Question 1

Find all the local maxima, local minima, and saddle points of the function.
-$$f ( x , y ) = x ^ { 2 } + 14 x + y ^ { 2 } - 4 y + 3$$

Accepted Answer

A)  f(7, -2) = 162, local maximum 
B)  f(-7, 2) = -50, local minimum 
C)  f(-7, -2) = -34, saddle point 
D)  f(7, 2) = 146, saddle point 
A)  f(7, -2) = 162, local maximum 
B)  f(-7, 2) = -50, local minimum 
C)  f(-7, -2) = -34, saddle point 
D)  f(7, 2) = 146, saddle point

Question 2

Solve the problem.
-$\text { Find the maximum value of } f ( x , y , z , w ) = x + y + z + w \text { subject to } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + w ^ { 2 } = 1 \text {. }$

Accepted Answer

A)  10 
B)  4 
C)  2 
D)  1 
A)  10 
B)  4 
C)  2 
D)  1

Question 3

Solve the problem.
-Find the derivative of the function $f ( x , y ) = e ^ { x y }$ at the point $( 0,2 )$ in the direction in which the function increases most rapidly.

Accepted Answer

A)  2 
B)  1 
C)  4 
D)  6 
A)  2 
B)  1 
C)  4 
D)  6

Question 4

Find the absolute maxima and minima of the function on the given domain.
-$$f ( x , y ) = 4 x ^ { 2 } + 7 y ^ { 2 } \text { on the disk bounded by the circle } x ^ { 2 } + y ^ { 2 } = 9$$

Accepted Answer

A)  Absolute maximum: 36 at (3, 0) and (-3, 0); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 99 at (3, 3); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 63 at (0, 3) and (0, -3); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 63 at (0, 3) and (0, -3); absolute minimum: 36 at (3, 0) and (-3, 0) 
A)  Absolute maximum: 36 at (3, 0) and (-3, 0); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 99 at (3, 3); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 63 at (0, 3) and (0, -3); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 63 at (0, 3) and (0, -3); absolute minimum: 36 at (3, 0) and (-3, 0)

Question 5

Find the absolute maxima and minima of the function on the given domain.
-$$f ( x , y ) = x ^ { 2 } + y ^ { 2 } \text { on the diamond-shaped region } | x | + | y | \leq 4$$

Accepted Answer

A)  Absolute maximum: 16 at (4, 0), (-4, 0), (0, 4), and (0, -4); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 8 at 2, 2 , 2, - 2 , - 2, 2 , and - 2, - 2 ; absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 16 at (4, 0) and (0, 4); absolute minimum: 0 at (-4, 0) and (0, -4) 
D)  Absolute maximum: 16 at (4, 0), (-4, 0), (0, 4), and (0, -4); absolute minimum: 8 at 2, 2 , 2, - 2 , - 2, 2 , and - 2, - 2 , 
A)  Absolute maximum: 16 at (4, 0), (-4, 0), (0, 4), and (0, -4); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 8 at 2, 2 , 2, - 2 , - 2, 2 , and - 2, - 2 ; absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 16 at (4, 0) and (0, 4); absolute minimum: 0 at (-4, 0) and (0, -4) 
D)  Absolute maximum: 16 at (4, 0), (-4, 0), (0, 4), and (0, -4); absolute minimum: 8 at 2, 2 , 2, - 2 , - 2, 2 , and - 2, - 2 ,

Question 6

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$
-$$f ( x , y , z ) = \ln ( x y ) ^ { z }$$

