Question 1

Find the maximum value of the function $f ( x , y ) = 6 - 4 x ^ { 2 } - y ^ { 2 }$ subject to the constraint that $4 x + y = 5$ .

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { \sqrt { 2 } }$ 
C) 1 
D)  $\sqrt { 2 }$ 
E) 2
F) $2 \sqrt { 2 }$
G)4  
H) $4 \sqrt { 2 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { \sqrt { 2 } }$ 
C) 1 
D)  $\sqrt { 2 }$ 
E) 2
F) $2 \sqrt { 2 }$
G)4  
H) $4 \sqrt { 2 }$

Question 2

Find the minimum value of the function $f ( x , y ) = x y$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } = 2$ .

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B) 1 
C)  $\frac { 3 } { 2 }$ 
D) .2 
E)  $- \frac { 1 } { 2 }$ 
F) $- 1$  
G) $- \frac { 3 } { 2 }$
H) $- 2$ 
A)  $\frac { 1 } { 2 }$ 
B) 1 
C)  $\frac { 3 } { 2 }$ 
D) .2 
E)  $- \frac { 1 } { 2 }$ 
F) $- 1$  
G) $- \frac { 3 } { 2 }$
H) $- 2$

Question 3

Let $f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }$ . Find the value of $f _ { x y } - f _ { y x }$ at the point $( 1,2 )$ .

Accepted Answer

A)  $- 32$ 
B)  $- 16$ 
C)  $- 8$ 
D)  $- 4$ 
E) 0
F)4
G)8
H)16 
A)  $- 32$ 
B)  $- 16$ 
C)  $- 8$ 
D)  $- 4$ 
E) 0
F)4
G)8
H)16

Question 4

Find an equation of the tangent plane to the surface with parametric equations $x = 1 + 2 u + 3 v$ , $y = 5 - u + 4 v$ , $z = 3 + 5 u - 7 v$ at the point $( 3,4,8 )$ .

Accepted Answer

The answer of Find an equation of the tangent plane...

Question 5

Let $z = \ln \left( 2 x ^ { 2 } + y \right)$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 1 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3 \text {, and } y ^ { \prime } ( 1 ) = 4$ . Find $d z / d t$ when $t = 1$ .

Accepted Answer

A) 3 
B) 4 
C) 6 
D) 8 
E) 9
F)10
G)12
H)16 
A) 3 
B) 4 
C) 6 
D) 8 
E) 9
F)10
G)12
H)16

Question 6

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)Does not exist 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)Does not exist

Question 7

If $f ( x , y ) = 4 - x ^ { 2 }$ , find $f _ { x } ( 1,1 )$ and $f _ { y } ( 1,1 )$ and interpret these numbers as slopes. Illustrate with sketches.

Accepted Answer

The answer of If \(f ( x , y )...

Question 8

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } y$ at the point $( - 1,1 )$ in the direction $( 3,4 )$ .

Accepted Answer

A)  $\frac { 2 } { 5 }$ 
B)  $\frac { 6 } { 5 }$ 
C) 2 
D) 6 
E)  $- \frac { 2 } { 5 }$ 
F) $- \frac { 6 } { 5 }$ 
G) $- 2$ 
H) $- 6$ 
A)  $\frac { 2 } { 5 }$ 
B)  $\frac { 6 } { 5 }$ 
C) 2 
D) 6 
E)  $- \frac { 2 } { 5 }$ 
F) $- \frac { 6 } { 5 }$ 
G) $- 2$ 
H) $- 6$

Question 9

Find the maximum value of the function $f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160$

Accepted Answer

A) 90 
B) 160 
C) 50 
D) 15 
E) 80
F)135
G)16
H)5 
A) 90 
B) 160 
C) 50 
D) 15 
E) 80
F)135
G)16
H)5

Question 10

Let $f ( x , y , z ) = x e ^ { y \sin z }$ . Find the gradient vector $\nabla f ( 1,1,0 )$ at the point $( x , y , z ) = ( 1,1,0 )$ .

