Question 1

The set of points on $z = x ^ { 2 } - 3 x y + y ^ { 2 }$ at which the tangent plane is parallel to the xy-plane is

Accepted Answer

A)  $\{ ( 0,0,3 ) \}$ 
B)  $\{ ( 0,0,2 ) \}$ 
C)  $\{ ( 0,0,1 ) \}$ 
D)  $\{ ( 0,0,0 ) \}$ 
E)  $\{ ( 0,0,5 ) \}$ 
A)  $\{ ( 0,0,3 ) \}$ 
B)  $\{ ( 0,0,2 ) \}$ 
C)  $\{ ( 0,0,1 ) \}$ 
D)  $\{ ( 0,0,0 ) \}$ 
E)  $\{ ( 0,0,5 ) \}$

Question 2

Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative maximum at

Accepted Answer

A)  $\left( \frac { \pi } { 5 } , \frac { \pi } { 5 } \right)$ 
B)  $\left( - \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$ 
C)  $\left( \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$ 
D)  $\left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$ 
E)  $\left( \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$ 
A)  $\left( \frac { \pi } { 5 } , \frac { \pi } { 5 } \right)$ 
B)  $\left( - \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$ 
C)  $\left( \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$ 
D)  $\left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$ 
E)  $\left( \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$

Question 3

The gradient of $f ( x , y , z ) = y ^ { 2 } + z ^ { 2 } - 4 x z$ at (-2, 1, 3) is

Accepted Answer

A)  $- 12 \mathbf { i } - 2 \mathbf { j } + 14 \mathbf { k }$ 
B)  $- 12 \mathbf { i } + 2 \mathbf { j } + 14 \mathbf { k }$ 
C)  $- 12 \mathbf { i } + 2 \mathbf { j } - 14 \mathbf { k }$ 
D)  $12 \mathbf { i } + 2 \mathbf { j } + 14 \mathbf { k }$ 
E)  $12 \mathbf { i } + 2 \mathbf { j } - 14 \mathbf { k }$ 
A)  $- 12 \mathbf { i } - 2 \mathbf { j } + 14 \mathbf { k }$ 
B)  $- 12 \mathbf { i } + 2 \mathbf { j } + 14 \mathbf { k }$ 
C)  $- 12 \mathbf { i } + 2 \mathbf { j } - 14 \mathbf { k }$ 
D)  $12 \mathbf { i } + 2 \mathbf { j } + 14 \mathbf { k }$ 
E)  $12 \mathbf { i } + 2 \mathbf { j } - 14 \mathbf { k }$

Question 4

The set of points on $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 50$ at which the tangent plane is parallel to the xy-plane is

Accepted Answer

A)  $\{ ( 0,0 , - 5 ) , ( 0,0,5 ) \}$ 
B)  $\{ ( 0,0,5 ) \}$ 
C)  $\{ ( 0,0 , - 5 ) \}$ 
D)  $\{ ( 0,0 , - 3 ) \}$ 
E)  $\{ ( 0,0,3 ) \}$ 
A)  $\{ ( 0,0 , - 5 ) , ( 0,0,5 ) \}$ 
B)  $\{ ( 0,0,5 ) \}$ 
C)  $\{ ( 0,0 , - 5 ) \}$ 
D)  $\{ ( 0,0 , - 3 ) \}$ 
E)  $\{ ( 0,0,3 ) \}$

Question 5

Let $f ( x , y ) = 6 x - 24 y - x ^ { 2 } + 2 y ^ { 3 }$ . Then f has a relative maximum at

Accepted Answer

A)  $( 3 , - 1 )$ 
B)  $( 3,1 )$ 
C)  $( 3 , - 2 )$ 
D)  $( 3,2 )$ 
E)  $( - 3,2 )$ 
A)  $( 3 , - 1 )$ 
B)  $( 3,1 )$ 
C)  $( 3 , - 2 )$ 
D)  $( 3,2 )$ 
E)  $( - 3,2 )$

Question 6

The gradient of $f ( x , y , z ) = z \tan ^ { - 1 } \left( \frac { y } { x } \right)$ at (1, 1, 3) is

Accepted Answer

A)  $\frac { 3 } { 2 } \mathbf { i } - \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
B)  $- \frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
C)  $\frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
D)  $- \frac { 3 } { 2 } \mathbf { i } - \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
E)  $- \frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } - \frac { \pi } { 4 } \mathbf { k }$ 
A)  $\frac { 3 } { 2 } \mathbf { i } - \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
B)  $- \frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
C)  $\frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
D)  $- \frac { 3 } { 2 } \mathbf { i } - \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$ 
E)  $- \frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } - \frac { \pi } { 4 } \mathbf { k }$

