Solution :
a). The level curves of the function :
are actually the curves
where k is a positive constant.
The equation is equivalent to
which is a family of ellipses.
We sketch the level curves for K =1,2,3 and 4.
If the shark always swim in the direction of maximum increase of blood concentration, its direction at any point would coincide with the gradient vector.
Then we know the shark's path is perpendicular to the level curves it intersects.
b). We have :
and
points in the direction of most rapid increase in concentration, which means
is tangent to the most rapid increase curve.
is a parametrization of the most
, then
is a tangent to the curve.
So then we have that
∴
Using separation of variables,
⇒ y = kx^2 for some constant k
but we know that
∴ The path of the shark will follow is along the parabola