Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

A)  $2 \cos ( x y ) [ y d x + x d y ]$ 
B)  $\cos ( x y ) [ - y d x + x d y ]$ 
C)  $\cos ( x y ) [ y d x + x d y ]$ 
D)  $\cos ( x y ) [ y d x - x d y ]$ 
E)  $- \cos ( x y ) [ y d x + x d y ]$

Question 3

Consider a rectangular solid with a square end of side length 10 inches and a length of 20 inches. If the measurements are accurate to within 0.1 inch, then the estimated error of the volume in cubic inches using differential is
​

Accepted Answer

A) 50 
B) 72.5 
C) 88.1 
D) 93.7 
E) 102.3

Question 4

The symmetric equations of the tangent line to the curve of intersection of the surface $z = x ^ { 2 } + y ^ { 2 }$ and the plane y = 2 at the point (1, 2, 5) are

Accepted Answer

A)  $z - 5 = 2 ( x - 1 ) , y = 2$ 
B)  $z - 5 = \frac { x - 1 } { - 2 } , y = 2$ 
C)  $2 z - 5 = \frac { x - 1 } { 2 } , y = 2$ 
D)  $2 z - 5 = \frac { x - 1 } { - 2 } , y = 2$ 
E)  $4 z - 5 = \frac { x - 1 } { 2 } , y = 2$

Question 5

Let $z = x \cos y + y \sin x$ . Then the differential $d ^ { z}$ is

Accepted Answer

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

Question 6

The domain of the function $f ( x , y ) = \sin ( x + y )$ 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)  $\{ ( x , y ) : - \infty < x < \infty , - \infty < y < \infty \}$

Question 7

Let $z = 2 x ^ { 2 } + x y - y ^ { 2 }$ . Then the change of z from (2, -1) to (2.1, -1.1) approximated by $d z$ is

Accepted Answer

A) 1.32 
B) 0.94 
C) 0.77 
D) 0.58 
E) 0.3

Question 8

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

Accepted Answer

A) 2.01 
B) 1.87 
C) 1.45 
D) 1.25 
E) 1.08

Question 9

Let $z = 3 x ^ { 2 } + x y - 2 y ^ { 3 }$ . Then the differential $d ^ { z}$ is

Accepted Answer

A)  $( 6 x + y ) d x - \left( x + 6 y ^ { 2 } \right) d y$ 
B)  $( 6 x + y ) d x + \left( x + 6 y ^ { 2 } \right) d y$ 
C)  $( 6 x - y ) d x + \left( x - 6 y ^ { 2 } \right) d y$ 
D)  $( 6 x + y ) d x - \left( x - 6 y ^ { 2 } \right) d y$ 
E)  $( 6 x + y ) d x + \left( x - 6 y ^ { 2 } \right) d y$

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Let $f ( x , y ) = \frac { x - 1 } { y + 2 }$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

Accepted Answer

A)  $\frac { ( 1 - x ) \Delta y } { ( y + 2 ) ( y + 2 + \Delta y ) }$ 
B)  $\frac { ( 1 + x ) \Delta y } { ( y + 2 ) ( y + 2 + \Delta y ) }$ 
C)  $\frac { ( 1 - x ) \Delta y } { ( y - 2 ) ( y + 2 + \Delta y ) }$ 
D)  $\frac { ( 1 + x ) \Delta y } { ( y - 2 ) ( y + 2 + \Delta y ) }$ 
E)  $\frac { ( 1 - x ) \Delta y } { ( y + 2 ) ( y + 2 - \Delta y ) }$

Question 13

Let $w = \ln ( x y z )$ where x = t, y = t + 1, and z = t + 2. When t = 1, $\frac { d z } { d t }$ is

Accepted Answer

A)  $- \frac { 11 } { 6 }$ 
B)  $\frac { 13 } { 6 }$ 
C)  $- \frac { 13 } { 6 }$ 
D)  $\frac { 17 } { 6 }$ 
E)  $\frac { 11 } { 6 }$

Question 14

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

Accepted Answer

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

Question 15

Let $z = \ln \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$ . Then the differential $d ^ { z }$ is

Accepted Answer

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

Question 16

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

Accepted Answer

A) Does not exist 
B) 2 
C)  $\frac { 1 } { 4 }$ 
D) 0 
E) 1

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

A)  $- \frac { 9 x y ^ { 2 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
B)  $\frac { 9 x y ^ { 2 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
C)  $- \frac { 9 y x ^ { 2 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
D)  $\frac { 9 y x ^ { 2 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$ 
E)  $\frac { 9 x ^ { 2 } y ^ { 2 } } { \left( \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } } \right) ^ { 3 } }$

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

Exam 13: Functions of Several Variables

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

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

Consider a rectangular solid with a square end of side length 10 inches and a length of 20 inches. If the measurements are accurate to within 0.1 inch, then the estimated error of the volume in cubic inches using differential is

The symmetric equations of the tangent line to the curve of intersection of the surface $z = x ^ { 2 } + y ^ { 2 }$ and the plane y = 2 at the point (1, 2, 5) are

