Question 1

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$ is

Accepted Answer

A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1 
A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1 A

Question 2

Let $f ( x , y ) = x y + 3$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

Accepted Answer

A)  $\Delta x \Delta y$ 
B)  $x - 3 \Delta y$ 
C)  $x + 3 \Delta y$ 
D)  $x \Delta y$ 
E)  $x ( \Delta y ) ^ { 2 }$ 
A)  $\Delta x \Delta y$ 
B)  $x - 3 \Delta y$ 
C)  $x + 3 \Delta y$ 
D)  $x \Delta y$ 
E)  $x ( \Delta y ) ^ { 2 }$ D

Question 3

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y ^ { 2 } } { x ^ { 4 } + y ^ { 4 } }$ is

Accepted Answer

A) Does not exist 
B) 2 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E) 1 
A) Does not exist 
B) 2 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E) 1 A

Question 4

The limit $\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { x y - 4 } { \sqrt { x y } - 2 }$ is

Accepted Answer

A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1 
A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1

Question 5

The limit $\lim _ { ( x , y ) \rightarrow ( 0.0 ) } \frac { x ^ { 3 } y } { x ^ { 4 } + y ^ { 4 } }$ is

Accepted Answer

A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1 
A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1

Question 6

Let $f ( x , y , z ) = \left\{ \begin{array} { c l } 
\frac { x - y - z } { x ^ { 2 } - ( y + z ) ^ { 2 } } & x ^ { 2 } \neq ( y + z ) ^ { 2 } \
\frac { 1 } { 4 } & x ^ { 2 } = ( y + z ) ^ { 2 }
\end{array} \right.$ . Then f is

Accepted Answer

A) Continuous at (2,1,1) 
B) Discontinuous at (2,1,1) because f is undefined at(2,1,1) 
C) Discontinuous at (2,1,1) because the limit of f at (2,1,1) does not exist 
D) Discontinuous at (2,1,1) because the limit of f at (2,1,1) is different from the function values at (2,1,1) 
E) Discontinuous at (2,1,1) for a reason that is different from the ones above 
A) Continuous at (2,1,1) 
B) Discontinuous at (2,1,1) because f is undefined at(2,1,1) 
C) Discontinuous at (2,1,1) because the limit of f at (2,1,1) does not exist 
D) Discontinuous at (2,1,1) because the limit of f at (2,1,1) is different from the function values at (2,1,1) 
E) Discontinuous at (2,1,1) for a reason that is different from the ones above

Question 7

Let $w = y z ^ { 2 } + x ^ { 2 } z + x y ^ { 2 }$ . Then the differential $d ^ { w }$ is

Accepted Answer

A)  $- \left( 2 x z + y ^ { 2 } \right) d x + \left( 2 x y + z ^ { 2 } \right) d y + \left( 2 y z + x ^ { 2 } \right) d z$ 
B)  $\left( 2 x z + y ^ { 2 } \right) d x - \left( 2 x y + z ^ { 2 } \right) d y - \left( 2 y z + x ^ { 2 } \right) d z$ 
C)  $\left( 2 x z + y ^ { 2 } \right) d x + \left( 2 x y + z ^ { 2 } \right) d y + \left( 2 y z + x ^ { 2 } \right) d z$ 
D)  $\left( 2 x z + y ^ { 2 } \right) d x - \left( 2 x y + z ^ { 2 } \right) d y + \left( 2 y z + x ^ { 2 } \right) d z$ 
E)  $\left( 2 x z + y ^ { 2 } \right) d x + \left( 2 x y + z ^ { 2 } \right) d y - \left( 2 y z + x ^ { 2 } \right) d z$ 
A)  $- \left( 2 x z + y ^ { 2 } \right) d x + \left( 2 x y + z ^ { 2 } \right) d y + \left( 2 y z + x ^ { 2 } \right) d z$ 
B)  $\left( 2 x z + y ^ { 2 } \right) d x - \left( 2 x y + z ^ { 2 } \right) d y - \left( 2 y z + x ^ { 2 } \right) d z$ 
C)  $\left( 2 x z + y ^ { 2 } \right) d x + \left( 2 x y + z ^ { 2 } \right) d y + \left( 2 y z + x ^ { 2 } \right) d z$ 
D)  $\left( 2 x z + y ^ { 2 } \right) d x - \left( 2 x y + z ^ { 2 } \right) d y + \left( 2 y z + x ^ { 2 } \right) d z$ 
E)  $\left( 2 x z + y ^ { 2 } \right) d x + \left( 2 x y + z ^ { 2 } \right) d y - \left( 2 y z + x ^ { 2 } \right) d z$

