Question

If the temperature on metal sheet is defined by f (x, y,z) = e ^2xcos(y − 2z) then find the maximum

rate of change of the function at the (4, −2, 0) and the direction in which this maximum rate of change

of temperature occurs.

Answer 1

$\displaystyle \nabla F = 2e^{2x}\cos(y - 2z) \vec{i} - e^{2x}\sin(y - 2z) \vec{j} + 2e^{2x}\sin(y - 2z) \vec{k} \\ \nabla F = (2e^{8}\cos(2), -e^{8}\sin(2), 2e^{8}\sin(2))\\ |\nabla F| = \sqrt{4e^{16}\cos^2(2) + 5e^{16}\sin^2(2)} = e^{8}\sqrt{\sin^2(2) + 4}\\ \textsf{The maximum rate of change is}\,\,\, e^{8}\sqrt{\sin^2(2) + 4}\\ \textsf{and it's direction is}\,\,\, 2e^{8}\cos(2) \vec{i} - e^{8}\sin(2)\vec{j} + 2e^{8}\sin(2)\vec{k}$