still having trouble with the calcululog and e part.......
but heres the asst. 8 stuff.
Write the differential equation expressing the hypothesis that the rate of
change of a population is proportional to the population P. Evaluate the
proportionality constant if it is known that the when the population is 4427
its rate of change is known to be 200. If this is the t=0 state of the
population, then approximately what will be the population at t = 1.1? What
then will be the population at t = 2.2?
0=t1
1.1=t 2
2.2=t 3
there is not enough information to determine the function equation, so the
derivitive will be used to approximate.
so 200/1=t' at (t=0)
so (y2-y1)/(x2-x1)
so (p2-p1)/(t2-t1)=
so @ t1, (p2-4427)/(1.1-0)=200/1
then p2=4647
so @ t2, (p3-4647)/(2.2-1.1)=200/1
then p3=4867
Good start. You have the right idea and you've put it together nicely, but your last step doesn't quite work.
Your first calculation sequence, ending with P = 4647, is correct. The second does not take account of the change in P ' when the population changes.
The rate of change of P would be denoted P '. To say that y is proportional to x is to say that there exists constant k such that y = k x.
So if the rate of change of P is proportional to P we would formulate this as P ' = k P.
Knowing that when P = 4427 we have P' = 200 we get
200 = k * 4427 so that
k = 200 / 4427.
Thus P ' = 200 / 4427 * P.
If you evaluate P ' at P = 4427 you get P ' = 200, as expected, and then since P ' * `dt = 200 * 1.1 = 220 you get P = 4647 at t = 1.1. This completely agrees with your first result.
However at this point we have a new value of P, which will give us a new value of P '. Your second calculation, where you get P = 4867, is based on P ' = 200 and doesn't take account of this change.
So if P = 4647, what is P', and how much change will that cause between t = 1.1 and t = 2.2? What therefore is your revised estimate of P when t = 2.2?
The velocity of an automobile coasting down a hill is given as a function of
clock time by v(t) = .00053 t^2 + .36 t + 1.6, with v in meters/sec when t
is in seconds. Determine the velocity of the vehicle for clock times t = 0,
9 and 18 sec and make a table of rate vs. clock time.
i will give you this on paper later this week.
OK. You might also want to give this a shot with Excel, which is great for setting up this sort of repeated calculation.