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Inverse function, 293, 310 trigonometric functions, 203, 227 Irrational numbers, 90 Irreducible quadratic factors, 340 Law of cosines, 207 Law of the mean, 137 Law of sines, 212 Laws of exponents, 125 Least-square principle, 193 Left-hand derivative, 103 Leibniz, Gottfried von, 253 L H pital s rule, 301 Limit of a function, 63, 68 in nite, 72 at in nity, 74 one-sided, 71 Limits of integration, 248 Line: point-slope equation, 28 slope-intercept equation, 29 Linear factors, 340 and roots of polynomials, 55 Logarithm: to the base a, 299 to the base e, 293 common, 299 natural, 285 Logarithmic differentiation, 289 m (slope), 25 Marginal cost, 155 Marginal pro t, 155 Mathematical induction, 100 Maximum: absolute, 111 relative, 110 Mean value, 255 Mean-value theorem: for derivative, 138 for integrals, 255 Midpoint formula, 11 Midpoint rule, 262 Minimum: absolute, 111 relative, 111 Natural logarithm, 285 tables, 352 Newton, Isaac, 253 Newton s method, 166 Normal line, 95 Null set, 20 Odd function, 54 One-one function, 309

transponder. A typical transponder bandwidth is 36 MHz, and allowing for a 4-MHz guardband between transponders, 12 such transponders can be accommodated in the 500-MHz bandwidth. By making use of polarization isolation, this number can be doubled. Polarization isolation refers to the fact that carriers, which may be on the same frequency but with opposite senses of polarization, can be isolated from one another by receiving antennas matched to the incoming polarization. With linear polarization, vertically and horizontally polarized carriers can be separated in this way, and with circular polarization, left-hand circular and right-hand circular polarizations can be separated. Because the carriers with opposite senses of polarization may overlap in frequency, this technique is referred to as frequency reuse. Figure 7.12 shows part of the frequency and polarization plan for a C-band communications satellite.

Figure 164 shows a program that implements the algorithm described in the previous section If you are running the program for the rst time read the instructions and press Enter or click the OK button to continue You will see the environment shown in Fig 165 Do not press the Cancel button (or Esc key) if you have never run the program before, since there are no memories for the robot to reload However, once there are memories you can always stop the program and run it at a later date and have the robot recall its previous experiences (by pressing the Cancel button or Esc key) Notice how the RestoreMemory subroutine uses the FilExists() function to avoid loading the memory les if they do not exist.

Figure 7.12 Section of an uplink frequency and polarization plan. Numbers refer to frequency in megahertz.

Frequency reuse also may be achieved with spot-beam antennas, and these may be combined with polarization reuse to provide an effective bandwidth of 2000 MHz from the actual bandwidth of 500 MHz. For one of the polarization groups, Fig. 7.13 shows the channeling scheme for the 12 transponders in more detail. The incoming, or uplink, frequency range is 5.925 to 6.425 GHz. The carriers may be received on one or more antennas, all having the same polarization. The input filter passes the full 500-MHz band to the common receiver while rejecting out-of-band noise and interference such as might be caused by image signals. There will be many modulated carriers within this 500-MHz passband, and all of these are amplified and frequencyconverted in the common receiver. The frequency conversion shifts the carriers to the downlink frequency band, which is also 500 MHz wide, extending from 3.7 to 4.2 GHz. At this point the signals are channelized into frequency bands which represent the individual transponder bandwidths. A transponder may handle one modulated carrier, such as a TV signal, or it may handle a number of separate carriers simultaneously, each modulated by its own telephony or other baseband channel.

The two intersection points are shown in Fig. 5-4. Notice that we could have solved 13x 2 + 54x + 45 = 0 by factoring the left side, (13x + 15)(x + 3) = 0 However, such a factorization is sometimes dif cult to discover, whereas the quadratic formula can be applied automatically.

The robot starts out with all of its needs at maximum value as indicated by the bar graphs on the right side of the screen in Fig 165 The robot begins exploring its environment and if it bumps into objects, gets too close to the re, or wanders into a briar patch, the appropriate information will be saved to the bad memory list Eventually, the robot will nd the garden and eat This will satisfy the hunger need, which will be indicated by the meter on the right of the screen gradually reducing to zero as the robot eats The action and the environmental conditions that led to the food will be saved in the good memory Notice that when the program is running, two counters at the top of the screen show the number of items stored in the good and bad memory lists.