Question 8

The domain of the function $f ( x , y ) = \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 x y + y ^ { 2 } }$ is

Accepted Answer

A)  $\{ ( x , y ) : x \neq - y \}$ 
B)  $\{ ( x , y ) : x \neq y \}$ 
C)  $\{ ( x , y ) : 2 x \neq y \}$ 
D)  $\{ ( x , y ) : x \neq 2 y \}$ 
E)  $\left\{ ( x , y ) : x ^ { 2 } \neq y ^ { 2 } \right\}$ 
A)  $\{ ( x , y ) : x \neq - y \}$ 
B)  $\{ ( x , y ) : x \neq y \}$ 
C)  $\{ ( x , y ) : 2 x \neq y \}$ 
D)  $\{ ( x , y ) : x \neq 2 y \}$ 
E)  $\left\{ ( x , y ) : x ^ { 2 } \neq y ^ { 2 } \right\}$

Question 9

Let $f ( x , y ) = \left\{ \begin{array} { c c } 
\frac { x y - 4 } { \sqrt { x y } - 2 } & ( x , y ) \neq ( 2,2 ) \
0 & ( x , y ) = ( 2,2 )
\end{array} \right.$ . Then f is

Accepted Answer

A) Continuous at (2,2) 
B) Discontinuous at (2,2)because f is undefined at (2,2) 
C) Discontinuous at (2,2)because the limit of f at (2,2) does not exist 
D) Discontinuous at (2,2)because the limit of f at (2,2) is different from the function value at (2,2) 
E) Discontinuous at (2,2) for a reason that is different from the ones above 
A) Continuous at (2,2) 
B) Discontinuous at (2,2)because f is undefined at (2,2) 
C) Discontinuous at (2,2)because the limit of f at (2,2) does not exist 
D) Discontinuous at (2,2)because the limit of f at (2,2) is different from the function value at (2,2) 
E) Discontinuous at (2,2) for a reason that is different from the ones above

Question 10

Let $f ( x , y ) = \cos x$ . Then the level curve for z = 0 is

Accepted Answer

A) A circle 
B) An ellipse 
C) A hyperbola 
D) A horizontal line 
E) A line parallel to the z-axis 
A) A circle 
B) An ellipse 
C) A hyperbola 
D) A horizontal line 
E) A line parallel to the z-axis

Question 11

Let $f ( x , y , z ) = z ^ { x y }$ . Then $f _ { y } ( x , y , z )$ is

Accepted Answer

A)  $x y z ^ { x y } \ln ( z )$ 
B)  $- y z ^ { x y } \ln ( z )$ 
C)  $x y ^ { x y } \ln ( z )$ 
D)  $- x z ^ { x y } \ln ( z )$ 
E)  $x z ^ { x y } \ln ( z )$ 
A)  $x y z ^ { x y } \ln ( z )$ 
B)  $- y z ^ { x y } \ln ( z )$ 
C)  $x y ^ { x y } \ln ( z )$ 
D)  $- x z ^ { x y } \ln ( z )$ 
E)  $x z ^ { x y } \ln ( z )$

Question 12

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

Accepted Answer

A)  $- \frac { 1 } { 5 }$ 
B)  $\frac { 2 } { 5 }$ 
C) 5 
D)  $- 5$ 
E)  $\frac{8}{5}$ 
A)  $- \frac { 1 } { 5 }$ 
B)  $\frac { 2 } { 5 }$ 
C) 5 
D)  $- 5$ 
E)  $\frac{8}{5}$

Question 13

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$ is

Accepted Answer

A) Does not exist 
B) 1 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E)  $- 1$ 
A) Does not exist 
B) 1 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E)  $- 1$

Question 14

Let $w = ( x + 2 y + 3 z ) ^ { 4 }$ , where x = s + t, y = s - t, z = st. When s = 0, t = 1, $w _ { s }$ is

