Question

Find the direction in which f(x,y) = e^(2x) [cos(3y) - sin(3y)] increases most rapidly at the point (0, pi/2), and find the maximal directional derivative at that point.

Answer 1

In the direction of the gradient
\[gradf(x,y)=\left\{2 e^{2 x} (\cos (3
y)-\sin (3 y)),e^{2 x} (-3
\sin (3 y)-3 \cos (3
y))\right\}

\]
evaluated at the given point gives
\]
gradf(0,\pi/2)={2,3}
\]
The maximum increase is the length of the gradient
\[\sqrt{ 2^2 + 3^2}=\sqrt{13}
\]