Deck 12: Multivariable Functions

Question 1

Evaluate $f ( x , y ) = x ^ { 2 } y$ at $( 2 , - 1 )$

Answer 1

-4

Question 2

Evaluate $f ( x , y ) = x \cos ( y )$ at $( 1,0 )$

Answer 2

1

Question 3

Evaluate $f ( x , y ) = x ^ { 2 } + y ^ { 3 }$ at $( 2 , - 2 )$

A) 12
B) 8
C) -4
D) 4

Answer 3

C

Question 4

Evaluate $f ( x , y , z ) = x ^ { 2 } y z$ at $( 1,2 , - 1 )$

Question 5

Evaluate $f ( x , y , z ) = x e ^ { x } \cos ( z )$ at $( 2,1,0 )$

Question 6

Evaluate $f ( x , y , z ) = e ^ { x } \sin ( y ) \cos ( z )$ at $\left( 0 , \frac { \pi } { 2 } , 0 \right)$

A) 1
B) 0
C) $e ^ { \frac { \pi } { 2 } }$
D) e

Question 7

Find the domain and range of $f ( x , y ) = \frac { x } { y + 2 }$

Question 8

Find the domain and range of $f ( x , y ) = \frac { \ln ( y ) } { \sqrt { x - 1 } }$

Question 9

Find the domain of $f ( x , y ) = \frac { x } { x ^ { 2 } - y ^ { 2 } }$

A) $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : x \neq y \right\}$
B) $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : x \neq - y \right\}$
C) $\operatorname { Dom } ( f ) = \left\{ ( x , y ) \in R ^ { 2 } : | x | \neq | y | \right\}$
D) $\operatorname { Dom } ( f ) = R ^ { 2 }$

Question 10

Find the range of $f ( x , y ) = \frac { x } { x ^ { 2 } - y ^ { 2 } }$

A) $R$
B) $[ 0 , \infty )$
C) $( - \infty , 0 )$
D) $( - 1,1 )$

Question 11

Find the domain and range of $f ( x , y , z ) = \frac { x } { y ^ { 2 } - z ^ { 2 } - 3 }$

Question 12

Find the domain and range of $f ( x , y , z ) = \frac { 1 } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 1 }$

Question 13

Find the range of $f ( x , y , z ) = \frac { 2 } { x ^ { 4 } + y ^ { 2 } + z ^ { 6 } + 4 }$

A) $R$
B) $\left[ 0 , \frac { 1 } { 2 } \right]$
C) $\left( 0 , \frac { 1 } { 2 } \right)$
D) $\left( 0 , \frac { 1 } { 2 } \right]$

Question 14

Find a function of two variables for the surface of revolution formed when the given function on the specified interval is revolved around the y-axis. $f ( x ) = x ^ { 2 }$ on $[ 0,2 ]$

Question 15

Determine if the given subset of $R ^ { 2 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y )$ so that $x > 0$ and $y < 0$

Question 16

Determine if the given subset of $R ^ { 2 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y )$ so that $1 \leq x < 2$ and $1 < y \leq 2$

Question 17

Find the compliment of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $x > 0$ and $y < 0$

Question 18

Find the compliment of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $1 \leq x < 2$ and $1 < y \leq 2$

Question 19

Find the boundary of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $x > 0$ and $y < 0$

Question 20

Find the boundary of the given subset of $R ^ { 2 }$ : all points $( x , y )$ so that $1 \leq x < 2$ and $1 < y \leq 2$

Question 21

Determine if the given subset of $R ^ { 2 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y )$ so that $2 \leq x \leq 12$ and $\pi \leq y \leq 7$

A) Open
B) Closed
C) Both open and closed
D) Neither open nor closed

Question 22

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x > 0$ , $y < 0$ , and $z < 0$

Question 23

Find the compliment of the set in $R ^ { 3 }$ : all points $( x , y , z )$ so that $x > 0$ , $y < 0$ , and $z < 0$

Question 24

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 0$

Question 25

Find the compliment of the set in $R ^ { 3 }$ : all points $( x , y , z )$ so that $x \geq 0$

Question 26

Find the boundary of the set in $R ^ { 3 }$ : all points $( x , y , z )$ so that $x \geq 0$

