Find the unit vector in the direction in which
f increases most rapidly at P and give the rate of change of f
in that direction; find the unit vector in the direction in which f
decreases most rapidly at P and give the rate of change of f in
that direction.
$f(x,y,z)$ = $ x^2ze^y+ xz^2$; $P(1,ln2,2)$
Ok so I know that the first step is to find the partial derivatives.
$\frac{∂}{∂x}$ $= 2xze^y+z^2 \mathbf i$ $\qquad$ $\frac{∂}{∂y}$ = $x^2ze^y \mathbf j$ $\qquad$$\frac{∂}{∂z}$ = $x^2e^y+2xz \mathbf k$
Then I plug in $\mathbf P(1,ln (2),2)$ into the partial derivatives and I get
$▽f$ = $(12i,4j,6k)$
How does one find where it increases most rapidly and decreases and the rate of change?
Best Answer
$u = $$\sqrt{12^2+4^2+6^2}$ = 14 $\quad$ $\color{purple}{Rate\quad of\quad Change}$
Fastest Increase : $\frac{6i+2j+3k}{7}$,$\quad$ 14
Fastest Decrease: $-\frac{6i+2j+3k}{7}$ $\quad$-14