Accepted Answer

A) i 
B) j 
C) k 
D)  $\mathbf { j } + \mathbf { k }$ 
E)  $\mathbf { i } + \mathbf { k }$ 
F) $\mathbf { i } + \mathbf { j }$ 
G) $\mathbf { i } + \mathbf { j } + \mathbf { k }$  
H)0 
A) i 
B) j 
C) k 
D)  $\mathbf { j } + \mathbf { k }$ 
E)  $\mathbf { i } + \mathbf { k }$ 
F) $\mathbf { i } + \mathbf { j }$ 
G) $\mathbf { i } + \mathbf { j } + \mathbf { k }$  
H)0

Question 11

Find the largest value of the directional derivative of the function $f ( x , y ) = \frac { y } { x + y }$ at the point $( x , y ) = ( 1,2 )$ .

Accepted Answer

A)  $\sqrt { 5 }$ 
B)  $\frac { 5 } { 9 }$ 
C)  $- \sqrt { 5 }$ 
D)  $- \frac { 5 } { 9 }$ 
E)  $\frac { \sqrt { 5 } } { 3 }$ 
F) $\frac { \sqrt { 5 } } { 9 }$

G) $- \frac { \sqrt { 5 } } { 3 }$

H) $- \frac { \sqrt { 5 } } { 9 }$ 
A)  $\sqrt { 5 }$ 
B)  $\frac { 5 } { 9 }$ 
C)  $- \sqrt { 5 }$ 
D)  $- \frac { 5 } { 9 }$ 
E)  $\frac { \sqrt { 5 } } { 3 }$ 
F) $\frac { \sqrt { 5 } } { 9 }$

G) $- \frac { \sqrt { 5 } } { 3 }$

H) $- \frac { \sqrt { 5 } } { 9 }$

Question 12

Find the absolute maximum and minimum values of $f ( x , y ) = x ^ { 2 } + 2 x y + 3 y ^ { 2 }$ over the region $D$ , where $D$ is a closed triangular region with vertices $( - 1 , - 2 )$ , $( - 1,1 )$ , and $( 2,1 )$ .

Accepted Answer

Absolute m

Question 13

Determine whether $f ( x , y ) = \left\{ \begin{array} { l l } 
\frac { 3 x ^ { 2 } - 2 y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \
0 & \text { if } ( x , y ) = ( 0,0 )
\end{array} \right.$ is continuous at $( 0,0 )$ .

Accepted Answer

The functi

Question 14

A boundary stripe 3 inches wide is painted around a rectangle whose dimensions are 100 feet by 200 feet. Use differentials to approximate the number of square feet of paint in the stripe.

Accepted Answer

A) 120 
B) 130 
C) 140 
D) 150 
E) 160
F)170
G)180
H)190 
A) 120 
B) 130 
C) 140 
D) 150 
E) 160
F)170
G)180
H)190

Question 15

Let $z = x ^ { 2 } + x y + y ^ { 2 }$ , $x = 2 u + v$ , and $y = u - 3 v$ . Show that $\frac { \delta ^ { 2 } z } { \delta u \delta v } = - 7$

Accepted Answer

To find the second mixed partial derivat

Question 16

Find the point at which the function $f ( x , y , z ) = 2 x + y - 2 z$ has the maximum value subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Accepted Answer

A)  $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
B) 
$\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$ 
C)  $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
D)  $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$ 
E)  $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
F) $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

G) $\left( - \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

H)None of these 
A)  $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
B) 
$\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$ 
C)  $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
D)  $\left( \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$ 
E)  $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , \frac { 4 } { 3 } \right)$ 
F) $\left( - \frac { 4 } { 3 } , - \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

G) $\left( - \frac { 4 } { 3 } , \frac { 2 } { 3 } , - \frac { 4 } { 3 } \right)$

H)None of these

Question 17

Find an equation of the tangent plant to the parametric surface $x = u - v , y = u + v , z = u ^ { 2 }$ at the point $( 0,2,1 )$ .