Question 27

Determine if the given subset of $R ^ { 3 }$ is open, closed, both open and closed, or neither open nor closed. All points $( x , y , z )$ so that $x \geq 1$ , $y < 0$ and $z > 0$

A) Open
B) Closed
C) Both open and closed
D) Neither open nor closed

Question 28

Evaluate the limit if it exists: $$\lim _ { ( x , y ) \rightarrow ( 1,2 ) } x ^ { 2 } - x y$$

Question 29

Evaluate the limit if it exists: $$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + y ^ { 2 } }$$

Question 30

Evaluate the limit if it exists: $$\lim _ { ( x , y ) \rightarrow ( 0.0 ) } ( y + 1 ) ^ { 2 } \cos ( x )$$

A) 2
B) 0
C) 1
D) $\pi$

Question 31

Where is the function continuous? $$f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$$

A) $R ^ { 2 }$
B) $x \neq 0$
C) $y \neq 0$
D) $( x , y ) \neq ( 0,0 )$

Question 32

Find $\frac { \partial f } { \partial x }$ for $f ( x , y ) = \cos ( x y )$

Question 33

Find $\frac { \partial f } { \partial y }$ for $f ( x , y ) = \cos ( x y )$

Question 34

Find $\frac { \partial f } { \partial x }$ for $f ( x , y ) = e ^ { x - y ^ { 2 } }$

Question 35

Find $\frac { \partial f } { \partial y }$ for $f ( x , y ) = e ^ { x - y ^ { 2 } }$

Question 36

Find $\frac { \partial f } { \partial x }$ for $f ( x , y ) = \sin \left( x ^ { 2 } - y ^ { 2 } \right)$

A) $- 2 y \cos \left( x ^ { 2 } - y ^ { 2 } \right)$
B) $2 x \cos \left( x ^ { 2 } - y ^ { 2 } \right)$
C) $2 y \sin \left( x ^ { 2 } - y ^ { 2 } \right)$
D) $2 x \sin \left( x ^ { 2 } - y ^ { 2 } \right)$

Question 37

Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = x ^ { 2 } + x y z + y ^ { 3 }$

Question 38

Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = \cos ( x y z )$

Question 39

Find $\frac { \partial f } { \partial z }$ for $f ( x , y , z ) = e ^ { x y ^ { 2 } z ^ { 3 } }$

A) $y ^ { 2 } z ^ { 3 } e ^ { x y ^ { 2 } z ^ { 3 } }$
B) $2 x y z ^ { 3 } e ^ { x y ^ { 2 } z ^ { 3 } }$
C) $3 x y ^ { 2 } z ^ { 2 } e ^ { x y ^ { 2 } z ^ { 3 } }$
D) $x y ^ { 2 } z ^ { 3 } e ^ { x y ^ { 2 } z ^ { 3 } }$

Question 40

For $f ( x , y ) = x ^ { 2 } - y ^ { 2 }$ find a line in the x direction tangent to the surface defined by f at (1,2).

Question 41

For $f ( x , y ) = x ^ { 2 } - y ^ { 2 }$ find a line in the y direction tangent to the surface defined by f at (1,2).

Question 42

For $f ( x , y ) = x ^ { 2 } - y ^ { 2 }$ find a line in the y direction tangent to the surface defined by f at (1,2).

Question 43

For $f ( x , y ) = x ^ { 2 } - y ^ { 2 }$ find the tangent plane to the surface defined by f at (1,2).

Question 44

For $f ( x , y ) = x ^ { 2 } y$ find a line in the x direction tangent to the surface defined by f at (1,2).

