Use Maple for much of this problem.

Under the Volterra model of predator prey interaction, we will show
later in the course that the population **x** of prey and **y** of predators
(each a function of time **t**) are related by

for suitable positive constants **a**, **b**, **c**, and **d**.

Suppose that , , , and and that at some time .

** a)** Find a value of the constant **K** so that the model
fits this data. (The Maple command

will evaluate an expression f at the point .)

** b)** Use Maple to compute the partial derivatives of **f** at the
point . (Recall that the Maple command

computes the partial derivatives symbolically.)

** c)**
Linear approximation along the level curve says that
. We can solve this equation to
estimate **y** in terms of **x** near the point
on the level curve. Use this technique to estimate the predator
population **y** as changes from to **x=1900**.

** d)**
Use the Maple fsolve command to more exactly solve for the value of
**y** that corresponds to **x=1900**. How do you explain the discrepancy
from your answer in part c)?
(The Maple command

will look numerically for solutions to in the range **y**
between 10 and 20. For example might be given by

If no value is returned by Maple , this may mean there is no solution in this range, or it may mean that the solution method failed to converge.

** e)**
Use implicit differentiation to calculate
when
.

** f)**
Repeat parts c) and d) using **x=1999** instead of **x=1900**. Compare the
quality of the linear approximation.