Accepted Answer

A)  $- 16$ 
B)  $- 24$ 
C) 12 
D)  $- 8$ 
E)  $- 1$ 
A)  $- 16$ 
B)  $- 24$ 
C) 12 
D)  $- 8$ 
E)  $- 1$

Question 15

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x + y - 1 } { \sqrt { x + y } - 1 }$ is

Accepted Answer

A) Does not exist 
B) 2 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E) 1 
A) Does not exist 
B) 2 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E) 1

Question 16

The limit $\lim _ { ( x , y , z ) \rightarrow ( 0,1,0 ) } \frac { x ^ { 2 } y ^ { 2 } + y ^ { 2 } z ^ { 2 } } { x ^ { 2 } + z ^ { 2 } }$ is

Accepted Answer

A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1 
A) Does not exist 
B) 2 
C) 4 
D) 0 
E) 1

Question 17

Let $x e ^ { y z } + y e ^ { x z } + x y z = 0$ . If Z is a function of x and y, then $Z_ { x } \left( X ,y , Z  \right)$ is

Accepted Answer

A)  $- \frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } - e ^ { x z } + 1 \right) }$ 
B)  $\frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } - 1 \right) }$ 
C)  $- \frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } + 1 \right) }$ 
D)  $\frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } + 1 \right) }$ 
E)  $- \frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } - 1 \right) }$ 
A)  $- \frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } - e ^ { x z } + 1 \right) }$ 
B)  $\frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } - 1 \right) }$ 
C)  $- \frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } + 1 \right) }$ 
D)  $\frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } + 1 \right) }$ 
E)  $- \frac { e ^ { y z } + y z e ^ { x z } + y z } { x y \left( e ^ { y z } + e ^ { x z } - 1 \right) }$

Question 18

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

Accepted Answer

A)  $x y \cos z + y \sin x$ 
B)  $x y \cos z - y \sin x$ 
C)  $- x y \cos z - y \sin x$ 
D)  $- x y \cos z + y \sin x$ 
E)  $x y \sin z - y \cos x$ 
A)  $x y \cos z + y \sin x$ 
B)  $x y \cos z - y \sin x$ 
C)  $- x y \cos z - y \sin x$ 
D)  $- x y \cos z + y \sin x$ 
E)  $x y \sin z - y \cos x$

Question 19

The domain of the function $f ( x , y ) = \frac { 2 x y } { x ^ { 2 } + y ^ { 2 } }$ is

Accepted Answer

A)  $\{ ( x , y ) : x \neq - y \}$ 
B)  $\{ ( x , y ) : x \neq y \}$ 
C)  $\{ ( x , y ) : ( x , y ) \neq ( 0,0 ) \}$ 
D)  $\{ ( x , y ) : x \neq 0 \}$ 
E)  $\{ ( x , y ) : y \neq 0 \}$ 
A)  $\{ ( x , y ) : x \neq - y \}$ 
B)  $\{ ( x , y ) : x \neq y \}$ 
C)  $\{ ( x , y ) : ( x , y ) \neq ( 0,0 ) \}$ 
D)  $\{ ( x , y ) : x \neq 0 \}$ 
E)  $\{ ( x , y ) : y \neq 0 \}$

Question 20

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

Accepted Answer

A)  $e ^ { x y z } ( y z d x - x z d y + x y d z )$ 
B)  $e ^ { x y z } ( y z d x + x z d y + x y d z )$ 
C)  $e ^ { x y z } ( y z d x + x z d y - x y d z )$ 
D)  $e ^ { x y z } ( y z d x - x z d y - x y d z )$ 
E)  $e ^ { x y z } ( - y z d x + x z d y + x y d z )$ 
A)  $e ^ { x y z } ( y z d x - x z d y + x y d z )$ 
B)  $e ^ { x y z } ( y z d x + x z d y + x y d z )$ 
C)  $e ^ { x y z } ( y z d x + x z d y - x y d z )$ 
D)  $e ^ { x y z } ( y z d x - x z d y - x y d z )$ 
E)  $e ^ { x y z } ( - y z d x + x z d y + x y d z )$

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$ is

Let $f ( x , y ) = x y + 3$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y ^ { 2 } } { x ^ { 4 } + y ^ { 4 } }$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { x y - 4 } { \sqrt { x y } - 2 }$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0.0 ) } \frac { x ^ { 3 } y } { x ^ { 4 } + y ^ { 4 } }$ is

