Question

The directional derivative of $f(x,y,z)=2x^2+3y^2+z^2$ at the point $P(2,1,3)$ in the direction of the vector $\overrightarrow{a}=\hat{i}-2\hat{k}$ is

• A. $-2.785$
• B. $-2.145$
• C. $-1.789$
• D. $1.000$

Answer 1

The correct option is C $-1.789$

$f(x,y,z)=2x^2+3y^2+z^2$
$▽f=\hat{i}\frac{\partial f}{\partial x}+\hat{j}\frac{\partial f}{\partial y}+\hat{k}\frac{\partial f}{\partial z}$
$=\hat{i}\left(4x\right)+\hat{j}\left(6y\right)+\hat{k}\left(2z\right)$
$=8\hat{i}+6\hat{j}+6\hat{k};$ at $P(2,1,3)$
Directional derivative in the direction of $\overrightarrow{a}=\hat{i}-2\hat{k}$ is
$=▽f.\frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}=(8\hat{i}+6\hat{j}+6\hat{k}).\frac{(\hat{i}-2\hat{k})}{\sqrt{1+4}}$
$=\frac{8-12}{\sqrt{5}}=-\frac{4}{\sqrt{5}}$
$=-1.789$