Exam 14: Directional Derivatives, Gradients, and Extrema

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } - 18 x y$ . Then f has a relative minimum at

An equation of the tangent plane to the surface $x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } + z ^ { \frac { 2 } { 3 } } = 14$ at (-8, 27, 1) is

Let $f ( x , y , z ) = z \tan ^ { - 1 } \left( \frac { y } { x } \right)$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 1,1,3 )$ is

Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative minimum at

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $3 x - y + 5 z = 1$ . Then the minimum value of w is

Let $f ( x , y ) = e ^ { x y } + 2 x ^ { 2 } y ^ { 2 }$ . Then the directional derivative of f at (0,1) in the direction of the unit vector $\mathbf { U }$ which is parallel to $\mathbf { v } = - 5 \mathbf { i } + 12 \mathbf { j }$ is

An equation of the tangent plane to the surface $16 z = 4 x ^ { 2 } + y ^ { 2 }$ at (2, 4, 2) is

Let $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 2,4 )$ is

Let $f ( x , y ) = 5 x - 3 y$ with constraint $x ^ { 2 } + y ^ { 2 } = 136$ . Then f has a relative minimum at

An equation of the tangent plane to the surface $z x ^ { 2 } - x y ^ { 2 } - y z ^ { 2 } = 18$ at (0, -2, 3) is

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then the set of saddle points of f is

The absolute maximum of $f ( x , y ) = 2 x + 2 y - x ^ { 2 } - y ^ { 2 } + 2$ on or inside the triangle with vertices (0, 0), (9, 0), and (0,9) is

An equation of the tangent plane to the surface $x ^ { 2 } = 12 y$ at (-6, 3, 1) is

An equation of the tangent plane to the surface $x ^ { 2 } - 3 y ^ { 2 } - 4 z ^ { 2 } = 2$ at (3, 1, 1) is

The gradient of $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } x$ at (1, 2, -1) is

Let $f ( x , y , z ) = x y z$ with constraint $x + y + z = 1$ . Assume that $x , y , z \geq 0$ . Then f has a relative maximum at

Let $f ( x , y ) = 2 x e ^ { y } + 3 y e ^ { x }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 0,0 )$ is

The symmetric equations of the normal line to the surface $x ^ { 2 } = 12 y$ at (-6, 3, 1) are

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $9 x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } = 36$ . Then the maximum value of w is

Let $f ( x , y , z ) = x ^ { 2 } y - x y z ^ { 2 }$ . Then the directional derivative of f at $P = ( 0,1,2 )$ in the direction of the unit vector $\mathbf { U }$ which is parallel to $\overrightarrow { P Q }$ where $Q = ( 1,4,3 )$ is