Question 1

At what points is the given function continuous?
-$$f ( x , y , z ) = \frac { z } { x ^ { 2 } + y ^ { 2 } - 5 }$$

Accepted Answer

A)  All $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \neq 0$ 
B)  All $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \neq 5$ 
C)  All $( x , y , z )$ 
D)  All $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \neq 25$ 
A)  All $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \neq 0$ 
B)  All $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \neq 5$ 
C)  All $( x , y , z )$ 
D)  All $( x , y , z )$ such that $x ^ { 2 } + y ^ { 2 } \neq 25$

Question 2

Find the absolute maxima and minima of the function on the given domain.
-$$f ( x , y ) = 5 x y ^ { 2 } + 8 x y \quad \text { on the trapezoidal region with vertices } ( 0,0 ) , ( 1,0 ) , ( 0,2 ) \text {, and } ( 1,1 )$$

Accepted Answer

A)  Absolute maximum: 16 at (0, 2); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 10 at (2, 0); absolute minimum: 5 at (1, 0) 
C)  Absolute maximum: 13 at (1, 1); absolute minimum: 5 at (1, 0) 
D)  Absolute maximum: 13 at (1, 1); absolute minimum: 0 at (0, 0) 
A)  Absolute maximum: 16 at (0, 2); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 10 at (2, 0); absolute minimum: 5 at (1, 0) 
C)  Absolute maximum: 13 at (1, 1); absolute minimum: 5 at (1, 0) 
D)  Absolute maximum: 13 at (1, 1); absolute minimum: 0 at (0, 0)

Question 3

Find the linearization of the function at the given point.
-$$f ( x , y , z ) = 7 x ^ { 2 } + 8 y ^ { 2 } + 7 z ^ { 2 } \text { at } ( 1 , - 2,3 )$$

Accepted Answer

A)  L(x, y, z) = 14x - 32y + 42z - 102 
B)  L(x, y, z) = 14x + 32y + 42z - 102 
C)  L(x, y, z) = 14x + 32y + 42z + 102 
D)  L(x, y, z) = 14x - 32y + 42z + 102 
A)  L(x, y, z) = 14x - 32y + 42z - 102 
B)  L(x, y, z) = 14x + 32y + 42z - 102 
C)  L(x, y, z) = 14x + 32y + 42z + 102 
D)  L(x, y, z) = 14x - 32y + 42z + 102

Question 4

Solve the problem.
-Find the point on the curve of intersection of the paraboloid $x ^ { 2 } + y ^ { 2 } + 2 z = 4$ and the plane $x - y + 2 z = 0$ that is closest to the origin.

Accepted Answer

A)  (1, 1, 1) 
B)  (1, -1, 1) 
C)  (-1, 1, 1) 
D)  (1, 1, -1) 
A)  (1, 1, 1) 
B)  (1, -1, 1) 
C)  (-1, 1, 1) 
D)  (1, 1, -1)

Question 5

Find the linearization of the function at the given point.
-$$f ( x , y ) = 4 x ^ { 2 } y ^ { 3 } \text { at } ( - 9,5 )$$

Accepted Answer

A)  L(x, y) = -9000x + 24,300y - 162,000 
B)  L(x, y) = 1000x - 2700y - 162,000 
C)  L(x, y) = -9000x - 2700y - 162,000 
D)  L(x, y) = 1000x + 24,300y - 162,000 
A)  L(x, y) = -9000x + 24,300y - 162,000 
B)  L(x, y) = 1000x - 2700y - 162,000 
C)  L(x, y) = -9000x - 2700y - 162,000 
D)  L(x, y) = 1000x + 24,300y - 162,000

Question 6

Sketch a typical level surface for the function.
-$$f ( x , y , z ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 7

At what points is the given function continuous?
-$$f ( x , y ) = e ^ { x + y }$$

Accepted Answer

A)  $\operatorname { All } ( x , y ) \neq ( 0,0 )$ 
B)  All $( x , y )$ 
C)  All $( x , y )$ satisfying $x + y \geq 0$ 
D)  All (x,y) in the first quadrant 
A)  $\operatorname { All } ( x , y ) \neq ( 0,0 )$ 
B)  All $( x , y )$ 
C)  All $( x , y )$ satisfying $x + y \geq 0$ 
D)  All (x,y) in the first quadrant

Question 8

Find the limit.
-$$\lim _ { P \rightarrow ( 4,4,1 ) } \ln \left( z \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$$

Accepted Answer

A)  $\ln \sqrt { 2 }$ 
B)  $\ln 4 \sqrt { 2 }$ 
C)  0 
D)  $\ln 16$ 
A)  $\ln \sqrt { 2 }$ 
B)  $\ln 4 \sqrt { 2 }$ 
C)  0 
D)  $\ln 16$

Question 9

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-$$f ( x , y ) = \frac { x ^ { 2 } y } { x ^ { 4 } + y ^ { 2 } }$$

Accepted Answer

The answer of Find two paths of approach from which...

