Directional Derivative

Q. 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

1. $-2.785$
2. $-2.145$
3. $-1.789$
4. $1.000$

Q. The value of the directional derivative of the $\varphi$ function $(x,y,z)=x{y}^2+y{z}^2+z{x}^2$ at the point $(2,-1,1)$ in the direction of the vector $p=i+2j+2k$ is

1. $1$
2. $0.95$
3. $0.93$
4. $0.9$

Q. Directional derivative of $\varphi =2xz-y^2$ at the piont $(1,3,2)$ becomes maximum in the direction of

1. $4\hat{i}+2\hat{j}-3\hat{k}$
2. $4\hat{i}-6\hat{j}+2\hat{k}$
3. $2\hat{i}-6\hat{j}+2\hat{k}$
4. $4\hat{i}-6\hat{j}-2\hat{k}$

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

1. $-18$
2. $-3\sqrt{6}$
3. $2\sqrt{6}$
4. $18$

Q. The derivative of $f(x,y)$ at point (1, 2) in the direction of vector $i+j$ is $2\sqrt{2}$ and in the direction of the vector $-2j$ is $-3$. Then the derivative of $f(x,y)$ in direction $-i-2j$ is

1. $2\sqrt{2}+3/2$
2. $-7/\sqrt{5}$
3. $-2\sqrt{2}-3/2$
4. $1/\sqrt{5}$

Q. The directional derivative of the following function at $(1,2)$ in the direction $(4\hat{i}+3\hat{j})$ is: $f(x,y)=x^2+y^2$

1. $4/5$
2. $4$
3. $2/5$
4. $1$

Q. A scalar field is given by $f=x^{2/3}+y^{2/3},$ where $x$ and $y$ are the Cartesian coordinates. The derivative of ${}^{\prime }{f}^{\prime }$ along the line $y=x$ directed away from the origin at the point (8, 8) is

1. $\frac{\sqrt{2}}{3}$
2. $\frac{\sqrt{3}}{2}$
3. $\frac{2}{\sqrt{3}}$
4. $\frac{3}{\sqrt{2}}$

Q. The magnitude of the directional derivative of the function $f(x,y)=x^2+3y^2$ in a direction normal to the circle $x^2+y^2=2,$ at the point $(1,1),$ is

1. $4\sqrt{2}$
2. $5\sqrt{2}$
3. $7\sqrt{2}$
4. $9\sqrt{2}$

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

1. $\frac{4}{\sqrt{5}}$
2. $-\frac{4}{\sqrt{5}}$
3. $\frac{\sqrt{5}}{4}$
4. $-\frac{\sqrt{5}}{4}$

Q. The directional derivative of the function f(x, y) = $x^2+y^2$ along a line directed from (0, 0) to (1, 1), evaluated at the point x = 1, y = 1 is

1. 2
2. $4\sqrt{2}$
3. $2\sqrt{2}$
4. $\sqrt{2}$

Q. The directional derivative of $\varphi =3x^{2}y-4y{z}^2+6z^{2}x$ at point (1, 1, 1) in the direction of line
$\frac{x-1}{2}=\frac{y-4}{2}=\frac{z}{3}$

1. $2\sqrt{17}$
2. $\frac{2}{\sqrt{17}}$
3. $\frac{29}{\sqrt{17}}$
4. $\frac{40}{\sqrt{17}}$