Question 7

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $x ^ { 2 } + 4 y ^ { 2 } = 16$ . Then the minimum value of w is

Accepted Answer

A) 24 
B) 23 
C) 19 
D) 17 
E) 16 
A) 24 
B) 23 
C) 19 
D) 17 
E) 16

Question 8

An equation of the tangent plane to the surface $y = e ^ { x } \cos z$ at (1, e, 0) is

Accepted Answer

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

Question 9

Let $f ( x , y ) = x ^ { 3 } + y ^ { 2 } - 6 x ^ { 2 } + y - 1$ . Then f has a relative minimum at

Accepted Answer

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

Question 10

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

Accepted Answer

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

Question 11

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

Accepted Answer

A)  $4 \sqrt { 13 }$ 
B)  $3 \sqrt { 13 }$ 
C)  $2 \sqrt { 13 }$ 
D)  $\sqrt { 13 }$ 
E) 13 
A)  $4 \sqrt { 13 }$ 
B)  $3 \sqrt { 13 }$ 
C)  $2 \sqrt { 13 }$ 
D)  $\sqrt { 13 }$ 
E) 13

Question 12

The symmetric equations of the normal line to the surface $16 z = 4 x ^ { 2 } + y ^ { 2 }$ at (2, 4, 2) are

Accepted Answer

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

Question 13

The absolute maximum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Accepted Answer

A) 5 
B) 4 
C) 3 
D) 2 
E) 1 
A) 5 
B) 4 
C) 3 
D) 2 
E) 1

Question 14

Let $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( - 1,2 )$ is

Accepted Answer

A) 3 
B)  $\frac { 3 \sqrt { 5 } } { 25 }$ 
C)  $\frac { 3 } { 5 }$ 
D)  $\frac { 1 } { 3 }$ 
E) 5 
A) 3 
B)  $\frac { 3 \sqrt { 5 } } { 25 }$ 
C)  $\frac { 3 } { 5 }$ 
D)  $\frac { 1 } { 3 }$ 
E) 5

Question 15

Let $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 3,4,0 )$ is

Accepted Answer

A) 5 
B) 3 
C) 2 
D) 1 
E)  $\frac { 1 } { 3 }$ 
A) 5 
B) 3 
C) 2 
D) 1 
E)  $\frac { 1 } { 3 }$

Question 16

Let $f ( x , y ) = x y$ with constraint $3 x ^ { 2 } + y ^ { 2 } = 6$ . Then f has a relative minimum at

Accepted Answer

A)  $( 1 , \sqrt { 3 } )$ 
B)  $( - 1 , \sqrt { 3 } )$ 
C)  $( 1 , - \sqrt { 3 } )$ 
D)  $( \pm 1 , \pm \sqrt { 3 } ) ( \pm 1 , \mp \sqrt { 3 } )$ 
E)  $( \pm 1 , \mp \sqrt { 3 } )$ 
A)  $( 1 , \sqrt { 3 } )$ 
B)  $( - 1 , \sqrt { 3 } )$ 
C)  $( 1 , - \sqrt { 3 } )$ 
D)  $( \pm 1 , \pm \sqrt { 3 } ) ( \pm 1 , \mp \sqrt { 3 } )$ 
E)  $( \pm 1 , \mp \sqrt { 3 } )$

Question 17

The absolute minimum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Accepted Answer

A) 1137 
B) -61 
C) -5 
D) 4 
E) 1 
A) 1137 
B) -61 
C) -5 
D) 4 
E) 1

Question 18

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint x = 1. Then f has a relative minimum at

Accepted Answer

A)  $\left( 1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$ 
B)  $\left( 1 , \frac { 1 } { 2 } , - \frac { 1 } { 4 } \right)$ 
C)  $( 1,0,1 )$ 
D)  $\left( 1 , - \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$ 
E)  $( 1,0,0 )$ 
A)  $\left( 1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$ 
B)  $\left( 1 , \frac { 1 } { 2 } , - \frac { 1 } { 4 } \right)$ 
C)  $( 1,0,1 )$ 
D)  $\left( 1 , - \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$ 
E)  $( 1,0,0 )$

