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The z-value at the absolute minimum point is zmin = 2xmin2 + 4xmin + 3 = 2 ( 1)2 + 4 ( 1) + 3 = 2 4 + 3 = 2 + 3 = 1 Now we know that the coordinates of the parabola s vertex are ( 1,5,1) As the basis for our next point, let s choose x = 3 We can plug it directly into the function to get z = 2x2 + 4x + 3 = 2 ( 3)2 + 4 ( 3) + 3 = 18 12 + 3 = 6 + 3 = 9 This gives us ( 3,5,9) as the coordinates of a second point on the curve Finally, let s set x = 1 Plugging it in, we obtain z = 2x2 + 4x + 3 = 2 12 + 4 1 + 3 =2+4+3=9 The third point on our curve is (1,5,9) We now have three points: ( 3,5,9), ( 1,5,1), and (1,5,9) Figure 18-7 shows these points, along with a graph of the parabola passing through them, as seen in the plane where y maintains a constant value of 5

( 3, 5, 9)

(1, 5, 9)

Most experienced project managers have learned PERT and CPM. Even so, they rarely use them. PERT and CPM are generally only implemented on projects where a xed-price contract is written before work starts, and that is done to ensure that the consulting rm makes money on the contract. It should not be that way. Good time management is crucial on all projects because it allows us to deliver the highest possible quality. That is what really matters.

FIG 5 19. The proportional characteristic of a continuous nonlinear controller displays variable damping.

+x ( 1, 5, 1)

Figure 18-7

Rather than reproduce what is easily available elsewhere, we will discuss the key issues in the sequence of steps that create a schedule.

Graph of a parabola in a plane parallel to the xz plane, such that y has a constant value of 5 On both axes, each increment represents 1 unit

Think about the graphs of higher-degree polynomial functions confined to specific planes in Cartesian xyz space For example, consider the cubic function in the plane where x = 2, such that x=2 y=t z = t3 or the quartic function in the plane where z = 7, such that x = 3t 4 + 6 y=t z = 7 Can you draw graphs of these curves

In Chap 13, we learned that the equation of a circle centered at the origin in the Cartesian xy plane can be written in the form x2 + y2 = r2 where r is the radius In Chap 16, we learned that the parametric equations for such a circle are x = r cos t and y = r sin t where t is the parameter Let s expand these notions to deal with any circle in xyz space that s centered on, and exists entirely in a plane perpendicular to, one of the three coordinate axes

Project management software programs are dangerous. You can create a full-color schedule that looks beautiful and convinces everyone, but will not work at all. Or you can create a schedule that works, but, as soon as one thing changes, you have to re-do the whole thing. However, if you follow the steps listed here, and address the issues we raise, you can create realistic schedules that can be adjusted as the project moves forward. At that point, the schedule in your project management software program becomes a real tool that helps the project.

It is not desirable shape of the parabolic sections shown in Fig 519 for fl to equal zero, since this would render the controller essentially insensitive to small signals, and offset would result A value of p in the vicinity of 01 would make the minimum gain of the controller 10/P A characteristic of this sort produces varying degrees of damping in the closed loop If a linear controller were used to regulate a given linear process, a certain proportional gain could be found which would produce uniform oscillations A straight line represent ing this gain, labeled zero damping, is superimposed on the curve in Fig 519 If the proportional gain of t he linear controller were halved, the closed loop would exhibit >a-amplitude damping The controller gain representing ji-amplitude damping is also indicated The nonlinear characteristic crosses both these contours of constant damping.