Question 1

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

Let $z = x ^ { 2 } - y ^ { 2 }$ , where x = 3r - s, y = r + 2s. When r = 1, s = 1, $Z _ { s }$ is

Accepted Answer

A)  $- 16$ 
B)  $\frac { 17 } { 6 }$ 
C) 4 
D) 6 
E)  $\frac { 11 } { 6 }$ 
A)  $- 16$ 
B)  $\frac { 17 } { 6 }$ 
C) 4 
D) 6 
E)  $\frac { 11 } { 6 }$

Question 3

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

Accepted Answer

A)  $\sin ( x y z ) [ y z d x + x z d y - x y d z ]$ 
B)  $- \sin ( x y z ) [ y z d x + x z d y - x y d z ]$ 
C)  $- \sin ( x y z ) [ y z d x + x z d y + x y d z ]$ 
D)  $- \sin ( x y z ) [ y z d x + x z d y + x y d z ]$. 
E)  $\sin ( x y z ) [ y z d x + x z d y + x y d z ]$ 
A)  $\sin ( x y z ) [ y z d x + x z d y - x y d z ]$ 
B)  $- \sin ( x y z ) [ y z d x + x z d y - x y d z ]$ 
C)  $- \sin ( x y z ) [ y z d x + x z d y + x y d z ]$ 
D)  $- \sin ( x y z ) [ y z d x + x z d y + x y d z ]$. 
E)  $\sin ( x y z ) [ y z d x + x z d y + x y d z ]$

Question 4

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

Let $z = x y \ln ( x )$ , where . When is

Accepted Answer

A)  $- 16$ 
B)  $- 2$ 
C) 1 
D)  $- \frac { 1 } { 2 }$ 
E)  $- 1$ 
A)  $- 16$ 
B)  $- 2$ 
C) 1 
D)  $- \frac { 1 } { 2 }$ 
E)  $- 1$

Question 6

The symmetric equations of the tangent line to the curve of intersection of the surface $z = 16 - x ^ { 2 } - y ^ { 2 }$ and the plane x = 1 at the point (1, 2, 11) are

Accepted Answer

A)  $2 z - 11 = \frac { y - 2 } { - \frac { 1 } { 4 } } , x = 1$ 
B)  $4 z - 11 = \frac { y - 2 } { \frac { 1 } { 4 } } , x = 1$ 
C)  $4 z - 11 = \frac { y - 2 } { - \frac { 1 } { 4 } } , x = 1$ 
D)  $z - 11 = \frac { y - 2 } { \frac { 1 } { 4 } } , x = 1$ 
E)  $z - 11 = - 4 ( y - 2 ) , x = 1$ 
A)  $2 z - 11 = \frac { y - 2 } { - \frac { 1 } { 4 } } , x = 1$ 
B)  $4 z - 11 = \frac { y - 2 } { \frac { 1 } { 4 } } , x = 1$ 
C)  $4 z - 11 = \frac { y - 2 } { - \frac { 1 } { 4 } } , x = 1$ 
D)  $z - 11 = \frac { y - 2 } { \frac { 1 } { 4 } } , x = 1$ 
E)  $z - 11 = - 4 ( y - 2 ) , x = 1$

Question 7

The domain of the function $f ( x , y ) = \frac { 2 x + y } { \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } }$ is

Accepted Answer

A) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \neq 1 \right\}$ B) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } > 1 \right\}$ C) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \geq 1 \right\}$ D) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } < 1 \right\}$ E) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$ A) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \neq 1 \right\}$ B) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } > 1 \right\}$ C) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \geq 1 \right\}$ D) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } < 1 \right\}$ E) $\left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$

Question 8

Let $f ( x , y ) = x ^ { 2 } - 2 x y$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

Accepted Answer

A)  $- \Delta x \Delta y$ 
B)  $- 2 \Delta x \Delta y$ 
C)  $- 2 x \Delta y$ 
D)  $x \Delta y$ 
E)  $2 x \Delta y$ 
A)  $- \Delta x \Delta y$ 
B)  $- 2 \Delta x \Delta y$ 
C)  $- 2 x \Delta y$ 
D)  $x \Delta y$ 
E)  $2 x \Delta y$

Question 9

Let $f ( x , y ) = x ^ { 2 } - 2 x y$ . Then $f ( x + \Delta x , y ) - f ( x , y )$ is

Accepted Answer

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

Question 10

Let $f ( x , y , z ) = \tan ^ { - 1 } \left( \frac { x y } { z } \right)$ . Then $f _ { x } ( x , y , z )$ is