Question 10

Find the derivative of the function at P0 in the direction of u.
-$$f ( x , y ) = \ln ( x + 9 y ) , \quad P _ { 0 } ( 9 , - 9 ) , \quad u = 6 i + 8 j$$

Accepted Answer

A)  $- \frac { 68 } { 45 }$ 
B)  $- \frac { 44 } { 45 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  $- \frac { 52 } { 45 }$ 
A)  $- \frac { 68 } { 45 }$ 
B)  $- \frac { 44 } { 45 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  $- \frac { 52 } { 45 }$

Question 11

Find the absolute maximum and minimum values of the function on the given curve.
-Function: $f ( x , y ) = x + y ;$ curve: $x ^ { 2 } + y ^ { 2 } = 49 , y \geq 0$. (Use the parametric equations $x = 7 \cos t , y = 7 \sin t$.)

Accepted Answer

A)  Absolute maximum: $7 \sqrt { 2 }$ at $t = \frac { \pi } { 4 }$; absolute minimum: $- 7 \sqrt { 2 }$ at $t = \frac { 3 \pi } { 4 }$ 
B)  Absolute maximum: $7 \sqrt { 2 }$ at $t = \frac { \pi } { 4 }$; absolute minimum: $- 7$ at $t = \pi$ 
C)  Absolute maximum: 7 at $t = \frac { \pi } { 2 } ;$ absolute minimum: $- 7 \sqrt { 2 }$ at $t = \frac { 3 \pi } { 4 }$ 
D)  Absolute maximum: 7 at $t = \frac { \pi } { 2 }$; absolute minimum: $- 7$ at $t = \pi$ 
A)  Absolute maximum: $7 \sqrt { 2 }$ at $t = \frac { \pi } { 4 }$; absolute minimum: $- 7 \sqrt { 2 }$ at $t = \frac { 3 \pi } { 4 }$ 
B)  Absolute maximum: $7 \sqrt { 2 }$ at $t = \frac { \pi } { 4 }$; absolute minimum: $- 7$ at $t = \pi$ 
C)  Absolute maximum: 7 at $t = \frac { \pi } { 2 } ;$ absolute minimum: $- 7 \sqrt { 2 }$ at $t = \frac { 3 \pi } { 4 }$ 
D)  Absolute maximum: 7 at $t = \frac { \pi } { 2 }$; absolute minimum: $- 7$ at $t = \pi$

Question 12

Give an appropriate answer.
-Given the function $f ( x , y )$ and the positive number $\varepsilon$ as in the formal definition of a limit, find a positive number $\delta$ as in the definition that insures $| f ( x , y ) - f ( 0,0 ) | < \varepsilon$.
$f ( x , y ) = x + y ; \varepsilon = 0.08$

Accepted Answer

The answer of Give an appropriate answer.
-Given the function \(f...

Question 13

Solve the problem.
-Find the point on the line of intersection of the planes x + y + z = 1 and 3x + 2y + z = 6 that is closest to the origin.

Accepted Answer

A)  $\left( \frac { 7 } { 3 } , \frac { 1 } { 3 } , - \frac { 5 } { 3 } \right)$ 
B)  $\left( \frac { 7 } { 3 } , - \frac { 1 } { 3 } , - \frac { 5 } { 3 } \right)$ 
C)  $\left( - \frac { 7 } { 3 } , \frac { 1 } { 3 } , \frac { 5 } { 3 } \right)$ 
D)  $\left( \frac { 7 } { 3 } , - \frac { 1 } { 3 } , \frac { 5 } { 3 } \right)$ 
A)  $\left( \frac { 7 } { 3 } , \frac { 1 } { 3 } , - \frac { 5 } { 3 } \right)$ 
B)  $\left( \frac { 7 } { 3 } , - \frac { 1 } { 3 } , - \frac { 5 } { 3 } \right)$ 
C)  $\left( - \frac { 7 } { 3 } , \frac { 1 } { 3 } , \frac { 5 } { 3 } \right)$ 
D)  $\left( \frac { 7 } { 3 } , - \frac { 1 } { 3 } , \frac { 5 } { 3 } \right)$

Question 14

Determine whether the given function satisfies a Laplace equation.
-f(x, y) = cos(x) sin(-y)

Accepted Answer

A) True 
 B)False

Question 15

Sketch a typical level surface for the function.
-$$f ( x , y , z ) = \ln \left( \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 4 } + \frac { z ^ { 2 } } { 8 } \right)$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 16

Use implicit differentiation to find the specified derivative at the given point.
-Find $\frac { d y } { d x }$ at the point $( - 1,1 )$ for $3 x - \frac { 3 } { y } + 2 x ^ { 2 } y ^ { 2 } = 0$.

