4.9 Further exercises Determine the equation of the circle with centre (2, 1) and radius 3. The equation of the circle is ( 4.5 Ellipse (0 < e < 1) 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 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 4.2 Circles Recall that a circle in 4.1 Conic sections Conic section is the collective name given to the shapes that we obtain by taking different plane slices through a double cone. The shapes that we obtain from these cross-sections are drawn below. It is thought that the Greek mathematician Menaechmus discovered the conic sections around 350 bc. Introduction This unit is an adapted extract from the course Pure mathematics
(M208) The idea of vectors and conics may be new to you. In this unit we look at some of the ways that we represent points, lines and planes in mathematics. In Section 1 we revise coordinate geometry in two-dimensional Euclidean space, Acknowledgements All written material contained within this unit originated at the Open University. 5.2 The identification topology Our aim is to show that the object that we produce when we identify some or all the edges of a polygon is a surface. Therefore, by the definition of a surface given in Section 2.5, we must show how it can be given the structure of a topological space, and that this space is Hausdorff. Furthermore, we must show that every point has 2.2.3 Surfaces with boundary Examples of surfaces with boundary are a cylinder and a Möbius band. Other examples are the following:
Surfaces with holes
We can obtain a surface with boundary by taking any surface without boundary and punching some holes in it by removing open discs. For example, Figure 19 shows a sphere with 3 Learning outcomes By the end of this unit you should be able to: explain the terms surface, surface in space, disc-like neighbourhood and half-disc-like neighbourhood; explain the terms n-fold torus, torus with n holes, Möbius band and Klein bottle; explain what is meant by the boundary of a surface, and determine the boundary number of a given surface with boundary; construct certa Studying mammals: A winning design Nature matters in conversation Engineering: The nature of problems Testing More working with charts, graphs and tables Darwin for young people Topic 5: Income Taxation and Labor Supply part 3 | Economics 2450A: Public Economics Reuters Breakingviews: Japan's post-election policy struggle 9.S915 Developmental Cognitive Neuroscience (MIT) 12,000 Miles to University on the back of a moped Algebra For the Real World
Exercise 58
Answer
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.
Author(s):
The term mammal encompasses a huge variety of animals, including humans. But what makes a mammal a mammal? This unit explores some of the features, such as reproduction, lactation and thermoregulation methods, that mammals have in common. It is the first in a series of 10 ‘Studying mammals’ units. First published on Thu, 21 Jul 2011 as Author(s):
This unit focuses on the substance of environmental responsibility – what matters. The question ‘What should constitute our prime focus of attention?’ can prompt different responses. We consider two points of contrast in differing focuses on what matters:
1 a distinction between nature and the environment
2 a distinction between nature/environment and related human interactionsAuthor(s):
Engineering is about extending the horizons of society by solving technical problems, ranging from the meeting of basic human needs for food and shelter to the generation of wealth by trade. Engineers see the problems more as challenges and opportunities than as difficulties. What they appear to be doing is solving problems, but in fact they are busy creating solutions, an altogether more imaginative activity.Author(s):
Kyle Xu
In this module, we show the testing results and compare those of different algorithm.
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Your course might not include any maths or technical content but, at some point during your studies, it’s likely that you’ll come across information represented in charts, graphs and tables. You’ll be expected to know how to interpret this information, and possibly encouraged to present your own findings in this way. This unit will help you to develop the skills you need to do this, and gain the confidence to use them. This unit can be used in conjunction with, and builds on the ‘Working
Several 'Darwin Now' projects have focussed on bringing young people together to discuss Darwin and his legacy. Events and conferences have been taking place both in Britain and in Egypt.
Raj Chetty
Fall 2012
Dec. 13 - Should the opposition win Japan's elections, Breakingviews' Andy Mukherjee believes leader Shinzo Abe's target of 3 percent inflation won't be easy, and getting the central bank on side will be key.
This course uses neuroscience methods to study the cognitive development of human infants and children. Case studies draw from research on face recognition, language, executive function, representations of objects, number and theory of mind.
Ahmed Mashadani rode from the University of Nottingham Malaysia campus to Nottingham UK in the summer holidays, in order to raise money for the Red Cross.
Students will solve real world and mathematical problem situations using simple algebraic concepts including variables and open sentences.
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