Question 19

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

Accepted Answer

A)  $\frac { x - 3 } { - 3 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { - 4 }$ 
B)  $\frac { x - 3 } { 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { 4 }$ 
C)  $\frac { x - 3 } { 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { - 4 }$ 
D)  $\frac { x - 3 } { - 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { 4 }$ 
E)  $\frac { x - 3 } { 3 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 4 }$ 
A)  $\frac { x - 3 } { - 3 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { - 4 }$ 
B)  $\frac { x - 3 } { 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { 4 }$ 
C)  $\frac { x - 3 } { 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { - 4 }$ 
D)  $\frac { x - 3 } { - 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { 4 }$ 
E)  $\frac { x - 3 } { 3 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 4 }$

Question 20

The symmetric equations of the normal line to the surface $3 z = x ^ { 2 } + y ^ { 2 } - 2$ at (-2, -4, 6) are

Accepted Answer

A)  $\frac { x + 2 } { 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { 3 }$ 
B)  $\frac { x + 2 } { - 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { 3 }$ 
C)  $\frac { x + 2 } { 4 } = \frac { y + 4 } { - 8 } = \frac { z - 6 } { 3 }$ 
D)  $\frac { x + 2 } { 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { - 3 }$ 
E)  $\frac { x + 2 } { - 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { - 3 }$ 
A)  $\frac { x + 2 } { 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { 3 }$ 
B)  $\frac { x + 2 } { - 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { 3 }$ 
C)  $\frac { x + 2 } { 4 } = \frac { y + 4 } { - 8 } = \frac { z - 6 } { 3 }$ 
D)  $\frac { x + 2 } { 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { - 3 }$ 
E)  $\frac { x + 2 } { - 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { - 3 }$

The set of points on $z = x ^ { 2 } - 3 x y + y ^ { 2 }$ at which the tangent plane is parallel to the xy-plane is

Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative maximum at

The gradient of $f ( x , y , z ) = y ^ { 2 } + z ^ { 2 } - 4 x z$ at (-2, 1, 3) is

The set of points on $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 50$ at which the tangent plane is parallel to the xy-plane is

Let $f ( x , y ) = 6 x - 24 y - x ^ { 2 } + 2 y ^ { 3 }$ . Then f has a relative maximum at

The gradient of $f ( x , y , z ) = z \tan ^ { - 1 } \left( \frac { y } { x } \right)$ at (1, 1, 3) is

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $x ^ { 2 } + 4 y ^ { 2 } = 16$ . Then the minimum value of w is

An equation of the tangent plane to the surface $y = e ^ { x } \cos z$ at (1, e, 0) is

Let $f ( x , y ) = x ^ { 3 } + y ^ { 2 } - 6 x ^ { 2 } + y - 1$ . Then f has a relative minimum at

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

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

The symmetric equations of the normal line to the surface $16 z = 4 x ^ { 2 } + y ^ { 2 }$ at (2, 4, 2) are

The absolute maximum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Let $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( - 1,2 )$ is

Let $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 3,4,0 )$ is

Let $f ( x , y ) = x y$ with constraint $3 x ^ { 2 } + y ^ { 2 } = 6$ . Then f has a relative minimum at

The absolute minimum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint x = 1. Then f has a relative minimum at

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

The symmetric equations of the normal line to the surface $3 z = x ^ { 2 } + y ^ { 2 } - 2$ at (-2, -4, 6) 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

The set of points on z=x2−3xy+y2z = x ^ { 2 } - 3 x y + y ^ { 2 }z=x2−3xy+y2 at which the tangent plane is parallel to the xy-plane is

Let f(x,y)=sin⁡(x+y)+sin⁡x+sin⁡yf ( x , y ) = \sin ( x + y ) + \sin x + \sin yf(x,y)=sin(x+y)+sinx+siny . Then f has a relative maximum at

The gradient of f(x,y,z)=y2+z2−4xzf ( x , y , z ) = y ^ { 2 } + z ^ { 2 } - 4 x zf(x,y,z)=y2+z2−4xz at (-2, 1, 3) is

The set of points on 4x2+y2+2z2=504 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 504x2+y2+2z2=50 at which the tangent plane is parallel to the xy-plane is

Let f(x,y)=6x−24y−x2+2y3f ( x , y ) = 6 x - 24 y - x ^ { 2 } + 2 y ^ { 3 }f(x,y)=6x−24y−x2+2y3 . Then f has a relative maximum at

