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Chapter 1: Q-24E (page 1)
Question: In Problems 23鈥�26, express the given power series as a series
with generic term Xk.
24.
The required term is .
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Mixing.Suppose a brine containing 0.2 kg of salt per liter runs into a tank initially filled with 500 L of water containing 5 kg of salt. The brine enters the tank at a rate of 5 L/min. The mixture, kept uniform by stirring, is flowing out at the rate of 5 L/min (see Figure 2.6).
(a)Find the concentration, in kilograms per liter, of salt in the tank after 10 min. [Hint:LetAdenote the number of kilograms of salt in the tank attminutes after the process begins and use the fact that
rate of increase inA=rate of input- rate of exit.
A further discussion of mixing problems is given in Section 3.2.]
(b)After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank (see Figure 2.7). What will be the concentration, in kilograms per liter, of salt in the tank 20 min after the leak develops? [Hint:Use the method discussed in Problems 31 and 32.]
Question: Let f(x)and g(x)be analytic at x0. Determine whether the following statements are always true or sometimes false:
(a) 3f(x)+g(x)is analytic at x0.
(b) f(x)/g(x)is analytic at x0.
(c) f'(x)is analytic at x0.
(d) is analytic at x0.
In Problems 14鈥�24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge鈥揔utta algorithm. (At the instructor鈥檚 discretion, other algorithms may be used.)鈥�
Using the vectorized Runge鈥揔utta algorithm, approximate the solution to the initial value problem at . Starting with , continue halving the step size until two successive approximations of both anddiffer by at most 0.1.
In Problems 14鈥�24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge鈥揔utta algorithm. (At the instructor鈥檚 discretion, other algorithms may be used.)鈥�
Using the vectorized Runge鈥揔utta algorithm for systems with, approximate the solution to the initial value problem at.
Compare this approximation to the actual solution.
Decide whether the statement made is True or False. The function is a solution to .
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