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 7 }$ 
C)  - 1 
D)  $- \frac { 1 } { 7 }$ 
A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 7 }$ 
C)  - 1 
D)  $- \frac { 1 } { 7 }$

Question 17

Use implicit differentiation to find the specified derivative at the given point.
-Find $\frac { d y } { d x }$ at the point $( 1,1 )$ for $3 x ^ { 2 } + 5 y ^ { 3 } + 2 x y = 0$.

Accepted Answer

A)  $- \frac { 8 } { 7 }$ 
B)  $- \frac { 8 } { 17 }$ 
C)  $\frac { 8 } { 17 }$ 
D)  $- \frac { 5 } { 7 }$ 
A)  $- \frac { 8 } { 7 }$ 
B)  $- \frac { 8 } { 17 }$ 
C)  $\frac { 8 } { 17 }$ 
D)  $- \frac { 5 } { 7 }$

Question 18

Find the linearization of the function at the given point.
-f(x, y) = -9x + 3y + 6 at (-2, 4)

Accepted Answer

A)  L(x, y) = -2x + 4y + 6 
B)  L(x, y) = 18x + 12y + 36 
C)  L(x, y) = -9x + 3y + 6 
D)  L(x, y) = 18x + 12y + 6 
A)  L(x, y) = -2x + 4y + 6 
B)  L(x, y) = 18x + 12y + 36 
C)  L(x, y) = -9x + 3y + 6 
D)  L(x, y) = 18x + 12y + 6

Question 19

Find the derivative of the function at P0 in the direction of u.
-$$f ( x , y ) = - 3 x ^ { 2 } - 10 y , \quad P _ { 0 } ( - 8 , \quad ) , \quad \mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j }$$

Accepted Answer

A)  56 
B)  $\frac { 328 } { 5 }$ 
C)  $\frac { 184 } { 5 }$ 
D)  $\frac { 232 } { 5 }$ 
A)  56 
B)  $\frac { 328 } { 5 }$ 
C)  $\frac { 184 } { 5 }$ 
D)  $\frac { 232 } { 5 }$

Question 20

Solve the problem.
-Find the extreme values of $f ( x , y , z ) = x + 2 y$ subject to $x + y + z = 1$ and $y ^ { 2 } + z ^ { 2 } = 4$.

Accepted Answer

A)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , \sqrt { 2 } ) ;$ minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , \sqrt { 2 } )$ 
B)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , - \sqrt { 2 } ) ;$ minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , - \sqrt { 2 }$, 
C)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , \sqrt { 2 } )$; minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , - \sqrt { 2 } )$ 
D)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , - \sqrt { 2 } ) ;$ minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , \sqrt { 2 } )$ 
A)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , \sqrt { 2 } ) ;$ minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , \sqrt { 2 } )$ 
B)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , - \sqrt { 2 } ) ;$ minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , - \sqrt { 2 }$, 
C)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , \sqrt { 2 } )$; minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , - \sqrt { 2 } )$ 
D)  Maximum: $1 + 2 \sqrt { 2 }$ at $( 1 , \sqrt { 2 } , - \sqrt { 2 } ) ;$ minimum: $1 - 2 \sqrt { 2 }$ at $( 1 , - \sqrt { 2 } , \sqrt { 2 } )$

At what points is the given function continuous? - $f ( x , y , z ) = \frac { z } { x ^ { 2 } + y ^ { 2 } - 5 }$

Find the absolute maxima and minima of the function on the given domain. - $f ( x , y ) = 5 x y ^ { 2 } + 8 x y \quad \text { on the trapezoidal region with vertices } ( 0,0 ) , ( 1,0 ) , ( 0,2 ) \text {, and } ( 1,1 )$

Find the linearization of the function at the given point. - $f ( x , y , z ) = 7 x ^ { 2 } + 8 y ^ { 2 } + 7 z ^ { 2 } \text { at } ( 1 , - 2,3 )$

Solve the problem. -Find the point on the curve of intersection of the paraboloid $x ^ { 2 } + y ^ { 2 } + 2 z = 4$ and the plane $x - y + 2 z = 0$ that is closest to the origin.