Accepted Answer

A)  $\mathrm { f } _ { \mathrm { X } } = \mathrm { z } \ln \frac { \mathrm { z } } { \mathrm { x } } ; \mathrm { f } _ { \mathrm { y } } = \mathrm { z } \ln \frac { \mathrm { z } } { \mathrm { y } } ; \mathrm { f } _ { \mathrm { Z } } = \ln ( \mathrm { xy } )$ 
B)  $f _ { X } = - \frac { z } { x } ; f _ { y } = - \frac { z } { y } ; f _ { z } = \ln ( x y )$ 
C)  $\mathrm { f } _ { \mathrm { X } } = \frac { \mathrm { z } } { \mathrm { x } } ; \mathrm { f } _ { \mathrm { y } } = \frac { \mathrm { z } } { \mathrm { y } } ; \mathrm { f } _ { \mathrm { Z } } = \mathrm { z } \ln ( \mathrm { xy } ) ^ { \mathrm { z } - 1 }$ 
D)  $\mathrm { f } _ { \mathrm { X } } = \frac { \mathrm { z } } { \mathrm { x } } ; \mathrm { f } _ { \mathrm { y } } = \frac { \mathrm { z } } { \mathrm { y } } ; \mathrm { f } _ { \mathrm { Z } } = \ln ( \mathrm { xy } )$ 
A)  $\mathrm { f } _ { \mathrm { X } } = \mathrm { z } \ln \frac { \mathrm { z } } { \mathrm { x } } ; \mathrm { f } _ { \mathrm { y } } = \mathrm { z } \ln \frac { \mathrm { z } } { \mathrm { y } } ; \mathrm { f } _ { \mathrm { Z } } = \ln ( \mathrm { xy } )$ 
B)  $f _ { X } = - \frac { z } { x } ; f _ { y } = - \frac { z } { y } ; f _ { z } = \ln ( x y )$ 
C)  $\mathrm { f } _ { \mathrm { X } } = \frac { \mathrm { z } } { \mathrm { x } } ; \mathrm { f } _ { \mathrm { y } } = \frac { \mathrm { z } } { \mathrm { y } } ; \mathrm { f } _ { \mathrm { Z } } = \mathrm { z } \ln ( \mathrm { xy } ) ^ { \mathrm { z } - 1 }$ 
D)  $\mathrm { f } _ { \mathrm { X } } = \frac { \mathrm { z } } { \mathrm { x } } ; \mathrm { f } _ { \mathrm { y } } = \frac { \mathrm { z } } { \mathrm { y } } ; \mathrm { f } _ { \mathrm { Z } } = \ln ( \mathrm { xy } )$

Question 7

Find the requested partial derivative.
-$\frac { \partial z } { \partial y }$ at $( x , y , z ) = ( 1,1,1 )$ if $z ^ { 3 } = z + x y - 1$ and $y ^ { 3 } = x + y - 1$

Accepted Answer

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

Question 8

Give an appropriate answer.
-$$f ( x , y ) = \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = - \left( \frac { 1 } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = - \left( \frac { 1 } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ 
B)  $\frac { \partial f } { \partial x } = \left( \frac { x } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = \left( \frac { y } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ 
C)  $\frac { \partial f } { \partial x } = - \left( \frac { x } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = - \left( \frac { y } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ 
D)  $\frac { \partial f } { \partial x } = - \left( \frac { x } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = - \left( \frac { y } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ 
A)  $\frac { \partial f } { \partial x } = - \left( \frac { 1 } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = - \left( \frac { 1 } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ 
B)  $\frac { \partial f } { \partial x } = \left( \frac { x } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = \left( \frac { y } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ 
C)  $\frac { \partial f } { \partial x } = - \left( \frac { x } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = - \left( \frac { y } { 2 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$ 
D)  $\frac { \partial f } { \partial x } = - \left( \frac { x } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right) ; \frac { \partial f } { \partial y } = - \left( \frac { y } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } } \right)$

Question 9

Solve the problem.
-The resistance $\mathrm { R }$ produced by wiring resistors of $\mathrm { R } _ { 1 } , \mathrm { R } _ { 2 }$, and $\mathrm { R } _ { 3 }$ ohms in parallel can be calculated from the formula $\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } } + \frac { 1 } { R _ { 3 } } .$
If $\mathrm { R } _ { 1 } , \mathrm { R } _ { 2 }$, and $\mathrm { R } _ { 3 }$ are measured to be 7 ohms, $5 \mathrm { ohms }$, and 6 ohms respectively, and if these measurements are accurate to within $0.05$ ohms, estimate the maximum possible error in computing $R$.

Accepted Answer

A)  0.014 
B)  0.017 
C)  0.020 
D)  0.010 
A)  0.014 
B)  0.017 
C)  0.020 
D)  0.010

Question 10

Provide an appropriate response.
-Find the extrema of f(x, y, z) = x + yz on the line defined by x = 8(6 + t), y = t - 8, and z = t + 6. Classify each extremum as a minimum or maximum. (Hint: w = f(x, y, z) is a differentiable function of t.)

