Question 1

Let $f ( x , y ) = 2 x ^ { 4 } + y ^ { 2 } - x ^ { 2 } - 2 y$ . Then, the set of saddle points of f is

Accepted Answer

A)  $\{ ( 0,2 , - 1 ) \}$ 
B)  $\{ ( 0,1,1 ) \}$ 
C)  $\{ ( 0,1 , - 1 ) \}$ 
D)  $\{ ( 0 , - 1,1 ) \}$ 
E)  $\{ ( 0 , - 1 , - 1 ) \}$ 
A)  $\{ ( 0,2 , - 1 ) \}$ 
B)  $\{ ( 0,1,1 ) \}$ 
C)  $\{ ( 0,1 , - 1 ) \}$ 
D)  $\{ ( 0 , - 1,1 ) \}$ 
E)  $\{ ( 0 , - 1 , - 1 ) \}$

Question 2

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then f has a relative minimum at

Accepted Answer

A)  $( 1,1 )$ 
B)  $( - 1,1 )$ 
C)  $( 1 , - 1 )$ 
D)  $( - 1 , - 1 )$ 
E)  $\left( 1 , \frac { 1 } { 2 } \right)$ 
A)  $( 1,1 )$ 
B)  $( - 1,1 )$ 
C)  $( 1 , - 1 )$ 
D)  $( - 1 , - 1 )$ 
E)  $\left( 1 , \frac { 1 } { 2 } \right)$

Question 3

Let $f ( x , y ) = x e ^ { y } + y e ^ { x }$ Then the directional derivative of f at (0,0) in the direction of the unit vector $\mathbf { U }$ which makes an angle of $\frac { \pi } { 6 }$ from the positive x-axis is

Accepted Answer

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

Question 4

Let $f ( x , y ) = 2 x \ln y$ Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 4,1 )$ is

Accepted Answer

A) 24 
B) 4 
C) 16 
D) 8 
E) 2 
A) 24 
B) 4 
C) 16 
D) 8 
E) 2

Question 5

The symmetric equations of the normal line to the surface $z x ^ { 2 } - x y ^ { 2 } - y z ^ { 2 } = 18$ at (0, -2, 3) are

Accepted Answer

A)  $\frac { x } { - 4 } = \frac { y + 2 } { - 9 } = \frac { z - 3 } { 12 }$ 
B)  $\frac { x } { - 4 } = \frac { y + 2 } { 9 } = \frac { z - 3 } { 12 }$ 
C)  $\frac { x } { 4 } = \frac { y + 2 } { - 9 } = \frac { z - 3 } { 12 }$ 
D)  $\frac { x } { 4 } = \frac { y + 2 } { 9 } = \frac { z - 3 } { 12 }$ 
E)  $\frac { x } { 9 } = \frac { y + 2 } { - 4 } = \frac { z - 3 } { 12 }$ 
A)  $\frac { x } { - 4 } = \frac { y + 2 } { - 9 } = \frac { z - 3 } { 12 }$ 
B)  $\frac { x } { - 4 } = \frac { y + 2 } { 9 } = \frac { z - 3 } { 12 }$ 
C)  $\frac { x } { 4 } = \frac { y + 2 } { - 9 } = \frac { z - 3 } { 12 }$ 
D)  $\frac { x } { 4 } = \frac { y + 2 } { 9 } = \frac { z - 3 } { 12 }$ 
E)  $\frac { x } { 9 } = \frac { y + 2 } { - 4 } = \frac { z - 3 } { 12 }$

Question 6

An equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) is

Accepted Answer

A)  $2 x - 2 y + z = 8$ 
B)  $2 x + 2 y + z = 8$ 
C)  $2 x + 2 y - z = 8$ 
D)  $2 x - 2 y - z = 8$ 
E)  $- 2 x + 2 y + z = 8$ 
A)  $2 x - 2 y + z = 8$ 
B)  $2 x + 2 y + z = 8$ 
C)  $2 x + 2 y - z = 8$ 
D)  $2 x - 2 y - z = 8$ 
E)  $- 2 x + 2 y + z = 8$

Question 7

Let $f ( x , y , z ) = x ^ { 2 } - y ^ { 2 }$ with constraint $x ^ { 2 } + 2 y ^ { 2 } + 3 z ^ { 2 } = 1$ . Then f has a relative minimum at