Accepted Answer

A)  $- \frac { x y } { z ^ { 2 } + x ^ { 2 } y ^ { 2 } }$ 
B)  $\frac { x y } { z ^ { 2 } + x ^ { 2 } y ^ { 2 } }$ 
C)  $\frac { x y } { x ^ { 2 } + z ^ { 2 } y ^ { 2 } }$ 
D)  $- \frac { x y } { x ^ { 2 } + z ^ { 2 } y ^ { 2 } }$ 
E)  $- \frac { x y } { x ^ { 2 } + x ^ { 2 } z ^ { 2 } }$ 
A)  $- \frac { x y } { z ^ { 2 } + x ^ { 2 } y ^ { 2 } }$ 
B)  $\frac { x y } { z ^ { 2 } + x ^ { 2 } y ^ { 2 } }$ 
C)  $\frac { x y } { x ^ { 2 } + z ^ { 2 } y ^ { 2 } }$ 
D)  $- \frac { x y } { x ^ { 2 } + z ^ { 2 } y ^ { 2 } }$ 
E)  $- \frac { x y } { x ^ { 2 } + x ^ { 2 } z ^ { 2 } }$

Question 11

Let $z = 2 x ^ { 2 } - 7 y$ , where x = sin t and y = cos t. When t =?, $\frac { d z } { d t }$ is

Accepted Answer

A)  $\pi$ 
B) 0 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 4 }$ 
E)  $\frac { \pi } { 5 }$ 
A)  $\pi$ 
B) 0 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 4 }$ 
E)  $\frac { \pi } { 5 }$

Question 12

Let $z = x y \ln ( x )$ , where $x = 2 s t$ , $y = t - s ^ { 3 }$ . When is

Accepted Answer

A)  $- 16$ 
B)  $- 2$ 
C)  $- 1$ 
D)  $- \frac { 1 } { 2 }$ 
E)  $- \frac { 11 } { 6 }$ 
A)  $- 16$ 
B)  $- 2$ 
C)  $- 1$ 
D)  $- \frac { 1 } { 2 }$ 
E)  $- \frac { 11 } { 6 }$

Question 13

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

Accepted Answer

A) Continuous at (0,0) 
B) Discontinuous at (0,0) because f is undefined at (0,0) 
C) Discontinuous at (0,0) because the limit of f at (0,0) does not exist 
D) Discontinuous at (0,0) because the limit of f at (0,0) is different from the function values at (0,0) 
E) Discontinuous at (0,0) for a reason that is different from the ones above 
A) Continuous at (0,0) 
B) Discontinuous at (0,0) because f is undefined at (0,0) 
C) Discontinuous at (0,0) because the limit of f at (0,0) does not exist 
D) Discontinuous at (0,0) because the limit of f at (0,0) is different from the function values at (0,0) 
E) Discontinuous at (0,0) for a reason that is different from the ones above

Question 14

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } + y } { x ^ { 2 } + y ^ { 2 } }$ 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 15

Let $f ( x , y ) = x + 2 y$ . Then $f ( x + \Delta x , y ) - f ( x , y )$ is

Accepted Answer

A)  $\Delta x + 2 y$ 
B)  $\Delta x$ 
C)  $\Delta x + 2 \Delta y$ 
D)  $\Delta x - 2 y$ 
E)  $\Delta x - 2 \Delta y$ 
A)  $\Delta x + 2 y$ 
B)  $\Delta x$ 
C)  $\Delta x + 2 \Delta y$ 
D)  $\Delta x - 2 y$ 
E)  $\Delta x - 2 \Delta y$

Question 16

The domain of the function $f ( x , y ) = \cos ( x + y )$ is

Accepted Answer

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

Question 17

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } + y ^ { 4 } } { x ^ { 2 } + y ^ { 2 } }$ 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 18

Let $z = \frac { x y } { x + y }$ . Then $\Delta z$, the change of z from (-1, 2) to (-0.9, 1.9), is

Accepted Answer

A) 1.04 
B) 0.87 
C) 0.63 
D) 0.29 
E) 0.15 
A) 1.04 
B) 0.87 
C) 0.63 
D) 0.29 
E) 0.15

Question 19

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

Accepted Answer

A)  $- \frac { 3 x y \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
B)  $\frac { 3 x \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
C)  $- \frac { 3 x \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
D)  $\frac { 3 y \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
E)  $- \frac { 3 y \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
A)  $- \frac { 3 x y \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
B)  $\frac { 3 x \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
C)  $- \frac { 3 x \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
D)  $\frac { 3 y \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
E)  $- \frac { 3 y \left( 6 x ^ { 2 } - y ^ { 3 } \right) } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$

