# ASSIGNMENT 2·2 ```ASSIGNMENT 2&middot;2
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete arguments and justifications. You will be graded on the correctness, clarity and elegance of
at the top of each page.
1. Consider the function f (x) = ln x. By the Mean Value Theorem, we know there exists a c in the interval
(1, 4) such that
f (4) − f (1)
.
f 0 (c) =
4−1
Find c.
2. Let f be a function such that f 0 (x) &lt; 0 on the interval (a, b). Prove, using the Mean Value Theorem,
that f is decreasing on the interval (a, b).
3. Let P (t) be a population at time t. Recall the Verhulst differential equation
P (t)
0
P (t) = rP (t) 1 −
,
K
where r and K are positive constants, describing the growth rate of the population under limited resources.
(a) Find P 00 (t).
(b) For what population is the growth rate maximized? Justify your answer.
(c) The constant K, called the carrying capacity, is the maximum sustainable population predicted by
the model. Describe how your answer in part (b) might be used to predict the carrying capacity for
the global human population.
```