Accepted Answer

A)  (24, -11, 3), minimum 
B)  (72, -5, 9), minimum; (0, -8, 0), minimum 
C)  (24, -11, 3), minimum; (72, -5, 9), maximum 
D)  (72, -5, 9), minimum 
A)  (24, -11, 3), minimum 
B)  (72, -5, 9), minimum; (0, -8, 0), minimum 
C)  (24, -11, 3), minimum; (72, -5, 9), maximum 
D)  (72, -5, 9), minimum

Question 11

Find the domain and range and describe the level curves for the function f(x,y).
-f(x, y) = ln (5x + 4y)

Accepted Answer

A)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $5 x + 4 y = c$ 
B)  Domain: all points in the xy-plane satisfying $5 x + 4 y \geq 0$; range: all real numbers; level curves: lines $5 x + 4 y = c$ 
C)  Domain: all points in the $x y$-plane satisfying $5 x + 4 y > 0$; range: all real numbers; level curves: lines $5 x + 4 y = c$ 
D)  Domain: all points in the $x y$-plane satisfying $5 x + 4 y > 0$; range: real numbers $z \geq 0$; level curves: lines $5 x + 4 y = c$ 
A)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $5 x + 4 y = c$ 
B)  Domain: all points in the xy-plane satisfying $5 x + 4 y \geq 0$; range: all real numbers; level curves: lines $5 x + 4 y = c$ 
C)  Domain: all points in the $x y$-plane satisfying $5 x + 4 y > 0$; range: all real numbers; level curves: lines $5 x + 4 y = c$ 
D)  Domain: all points in the $x y$-plane satisfying $5 x + 4 y > 0$; range: real numbers $z \geq 0$; level curves: lines $5 x + 4 y = c$

Question 12

Find the limit.
-$$\lim _ { \mathrm { P } \rightarrow ( 2,0 , - 2 ) } \mathrm { e } ^ { 2 } \sin ^ { - 1 } \left( - \frac { \mathrm { z } } { \mathrm { x } } \right)$$

Accepted Answer

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

Question 13

Find the requested partial derivative.
-$( \partial \mathrm { w } / \partial \mathrm { z } ) _ { \mathrm { x } }$ at $( \mathrm { x } , \mathrm { y } , \mathrm { z } , \mathrm { w } ) = ( 1,2,9,302 )$ if $\mathrm { w } = \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } + 12 \mathrm { xyz }$ and $\mathrm { z } = \mathrm { x } ^ { 3 } + \mathrm { y } ^ { 3 }$

Accepted Answer

A)  $\frac { 308 } { 3 }$ 
B)  $\frac { 154 } { 3 }$ 
C)  154 
D)  77 
A)  $\frac { 308 } { 3 }$ 
B)  $\frac { 154 } { 3 }$ 
C)  154 
D)  77

Question 14

Use implicit differentiation to find the specified derivative at the given point.
-Find $\frac { \partial z } { \partial y }$ at the point $( 8,1 , - 1 )$ for $\ln \left( \frac { y z } { x } \right) - e ^ { x y + z ^ { 2 } } = 0$.

Accepted Answer

A)  $\frac { 1 - 2 \mathrm { e } ^ { 9 } } { 1 - 8 \mathrm { e } ^ { 9 } }$ 
B)  $\frac { 2 \mathrm { e } ^ { 9 } - 1 } { 1 - 8 \mathrm { e } ^ { 9 } }$ 
C)  $\frac { 8 \mathrm { e } ^ { 9 } - 1 } { 1 - 2 \mathrm { e } ^ { 9 } }$ 
D)  $\frac { 1 - 8 \mathrm { e } ^ { 9 } } { 1 - 2 \mathrm { e } ^ { 9 } }$ 
A)  $\frac { 1 - 2 \mathrm { e } ^ { 9 } } { 1 - 8 \mathrm { e } ^ { 9 } }$ 
B)  $\frac { 2 \mathrm { e } ^ { 9 } - 1 } { 1 - 8 \mathrm { e } ^ { 9 } }$ 
C)  $\frac { 8 \mathrm { e } ^ { 9 } - 1 } { 1 - 2 \mathrm { e } ^ { 9 } }$ 
D)  $\frac { 1 - 8 \mathrm { e } ^ { 9 } } { 1 - 2 \mathrm { e } ^ { 9 } }$

Question 15

Use implicit differentiation to find the specified derivative at the given point.
-Find $\frac { d y } { d x }$ at the point $( 1 , - 1 )$ for $- 5 x y ^ { 2 } + 6 x ^ { 2 } y - 4 x = 0$.