A) $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,4 \rangle$
B) $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,2 \rangle$
C) $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 0,0,1 \rangle$
D) $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 4,1,0 \rangle$

Question 45

Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = x y$ and $\frac { \partial f } { \partial y } = x y$

Question 46

Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = 2 x y ^ { 3 }$ and $\frac { \partial f } { \partial y } = 3 x ^ { 2 } y ^ { 2 }$

Question 47

Find a function with the given partial derivatives $\frac { \partial f } { \partial x } = y \cos ( x y )$ and $\frac { \partial f } { \partial y } = x \cos ( x y )$

A) $f ( x , y ) = \cos ( x y )$
B) $f ( x , y ) = \sin ( x y )$
C) $f ( x , y ) = - \sin ( x y )$
D) $f ( x , y ) = - \cos ( x y )$

Question 48

Solve the exact equation $y \cos ( x y ) + x \cos ( x y ) \frac { d y } { d x } = 0$

Question 49

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \left\langle \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\rangle$

Question 50

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = \cos ( x ) + \sin \left( y ^ { 2 } \right) , P = ( \pi , \sqrt { \pi } ) , \vec { u } = \left\langle \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right\rangle$

Question 51

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 ) , \vec { u } = \left\langle \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right\rangle$

A) $\frac { \sqrt { 3 } + 1 } { 2 }$
B) $2 \sqrt { 3 } - 4$
C) $\frac { \sqrt { 3 } - 1 } { 2 }$
D) $- 1$

Question 52

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = x y z ^ { 2 } , P = ( 1,2,1 ) , \vec { u } = \left\langle \frac { 1 } { 2 } , 0 , \frac { \sqrt { 3 } } { 2 } \right\rangle$

Question 53

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , P = ( 1,2,3 ) , \vec { u } = \left\langle \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } , 0 \right\rangle$

Question 54

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \left\langle \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } , \frac { 1 } { \sqrt { 3 } } \right\rangle$

A) $10$
B) $10 \sqrt { 3 }$
C) $\frac { 10 } { 3 }$
D) $\frac { 10 } { \sqrt { 3 } }$

Question 55

Find the line tangent to the surface given by the function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \left\langle \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\rangle$

Question 56

Find the line tangent to the surface given by the function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = \cos ( x ) + \sin \left( y ^ { 2 } \right) , P = ( \pi , \sqrt { \pi } ) , \vec { u } = \left\langle \frac { \sqrt { 2 } } { 2 } , \frac { \sqrt { 2 } } { 2 } \right\rangle$

Question 57

Find the line tangent to the surface given by the function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 ) , \vec { u } = \left\langle \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } \right\rangle$

A) $\vec { r } ( t ) = \left\langle 2 + \frac { 1 } { 2 } t , 2 + \frac { \sqrt { 3 } } { 2 } t , 3 + \left( 2 + \frac { \sqrt { 3 } } { 2 } \right) t \right\rangle$
B) $\vec { r } ( t ) = \left\langle 1 + \frac { \sqrt { 3 } } { 2 } t , 2 + \frac { 1 } { 2 } t , - 1 + ( 2 \sqrt { 3 } - 4 ) _ { t } \right\rangle$
C) $\vec { r } ( t ) = \left\langle 1 + \frac { \sqrt { 3 } } { 2 } t , 2 + \frac { 1 } { 2 } t , - 1 - \left( \frac { \sqrt { 3 } + 1 } { 2 } \right) t \right\rangle$
D) $\vec { r } ( t ) = \left\langle 1 + \frac { \sqrt { 3 } } { 2 } t , 2 + \frac { 1 } { 2 } t , - 1 - t \right\rangle$

Question 58

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \langle 1 , \sqrt { 3 } \rangle$

Question 59

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = \cos ( x ) + \sin \left( y ^ { 2 } \right) , P = ( \pi , \sqrt { \pi } ) , \vec { u } = \langle \sqrt { 2 } , \sqrt { 2 } \rangle$

Question 60

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 ) , \vec { u } = \langle \sqrt { 3 } , 1 \rangle$

A) $\frac { \sqrt { 3 } + 1 } { 2 }$
B) $2 \sqrt { 3 } - 4$
C) $\frac { - \sqrt { 3 } - 1 } { 2 }$
D) $- 1$

Question 61

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = x y z ^ { 2 } , P = ( 1,2,1 ) , \vec { u } = \langle 1 , \sqrt { 3 } , 0 \rangle$

Question 62

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , P = ( 1,2,3 ) , \vec { u } = \langle \sqrt { 2 } , \sqrt { 2 } , 0 \rangle$

Question 63

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \langle 1,1,1 \rangle$

A) $10$
B) $10 \sqrt { 3 }$
C) $\frac { 10 } { 3 }$
D) $\frac { 10 } { \sqrt { 3 } }$