Question 20

Let $\frac { y ^ { 2 } z ^ { 2 } + x ^ { 2 } z ^ { 2 } + x ^ { 2 } y ^ { 2 } } { x ^ { 2 } y ^ { 2 } z ^ { 2 } } - \frac { 49 } { 36 } = 0$ . If Z is a function of x and y, then $z _ { y } ( - 1,2,3 )$ is

Accepted Answer

A)  $- \frac { 27 } { 2 }$ 
B)  $\frac { 27 } { 4 }$ 
C)  $- \frac { 27 } { 8 }$ 
D)  $\frac { 27 } { 8 }$ 
E)  $- \frac { 27 } { 4 }$ 
A)  $- \frac { 27 } { 2 }$ 
B)  $\frac { 27 } { 4 }$ 
C)  $- \frac { 27 } { 8 }$ 
D)  $\frac { 27 } { 8 }$ 
E)  $- \frac { 27 } { 4 }$

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

Let $z = x ^ { 2 } - y ^ { 2 }$ , where x = 3r - s, y = r + 2s. When r = 1, s = 1, $Z _ { s }$ is

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

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

Let $z = x y \ln ( x )$ , where . When is

The symmetric equations of the tangent line to the curve of intersection of the surface $z = 16 - x ^ { 2 } - y ^ { 2 }$ and the plane x = 1 at the point (1, 2, 11) are

The domain of the function $f ( x , y ) = \frac { 2 x + y } { \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } }$ is

Let $f ( x , y ) = x ^ { 2 } - 2 x y$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

Let $f ( x , y ) = x ^ { 2 } - 2 x y$ . Then $f ( x + \Delta x , y ) - f ( x , y )$ is

Let $f ( x , y , z ) = \tan ^ { - 1 } \left( \frac { x y } { z } \right)$ . Then $f _ { x } ( x , y , z )$ is

Let $z = 2 x ^ { 2 } - 7 y$ , where x = sin t and y = cos t. When t =?, $\frac { d z } { d t }$ is

Let $z = x y \ln ( x )$ , where $x = 2 s t$ , $y = t - s ^ { 3 }$ . When is

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

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

Let $f ( x , y ) = x + 2 y$ . Then $f ( x + \Delta x , y ) - f ( x , y )$ is

The domain of the function $f ( x , y ) = \cos ( x + y )$ is

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

Let $z = \frac { x y } { x + y }$ . Then $\Delta z$ , the change of z from (-1, 2) to (-0.9, 1.9), is

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

Let $\frac { y ^ { 2 } z ^ { 2 } + x ^ { 2 } z ^ { 2 } + x ^ { 2 } y ^ { 2 } } { x ^ { 2 } y ^ { 2 } z ^ { 2 } } - \frac { 49 } { 36 } = 0$ . If Z is a function of x and y, then $z _ { y } ( - 1,2,3 )$ is

Preparing for Calculus

Limits and Continuity

The Derivative

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The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

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Exam 13: Functions of Several Variables

The limit lim⁡(x,y)→(0,0)x2−y2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2x2−y2​ is

Let z=x2−y2z = x ^ { 2 } - y ^ { 2 }z=x2−y2 , where x = 3r - s, y = r + 2s. When r = 1, s = 1, ZsZ _ { s }Zs​ is

Let w=cos⁡(xyz)w = \cos ( x y z )w=cos(xyz) . Then the differential dwd ^ { w }dw is

The limit lim⁡(x,y)→(0,0)4xy2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 4 x y } { 2 x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​2x2+y24xy​ is

Let z=xyln⁡(x)z = x y \ln ( x )z=xyln(x) , where . When is

The symmetric equations of the tangent line to the curve of intersection of the surface z=16−x2−y2z = 16 - x ^ { 2 } - y ^ { 2 }z=16−x2−y2 and the plane x = 1 at the point (1, 2, 11) are

The domain of the function f(x,y)=2x+y1−x2−y2f ( x , y ) = \frac { 2 x + y } { \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } }f(x,y)=1−x2−y2​2x+y​ is

Let f(x,y)=x2−2xyf ( x , y ) = x ^ { 2 } - 2 x yf(x,y)=x2−2xy . Then f(x,y+Δy)−f(x,y)f ( x , y + \Delta y ) - f ( x , y )f(x,y+Δy)−f(x,y) is

Let f(x,y)=x2−2xyf ( x , y ) = x ^ { 2 } - 2 x yf(x,y)=x2−2xy . Then f(x+Δx,y)−f(x,y)f ( x + \Delta x , y ) - f ( x , y )f(x+Δx,y)−f(x,y) is

