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Connect an initially uncharged capacitor to a battery with voltage V0 through a resistor at t 0. The circuit is shown in Fig. 7-3(a).

Example 7.6. The triode ampli er of Fig. 4-14 has VGG ; VPP ; RG ; and RL , as given in Example 4.7. If the plate characteristics of the triode are given by Fig. 7-8 and vS 2 sin !t V, graphically nd vP and iP .

15.9. Given an acute angle, construct (a) its supplement; (b) its complement; (c) half its supplement; (d) half its complement. (15.4) 15.10. By actual construction, illustrate that the difference between the measures of the supplement and complement of an acute angle equals 90 . (15.4) 15.11. Construct a right triangle given its (a) legs; (b) hypotenuse and a leg; (c) leg and an acute angle adjacent to the leg; (d) leg and an acute angle opposite the leg; (e) hypotenuse and an acute angle. (15.5) 15.12. Construct an isosceles triangle given (a) an arm and a vertex angle; (b) an arm and a base angle; (c) an arm and the altitude to the base; (d) the base and the altitude to the base. (15.5) 15.13. Construct an isosceles right triangle given (a) a leg; (b) the hypotenuse; (c) the altitude to the hypotenuse. (15.5)

Fig. 7-3

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_2 _4 _6 _8 _ 10

Deductive reasoning enables us to derive true or acceptably true conclusions from statements which are true or accepted as true. It consists of three steps as follows: 1. Making a general statement referring to a whole set or class of things, such as the class of dogs: All dogs are quadrupeds (have four feet). 2. Making a particular statement about one or some of the members of the set or class referred to in the general statement: All greyhounds are dogs. 3. Making a deduction that follows logically when the general statement is applied to the particular statement: All greyhounds are quadrupeds. Deductive reasoning is called syllogistic reasoning because the three statements together constitute a syllogism. In a syllogism the general statement is called the major premise, the particular statement is the minor premise, and the deduction is the conclusion. Thus, in the above syllogism: 1. The major premise is: All dogs are quadrupeds. 2. The minor premise is: All greyhounds are dogs. 3. The conclusion is: All greyhounds are quadrupeds. Using a circle, as in Fig. 2-1, to represent each set or class will help you understand the relationships involved in deductive reasoning. 1. Since the major premise or general statement states that all dogs are quadrupeds, the circle representing dogs must be inside that for quadrupeds. 2. Since the minor premise or particular statement states that all greyhounds are dogs, the circle representing greyhounds must be inside that for dogs.

For t > 0, KVL around the loop gives Ri v V0 which, after substituting i C dv=dt , becomes dv 1 1 v V dt RC RC 0 with the initial condition v 0 v 0 0 5b t>0 5a

11.3 10 8.1

Fig. 2-1

The solution should satisfy both (5a) and (5b). The particular solution (or forced response) vp t V0 satis es (5a) but not (5b). The homogeneous solution (or natural response) vh t Ae t=RC can be added and its magnitude A can be adjusted so that the total solution (6a) satis es both (5a) and (5b). v t vp t vh t V0 Ae t=RC From the initial condition, v 0 V0 A 0 or A V0 . Thus the total solution is 6a

3. The conclusion is obvious. Since the circle of greyhounds must be inside the circle of quadrupeds, the only possible conclusion is that greyhounds are quadrupeds.

EXAMPLE 7.3 A 4-mF capacitor with an initial voltage of v 0 2 V is connected to a 12-V battery through a resistor R 5 k

Q _ 12

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