Accepted Answer

A)  $( \pm 1,0,0 )$ 
B)  $( - 1,0,0 )$ 
C)  $( 1,0,0 )$ 
D)  $\left( \frac { 1 } { 2 } , 0 , \frac { 1 } { 2 } \right)$ 
E)  $\left( \frac { 1 } { 2 } , 0 , - \frac { 1 } { 2 } \right)$ 
A)  $( \pm 1,0,0 )$ 
B)  $( - 1,0,0 )$ 
C)  $( 1,0,0 )$ 
D)  $\left( \frac { 1 } { 2 } , 0 , \frac { 1 } { 2 } \right)$ 
E)  $\left( \frac { 1 } { 2 } , 0 , - \frac { 1 } { 2 } \right)$

Question 8

An equation of the tangent plane to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) is

Accepted Answer

A)  $8 x - 4 y + 12 z = 52$ 
B)  $8 x - 4 y - 12 z = 52$ 
C)  $8 x + 4 y + 12 z = 52$ 
D)  $- 8 x - 4 y + 12 z = 52$ 
E)  $- 8 x + 4 y + 12 z = 52$ 
A)  $8 x - 4 y + 12 z = 52$ 
B)  $8 x - 4 y - 12 z = 52$ 
C)  $8 x + 4 y + 12 z = 52$ 
D)  $- 8 x - 4 y + 12 z = 52$ 
E)  $- 8 x + 4 y + 12 z = 52$

Question 9

Let $f ( x , y ) = \frac { 1 } { x } - \frac { 64 } { y } + x y$ . Then f has a relative maximum at

Accepted Answer

A)  $\left( - \frac { 1 } { 4 } , 16 \right)$ 
B)  $\left( \frac { 1 } { 4 } , 16 \right)$ 
C)  $\left( \frac { 1 } { 4 } , 32 \right)$ 
D)  $\left( - \frac { 1 } { 4 } , - 16 \right)$ 
E)  $\left( \frac { 1 } { 4 } , - 16 \right)$ 
A)  $\left( - \frac { 1 } { 4 } , 16 \right)$ 
B)  $\left( \frac { 1 } { 4 } , 16 \right)$ 
C)  $\left( \frac { 1 } { 4 } , 32 \right)$ 
D)  $\left( - \frac { 1 } { 4 } , - 16 \right)$ 
E)  $\left( \frac { 1 } { 4 } , - 16 \right)$

Question 10

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative minimum at

Accepted Answer

A)  $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$ 
B)  $( - \sqrt { 3 } , - \sqrt { 3 } , - \sqrt { 3 } )$ 
C)  $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$ 
D)  $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$ 
E)  $( \sqrt { 3 } , - \sqrt { 3 } , - \sqrt { 3 } )$ 
A)  $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$ 
B)  $( - \sqrt { 3 } , - \sqrt { 3 } , - \sqrt { 3 } )$ 
C)  $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$ 
D)  $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$ 
E)  $( \sqrt { 3 } , - \sqrt { 3 } , - \sqrt { 3 } )$

Question 11

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

Accepted Answer

A)  $( 1 , \pm 1 , \pm 1 )$ 
B)  $( - 1 , \pm 1 , \mp 1 )$ 
C)  $( 1,1,1 ) , ( - 1 , - 1 , - 1 )$ 
D)  $( 1 , - 1 , - 1 ) , ( - 1,1 , - 1 )$ 
E)  $( 1 , \pm 1 , \pm 1 ) , ( - 1 , \pm 1 , \mp 1 )$ 
A)  $( 1 , \pm 1 , \pm 1 )$ 
B)  $( - 1 , \pm 1 , \mp 1 )$ 
C)  $( 1,1,1 ) , ( - 1 , - 1 , - 1 )$ 
D)  $( 1 , - 1 , - 1 ) , ( - 1,1 , - 1 )$ 
E)  $( 1 , \pm 1 , \pm 1 ) , ( - 1 , \pm 1 , \mp 1 )$

Question 12

Let $f ( x , y ) = x ^ { 2 } + y ^ { 3 }$ . Then f has a relative minimum at

Accepted Answer

A)  $( 2,3 )$ 
B)  $( - 2,3 )$ 
C)  $( 0,0 )$ 
D)  $( - 2 , - 3 )$ 
E)  $( 2 , - 3 )$ 
A)  $( 2,3 )$ 
B)  $( - 2,3 )$ 
C)  $( 0,0 )$ 
D)  $( - 2 , - 3 )$ 
E)  $( 2 , - 3 )$

