Exam 13: Functions of Several Variables

Let $f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }$ . Then $f _ { x x } ( x , y )$ is

Let $w = \tan ( x y z )$ . Then the differential $d ^ { w }$ is

Let $z = e ^ { x y }$ , where x = t cos t and y = t sin t.When t = 0, $\frac { d z } { d t }$ is

Let $f ( x , y ) = \frac { x } { y }$ . Then $f ( x + \Delta x , y ) - f ( x , y )$ is

Let $f ( x , y ) = 5 ^ { x ^ { 2 } } y \sin ( x - y )$ . Then $f _ { y } ( x , y )$ is

Let $f ( x , y ) = \left\{ \begin{array} { c c } \frac { \sin ( x - y ) } { x - y } & x \neq y \\1 & x = y\end{array} \right.$ . Then f is

Let $z = x ^ { y }$ . Then the differential $d ^ { z }$ is

Let $f ( x , y ) = x \ln ( y ) - 4 x y + x$ . Then $f _ { y } ( x , y )$ is

Let $z ^ { 3 } - x y + 2 y z + y ^ { 3 } - 3 x y = 0$ . If Z is a function of x and y, then $z _ { y } ( 1,1,1 )$ is

Let $z = \ln \left( x ^ { 2 } + y ^ { 2 } \right)$ , where $x = e ^ { t }$ and $y = e ^ { - t }$ .When is

Let $f ( x , y ) = \sin \left( x y + y ^ { 2 } \right)$ . Then $f _ { x x } ( x , y )$ is

Let $f ( x , y ) = \ln \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$ . Then $f _ { y y } ( x , y )$ is

Let $f ( x , y , z ) = \left\{ \begin{array} { c c } \frac { 1 } { x ^ { 2 } + z ^ { 2 } } & ( x , z ) \neq ( 0,0 ) \\0 & ( x , z ) = ( 0,0 )\end{array} \right.$ . Then f is

Let $f ( x , y ) = x ^ { 2 } - y ^ { 2 }$ . Then the level curve for z = 3 is

Let $z = x ^ { 2 } + y ^ { 2 }$ . Then the change of z from (1, 3) to (1.1, 3.2) approximated by $d ^ { z }$ is

Let $x e ^ { y } + y e ^ { x } - x y = 0$ . If y is a function of x, then $\frac { d y } { d x }$ is

Let $x ^ { 2 } y - y ^ { 2 } x + x y - 5 = 0$ . If y is a function of x, then $\frac { d y } { d x }$ is

Let $z = \ln \left( \frac { x } { y } \right)$ . Then the differential $d ^ { z }$ is

Let $f ( x , y ) = \sin \left( x y + y ^ { 2 } \right)$ . Then $f _ {y y } ( x , y )$ is

Let $z = x ^ { 2 } + 2 x y + y ^ { 2 }$ , where x = t sin t and y = t cos t. When t = 0, $\frac { d z } { d t }$ is