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taken directly from Figs 14-1B and 14-2B Now let s look at the ratio function y = 10x /10 x Using the algebraic rules for exponents, we can rewrite this as y = 10[x ( x)] which simplifies to y = 102x The domain encompasses all real numbers The range is the set of all positive real numbers The graph is the solid black curve in Fig B-15 4 Figure B-16 shows the same graphs as Fig B-15 However, in this illustration, the y axis is logarithmic, spanning the three orders of magnitude from 01 to 100 The dashed gray lines are the graphs of y = 10x and y = 10 x

FIG 10.20. The temperature-control loop must be disabled at both the start and finish of the reaction.

Figure B-16

x 1 0 1

In Part Two we give you everything you need to succeed in each stage of the project life cycle. We introduced the cycle and explained its value in 4. Now, let us look at the beginning and end of each stage. The concept stage begins with a bright idea. If the idea is a good one, we de ne project goals and move on to the analysis stage. If the project is not worth doing, the gate at the end of the concept stage is our chance to abandon the project before we waste time and money. The analysis stage begins with a general outline of project goals and scope. We fully develop the de nition of the output of the project: the

542 Worked-Out Solutions to Exercises: 11-19

The solid black line is the graph of y = 102x 5 Figure B-17 is the graph of the function y = 10(1/x) for values of x ranging from 10 to 10 When we input x = 0, we get 101/0, which is undefined For any other real value of x, the output value y is a positive real, so the domain is the set of all nonzero reals No matter how large we want y to be when y > 1, we can always find some value of x that will produce it No matter how small we want y to be when 0 < y < 1, we can always find some value of x that will produce it However, we can t find any value for x that will give us y = 1 For that to happen, we must raise 10 to the zeroth power, meaning that we must find some x such that 1/x = 0 No such x exists, so the range of the function is the set of all positive reals except 1 The graph has a horizontal asymptote whose equation is y = 1, and a vertical asymptote corresponding to the y axis The open circle at (0,0) tells us that this point is not part of the graph 6 Suppose that there s a real number x that satisfies the equation ln 0 = x

detailed requirements for the product, service, system, or changes to a system that we will deliver when the project ends The analysis stage ends with a full de nition of project value and goals The design stage can start when we know all the requirements for what we are making Then we plan how to make it, that is, we create the project management plan And the team de nes exactly what we are making, that is, the product or service speci cation The design stage ends when all plans, including the design speci cation, the project schedule, the budget, the teamwork and communications, the quality, and the risk plans are all in place If they are approved, we go through the gate into the development stage The development stage is the rst part of the life cycle where we are doing, rather than planning.

y 10