Accepted Answer

A)  $- \frac { 21 } { 16 }$ 
B)  $\frac { 15 } { 11 }$ 
C)  $\frac { 21 } { 16 }$ 
D)  $- \frac { 21 } { 11 }$ 
A)  $- \frac { 21 } { 16 }$ 
B)  $\frac { 15 } { 11 }$ 
C)  $\frac { 21 } { 16 }$ 
D)  $- \frac { 21 } { 11 }$

Question 16

Find the linearization of the function at the given point.
-$$f ( x , y ) = e ^ { 4 x + 2 y } \text { at } ( 0,0 )$$

Accepted Answer

A)  L(x, y) = 2x + 4y 
B)  L(x, y) = 2x + 4y + 1 
C)  L(x, y) = 4x + 2y + 1 
D)  L(x, y) = 4x + 2y 
A)  L(x, y) = 2x + 4y 
B)  L(x, y) = 2x + 4y + 1 
C)  L(x, y) = 4x + 2y + 1 
D)  L(x, y) = 4x + 2y

Question 17

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$
-$f ( x , y , z ) = x z \sqrt { x + y }$

Accepted Answer

A)  $f _ { x } = z \left( \sqrt { x + y } - \frac { x } { 2 \sqrt { x + y } } \right) ; f _ { y } = - \frac { x y } { 2 \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$ 
B)  $f _ { x } = z \left( \sqrt { x + y } - \frac { x } { \sqrt { x + y } } \right) ; f _ { y } = - \frac { x y } { \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$ 
C)  $f _ { x } = z \left( \sqrt { x + y } + \frac { x } { \sqrt { x + y } } \right) ; f _ { y } = \frac { x z } { \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$ 
D)  $f _ { X } = z \left( \sqrt { x + y } + \frac { x } { 2 \sqrt { x + y } } \right) ; f _ { y } = \frac { x z } { 2 \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$ 
A)  $f _ { x } = z \left( \sqrt { x + y } - \frac { x } { 2 \sqrt { x + y } } \right) ; f _ { y } = - \frac { x y } { 2 \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$ 
B)  $f _ { x } = z \left( \sqrt { x + y } - \frac { x } { \sqrt { x + y } } \right) ; f _ { y } = - \frac { x y } { \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$ 
C)  $f _ { x } = z \left( \sqrt { x + y } + \frac { x } { \sqrt { x + y } } \right) ; f _ { y } = \frac { x z } { \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$ 
D)  $f _ { X } = z \left( \sqrt { x + y } + \frac { x } { 2 \sqrt { x + y } } \right) ; f _ { y } = \frac { x z } { 2 \sqrt { x + y } } ; f _ { z } = x \sqrt { x + y }$

Question 18

Find the requested partial derivative.
-$$( \partial w / \partial z ) _ { x , y } \text { at } ( x , y , z , w ) = ( 1,2,9,266 ) \text { if } w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 10 x y z$$

Accepted Answer

A)  28 
B)  33 
C)  38 
D)  48 
A)  28 
B)  33 
C)  38 
D)  48

Question 19

Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = e ^ { x + y }$$

Accepted Answer

A)  Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: lines x + y = c 
B)  Domain: all points in the xy-plane; range: real numbers z > 0; level curves: lines x + y = c 
C)  Domain: all points in the first quadrant of the xy-plane; range: real numbers z > 0; level curves: lines x + y = c 
D)  Domain: all points in the xy-plane; range: all real numbers; level curves: lines x + y = c 
A)  Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: lines x + y = c 
B)  Domain: all points in the xy-plane; range: real numbers z > 0; level curves: lines x + y = c 
C)  Domain: all points in the first quadrant of the xy-plane; range: real numbers z > 0; level curves: lines x + y = c 
D)  Domain: all points in the xy-plane; range: all real numbers; level curves: lines x + y = c

Question 20

Give an appropriate answer.
-Given the function $f ( x , y , z )$ and the positive number $\varepsilon$ as in the formal definition of a limit, find a positive number $\delta$ as in the definition that insures $| f ( x , y , z ) - f ( 0,0,0 ) | < \varepsilon$.
$\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { x } + \mathrm { y } - \mathrm { z } ; \varepsilon = 0.09$

Accepted Answer

Let @#LAT-DLM&. Then if @#LAT-DLM&, and @#LAT-DLM& @#LAT-DLM&. MULTIPL

Find all the local maxima, local minima, and saddle points of the function. - $f ( x , y ) = x ^ { 2 } + 14 x + y ^ { 2 } - 4 y + 3$

Solve the problem. - $\text { Find the maximum value of } f ( x , y , z , w ) = x + y + z + w \text { subject to } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + w ^ { 2 } = 1 \text {. }$

Solve the problem. -Find the derivative of the function $f ( x , y ) = e ^ { x y }$ at the point $( 0,2 )$ in the direction in which the function increases most rapidly.

