The slope field for a population modeled by is shown in the figure below.

graph of a slope field

(a) On a print-out of the slope field, sketch three non-zero solution curves showing different types of behavior for the population . Give an initial condition that will produce each:

, and

(b) Is there a stable value of the population? If so, give the value; if not, enter none:

Stable value =

(c) Considering the shape of solutions for the population, give any intervals for which the following are true. If no such interval exists, enter none, and if there are multiple intervals, give them as a list. (Thus, if solutions are increasing when is between 1 and 3, enter (1,3) for that answer; if they are decreasing when is between 1 and 2 or between 3 and 4, enter (1,2),(3,4). Note that your answers may reflect the fact that is a population.)

is increasing when is in
is decreasing when is in

Think about what these conditions mean for the population, and be sure that you are able to explain that.

In the long-run, what is the most likely outcome for the population?

(Enter infinity if the population grows without bound.)

Are there any inflection points in the solutions for the population? If so, give them as a comma-separated list (e.g., 1,3); if not, enter none.

Inflection points are at

Be sure you can explain what the meaning of the inflection points is for the population.

(d) Sketch a graph of against . Use your graph to answer the following questions.

When is positive?
When is in
When is negative?
When is in
(Give your answers as intervals or a list of intervals.)

When is zero?
(If there is more than one answer, give a list of answers, e.g., 1,2.)

When is at a maximum?

Be sure that you can see how the shape of your graph of explains the shape of solution curves to the differential equation.