Question 64

Find the equation of the plane tangent to the surface given by the function at the specified point $P$ $f ( x , y ) = x ^ { 2 } + x y , P = ( 1,2 ) , \vec { u } = \left\langle \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\rangle$

Question 65

Find the equation of the plane tangent to the surface given by the function at the specified point $P$ $f ( x , y ) = \cos ( x ) + \sin \left( y ^ { 2 } \right) , P = ( \pi , \sqrt { \pi } ) ,$

Question 66

Find the equation of the plane tangent to the surface given by the given function at the specified point $P$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 )$

A) $z = - 1 - y - 2$
B) $z = - x - y + 7$
C) $z = - x - y + 2$
D) $z = - x - y$

Question 67

Find $\frac { d z } { d t }$ for $z = \sin \left( x ^ { 2 } y \right) , x = t , y = e ^ { t }$

Question 68

Find $\frac { d z } { d t }$ for $z = x e ^ { y } , x = \cos ( t ) , y = e ^ { t }$

Question 69

Find $\frac { d z } { d t }$ for $z = \frac { x } { y } , x = t ^ { 2 } , y = e ^ { t }$

A) $- 2 t e ^ { - t }$
B) $2 t e ^ { - t } - t ^ { 2 } e ^ { - t }$
C) $t e ^ { - t } - t ^ { 2 } e ^ { - 2 t }$
D) $2 t e ^ { - t }$

Question 70

Find $\frac { d z } { d t }$ for $z = x y ^ { 2 } , x = t s , y = t ^ { 2 } s$

Question 71

Find $\frac { d z } { d t }$ for $z = \sin \left( x ^ { 2 } y \right) , x = \frac { s } { t } , y = t ^ { 2 } e ^ { s t }$

Question 72

Find $\frac { d z } { d t }$ for $z = \frac { x } { y } , x = t ^ { 2 } s ^ { 2 } , y = s e ^ { t }$

A) $- 2 s e ^ { - t }$
B) $2 s t e ^ { - t } - s t ^ { 2 } e ^ { - t }$
C) $2 s t e ^ { - t } - s t ^ { 2 } e ^ { - 2 t }$
D) $2 t e ^ { - t } - t ^ { 2 } e ^ { - t }$

Question 73

Find $\nabla f$ for $f ( x , y ) = x ^ { 2 } y$

Question 74

Find $\nabla f$ for $f ( x , y ) = e ^ { x \sin ( y ) }$

Question 75

Find $\nabla f$ for $f ( x , y ) = \frac { \tan ( x ) } { y }$ .

A) $< - \frac { \tan ( x ) } { y ^ { 2 } } , \frac { \sec ^ { 2 } ( x ) } { y ^ { 2 } } >$
B) $< \frac { \tan ( x ) } { y } , - \frac { \tan ( x ) } { y ^ { 2 } } >$
C) $< \frac { \sec ^ { 2 } ( x ) } { y } , - \frac { \tan ( x ) } { y ^ { 2 } } >$
D) $< \frac { \sec ^ { 2 } ( x ) } { y ^ { 2 } } , \frac { \tan ( x ) } { y } >$

Question 76

Find a function of two variables with the given gradient, $\nabla f = \left\langle 2 x y ^ { 3 } , 3 x ^ { 2 } y ^ { 2 } \right\rangle$

Question 77

Find the direction in which the given function increases most rapidly at the given point. $f ( x , y , z ) = x e ^ { z } - y e ^ { x } , P = ( 0,2,0 )$

Question 78

Find the direction in which the given function decreases most rapidly at the given point. $f ( x , y , z ) = x e ^ { z } - y e ^ { x } , P = ( 0,2,0 )$

Question 79

Find the rate of change in the direction in which the given function increases most rapidly at the given point. $f ( x , y , z ) = x e ^ { z } - y e ^ { x } , P = ( 0,2,0 )$

Question 80

Find the directional derivative of the given function at the specified point $P$ in the direction $\vec { u }$ $f ( x , y ) = x e ^ { y } , P = ( 2,0 ) , \vec { u } = \left\langle \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\rangle$