The logistic equation for the population (in thousands) of a certain species is given by:

dp/dt= 3p- 2p²

a. sketch the direction field by using either a computer software package or the method of isoclines.
b. if the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population lim t →+[infinity] p(t)?
c. if p(0) = 0.8, what is lim t →+[infinity] p(t)?
d. can a population of 2000 ever decline to 800?

a.

b. 1.5

c. 1.5

d. No

Step-by-step explanation:

a. First, let's solve the differential equation:

Divide both sides by  and multiply both sides by dt:

Integrate both sides:

Evaluate the integrals and simplify:

Where C1 is an arbitrary constant

I sketched the direction field using a computer software. You can see it in the picture that I attached you.

b. First let's find the constant C1 for the initial condition given:

Solving for C1:

Now, let's evaluate the limit:

The expression tends to zero as x approaches ∞ . Hence:

c. As we did before, let's find the constant C1 for the initial condition given:

Solving for C1:

Now, let's evaluate the limit:

The expression tends to zero as x approaches ∞ . Hence:

d. To figure out that, we need to do the same procedure as we did before. So,  let's find the constant C1 for the initial condition given:

Solving for C1:

Can a population of 2000 ever decline to 800? well, let's find the limit of the function when it approaches to ∞:

The expression tends to zero as x approaches ∞ . Hence:

Therefore, a population of 2000 never will decline to 800.

step-by-step explanation:

this woman was an important writer and propagandist during the american revolution. she was a member of the committee of correspondence (in fact the committee was formed in her home) and was an advisor to many of the revolutionary leaders, including george washington and john adams. who was this revolutionary woman?