Question

Find the maximum rate of change of\(f\)at the given point and the direction in which it occurs.

\(f(x,y) = \sin (xy),\quad (1,0)\)

Short answer

Maximum rate of change occurs in the direction \(\langle 0,1\rangle \) Magnitude of Maximum rate of change is 1.

Step-by-step solution

Solve it

Given that

\(\begin{aligned}{l}f(x,y) = \sin (xy)\nabla f(x,y)\\ = \left\langle {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right\rangle \\ = \langle y\cos (xy),x\cos (xy)\rangle \\\nabla f(1,0) = \langle 0,\cos (0)\rangle \\ = \langle 0,1\rangle \end{aligned}\)

Magnitude of maximum rate

Find Magnitude

\(|\nabla f(1,0)| = \sqrt {{0^2} + {1^2}} = 1\)