Accepted Answer

A)  $x + y + z = 3$ 
B)  $x + y - z = 1$ 
C)  $x + y + 2 z = 4$ 
D)  $x + y - 2 z = 0$ 
E)  $x + 2 y + z = 5$ 
F) $x - 2 y + z = - 3$

G) $2 x + y + z = 3$

H) $2 x + y - z = 1$ 
A)  $x + y + z = 3$ 
B)  $x + y - z = 1$ 
C)  $x + y + 2 z = 4$ 
D)  $x + y - 2 z = 0$ 
E)  $x + 2 y + z = 5$ 
F) $x - 2 y + z = - 3$

G) $2 x + y + z = 3$

H) $2 x + y - z = 1$

Question 18

Let $f ( x , y , z ) = x ^ { 2 } y - y z ^ { 3 } + z$ and $P ( 1 , - 2,0 )$ . Calculate $D _ { u } f ( P )$ where $\mathbf { u }$ is a unit vector making an angle $\theta = 45 ^ { \circ }$ with $\nabla f ( P )$ .

Accepted Answer

The answer of Let \(f ( x , y ,...

Question 19

Find an equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at the point $( 4,1,1 )$ .

Accepted Answer

A)  $2 x + y - z = 1$ 
B)  $x + 2 y + 3 z = 9$ 
C)  $x - 2 y + 4 z = 0$ 
D)  $4 x - y + 6 z = 1$ 
E)  $x + 2 y + 2 z = 8$ 
F) $3 x + 2 y + z = 1$

G) $x + y + z = 6$

H) $2 x + y + z = 10$ 
A)  $2 x + y - z = 1$ 
B)  $x + 2 y + 3 z = 9$ 
C)  $x - 2 y + 4 z = 0$ 
D)  $4 x - y + 6 z = 1$ 
E)  $x + 2 y + 2 z = 8$ 
F) $3 x + 2 y + z = 1$

G) $x + y + z = 6$

H) $2 x + y + z = 10$

Question 20

Find the maximum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Accepted Answer

A)  $- 4$ 
B)  $- 5$ 
C)  $- 6$ 
D)  $- 7$ 
E) 4
F)5
G)6
H)7 
A)  $- 4$ 
B)  $- 5$ 
C)  $- 6$ 
D)  $- 7$ 
E) 4
F)5
G)6
H)7

Find the maximum value of the function $f ( x , y ) = 6 - 4 x ^ { 2 } - y ^ { 2 }$ subject to the constraint that $4 x + y = 5$ .

Find the minimum value of the function $f ( x , y ) = x y$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } = 2$ .

Let $f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }$ . Find the value of $f _ { x y } - f _ { y x }$ at the point $( 1,2 )$ .

Find an equation of the tangent plane to the surface with parametric equations $x = 1 + 2 u + 3 v$ , $y = 5 - u + 4 v$ , $z = 3 + 5 u - 7 v$ at the point $( 3,4,8 )$ .

Let $z = \ln \left( 2 x ^ { 2 } + y \right)$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 1 , y ( 1 ) = 2 , x ^ { \prime } ( 1 ) = 3 \text {, and } y ^ { \prime } ( 1 ) = 4$ . Find $d z / d t$ when $t = 1$ .

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$

If $f ( x , y ) = 4 - x ^ { 2 }$ , find $f _ { x } ( 1,1 )$ and $f _ { y } ( 1,1 )$ and interpret these numbers as slopes. Illustrate with sketches.

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } y$ at the point $( - 1,1 )$ in the direction $( 3,4 )$ .

Find the maximum value of the function $f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160$

Let $f ( x , y , z ) = x e ^ { y \sin z }$ . Find the gradient vector $\nabla f ( 1,1,0 )$ at the point $( x , y , z ) = ( 1,1,0 )$ .

Find the largest value of the directional derivative of the function $f ( x , y ) = \frac { y } { x + y }$ at the point $( x , y ) = ( 1,2 )$ .

Find the absolute maximum and minimum values of $f ( x , y ) = x ^ { 2 } + 2 x y + 3 y ^ { 2 }$ over the region $D$ , where $D$ is a closed triangular region with vertices $( - 1 , - 2 )$ , $( - 1,1 )$ , and $( 2,1 )$ .

Determine whether $f ( x , y ) = \left\{ \begin{array} { l l } \frac { 3 x ^ { 2 } - 2 y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\0 & \text { if } ( x , y ) = ( 0,0 )\end{array} \right.$ is continuous at $( 0,0 )$ .

