This problem models pollution effects in the Great Lakes. We assume pollutants are flowing into a lake at a constant rate of I k
g/year, and that water is flowing out at a constant rate of F km3/year. We also assume that the pollutants are uniformly distributed throughout the lake. If C(t) denotes the concentration (in kg/km3) of pollutants at time t (in years), then C(t) satisfies the differential equation dC dt = − F V C + I V where V is the volume of the lake (in km3). We assume that (pollutant-free) rain and streams flowing into the lake keep the volume of water in the lake constant. (a) Suppose that the concentration at time t = 0 is C0. Determine the concentration at any time t by solving the differential equation.
1 answer:
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Part 1: You can simplify to
Presumably, the sequence starts at <em>n</em> = 1. It's easy to see that the sequence is strictly decreasing, since larger values of <em>n</em> make either fraction smaller.
(a) So, the sequence is bounded above by its first value,
(b) And because both fractions in converge to 0, while remaining positive for any natural number <em>n</em>, the sequence is bounded below by 0,
(c) Finally, is bounded above and below, so it is a bounded sequence.
Part 2: Yes, is monotonic and strictly decreasing.
Part 3:
(a) I assume the choices are between convergent and divergent. Any monotonic and bounded sequence is convergent.
(b) Since is decreasing and bounded below by 0, its limit as <em>n</em> goes to infinity is 0.
Part 4:
(a) We have
and the (-1)ⁿ makes this limit alternate between -10 and 10. So the sequence is bounded but clearly not monotonic, and hence divergent.
(b) Taking the limit gives
so the sequence is unbounded and divergent. It should also be easy to see or establish that the sequence is strictly increasing and thus monotonic.
For the next three, I'm guessing the options here are something to the effect of "does", "may", or "does not".
(c) may : the sequence in (a) demonstrates that a bounded sequence need not converge
(d) does not : a monotonic sequence has to be bounded in order to converge, otherwise it grows to ± infinity.
(e) does : this is true and is known as the monotone convergence theorem.