In this unit, you have been introduced to a number of ways of representing data graphically and of summarizing data numerically. We began by looking at some data sets and considering informally the kinds of questions they might be used to answer.

An important first stage in any assessment of a collection of data, preceding any numerical analysis, is to represent the data, if possible, in some informative diagrammatic way. Useful graphical representations that you have met in this unit i
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In this subsection we consider, briefly, some problems that can arise with certain ways of drawing bar charts and pie charts.

Figure 5 shows what is essentially the same bar chart as Author(s): The Open University

The data set in Table 2 comprises the figures published by the US Labor Department for the composition of its workforce in 1986. It shows the average numbers over the year of male and female workers in the various different employment categories and is typical of the kind of data publ
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In several different parts of the world, footprints from prehistoric human civilisations have been found preserved in either sand or volcanic ash. From these tracks it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running by using the following three formulas:

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One of the advantages of identifying the general features of a calculation and then describing it mathematically is that the formula can then be used in either a computer or a calculator program to work out many different calculations quickly and efficiently. Many utility suppliers (gas, water, electricity, telephone) have tariffs based on a fixed daily (or monthly or quarterly) charge and a further charge based on how much you have used during the billing period.

For example in 2005, a
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A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if

1. the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;

2. the directrix d is the line with equation xÂ =Â a/e.

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An ellipse with eccentricity e (where 0 < e < 1) is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d. We obtain such an ellipse in standard form if

1. the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;

2. the directrix d is the line with equation xÂ =Â a
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4.2 Circles

Recall that a circle in 2 is the set of points (x, y) that lie at a fixed distance, called the radius, from a fixed point, called the centre of the circle. We can use the techniques of coordinate geometry to find the equation of a circle with a given centre and radius.
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3.3 Equation of a plane in three-dimensional Euclidean space

We stated in Section 1.7 that the general form of the equation of a plane in 3 is

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3.1 Definition, properties and some applications

In the previous section we saw how to add two vectors and how to multiply a vector by a scalar, but we did not consider how to multiply two vectors. There are two different ways in which we can multiply two vectors, known as the dot product (or scalar product) and the vector product. They are given these names because the result of the first is a scalar and the result of the second is a vector. (We shall not consider vector products in this course.)

In the audio sec
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2.4 Components and the arithmetic of vectors

We introduce now a different method of representing vectors, which will make the manipulation of vectors much easier. Thus we shall avoid having to solve problems involving vectors by drawing the vectors and making measurements, which is very time-consuming and never very accurate.

We can think of a vector as a translation, that is, as representing a movement by a certain amount in a given direction. Then we can use the Cartesian axes in the plane or in Author(s): The Open University

2.2 Multiplication by a scalar

In the collection of vectors sketched in Section 2.1, although v is not equal to c, the vectors v and c are closely related: c is a vector in the same direction as v, but it is twice as long as v. Thus it is natural to write 2v for c, since we can think of a journey repre
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2.1 Definitions

In this section we introduce an alternative way of describing points in the plane 2 or in three-dimensional space 3;
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1.8 Intersection of two planes

We saw earlier that two arbitrary lines in 2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in
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1.3 Parallel and perpendicular lines

We often wish to know whether two lines are parallel (that is, they never meet) or perpendicular (that is, they meet at right angles).

Two distinct lines, yÂ =Â m1x + c1 and yÂ =Â m2x + c2, are parallel if and only if they have the same gradient; that is, if and only if m1Â =Â m2. For example, the lines yÂ =Â âˆ’2x + 7 and
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Learning outcomes

By the end of this unit you should be able to:

• Section 1

• recognise the equation of a line in the plane;

• determine the point of intersection of two lines in the plane, if it exists;

• recognise the one-one correspondence between the set of points in three-dimensional space and the set of ordered triples of real numbers;

• recognise the equation of a plane in three dimensions.

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4 OpenMark quiz

Now try the quiz, and see if there are any areas you need to work on.

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3.4 Decreasing by a percentage

Discount can be calculated in the same way as an increase by a percentage. For example, Â£8 with 15% discount means you actually pay

Â Â Â£8 less (15% of Â£8)

Â Â 15% of 8 = Ã— 8 = Author(s): The Open University

7.4 Elixirs of the nervous system: neurotrophins

According to Section 7.2 axons obtain an elixir from targets at their synapses.

Confirmation that there is indeed an elixir came from a series of events that reveals how much of science really works. Elmer Bucker, working with Hamburger in the mid-1940s, had removed a limb bud from a chick and replaced it with a tumour from
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Climbing Droplet
By: Vivienne Self propulsion of a droplet on an incline
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