Let f(x,y,z)=tan⁡−1(xyz)f ( x , y , z ) = \tan ^ { - 1 } \left( \frac { x y } { z } \right)f(x,y,z)=tan−1(zxy​) . Then fx(x,y,z)f _ { x } ( x , y , z )fx​(x,y,z) is

Let z=2x2−7yz = 2 x ^ { 2 } - 7 yz=2x2−7y , where x = sin t and y = cos t. When t =?, dzdt\frac { d z } { d t }dtdz​ is

Let z=xyln⁡(x)z = x y \ln ( x )z=xyln(x) , where x=2stx = 2 s tx=2st , y=t−s3y = t - s ^ { 3 }y=t−s3 . When is

Let f(x,y)={xyx2+y2(x,y)≠(0,0)0(x,y)=(0,0)f ( x , y ) = \left\{ \begin{array} { c c } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } & ( x , y ) \neq ( 0,0 ) \\0 & ( x , y ) = ( 0,0 )\end{array} \right.f(x,y)={x2+y2​xy​0​(x,y)=(0,0)(x,y)=(0,0)​ . Then f is

The limit lim⁡(x,y)→(0,0)x2+yx2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } + y } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2x2+y​ is

Let f(x,y)=x+2yf ( x , y ) = x + 2 yf(x,y)=x+2y . Then f(x+Δx,y)−f(x,y)f ( x + \Delta x , y ) - f ( x , y )f(x+Δx,y)−f(x,y) is

The domain of the function f(x,y)=cos⁡(x+y)f ( x , y ) = \cos ( x + y )f(x,y)=cos(x+y) is

The limit lim⁡(x,y)→(0,0)x2+y4x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } + y ^ { 4 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2x2+y4​ is

Let z=xyx+yz = \frac { x y } { x + y }z=x+yxy​ . Then Δz\Delta zΔz , the change of z from (-1, 2) to (-0.9, 1.9), is

Let f(x,y)=3x2−2y3f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }f(x,y)=3x2−2y3​ . Then fwy(x,y)f _ { w y } ( x , y )fwy​(x,y) is

Let y2z2+x2z2+x2y2x2y2z2−4936=0\frac { y ^ { 2 } z ^ { 2 } + x ^ { 2 } z ^ { 2 } + x ^ { 2 } y ^ { 2 } } { x ^ { 2 } y ^ { 2 } z ^ { 2 } } - \frac { 49 } { 36 } = 0x2y2z2y2z2+x2z2+x2y2​−3649​=0 . If Z is a function of x and y, then zy(−1,2,3)z _ { y } ( - 1,2,3 )zy​(−1,2,3) 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 $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$ is

Let $z = x ^ { 2 } - y ^ { 2 }$ , where x = 3r - s, y = r + 2s. When r = 1, s = 1, $Z _ { s }$ is

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

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

Let $z = x y \ln ( x )$ , where . When is

The symmetric equations of the tangent line to the curve of intersection of the surface $z = 16 - x ^ { 2 } - y ^ { 2 }$ and the plane x = 1 at the point (1, 2, 11) are

The domain of the function $f ( x , y ) = \frac { 2 x + y } { \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } }$ is

Let $f ( x , y ) = x ^ { 2 } - 2 x y$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

Let $f ( x , y ) = x ^ { 2 } - 2 x y$ . Then $f ( x + \Delta x , y ) - f ( x , y )$ is

Let $f ( x , y , z ) = \tan ^ { - 1 } \left( \frac { x y } { z } \right)$ . Then $f _ { x } ( x , y , z )$ is

Let $z = 2 x ^ { 2 } - 7 y$ , where x = sin t and y = cos t. When t =?, $\frac { d z } { d t }$ is

Let $z = x y \ln ( x )$ , where $x = 2 s t$ , $y = t - s ^ { 3 }$ . When is

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

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

Let $f ( x , y ) = x + 2 y$ . Then $f ( x + \Delta x , y ) - f ( x , y )$ is

The domain of the function $f ( x , y ) = \cos ( x + y )$ is

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

Let $z = \frac { x y } { x + y }$ . Then $\Delta z$ , the change of z from (-1, 2) to (-0.9, 1.9), is

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

Let $\frac { y ^ { 2 } z ^ { 2 } + x ^ { 2 } z ^ { 2 } + x ^ { 2 } y ^ { 2 } } { x ^ { 2 } y ^ { 2 } z ^ { 2 } } - \frac { 49 } { 36 } = 0$ . If Z is a function of x and y, then $z _ { y } ( - 1,2,3 )$ is