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(a) Use the trapezoidal rule [Problem 31.9(a)], with n = 10, to approximate the integral, and compare the result with the exact answer obtained by the fundamental theorem. [Hint: You may assume the formula 13 + 23 + + n3 = (n(n + 1)/2)2 .] (b) GC Approximate the integral by the midpoint rule, with n = 10. (c) GC Approximate the integral by Simpson s rule, with n = 10.

The triode input characteristics of Fig. 4-13(a) show that operation with a positive grid voltage results in ow of grid current; however, with a negative grid voltage (the common application), negligible grid current ows and the plate characteristics are reasonably approximated by a three-halves-power relationship involving a linear combination of plate and grid voltages: iP vP vG 3=2 4:9

where denotes the perveance (a constant that depends upon the mechanical design of the tube) and is the ampli cation factor, a constant whose signi cance is elucidated in 7 when small-signal ampli cation of the triode is addressed. To establish a range of triode operation favorable to the signal to be ampli ed, a quiescent point must be determined by dc bias circuitry. The basic triode ampli er of Fig. 4-14 has a grid power supply VGG of such polarity as to maintain vG negative (the more common mode of operation). With no input signal vS 0 , application of KVL around the grid loop of Fig. 4-14 yields the equation of the grid bias line, iG VGG vG RG RG 4:10

rule is usually much more accurate than the midpoint rule or the trapezoidal rule. If f has a continuous fourth derivative on [a, b], then b the error in approxiamating a f (x)dx by Simpson s rule is at most ((b a)5 /180n4 )M4 , where M4 is the maximum of |f (4) (x)| on [a, b] and n is the number of subintervals.

which can be solved simultaneously with (4.7) or plotted as indicated on Fig. 4-13(a) to determine the quiescent values IGQ and VGQ . If VGG is of the polarity indicated in Fig. 4-14, the grid is negatively biased, giving the Q point labeled Qn . At that point, IGQ % 0 and VGQ % VGG ; these approximate solutions su ce in the case of negative grid bias. However, if the polarity of VGG were reversed, the grid would have a positive bias, and the quiescent point Qp would give IGQ > 0 and VGQ < VGG .

CHAP. 4]

on [0, 2]

Solved Problem 17.1 Two separate business organizations, a partner ship and a corporation, were formed on January 1, 2001. 1. The initial investments of the partners, Blue and Gray, were $25,000 and $20,000, respectively. 2. The Green Corporation has ve stockholders, each owning 90 shares of $100-par common stock. At the end of the calendar year, the net income of each company was $15,000. (a) For each organization, show the proper entry to close the In come Summary account. (b) Prepare a capital statement for the partner ship and a stockholders equity statement for the corporation, as of De cember 31, 2001. Solution: (a) Partnership Entry: Income Summary Blue, Capital Gray, Capital Corporation Entry: Income Summary Retained Earnings

Voltage summation around the plate circuit of Fig. 4-14 leads to the equation of the dc load line iP VPP vP RL RL 4:11

which, when plotted on the plate characteristics of Fig. 4-13(b), yields the quiescent values VPQ and IPQ at its intersection with the curve vG VGQ .

14.8 A farmer wishes to fence in a rectangular eld. If north-south fencing costs $3 per yard, and east west fencing costs $2 per yard, what are the dimensions of the eld of maximum area that can be fenced in for $600 14.9 A farmer has to fence in a rectangular eld alongside a straight-running stream. If the farmer has 120 yards of fencing, and the side of the eld alongside the stream does not have to be fenced, what dimensions of the eld will yield the largest area