying mammals: Food for thought, we will examine culture, tool use and social structure in both apes and humans to gain an understanding of where we come from and why we behave as we do. This is the tenth course in the Studying mammals series. Author(s): Creator not set

Question 5

Information on the different albedos of various types of surface was given in Author(s): The Open University

I believe that the principal task for an environmental pragmatism is not to reengage the … debates in environmental ethics but rather to impress upon environmental philosophers the need to take up the largely empirical question of what morally motivates humans to change their attitudes, behaviours, and policy preferences toward those more supportive of long-term environmental sustainability.

(Light, 2002, p. 446)

Author(s): The Open University

By framing, I mean the structures and pre-assumptions that we consciously or unconsciously apply to a situation in order to make sense of it. So are there any differences between the way in which we frame nature in caring for environment and the way in which we frame it to provide accountability? What significance might this have, and what tools might be used to bridge the responsibilities of caring and accountability?

Caring for environment makes manifest the informal aspects of
Author(s): The Open University

Before you read on, I would like you to dwell for just a moment on the significance of this shift from direct investment by Western firms to the establishment of subcontracting ties with overseas partners. Aside from outside firms being able to pa
Author(s): The Open University

We are also guilty of importing exotic species, some of which, like the rhododendron (imported from Asia to Europe), have run riot in the absence of natural predators or primary consumers, and so have tended to out-compete native plants. Sometimes introductions have been accidental; rats and many disease-causing organisms have spread around the world via relatively modern transportation such as sailing ships. However, deliberate introductions, such as the rhododendron, have been made with wor
Author(s): The Open University

The UN's 1948 Universal Declaration of Human Rights asserted that the ‘recognition of the inherent dignity and of the equal and inalienable rights of all members of the human family is the foundation of freedom, justice and peace in the world’. It further affirmed that human rights should be protected by the rule of law, that they were ‘essential to promote the development of friendly relations between nations’, that these fundamental human rights include the equal rights between men
Author(s): The Open University

Approaches to software development
This free course, Approaches to software development, presents an engineering approach to the development of software systems – a software engineering approach. The course pays particular attention to issues of software quality, in terms of both product (what is built) and process (how we build it). First published on Mon, 18 Jun 2018 as Author(s): Creator not set

The advancement of democracy in contemporary Western nation-states and the intensification of globalisation processes have encouraged the re-emergence of nationalist movements representing oppressed or silenced nations that demand the right to self-determination. In the case of ethnic groups formed by people of immigrant origin, democracy has provided them with the tools to pursue the right to develop and practice their indigenous culture and language alongside those of the host country. One
Author(s): The Open University

Problem ownership is a tricky issue. It's also an issue that good leaders get right instinctively, and poor leaders get wrong consistently. The point is that there are two distinct classes of problems faced by leaders. The first consists of problems which are owned by the group members. Examples include when some additional resources are required, when instructions are not understood or when members complain that something is wrong. Under these conditions the leader's function is to provide p
Author(s): The Open University

1. There are many different types of projects; all have specific objectives, constraints (such as budgets and schedules) and a group or team responsible for the completion of the project.

2. Project teams are effective when both task and relationship behaviours are competently handled. The main task-oriented behaviours are:

• estimating and planning;

• assembling a team;

• reporting a
Author(s): The Open University