Answer to Question #219988 in Calculus for Hari

Question

If the temperature on metal sheet is defined by f (x, y,z) = e2x

cos(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

$$\nabla f=\dfrac{\partial f}{\partial x}\vec i +\dfrac{\partial f}{\partial y}\vec j +\dfrac{\partial f}{\partial z}\vec k$$

$$=(2e^{2x}\cos(y-2z))\vec i +(-e^{2x}\sin(y-2z))\vec j +(2e^{2x}\sin(y-2z))\vec k$$

$$\nabla|_{(4, −2, 0)}=2e^{8}\cos(2)\vec i +e^{8}\sin(2)\vec j -2e^{8}\sin(2)\vec k$$

The maximum rate of change is the magnitude of the gradient

$$|\nabla|_{(4, −2, 0)}|=\sqrt{(2e^{8}\cos(2))^2+(e^{8}\sin(2))^2+(-2e^{8}\sin(2))^2}$$

$$=e^8\sqrt{4+\sin^2(2)}$$

The fastest increase is in the direction of the gradient

$$\nabla|_{(4, −2, 0)}=2e^{8}\cos(2)\vec i +e^{8}\sin(2)\vec j -2e^{8}\sin(2)\vec k$$