Question 13

The symmetric equations of the normal line to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) are

Accepted Answer

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

Question 14

Let $f ( x , y ) = \frac { 2 x + 2 y + 1 } { x ^ { 2 } + y ^ { 2 } + 1 }$ . Then f has a relative maximum at

Accepted Answer

A)  $\left( \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$ 
B)  $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $\left( \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
D)  $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ 
E)  $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ 
A)  $\left( \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$ 
B)  $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
C)  $\left( \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ 
D)  $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ 
E)  $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

Question 15

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative maximum at

Accepted Answer

A)  $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$ 
B)  $( \sqrt { 3 } , - \sqrt { 3 } , \sqrt { 3 } )$ 
C)  $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$ 
D)  $( - \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$ 
E)  $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$ 
A)  $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$ 
B)  $( \sqrt { 3 } , - \sqrt { 3 } , \sqrt { 3 } )$ 
C)  $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$ 
D)  $( - \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$ 
E)  $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$

Question 16

The absolute maximum of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 x$ on or inside the rectangle with vertices (0, -3), (5, -3), (5, 3), and (0, 3) is

Accepted Answer

A) 19 
B) 16 
C) 14 
D) 12 
E) 8 
A) 19 
B) 16 
C) 14 
D) 12 
E) 8

Question 17

Let $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ with constraint $3 x + y = 3$ . Then f has a relative minimum at

Accepted Answer

A)  $\left( \frac { 9 } { 10 } , \frac { 3 } { 10 } \right)$ 
B)  $\left( - \frac { 9 } { 10 } , \frac { 3 } { 10 } \right)$ 
C)  $\left( \frac { 9 } { 10 } , - \frac { 3 } { 10 } \right)$ 
D)  $\left( \frac { 3 } { 10 } , \frac { 9 } { 10 } \right)$ 
E)  $\left( \frac { 3 } { 10 } , - \frac { 9 } { 10 } \right)$ 
A)  $\left( \frac { 9 } { 10 } , \frac { 3 } { 10 } \right)$ 
B)  $\left( - \frac { 9 } { 10 } , \frac { 3 } { 10 } \right)$ 
C)  $\left( \frac { 9 } { 10 } , - \frac { 3 } { 10 } \right)$ 
D)  $\left( \frac { 3 } { 10 } , \frac { 9 } { 10 } \right)$ 
E)  $\left( \frac { 3 } { 10 } , - \frac { 9 } { 10 } \right)$

Question 18

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $9 x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } = 36$ . Then the minimum value of w is

Accepted Answer

A) 4 
B) 2 
C) 1 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 1 } { 3 }$ 
A) 4 
B) 2 
C) 1 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 1 } { 3 }$

Question 19

The symmetric equations of the normal line to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) are

Accepted Answer

A)  $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { 3 }$ 
B)  $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { 2 }$ 
C)  $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { - 2 }$ 
D)  $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { - 3 }$ 
E)  $\frac { x - 1 } { - 2 } = y + 2 = \frac { z - 3 } { - 3 }$ 
A)  $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { 3 }$ 
B)  $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { 2 }$ 
C)  $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { - 2 }$ 
D)  $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { - 3 }$ 
E)  $\frac { x - 1 } { - 2 } = y + 2 = \frac { z - 3 } { - 3 }$

Question 20

The symmetric equations of the normal line to the surface $x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } + z ^ { \frac { 2 } { 3 } } = 14$ at (-8, 27, 1) are

Accepted Answer

A)  $\frac { x + 8 } { 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { - 6 }$ 
B)  $\frac { x + 8 } { - 3 } = \frac { y - 27 } { 2 } = \frac { z - 1 } { - 6 }$ 
C)  $\frac { x + 8 } { 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { 6 }$ 
D)  $\frac { x + 8 } { 3 } = \frac { y - 27 } { 2 } = \frac { z - 1 } { 6 }$ 
E)  $\frac { x + 8 } { - 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { 6 }$ 
A)  $\frac { x + 8 } { 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { - 6 }$ 
B)  $\frac { x + 8 } { - 3 } = \frac { y - 27 } { 2 } = \frac { z - 1 } { - 6 }$ 
C)  $\frac { x + 8 } { 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { 6 }$ 
D)  $\frac { x + 8 } { 3 } = \frac { y - 27 } { 2 } = \frac { z - 1 } { 6 }$ 
E)  $\frac { x + 8 } { - 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { 6 }$