Let $z = x \cos y + y \sin x$ . Then the differential $d ^ { z}$ is

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

Let $z = 2 x ^ { 2 } + x y - y ^ { 2 }$ . Then the change of z from (2, -1) to (2.1, -1.1) approximated by $d z$ is

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

Let $z = 3 x ^ { 2 } + x y - 2 y ^ { 3 }$ . Then the differential $d ^ { z}$ is

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

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

Let $f ( x , y ) = \frac { x - 1 } { y + 2 }$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

Let $w = \ln ( x y z )$ where x = t, y = t + 1, and z = t + 2. When t = 1, $\frac { d z } { d t }$ is

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

Let $z = \ln \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$ . Then the differential $d ^ { z }$ is

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

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

Let $f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }$ . Then $f _ { x y } ( x , y )$ 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 y = 2 at the point (1, 2, 11) are

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

Exam 13: Functions of Several Variables

Let w=eyz+exz+exyw = e ^ { y z } + e ^ { x z } + e ^ { x y }w=eyz+exz+exy . Then the differential dwd ^ { w }dw is

Let z=sin⁡(xy) z = \sin ( x y ) z=sin(xy) . Then the differential dzd ^ { z }dz is

Consider a rectangular solid with a square end of side length 10 inches and a length of 20 inches. If the measurements are accurate to within 0.1 inch, then the estimated error of the volume in cubic inches using differential is ​

The symmetric equations of the tangent line to the curve of intersection of the surface z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and the plane y = 2 at the point (1, 2, 5) are

Let z=xcos⁡y+ysin⁡xz = x \cos y + y \sin xz=xcosy+ysinx . Then the differential dzd ^ { z}dz is

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

Let z=2x2+xy−y2z = 2 x ^ { 2 } + x y - y ^ { 2 }z=2x2+xy−y2 . Then the change of z from (2, -1) to (2.1, -1.1) approximated by dzd zdz is

Let z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 . Then Δz\Delta zΔz , the change of z from (1, 3) to (1.1, 3.2) , is

Let z=3x2+xy−2y3z = 3 x ^ { 2 } + x y - 2 y ^ { 3 }z=3x2+xy−2y3 . Then the differential dzd ^ { z}dz is

Let z=ex2+y2z = e ^ { x ^ { 2 } + y ^ { 2 } }z=ex2+y2 . Then the differential dzd ^ { z }dz is

Let w=ex2+y2+z2w = e ^ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }w=ex2+y2+z2 . Then the differential dwd ^ { w }dw is

Let f(x,y) =x−1y+2f ( x , y ) = \frac { x - 1 } { y + 2 }f(x,y) =y+2x−1​ . 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 w=ln⁡(xyz) w = \ln ( x y z ) w=ln(xyz) where x = t, y = t + 1, and z = t + 2. When t = 1, dzdt\frac { d z } { d t }dtdz​ is

Let f(x,y) =xyf ( x , y ) = \frac { x } { y }f(x,y) =yx​ . 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 z=ln⁡(x2+y2) z = \ln \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right) z=ln(x2+y2​) . Then the differential dzd ^ { z }dz is

The limit lim⁡(x,y,z) →(2,1,1) x−y−zx2−(y+z) 2\lim _ { ( x , y , z ) \rightarrow ( 2,1,1 ) } \frac { x - y - z } { x ^ { 2 } - ( y + z ) ^ { 2 } }lim(x,y,z) →(2,1,1) ​x2−(y+z) 2x−y−z​

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

Let f(x,y) =3x2−2y3f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }f(x,y) =3x2−2y3​ . Then fxy(x,y) f _ { x y } ( x , y ) fxy​(x,y) 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 y = 2 at the point (1, 2, 11) are

Let f(x,y) =ln⁡(x2+y2) f ( x , y ) = \ln \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right) f(x,y) =ln(x2+y2​) . Then fxx(x,y) f _ { x x } ( x , y ) fxx​(x,y) is

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

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

Consider a rectangular solid with a square end of side length 10 inches and a length of 20 inches. If the measurements are accurate to within 0.1 inch, then the estimated error of the volume in cubic inches using differential is

The symmetric equations of the tangent line to the curve of intersection of the surface $z = x ^ { 2 } + y ^ { 2 }$ and the plane y = 2 at the point (1, 2, 5) are

Let $z = x \cos y + y \sin x$ . Then the differential $d ^ { z}$ is

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

Let $z = 2 x ^ { 2 } + x y - y ^ { 2 }$ . Then the change of z from (2, -1) to (2.1, -1.1) approximated by $d z$ is

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

Let $z = 3 x ^ { 2 } + x y - 2 y ^ { 3 }$ . Then the differential $d ^ { z}$ is

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

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

Let $f ( x , y ) = \frac { x - 1 } { y + 2 }$ . Then $f ( x , y + \Delta y ) - f ( x , y )$ is

Let $w = \ln ( x y z )$ where x = t, y = t + 1, and z = t + 2. When t = 1, $\frac { d z } { d t }$ is

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

Let $z = \ln \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$ . Then the differential $d ^ { z }$ is

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

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

Let $f ( x , y ) = \sqrt { 3 x ^ { 2 } - 2 y ^ { 3 } }$ . Then $f _ { x y } ( x , y )$ 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 y = 2 at the point (1, 2, 11) are

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