Find the absolute maxima and minima of the function on the given domain. - $f ( x , y ) = 4 x ^ { 2 } + 7 y ^ { 2 } \text { on the disk bounded by the circle } x ^ { 2 } + y ^ { 2 } = 9$

Find the absolute maxima and minima of the function on the given domain. - $f ( x , y ) = x ^ { 2 } + y ^ { 2 } \text { on the diamond-shaped region } | x | + | y | \leq 4$

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = \ln ( x y ) ^ { z }$

Find the requested partial derivative. - $\frac { \partial z } { \partial y }$ at $( x , y , z ) = ( 1,1,1 )$ if $z ^ { 3 } = z + x y - 1$ and $y ^ { 3 } = x + y - 1$

Give an appropriate answer. - $f ( x , y ) = \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$

Provide an appropriate response. -Find the extrema of f(x, y, z) = x + yz on the line defined by x = 8(6 + t), y = t - 8, and z = t + 6. Classify each extremum as a minimum or maximum. (Hint: w = f(x, y, z) is a differentiable function of t.)

Find the domain and range and describe the level curves for the function f(x,y). -f(x, y) = ln (5x + 4y)

Find the limit. - $\lim _ { \mathrm { P } \rightarrow ( 2,0 , - 2 ) } \mathrm { e } ^ { 2 } \sin ^ { - 1 } \left( - \frac { \mathrm { z } } { \mathrm { x } } \right)$

Use implicit differentiation to find the specified derivative at the given point. -Find $\frac { \partial z } { \partial y }$ at the point $( 8,1 , - 1 )$ for $\ln \left( \frac { y z } { x } \right) - e ^ { x y + z ^ { 2 } } = 0$ .

Use implicit differentiation to find the specified derivative at the given point. -Find $\frac { d y } { d x }$ at the point $( 1 , - 1 )$ for $- 5 x y ^ { 2 } + 6 x ^ { 2 } y - 4 x = 0$ .

Find the linearization of the function at the given point. - $f ( x , y ) = e ^ { 4 x + 2 y } \text { at } ( 0,0 )$

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = x z \sqrt { x + y }$

Find the requested partial derivative. - $( \partial w / \partial z ) _ { x , y } \text { at } ( x , y , z , w ) = ( 1,2,9,266 ) \text { if } w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 10 x y z$

Find the domain and range and describe the level curves for the function f(x,y). - $f ( x , y ) = e ^ { x + y }$

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Exam 15: Partial Derivatives

Find all the local maxima, local minima, and saddle points of the function. - f(x,y)=x2+14x+y2−4y+3f ( x , y ) = x ^ { 2 } + 14 x + y ^ { 2 } - 4 y + 3f(x,y)=x2+14x+y2−4y+3

Solve the problem. -Find the derivative of the function f(x,y)=exyf ( x , y ) = e ^ { x y }f(x,y)=exy at the point (0,2)( 0,2 )(0,2) in the direction in which the function increases most rapidly.

Find the absolute maxima and minima of the function on the given domain. - f(x,y)=4x2+7y2 on the disk bounded by the circle x2+y2=9f ( x , y ) = 4 x ^ { 2 } + 7 y ^ { 2 } \text { on the disk bounded by the circle } x ^ { 2 } + y ^ { 2 } = 9f(x,y)=4x2+7y2 on the disk bounded by the circle x2+y2=9

Find the absolute maxima and minima of the function on the given domain. - f(x,y)=x2+y2 on the diamond-shaped region ∣x∣+∣y∣≤4f ( x , y ) = x ^ { 2 } + y ^ { 2 } \text { on the diamond-shaped region } | x | + | y | \leq 4f(x,y)=x2+y2 on the diamond-shaped region ∣x∣+∣y∣≤4

Find fx,fy, and fZ\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } } Find fx​,fy​, and fZ​ - f(x,y,z)=ln⁡(xy)zf ( x , y , z ) = \ln ( x y ) ^ { z }f(x,y,z)=ln(xy)z

Find the requested partial derivative. - ∂z∂y\frac { \partial z } { \partial y }∂y∂z​ at (x,y,z)=(1,1,1)( x , y , z ) = ( 1,1,1 )(x,y,z)=(1,1,1) if z3=z+xy−1z ^ { 3 } = z + x y - 1z3=z+xy−1 and y3=x+y−1y ^ { 3 } = x + y - 1y3=x+y−1

Give an appropriate answer. - f(x,y)=1x2+y2f ( x , y ) = \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } } }f(x,y)=x2+y2​1​