Let $f ( x , y ) = 2 x ^ { 4 } + y ^ { 2 } - x ^ { 2 } - 2 y$ . Then, the set of saddle points of f is

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then f has a relative minimum at

Let $f ( x , y ) = x e ^ { y } + y e ^ { x }$ Then the directional derivative of f at (0,0) in the direction of the unit vector $\mathbf { U }$ which makes an angle of $\frac { \pi } { 6 }$ from the positive x-axis is

Let $f ( x , y ) = 2 x \ln y$ Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 4,1 )$ is

The symmetric equations of the normal line to the surface $z x ^ { 2 } - x y ^ { 2 } - y z ^ { 2 } = 18$ at (0, -2, 3) are

An equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) is

Let $f ( x , y , z ) = x ^ { 2 } - y ^ { 2 }$ with constraint $x ^ { 2 } + 2 y ^ { 2 } + 3 z ^ { 2 } = 1$ . Then f has a relative minimum at

An equation of the tangent plane to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) is

Let $f ( x , y ) = \frac { 1 } { x } - \frac { 64 } { y } + x y$ . Then f has a relative maximum at

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative minimum at

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

Let $f ( x , y ) = x ^ { 2 } + y ^ { 3 }$ . Then f has a relative minimum at

The symmetric equations of the normal line to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) are

Let $f ( x , y ) = \frac { 2 x + 2 y + 1 } { x ^ { 2 } + y ^ { 2 } + 1 }$ . Then f has a relative maximum at

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative maximum at

The absolute maximum of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 x$ on or inside the rectangle with vertices (0, -3), (5, -3), (5, 3), and (0, 3) is

Let $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ with constraint $3 x + y = 3$ . Then f has a relative minimum at

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $9 x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } = 36$ . Then the minimum value of w is

The symmetric equations of the normal line to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) are

The symmetric equations of the normal line to the surface $x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } + z ^ { \frac { 2 } { 3 } } = 14$ at (-8, 27, 1) are

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

Functions of Several Variables

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Exam 14: Directional Derivatives, Gradients, and Extrema

Let f(x,y)=2x4+y2−x2−2yf ( x , y ) = 2 x ^ { 4 } + y ^ { 2 } - x ^ { 2 } - 2 yf(x,y)=2x4+y2−x2−2y . Then, the set of saddle points of f is

Let f(x,y)=x3+y3+3y2−3x−9y+2f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2f(x,y)=x3+y3+3y2−3x−9y+2 . Then f has a relative minimum at

Let f(x,y)=xey+yexf ( x , y ) = x e ^ { y } + y e ^ { x }f(x,y)=xey+yex Then the directional derivative of f at (0,0) in the direction of the unit vector U\mathbf { U }U which makes an angle of π6\frac { \pi } { 6 }6π​ from the positive x-axis is

Let f(x,y)=2xln⁡yf ( x , y ) = 2 x \ln yf(x,y)=2xlny Then the maximum value of the directional derivative Duf(4,1)D _ { \mathrm { u } } f ( 4,1 )Du​f(4,1) is

The symmetric equations of the normal line to the surface zx2−xy2−yz2=18z x ^ { 2 } - x y ^ { 2 } - y z ^ { 2 } = 18zx2−xy2−yz2=18 at (0, -2, 3) are

An equation of the tangent plane to the surface x+y+z=4\sqrt { x } + \sqrt { y } + \sqrt { z } = 4x​+y​+z​=4 at (1, 1, 4) is

Let f(x,y,z)=x2−y2f ( x , y , z ) = x ^ { 2 } - y ^ { 2 }f(x,y,z)=x2−y2 with constraint x2+2y2+3z2=1x ^ { 2 } + 2 y ^ { 2 } + 3 z ^ { 2 } = 1x2+2y2+3z2=1 . Then f has a relative minimum at

An equation of the tangent plane to the surface 4x2+y2+2z2=264 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 264x2+y2+2z2=26 at (1, -2, 3) is

