cal_a8_quiz2

dp/dt = k * P, not k * P * t, and not k * P * `dt (it's not completely clear which you intend).

You get dp/dt = .082 * 2433, which gives you the rate of change of p with respect to t at t = 0.

Then you multiply dp/dt by the time interva `dt (not by the clock time t, though since the initial instant is t = 0, the clock time and the time interval are identical, for the first interval only). This provides an estimate of the change in p.

Adding the change in p to the original value of p we get the estimated new value of p, i.e., the value at t = 1.9.

Base on our estimate of p, we evaluate dp/dt again, using the t = 1.9 values.

Then we proceed, based on this value, to make an estimate for the new interval, based on `dt = 1.9.

This estimate of dp/dt is based on the rate of change at t = 0 and the clock time after two intervals of duration 1.9 units. We wouldn't ordinarily mix the values in this way.

This is unlikely to lead to a good estimate.

I'm also appending a similar problem with a worked-out solution, for your reference:

Write the differential equation expressing the hypothesis that the rate of

change of an investment is proportional to the principal P.

y is proportional to x if there exists a constant

number k, called the proportionality constant, such that y = k x.

The rate of change of quantity P with respect to clock

time t is dP/dt.

The differential equation is therefore

dP/dt = k P

Evaluate the proportionality constant if it is known that the when the

principal is 3353 its rate of change is 400.

The principal in the equation above is P and its rate

of change is dP/dt.  Thus dP/dt = 400 when P = 3353 so the equation dP/dt =

k P becomes

400 = k * 3353, so that the value of the

proportionality constant k is

k = 400 / 3353 = .12, approx.

Thus the rate of change of the principle is given in

terms of the principle by

dP/dt = .12 P.

If this is the t=0 value of the principal, then approximately what will be

the principal at t = 1.1?

The rate of change at t = 0 is 400.  Assuming that

this rate continues over the interval from t = 0 to t = 1.1, the change in

principle will be

`dP = change in principle = rate of change * `dt = 400

* 1.1 = 440.

That is, the principle will increase by about 440. 

Having started at P = 3353, the new principle will be

P(1.1) = P(0) + `dP = 3353 + 440 = 3793.

  What then will be the principal at t

= 2.2?

Assuming that the principle at t = 1.1 is 3793, we have

new rate of change of principle = dP/dt = .12 P = .12 *

3793 = 456.

If the principle changes at this rate during the

interval from t = 1.1 to t = 1.2 the principle will increase by

`dP = change in principle = rate of change * `dt = 456

* 1.1 = 500, approx..

Thus our new principle at t = 2.2 is

P(2.2) = P(1.1) + `dP = 3793 + 500 = 4293.

These approximations are based on the assumption that

the rate of change of principle remains constant for the 1.1-unit time

intervals.  In this case the rate increases with population, so these

approximations are underestimates.

(if you are bothered by the fact that k is approximated

to only two significant figures, while the changes in principal are calculated

to 3 significant figures, you are correct; significant figures should be

considered here but in this solution the instructor has chosen not to address

this issue)

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