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5 Hyperbolic functions

In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different.

Click the link below to open Section 4 (5 pages, 104KB).

Section 4
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1 Overview

A fundamental concept in mathematics is that of a function.

Consider, for example, the function f defined by

This is an example of a real function, because it associates with a given real number
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Keep on learning

Study another free course

There are more than 800 courses on OpenLearn for you t
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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, £8 with 15% discount means you actually pay

  £8 less (15% of £8)

  15% of 8 = × 8 =
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3.2.1 Try some yourself

Activity 20

Convert each of the following to percentages. Round off the percentages to whole numbers.

  • (a)

    • (i) 0.8

    • (ii) 0.
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3.2 Converting to a percentage

Fractions and decimals can also be converted to percentages, by multiplying by 100%.

So, for example, 0.17, 0.3 and can be expressed as percentages as follows:

  0.17 × 100% = 17%;

<
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1.4.1 Try some yourself

Activity 5

Convert each of the following fraction ratios to decimal ratios.

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References

Ahmed, A. (1987) Better Mathematics, London, HMSO.
DfEE (2001) Key Stage 3 National Strategy: Framework for Teaching Mathematics: Years 7, 8 and 9, London, DfEE.
NCTM (1989) Curriculum and Evaluation Standards for School Mathematics Reston VA, National Council of Teachers of Mathematics.

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Keep on learning

Study another free course

There are more than 800 courses on OpenLearn for you t
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Conclusion

In this course you have been introduced to the difference between mathematical content and processes. You have worked on the do–talk–record (DTR) framework for learning mathematics.


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Energy efficiency: hospitals
What strategies can hospitals adopt to reduce their impact on the local and global environment? How might they reduce the carbon footprint of vehicles transporting patients, visitors and goods? This album visits three hospitals which have adopted green approaches to energy efficiency, by cutting electricity consumption, reducing traffic volume, and even generating their own renewable energy. This material forms part of The Open University course T206 Energy for a sustainable future.Author(s): The OpenLearn team

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Studying mammals: Return to the water
Some of the most unusual and versatile of all the mammals are the groups that live, feed and reproduce underwater. In this free course, Studying mammals: Return to the water, we will see how these formerly land-based mammals adapted to a return to the water, discussing such challenges as breathing, movement and communication. This is the seventh course in the Studying mammals series. Author(s): Creator not set

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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

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Surface water
Water is arguably the most important physical resource as it is the one that is essential to human survival. Understanding the global water cycle and how we use water is essential to planning a sustainable source of water for the future. In the UK there are areas where water supplies are limited, as shown by recent droughts. Globally, there are many areas that do not have enough water to support the current population adequately. Decisions will have to be made on the best way to use water in a w
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Introduction

This free course provides a sample of postgraduate study in Technology http://www.open.ac.uk/ postgraduate/ find/ technology.

This course comprises: the transformation process mo
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The science of nuclear energy
This free course, The science of nuclear energy, will delve into the science behind nuclear power and explain what happens inside a nuclear reactor and what it means for an element to be radioactive. It will explore some of the risks of producing nuclear power and examine the arguments for and against including it in future energy planning as well as looking at other potential future solutions. Author(s): Creator not set

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Understanding water quality
Water is arguably the most important physical resource as it is the one that is essential to human survival. Understanding the global water cycle and how we use water is essential to planning a sustainable source of water for the future. In the UK there are areas where water supplies are limited, shown by recent droughts. Globally, there are many areas that do not have enough water to support the current population adequately. Decisions will have to be made on the best way to use water in a worl
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References

Edwards, P. N. and Schneider, S. H. (2001) Self-governance and peer review in science-for-policy: the case of the IPCC Second Assessment Report, in Miller, C. and Edwards, P. N. (eds) Changing the Atmosphere: Expert Knowledge and Environmental Governance. Cambridge, MA: MIT Press. Available from: http://stephenschneider.stanford.edu/Mediarology/Mediarology.html (accessed 10 May 2007).
IPCC (2000) Land Use, L
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1.8 End of section questions

Question 5

1.2.8 In praise of cheap offshore labour? continued

There are two points which are central to this line of thinking. One, according to Wolf (2004), is that the whole process, as odd as it may sound, is about mutual exploitation. Outside firms do indeed exploit the poor by taking advantage of the profitable opportunities that a pool of cheap labour represents. But Indonesian or Chinese workers, for instance, could be said to exploit the incoming firms by extracting higher pay from them and taking advantage of opportunities that previousl
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