A boundary stripe 3 inches wide is painted around a rectangle whose dimensions are 100 feet by 200 feet. Use differentials to approximate the number of square feet of paint in the stripe.

Let $z = x ^ { 2 } + x y + y ^ { 2 }$ , $x = 2 u + v$ , and $y = u - 3 v$ . Show that $\frac { \delta ^ { 2 } z } { \delta u \delta v } = - 7$

Find the point at which the function $f ( x , y , z ) = 2 x + y - 2 z$ has the maximum value subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Find an equation of the tangent plant to the parametric surface $x = u - v , y = u + v , z = u ^ { 2 }$ at the point $( 0,2,1 )$ .

Let $f ( x , y , z ) = x ^ { 2 } y - y z ^ { 3 } + z$ and $P ( 1 , - 2,0 )$ . Calculate $D _ { u } f ( P )$ where $\mathbf { u }$ is a unit vector making an angle $\theta = 45 ^ { \circ }$ with $\nabla f ( P )$ .

Find an equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at the point $( 4,1,1 )$ .

Find the maximum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

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Exam 11: Partial Derivatives

Find the maximum value of the function f(x,y)=6−4x2−y2f ( x , y ) = 6 - 4 x ^ { 2 } - y ^ { 2 }f(x,y)=6−4x2−y2 subject to the constraint that 4x+y=54 x + y = 54x+y=5 .

Find the minimum value of the function f(x,y)=xyf ( x , y ) = x yf(x,y)=xy subject to the constraint that x2+y2=2x ^ { 2 } + y ^ { 2 } = 2x2+y2=2 .

Let f(x,y)=(x3+y4)5f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }f(x,y)=(x3+y4)5 . Find the value of fxy−fyxf _ { x y } - f _ { y x }fxy​−fyx​ at the point (1,2)( 1,2 )(1,2) .

Find an equation of the tangent plane to the surface with parametric equations x=1+2u+3vx = 1 + 2 u + 3 vx=1+2u+3v , y=5−u+4vy = 5 - u + 4 vy=5−u+4v , z=3+5u−7vz = 3 + 5 u - 7 vz=3+5u−7v at the point (3,4,8)( 3,4,8 )(3,4,8) .

Evaluate lim⁡(x,y)→(0,0)sin⁡(x2+y2)x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2sin(x2+y2)​

If f(x,y)=4−x2f ( x , y ) = 4 - x ^ { 2 }f(x,y)=4−x2 , find fx(1,1)f _ { x } ( 1,1 )fx​(1,1) and fy(1,1)f _ { y } ( 1,1 )fy​(1,1) and interpret these numbers as slopes. Illustrate with sketches.

Find the second directional derivative of f(x,y)=x2yf ( x , y ) = x ^ { 2 } yf(x,y)=x2y at the point (−1,1)( - 1,1 )(−1,1) in the direction (3,4)( 3,4 )(3,4) .

Find the maximum value of the function f(x,y)=10x+30y−x2−y2−160f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160f(x,y)=10x+30y−x2−y2−160

Let f(x,y,z)=xeysin⁡zf ( x , y , z ) = x e ^ { y \sin z }f(x,y,z)=xeysinz . Find the gradient vector ∇f(1,1,0)\nabla f ( 1,1,0 )∇f(1,1,0) at the point (x,y,z)=(1,1,0)( x , y , z ) = ( 1,1,0 )(x,y,z)=(1,1,0) .

Find the largest value of the directional derivative of the function f(x,y)=yx+yf ( x , y ) = \frac { y } { x + y }f(x,y)=x+yy​ at the point (x,y)=(1,2)( x , y ) = ( 1,2 )(x,y)=(1,2) .

Find the absolute maximum and minimum values of f(x,y)=x2+2xy+3y2f ( x , y ) = x ^ { 2 } + 2 x y + 3 y ^ { 2 }f(x,y)=x2+2xy+3y2 over the region DDD , where DDD is a closed triangular region with vertices (−1,−2)( - 1 , - 2 )(−1,−2) , (−1,1)( - 1,1 )(−1,1) , and (2,1)( 2,1 )(2,1) .

