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Refer to the text in this chapter if necessary A good score is at least 18 correct Answers are in the back of the book 1 As the size of the plates in a capacitor increases, all other things being equal, (a) the value of X C increases negatively (b) the value of X C decreases negatively (c) the value of X C does not change (d) we cannot say what happens to X C without more data 2 If the dielectric material between the plates of a capacitor is changed, all other things being equal, (a) the value of X C increases negatively (b) the value of X C decreases negatively (c) the value of X C does not change (d) we cannot say what happens to X C without more data 3 As the frequency of a wave gets lower, all other things being equal, the value of XC for a capacitor (a) increases negatively (b) decreases negatively (c) does not change (d) depends on the current 4 What is the reactance of a 330-pF capacitor at 800 kHz (a) 166 (b) 000166 (c) 603 (d) 603 k 5 Suppose a capacitor has a reactance of 450 at 377 Hz What is its capacitance (a) 939 F (b) 939 F (c) 742 F (d) 742 F 6 Suppose a 47- F capacitor has a reactance of 47 What is the frequency (a) 72 Hz (b) 72 MHz (c) 0000072 Hz (d) 72 Hz

Three Relations y 6 ( 2,4) ( 1,1) 2 (2,4) (1,1) x 6 4 (0,0) 4 6 2 2 2 4 6

.

y 6 4 2 (2,4) (1,1) x 6 4 (0,0) 4 6 2 2 2 4 6

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7 Suppose a capacitor has X C = 8800 at f = 830 kHz What is C (a) 218 F (b) 218 pF (c) 000218 F (d) 218 pF 8 Suppose a capacitor has C = 166 pF at f = 400 kHz What is X C (a) 24 k (b) 24 (c) 24 10 6 (d) 24 M 9 Suppose a capacitor has C = 4700 F and XC = 33 What is f (a) 10 Hz (b) 10 Hz (c) 10 kHz (d) 10 kHz 10 Each point in the RC plane (a) corresponds to a unique inductance (b) corresponds to a unique capacitance (c) corresponds to a unique combination of resistance and capacitance (d) corresponds to a unique combination of resistance and reactance 11 If R increases in an RC circuit, but X C is always zero, the vector in the RC plane will (a) rotate clockwise (b) rotate counterclockwise (c) always point straight toward the right (d) always point straight down 12 If the resistance R increases in an RC circuit, but the capacitance and the frequency are nonzero and constant, then the vector in the RC plane will (a) get longer and rotate clockwise (b) get longer and rotate counterclockwise (c) get shorter and rotate clockwise (d) get shorter and rotate counterclockwise 13 Each complex impedance value R jX C (a) represents a unique combination of resistance and capacitance (b) represents a unique combination of resistance and reactance (c) represents a unique combination of resistance and frequency (d) All of the above are true

Simplex is not a polynomial time algorithm Certain rare kinds of linear programs cause it to go from one corner of the feasible region to a better corner and then to a still better one, and so on for an exponential number of steps For a long time, linear programming was considered a paradox, a problem that can be solved in practice, but not in theory! Then, in 1979, a young Soviet mathematician called Leonid Khachiyan came up with the ellipsoid algorithm, one that is very different from simplex, extremely simple in its conception (but sophisticated in its proof) and yet one that solves any linear program in polynomial time Instead of chasing the solution from one corner of the polyhedron to the next, Khachiyan s algorithm con nes it to smaller and smaller ellipsoids (skewed highdimensional balls) When this algorithm was announced, it became a kind of mathematical Sputnik, a splashy achievement that had the US establishment worried, in the height of the Cold War, about the possible scienti c superiority of the Soviet Union The ellipsoid algorithm turned out to be an important theoretical advance, but did not compete well with simplex in practice The paradox of linear programming deepened: A problem with two algorithms, one that is ef cient in theory, and one that is ef cient in practice! A few years later Narendra Karmarkar, a graduate student at UC Berkeley, came up with a completely different idea, which led to another provably polynomial algorithm for linear programming Karmarkar s algorithm is known as the interior point method, because it does just that: it dashes to the optimum corner not by hopping from corner to corner on the surface of the polyhedron like simplex does, but by cutting a clever path in the interior of the polyhedron And it does perform well in practice But perhaps the greatest advance in linear programming algorithms was not Khachiyan s theoretical breakthrough or Karmarkar s novel approach, but an unexpected consequence of the latter: the erce competition between the two approaches, simplex and interior point, resulted in the development of very fast code for linear programming.

Figure 14-10 Cartesian graph of the relation y = (+x)2

The process for graphing a function is the same as it is for graphing a relation Remember, a function is nothing more than a relation with special properties! In this section, we ll look at the graphs of the three functions we evaluated in Chap 13

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