Let $f ( x , y , z ) = \left\{ \begin{array} { c l } \frac { x - y - z } { x ^ { 2 } - ( y + z ) ^ { 2 } } & x ^ { 2 } \neq ( y + z ) ^ { 2 } \\\frac { 1 } { 4 } & x ^ { 2 } = ( y + z ) ^ { 2 }\end{array} \right.$ . Then f is

Let $w = y z ^ { 2 } + x ^ { 2 } z + x y ^ { 2 }$ . Then the differential $d ^ { w }$ is

The domain of the function $f ( x , y ) = \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 x y + y ^ { 2 } }$ is

Let $f ( x , y ) = \left\{ \begin{array} { c c } \frac { x y - 4 } { \sqrt { x y } - 2 } & ( x , y ) \neq ( 2,2 ) \\0 & ( x , y ) = ( 2,2 )\end{array} \right.$ . Then f is

Let $f ( x , y ) = \cos x$ . Then the level curve for z = 0 is

Let $f ( x , y , z ) = z ^ { x y }$ . Then $f _ { y } ( x , y , z )$ 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 $z _ { x } ( 1,1,1 )$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$ is

Let $w = ( x + 2 y + 3 z ) ^ { 4 }$ , where x = s + t, y = s - t, z = st. When s = 0, t = 1, $w _ { s }$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x + y - 1 } { \sqrt { x + y } - 1 }$ is

The limit $\lim _ { ( x , y , z ) \rightarrow ( 0,1,0 ) } \frac { x ^ { 2 } y ^ { 2 } + y ^ { 2 } z ^ { 2 } } { x ^ { 2 } + z ^ { 2 } }$ is

Let $x e ^ { y z } + y e ^ { x z } + x y z = 0$ . If Z is a function of x and y, then $Z_ { x } \left( X ,y , Z \right)$ is

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

The domain of the function $f ( x , y ) = \frac { 2 x y } { x ^ { 2 } + y ^ { 2 } }$ is

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

Preparing for Calculus

Limits and Continuity

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Exam 13: Functions of Several Variables

The limit lim⁡(x,y)→(0,0)y4+3x2y2+2xy3(x2+y2)2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }lim(x,y)→(0,0)​(x2+y2)2y4+3x2y2+2xy3​ is

Let f(x,y)=xy+3f ( x , y ) = x y + 3f(x,y)=xy+3 . Then f(x,y+Δy)−f(x,y)f ( x , y + \Delta y ) - f ( x , y )f(x,y+Δy)−f(x,y) is

The limit lim⁡(x,y)→(0,0)x2y2x4+y4\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y ^ { 2 } } { x ^ { 4 } + y ^ { 4 } }lim(x,y)→(0,0)​x4+y4x2y2​ is

The limit lim⁡(x,y)→(2,2)xy−4xy−2\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { x y - 4 } { \sqrt { x y } - 2 }lim(x,y)→(2,2)​xy​−2xy−4​ is

The limit lim⁡(x,y)→(0.0)x3yx4+y4\lim _ { ( x , y ) \rightarrow ( 0.0 ) } \frac { x ^ { 3 } y } { x ^ { 4 } + y ^ { 4 } }lim(x,y)→(0.0)​x4+y4x3y​ is

Let w=yz2+x2z+xy2w = y z ^ { 2 } + x ^ { 2 } z + x y ^ { 2 }w=yz2+x2z+xy2 . Then the differential dwd ^ { w }dw is

The domain of the function f(x,y)=x2−y2x2−2xy+y2f ( x , y ) = \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 x y + y ^ { 2 } }f(x,y)=x2−2xy+y2x2−y2​ is

Let f(x,y)={xy−4xy−2(x,y)≠(2,2)0(x,y)=(2,2)f ( x , y ) = \left\{ \begin{array} { c c } \frac { x y - 4 } { \sqrt { x y } - 2 } & ( x , y ) \neq ( 2,2 ) \\0 & ( x , y ) = ( 2,2 )\end{array} \right.f(x,y)={xy​−2xy−4​0​(x,y)=(2,2)(x,y)=(2,2)​ . Then f is

Let f(x,y)=cos⁡xf ( x , y ) = \cos xf(x,y)=cosx . Then the level curve for z = 0 is