A boundary stripe 3 inches wide is painted around a rectangle whose dimensions are 100 feet by 200 feet. Use differentials to approximate the number of square feet of paint in the stripe.

Let z=x2+xy+y2z = x ^ { 2 } + x y + y ^ { 2 }z=x2+xy+y2 , x=2u+vx = 2 u + vx=2u+v , and y=u−3vy = u - 3 vy=u−3v . Show that δ2zδuδv=−7\frac { \delta ^ { 2 } z } { \delta u \delta v } = - 7δuδvδ2z​=−7

Find the point at which the function f(x,y,z)=2x+y−2zf ( x , y , z ) = 2 x + y - 2 zf(x,y,z)=2x+y−2z has the maximum value subject to the constraint that x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4 .

Find an equation of the tangent plant to the parametric surface x=u−v,y=u+v,z=u2x = u - v , y = u + v , z = u ^ { 2 }x=u−v,y=u+v,z=u2 at the point (0,2,1)( 0,2,1 )(0,2,1) .

Find an equation of the tangent plane to the surface x+y+z=4\sqrt { x } + \sqrt { y } + \sqrt { z } = 4x​+y​+z​=4 at the point (4,1,1)( 4,1,1 )(4,1,1) .

Find the maximum value of the function f(x,y,z)=2x+y−2zf ( x , y , z ) = 2 x + y - 2 zf(x,y,z)=2x+y−2z subject to the constraint that x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4 .

Functions and Models

Limits and Derivatives

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Find the maximum value of the function $f ( x , y ) = 6 - 4 x ^ { 2 } - y ^ { 2 }$ subject to the constraint that $4 x + y = 5$ .

Find the minimum value of the function $f ( x , y ) = x y$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } = 2$ .

Let $f ( x , y ) = \left( x ^ { 3 } + y ^ { 4 } \right) ^ { 5 }$ . Find the value of $f _ { x y } - f _ { y x }$ at the point $( 1,2 )$ .

Find an equation of the tangent plane to the surface with parametric equations $x = 1 + 2 u + 3 v$ , $y = 5 - u + 4 v$ , $z = 3 + 5 u - 7 v$ at the point $( 3,4,8 )$ .

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$

If $f ( x , y ) = 4 - x ^ { 2 }$ , find $f _ { x } ( 1,1 )$ and $f _ { y } ( 1,1 )$ and interpret these numbers as slopes. Illustrate with sketches.

Find the second directional derivative of $f ( x , y ) = x ^ { 2 } y$ at the point $( - 1,1 )$ in the direction $( 3,4 )$ .

Find the maximum value of the function $f ( x , y ) = 10 x + 30 y - x ^ { 2 } - y ^ { 2 } - 160$

Let $f ( x , y , z ) = x e ^ { y \sin z }$ . Find the gradient vector $\nabla f ( 1,1,0 )$ at the point $( x , y , z ) = ( 1,1,0 )$ .

Find the largest value of the directional derivative of the function $f ( x , y ) = \frac { y } { x + y }$ at the point $( x , y ) = ( 1,2 )$ .

Find the absolute maximum and minimum values of $f ( x , y ) = x ^ { 2 } + 2 x y + 3 y ^ { 2 }$ over the region $D$ , where $D$ is a closed triangular region with vertices $( - 1 , - 2 )$ , $( - 1,1 )$ , and $( 2,1 )$ .

Let $z = x ^ { 2 } + x y + y ^ { 2 }$ , $x = 2 u + v$ , and $y = u - 3 v$ . Show that $\frac { \delta ^ { 2 } z } { \delta u \delta v } = - 7$

Find the point at which the function $f ( x , y , z ) = 2 x + y - 2 z$ has the maximum value subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Find an equation of the tangent plant to the parametric surface $x = u - v , y = u + v , z = u ^ { 2 }$ at the point $( 0,2,1 )$ .

Find an equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at the point $( 4,1,1 )$ .

Find the maximum value of the function $f ( x , y , z ) = 2 x + y - 2 z$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .