Learning outcomes After studying this Unit you should be able to: calculate the mean, weighted mean and the median of a batch of numbers; calculate a percentage and a percentage price increase; use a weighted mean to find an average percentage price increase (given the weights); use the statistical facilities of your calculator to find the mean, the median and (given the weights) the weighted mean of a batch of numbers; calculate a perce
6.5.1 Another ‘making a lawn’ solution Suppose you have some friends who are planning to put a new lawn in their garden. The lawn is to be 12 m by 14 m and they have a choice of either laying turf or sowing grass seed. You have been asked to help them decide between the two. 3.2 Using formulas Formulas are important because they describe general relationships, rather than specific numerical ones. For example, the tins of paint formula applies to every wall. To use such a formula you need to substitute specific values for the general terms, as the following examples show. Introduction Do you want to improve your ability to subtract one number from another, especially if decimals are involved, without having to rely on a calculator? This unit will help you get to grips with subtraction and give you some practice in doing it. You can start with some practice in subtracting small numbers in your head if you want to. Then we will show you how to subtract bigger numbers on paper. Finally we look at how to subtract decimal numbers. You don’t need to complete the 8 Dealing with remainders How do you deal with divisions where there is a remainder and there are no more digits you can carry to? In most cases you will need to express your answer as a decimal number rather than as a whole number plus a remainder. Take the example of 518 divided by 8. 3.5 Maths in archaeology 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: 4.6 Hyperbola (e > 1) 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 the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0; the directrix d is the line with equation x = a/e.
4.2 Circles Recall that a circle in 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 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): 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 1.8 Intersection of two planes We saw earlier that two arbitrary lines in 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, y = m1x + c1 and y = m2x + c2, are parallel if and only if they have the same gradient; that is, if and only if m1 = m2. For example, the lines y = −2x + 7 and 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. 2.1.3 Concept cards Another way to tackle unfamiliar words is to start a ‘concept card’ system, using index cards. When you meet a word which seems important, take a new card and write the word at the top, followed by any useful information you have found. File the cards alphabetically and add details as you come across new information. (It is worth getting an index card box anyway, then you can try out various ways of using it to organise your studies.) 6 Phylogeny and cladistic analysis In Section 3.3 the point was made that many physiologists consider that desert birds are successful because of their avian physiology, not because of any specific adaptations. While Williams and Tieleman's research on hoopoe larks demonstrated that desert species are capable of flexibility in metabolic rate and evaporative water loss, it suggested that adaptation is important too. The selective advantages of lowered BMR and TEWL for desert birds include reduced energy demand, and lower produc The Lost Art of Finding Our Way The architecture of refugee protection 21M.220 Early Music (MIT)
Example 18 Making a lawn
Example 8
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.
2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in
http://www.hup.harvard.edu/catalog.php?isbn=9780674072824
Long before GPS, Google Earth, and global transit, humans traveled vast distances using only environmental clues and simple instruments. In his new book, THE LOST ART OF FINDING OUT WAY, John Huth asks what is lost when modern technology substitutes for our innate capacity to find our way. Encyclopedic in breadth, weaving together astronomy, meteorology, oceanography, and ethnography, the book puts us in the shoes, ships, and sleds of ear
Annual Harrell-Bond Lecture 2012. Lecture by Professor Mariano-Florentino Cuéllar (Co-Director of the Stanford Center for International Security and Cooperation)recorded on 7 November 2012 at the Oxford Museum of Natural History.
This class covers the history of Western music from antiquity until approximately 1680, about 2000 years worth of music. Rather than cover each topic at the same level of depth, we will focus on four topics in particular and glue them together with a broad overview of other topics. The four topics chosen for this term are (1) chant structure, performance, and development; (2) 14th century music of Italy and France; (3) Elizabethan London; and (4) Venice in the Baroque era.
The class will also in