Let f(x,y,z)=zxyf ( x , y , z ) = z ^ { x y }f(x,y,z)=zxy . Then fy(x,y,z)f _ { y } ( x , y , z )fy​(x,y,z) is

Let z3−xy+2yz+y3−3xy=0z ^ { 3 } - x y + 2 y z + y ^ { 3 } - 3 x y = 0z3−xy+2yz+y3−3xy=0 . If Z is a function of x and y, then zx(1,1,1)z _ { x } ( 1,1,1 )zx​(1,1,1) is

The limit lim⁡(x,y)→(0,0)xyx2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } }lim(x,y)→(0,0)​x2+y2​xy​ is

Let w=(x+2y+3z)4w = ( x + 2 y + 3 z ) ^ { 4 }w=(x+2y+3z)4 , where x = s + t, y = s - t, z = st. When s = 0, t = 1, wsw _ { s }ws​ is

The limit lim⁡(x,y)→(0,0)x+y−1x+y−1\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x + y - 1 } { \sqrt { x + y } - 1 }lim(x,y)→(0,0)​x+y​−1x+y−1​ is

The limit lim⁡(x,y,z)→(0,1,0)x2y2+y2z2x2+z2\lim _ { ( x , y , z ) \rightarrow ( 0,1,0 ) } \frac { x ^ { 2 } y ^ { 2 } + y ^ { 2 } z ^ { 2 } } { x ^ { 2 } + z ^ { 2 } }lim(x,y,z)→(0,1,0)​x2+z2x2y2+y2z2​ is

Let xeyz+yexz+xyz=0x e ^ { y z } + y e ^ { x z } + x y z = 0xeyz+yexz+xyz=0 . If Z is a function of x and y, then Zx(X,y,Z)Z_ { x } \left( X ,y , Z \right)Zx​(X,y,Z) is

Let f(x,y,z)=xysin⁡z−yzsin⁡xf ( x , y , z ) = x y \sin z - y z \sin xf(x,y,z)=xysinz−yzsinx . Then fz(x,y,z)f _ { z } ( x , y , z )fz​(x,y,z) is

The domain of the function f(x,y)=2xyx2+y2f ( x , y ) = \frac { 2 x y } { x ^ { 2 } + y ^ { 2 } }f(x,y)=x2+y22xy​ is

Let w=exyzw = e ^ { x y z }w=exyz . Then the differential dwd ^ { w }dw is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$ is

Let $f ( x , y ) = x y + 3$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y ^ { 2 } } { x ^ { 4 } + y ^ { 4 } }$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { x y - 4 } { \sqrt { x y } - 2 }$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0.0 ) } \frac { x ^ { 3 } y } { x ^ { 4 } + y ^ { 4 } }$ is

Let $w = y z ^ { 2 } + x ^ { 2 } z + x y ^ { 2 }$ . Then the differential $d ^ { w }$ is

The domain of the function $f ( x , y ) = \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } - 2 x y + y ^ { 2 } }$ is

Let $f ( x , y ) = \left\{ \begin{array} { c c } \frac { x y - 4 } { \sqrt { x y } - 2 } & ( x , y ) \neq ( 2,2 ) \\0 & ( x , y ) = ( 2,2 )\end{array} \right.$ . Then f is

Let $f ( x , y ) = \cos x$ . Then the level curve for z = 0 is

Let $f ( x , y , z ) = z ^ { x y }$ . Then $f _ { y } ( x , y , z )$ 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 $z _ { x } ( 1,1,1 )$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$ is

Let $w = ( x + 2 y + 3 z ) ^ { 4 }$ , where x = s + t, y = s - t, z = st. When s = 0, t = 1, $w _ { s }$ is

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x + y - 1 } { \sqrt { x + y } - 1 }$ is

The limit $\lim _ { ( x , y , z ) \rightarrow ( 0,1,0 ) } \frac { x ^ { 2 } y ^ { 2 } + y ^ { 2 } z ^ { 2 } } { x ^ { 2 } + z ^ { 2 } }$ is

Let $x e ^ { y z } + y e ^ { x z } + x y z = 0$ . If Z is a function of x and y, then $Z_ { x } \left( X ,y , Z \right)$ is

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

The domain of the function $f ( x , y ) = \frac { 2 x y } { x ^ { 2 } + y ^ { 2 } }$ is

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