Let f(x,y)=1x−64y+xyf ( x , y ) = \frac { 1 } { x } - \frac { 64 } { y } + x yf(x,y)=x1​−y64​+xy . Then f has a relative maximum at

Let f(x,y,z)=x+y+zf ( x , y , z ) = x + y + zf(x,y,z)=x+y+z with constraint x2+y2+z2=9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9x2+y2+z2=9 . Then f has a relative minimum at

Let f(x,y,z)=x2+y2+z2f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }f(x,y,z)=x2+y2+z2 with constraint xyz = 1. Then f has a relative minimum at

Let f(x,y)=x2+y3f ( x , y ) = x ^ { 2 } + y ^ { 3 }f(x,y)=x2+y3 . Then f has a relative minimum at

The symmetric equations of the normal line to the surface x+y+z=4\sqrt { x } + \sqrt { y } + \sqrt { z } = 4x​+y​+z​=4 at (1, 1, 4) are

Let f(x,y)=2x+2y+1x2+y2+1f ( x , y ) = \frac { 2 x + 2 y + 1 } { x ^ { 2 } + y ^ { 2 } + 1 }f(x,y)=x2+y2+12x+2y+1​ . Then f has a relative maximum at

Let f(x,y,z)=x+y+zf ( x , y , z ) = x + y + zf(x,y,z)=x+y+z with constraint x2+y2+z2=9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9x2+y2+z2=9 . Then f has a relative maximum at

The absolute maximum of f(x,y)=x2+y2+xy−6xf ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 xf(x,y)=x2+y2+xy−6x on or inside the rectangle with vertices (0, -3), (5, -3), (5, 3), and (0, 3) is

Let f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2 with constraint 3x+y=33 x + y = 33x+y=3 . Then f has a relative minimum at

Let w=x2+y2+z2w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }w=x2+y2+z2 with constraint 9x2+4y2+z2=369 x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } = 369x2+4y2+z2=36 . Then the minimum value of w is

The symmetric equations of the normal line to the surface 4x2+y2+2z2=264 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 264x2+y2+2z2=26 at (1, -2, 3) are

The symmetric equations of the normal line to the surface x23+y23+z23=14x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } + z ^ { \frac { 2 } { 3 } } = 14x32​+y32​+z32​=14 at (-8, 27, 1) are

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

Functions of Several Variables

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Let $f ( x , y ) = 2 x ^ { 4 } + y ^ { 2 } - x ^ { 2 } - 2 y$ . Then, the set of saddle points of f is

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then f has a relative minimum at

Let $f ( x , y ) = x e ^ { y } + y e ^ { x }$ Then the directional derivative of f at (0,0) in the direction of the unit vector $\mathbf { U }$ which makes an angle of $\frac { \pi } { 6 }$ from the positive x-axis is

Let $f ( x , y ) = 2 x \ln y$ Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 4,1 )$ is

The symmetric equations of the normal line to the surface $z x ^ { 2 } - x y ^ { 2 } - y z ^ { 2 } = 18$ at (0, -2, 3) are

An equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) is

Let $f ( x , y , z ) = x ^ { 2 } - y ^ { 2 }$ with constraint $x ^ { 2 } + 2 y ^ { 2 } + 3 z ^ { 2 } = 1$ . Then f has a relative minimum at

An equation of the tangent plane to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) is

Let $f ( x , y ) = \frac { 1 } { x } - \frac { 64 } { y } + x y$ . Then f has a relative maximum at

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative minimum at

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

Let $f ( x , y ) = x ^ { 2 } + y ^ { 3 }$ . Then f has a relative minimum at

The symmetric equations of the normal line to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) are

Let $f ( x , y ) = \frac { 2 x + 2 y + 1 } { x ^ { 2 } + y ^ { 2 } + 1 }$ . Then f has a relative maximum at

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative maximum at

The absolute maximum of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 x$ on or inside the rectangle with vertices (0, -3), (5, -3), (5, 3), and (0, 3) is

Let $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ with constraint $3 x + y = 3$ . Then f has a relative minimum at

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $9 x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } = 36$ . Then the minimum value of w is

The symmetric equations of the normal line to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) are

The symmetric equations of the normal line to the surface $x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } + z ^ { \frac { 2 } { 3 } } = 14$ at (-8, 27, 1) are