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It is important to understand that nothing lives forever. When an animal or a person is mature and adding value to society, we tend to forget that the value that that animal or person o ers is separate from the organism. There is a simple reason for this: in our society, we do not like to think about old age and death. But the fact is, we will all grow old and die, and the products and services we build will all become outmoded and break down. When they do, they leave a void. That void is the value the system used to have, that is, the need it used to ful ll. That value and need is an element of the larger system: the business, society, or ecosystem to which the system contributed. In ecology, it is called the niche of the organism. In project management, de ning that niche, that crucial set of functions the business needs, is the key to designing a new product or service to replace or improve upon the old system. In nature, evolution lls this role. If the ecosystem has not changed, then evolutionary pressure will tend towards stability, and the children will be much like the parents. Nature will replace each organism with a similar organism in the next generation. But if the ecosystem is changing, or if there is a missing part, evolutionary pressure causes adaptation through the preference for new variation. For example, on the Galapagos Islands, where Charles Darwin rst studied, very few species of nches arrived by chance from the mainland. The nches di erentiated widely, ful lling many di erent roles usually held by many other birds and mammals with their evolution producing appropriate tools, such as specialized beaks, to allow them to function in their niches.

524 Worked-Out Solutions to Exercises: 11-19

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The living space of our whole planet is the ecosphere, the ball of water and air supported by earth and lighted by the sun, where life can ourish It is a system, and the measures of quality in the system are: biomass, the total weight of life on earth; biodiversity, the number of di erent species; and ecosystem stability, slow and cyclical uctuations in environmental factors When a problem arises, nature does not just x the problem; it improves the whole system Let us take a look at a case study About two billion years ago, the ecosphere had a major problem Plants had been producing a waste product called oxygen for about two billion years, and it had increased to toxic levels Too much oxygen kills plants It also causes spontaneous res around the entire planet, which create rapid uctuations in temperature.

Adding 9 to each side, we get x 2 + 9 = 9y 2 /4 Multiplying through by 4, and then transposing the left- and right-hand sides of the equation, we come up with 9y 2 = 4x 2 + 36 Dividing through by 9 yields y 2 = 4x 2 /9 + 4 After we take the complete square root of both sides, we have y = (4x 2 /9 + 4)1/2 Replacing the variable y by the relation notation f 1 (x), we conclude that f 1 (x) = (4x 2 /9 + 4)1/2 6 With the information stated in Problem 5, along with the graph of the relation shown in Fig 12-2, we can see that the real-number domain of f is the set of all reals greater than or equal to 2, or smaller than or equal to 2 Another way of stating this is to say that the domain of f is the set of all reals except those in the open interval ( 2,2) The range of f is clearly the set of all reals 7 Figure B-3 shows the graph of the original relation f (x) = (9x 2 /4 9)1/2 as a pair of gray curves, and the graph of the inverse relation f 1 (x) = (4x 2 /9 + 4)1/2 as a pair of black curves The point reflector is the dashed line The real-number domain of f 1 is the same as the real-number range of f ; that s the set of all reals The real-number range of f 1 is identical to the real-number domain of f ; that s the set of all reals except those in the open interval ( 2,2) 8 Figure B-4 shows the graph of the original relation with its domain restricted to the reals greater than or equal to 2 ( gray curve) and its inverse (black curve) The curve for the inverse relation is a half-hyperbola that opens upward and intersects the y axis at (0,2) A movable vertical line never intersects the black curve at more than one point Therefore, the inverse relation f 1 (x) = (4x 2 /9 + 4)1/2 is a true function of x.

-k 96 k ( WHCH/WCCC)(CC/A~C -k 35)

12

4 6

To bring the ecosphere back into balance, it was necessary to develop a better way of getting rid of that toxic waste product: oxygen The primary component of the niche that needed to be lled could be stated as, A new organism is needed to use up oxygen But that one feature was not a full speci cation What was the best thing to do with the oxygen How about if it was combined with carbon, to create carbon dioxide Plants like carbon dioxide But where will these new creatures get the carbon Well, they could eat plants But, to eat enough plants, they will need to be able to run around And if they eat plants, they will get nitrogen as well as carbon What can they do with the nitrogen Mix it with other chemicals and dump it out the back And so, a speci cation was born.

Figure B-3

y 6 4