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( f1 / f2)(x) = f1 (x) / f2 (x) = (x 1)/x2 = x 1 x 2 and ( f2 / f1)(x) = f2 (x) / f1 (x) = x2/(x 1) = x2 (x 1) 1 For f1 and f3, we have ( f1 / f3)(x) = f1(x) / f3 (x) = (x 1) / x1/2 = x1/2 x 1/2 and ( f3 / f1)(x) = f3 (x) / f1 (x) = x1/2/(x 1) = x1/2(x 1) 1 For f2 and f3, we have ( f2 / f3)(x) = f2(x) / f3 (x) = x2/x1/2 = x3/2 and (f3 / f2)(x) = f3(x) / f2(x) = x1/2/x2 = x 3/2

Spheres can be made out of the round by increasing or decreasing the axial radii in the x, y, and z directions individually

Alternative equation for a sphere centered at the origin Once again, consider the general equation of a perfect sphere centered at the origin That equation, in standard form, is

Since we need to create many project documents, it is good to have a list of all of them with de nitions. We present this in Table 8-1. Here are a few notes about project documents: Table 8-1 includes only project management documents and deliverables to the customer. Technical plans and design documents, such as the context diagram and data ow diagrams for computer systems, blueprints and engineering diagrams for construction, storyboards for movies, and so forth, should also be de ned and tracked. Feel free to modify document names or add or change documents to suit the needs of your project and your company. This list was designed to be both comprehensive and generic. You can use it as a checklist to make sure that you have not left out any crucial documents. You can also make a checklist appropriate to your own projects at your company. Document status and ownership should be tracked, either in a project tracking tool or in some other form. Based on Table 8-1, you can create a document completion tracking checklist.

The prime function of a controller is that of regulation. The controller is intended to change its output as often and as much as necessary to keep the controIled variable at the set point. Every process is subject to variations in load. In a well-regulated loop, the manipulated variable will be driven to balance the load. Consequently, the load is often measured in terms of the corresponding value of controller output. In the equation describing the proportional controller, the bias b equals the output when the error is zero. This bias may be fixed at the normal value of output, usually 50 percent, or it may be adjusted by hand to match the current load. This adjustment is called manual reset. But because of the proportional relationship between input and output, a change in output by any amount cannot be gained without a corresponding change in error. Should the output of the proportional con-

x2 + y2 + z2 = r2 where r is the radius If we divide through by r2, we get x2/r2 + y2/r2 + z2/r2 = 1 This equation tells us that the radius is always the same, whether we measure it in the direction of the x axis, y axis, or z axis To emphasize the fact that we can, if desired, change any of all of these axial radii, let s rewrite the above equation as x2/a2 + y2/b2 + z2/c2 = 1 where a, b, and c are positive real numbers representing the radii along the x, y, and z axes, respectively In the case of a perfect sphere, we have a=b=c If these three positive real-number constants a, b, and c are not all the same, then we have a distorted sphere

Oblate sphere centered at the origin Suppose that we take a perfect sphere and then shorten one of the three axial radii This process gives us an object called an oblate sphere It s flattened, like a soft rubber ball when pressed between our hands Figure 17-5 shows an example This is what we get if we take the sphere from Fig 17-3 and reduce the axial radius b (the one that goes along the y axis), while leaving the axial radii a and c unchanged The center of the object is still at the origin, but we can no longer say that all the points on its surface are equidistant from the origin The general equation for an oblate sphere centered at the origin is