2021-12-22

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

maximum rate of change =
direction vector =

Heather Fulton

Expert

$\mathrm{\nabla }f={⟨\frac{x}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}},\frac{y}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}},\frac{z}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}⟩}_{\left(9,7,-2\right)}$
$\mathrm{\nabla }f=⟨\frac{9}{\sqrt{{9}^{2}+{7}^{2}+{2}^{2}}},\frac{7}{\sqrt{{9}^{2}+{7}^{2}+{2}^{2}}},\frac{-2}{\sqrt{{9}^{2}+{7}^{2}+{2}^{2}}}⟩=\frac{⟨9,7,-2⟩}{\sqrt{134}}$
Maximum rate of change $=|\mathrm{\nabla }|f=\frac{\sqrt{{9}^{2}+{7}^{2}+{2}^{2}}}{\sqrt{134}}=1$
direction $=\frac{\mathrm{\nabla }f}{|\mathrm{\nabla }|f}=\frac{⟨9,7,-2⟩}{\sqrt{134}}$

Jordan Mitchell

Expert

The formula is just him finding the derivative with respect to whatever slot the equation is in (f gradient = fx',fy',fz') and so forth.

karton

Expert