The gradient of f(x,y,z)=ztan⁡−1(yx)f ( x , y , z ) = z \tan ^ { - 1 } \left( \frac { y } { x } \right)f(x,y,z)=ztan−1(xy​) at (1, 1, 3) is

Let w=x2+y2+z2w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }w=x2+y2+z2 with constraint x2+4y2=16x ^ { 2 } + 4 y ^ { 2 } = 16x2+4y2=16 . Then the minimum value of w is

An equation of the tangent plane to the surface y=excos⁡zy = e ^ { x } \cos zy=excosz at (1, e, 0) is

Let f(x,y)=x3+y2−6x2+y−1f ( x , y ) = x ^ { 3 } + y ^ { 2 } - 6 x ^ { 2 } + y - 1f(x,y)=x3+y2−6x2+y−1 . Then f has a relative minimum at

Let f(x,y,z)=x2+4y2+16z2f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }f(x,y,z)=x2+4y2+16z2 with constraint xyz = 1. Then f has a relative minimum at

Let f(x,y)=4xy+x2+2y2f ( x , y ) = 4 x y + x ^ { 2 } + 2 y ^ { 2 }f(x,y)=4xy+x2+2y2 . Then the maximum value of the directional derivative Duf(2,1)D _ { \mathrm { u } } f ( 2,1 )Du​f(2,1) is

The symmetric equations of the normal line to the surface 16z=4x2+y216 z = 4 x ^ { 2 } + y ^ { 2 }16z=4x2+y2 at (2, 4, 2) are

The absolute maximum of f(x,y)=2x2+y2−4x−4y+1f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1f(x,y)=2x2+y2−4x−4y+1 on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Let f(x,y)=xyx2+y2f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }f(x,y)=x2+y2xy​ . Then the maximum value of the directional derivative Duf(−1,2)D _ { \mathrm { u } } f ( - 1,2 )Du​f(−1,2) is

Let f(x,y,z)=x2+y2+z2f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }f(x,y,z)=x2+y2+z2​ . Then the maximum value of the directional derivative Duf(3,4,0)D _ { \mathrm { u } } f ( 3,4,0 )Du​f(3,4,0) is

Let f(x,y)=xyf ( x , y ) = x yf(x,y)=xy with constraint 3x2+y2=63 x ^ { 2 } + y ^ { 2 } = 63x2+y2=6 . Then f has a relative minimum at

The absolute minimum of f(x,y)=2x2+y2−4x−4y+1f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1f(x,y)=2x2+y2−4x−4y+1 on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Let f(x,y,z)=x2+4y2+16z2f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }f(x,y,z)=x2+4y2+16z2 with constraint x = 1. Then f has a relative minimum at

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

The symmetric equations of the normal line to the surface 3z=x2+y2−23 z = x ^ { 2 } + y ^ { 2 } - 23z=x2+y2−2 at (-2, -4, 6) 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

The set of points on $z = x ^ { 2 } - 3 x y + y ^ { 2 }$ at which the tangent plane is parallel to the xy-plane is

Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative maximum at

The gradient of $f ( x , y , z ) = y ^ { 2 } + z ^ { 2 } - 4 x z$ at (-2, 1, 3) is

The set of points on $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 50$ at which the tangent plane is parallel to the xy-plane is

Let $f ( x , y ) = 6 x - 24 y - x ^ { 2 } + 2 y ^ { 3 }$ . Then f has a relative maximum at

The gradient of $f ( x , y , z ) = z \tan ^ { - 1 } \left( \frac { y } { x } \right)$ at (1, 1, 3) is

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $x ^ { 2 } + 4 y ^ { 2 } = 16$ . Then the minimum value of w is

An equation of the tangent plane to the surface $y = e ^ { x } \cos z$ at (1, e, 0) is

Let $f ( x , y ) = x ^ { 3 } + y ^ { 2 } - 6 x ^ { 2 } + y - 1$ . Then f has a relative minimum at

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

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

The symmetric equations of the normal line to the surface $16 z = 4 x ^ { 2 } + y ^ { 2 }$ at (2, 4, 2) are

The absolute maximum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Let $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( - 1,2 )$ is

Let $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 3,4,0 )$ is

Let $f ( x , y ) = x y$ with constraint $3 x ^ { 2 } + y ^ { 2 } = 6$ . Then f has a relative minimum at

The absolute minimum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint x = 1. Then f has a relative minimum at

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

The symmetric equations of the normal line to the surface $3 z = x ^ { 2 } + y ^ { 2 } - 2$ at (-2, -4, 6) are