Find the linearization of the function at the given point. - $f ( x , y ) = 4 x ^ { 2 } y ^ { 3 } \text { at } ( - 9,5 )$

Sketch a typical level surface for the function. - $f ( x , y , z ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }$

At what points is the given function continuous? - $f ( x , y ) = e ^ { x + y }$

Find the limit. - $\lim _ { P \rightarrow ( 4,4,1 ) } \ln \left( z \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). - $f ( x , y ) = \frac { x ^ { 2 } y } { x ^ { 4 } + y ^ { 2 } }$

Find the derivative of the function at P0 in the direction of u. - $f ( x , y ) = \ln ( x + 9 y ) , \quad P _ { 0 } ( 9 , - 9 ) , \quad u = 6 i + 8 j$

Find the absolute maximum and minimum values of the function on the given curve. -Function: $f ( x , y ) = x + y ;$ curve: $x ^ { 2 } + y ^ { 2 } = 49 , y \geq 0$ . (Use the parametric equations $x = 7 \cos t , y = 7 \sin t$ .)

Give an appropriate answer. -Given the function $f ( x , y )$ and the positive number $\varepsilon$ as in the formal definition of a limit, find a positive number $\delta$ as in the definition that insures $| f ( x , y ) - f ( 0,0 ) | < \varepsilon$ . $f ( x , y ) = x + y ; \varepsilon = 0.08$

Solve the problem. -Find the point on the line of intersection of the planes x + y + z = 1 and 3x + 2y + z = 6 that is closest to the origin.

Determine whether the given function satisfies a Laplace equation. -f(x, y) = cos(x) sin(-y)

Sketch a typical level surface for the function. - $f ( x , y , z ) = \ln \left( \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 4 } + \frac { z ^ { 2 } } { 8 } \right)$

Use implicit differentiation to find the specified derivative at the given point. -Find $\frac { d y } { d x }$ at the point $( - 1,1 )$ for $3 x - \frac { 3 } { y } + 2 x ^ { 2 } y ^ { 2 } = 0$ .

Use implicit differentiation to find the specified derivative at the given point. -Find $\frac { d y } { d x }$ at the point $( 1,1 )$ for $3 x ^ { 2 } + 5 y ^ { 3 } + 2 x y = 0$ .

Find the linearization of the function at the given point. -f(x, y) = -9x + 3y + 6 at (-2, 4)

Find the derivative of the function at P0 in the direction of u. - $f ( x , y ) = - 3 x ^ { 2 } - 10 y , \quad P _ { 0 } ( - 8 , \quad ) , \quad \mathbf { u } = 3 \mathbf { i } - 4 \mathbf { j }$

Solve the problem. -Find the extreme values of $f ( x , y , z ) = x + 2 y$ subject to $x + y + z = 1$ and $y ^ { 2 } + z ^ { 2 } = 4$ .

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Exam 15: Partial Derivatives

At what points is the given function continuous? - f(x,y,z)=zx2+y2−5f ( x , y , z ) = \frac { z } { x ^ { 2 } + y ^ { 2 } - 5 }f(x,y,z)=x2+y2−5z​

Find the linearization of the function at the given point. - f(x,y,z)=7x2+8y2+7z2 at (1,−2,3)f ( x , y , z ) = 7 x ^ { 2 } + 8 y ^ { 2 } + 7 z ^ { 2 } \text { at } ( 1 , - 2,3 )f(x,y,z)=7x2+8y2+7z2 at (1,−2,3)

Solve the problem. -Find the point on the curve of intersection of the paraboloid x2+y2+2z=4x ^ { 2 } + y ^ { 2 } + 2 z = 4x2+y2+2z=4 and the plane x−y+2z=0x - y + 2 z = 0x−y+2z=0 that is closest to the origin.

Find the linearization of the function at the given point. - f(x,y)=4x2y3 at (−9,5)f ( x , y ) = 4 x ^ { 2 } y ^ { 3 } \text { at } ( - 9,5 )f(x,y)=4x2y3 at (−9,5)

Sketch a typical level surface for the function. - f(x,y,z)=e(x2+y2+z2)f ( x , y , z ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }f(x,y,z)=e(x2+y2+z2)

At what points is the given function continuous? - f(x,y)=ex+yf ( x , y ) = e ^ { x + y }f(x,y)=ex+y

Find the limit. - lim⁡P→(4,4,1)ln⁡(zx2+y2)\lim _ { P \rightarrow ( 4,4,1 ) } \ln \left( z \sqrt { x ^ { 2 } + y ^ { 2 } } \right)limP→(4,4,1)​ln(zx2+y2​)

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). - f(x,y)=x2yx4+y2f ( x , y ) = \frac { x ^ { 2 } y } { x ^ { 4 } + y ^ { 2 } }f(x,y)=x4+y2x2y​

Find the derivative of the function at P0 in the direction of u. - f(x,y)=ln⁡(x+9y),P0(9,−9),u=6i+8jf ( x , y ) = \ln ( x + 9 y ) , \quad P _ { 0 } ( 9 , - 9 ) , \quad u = 6 i + 8 jf(x,y)=ln(x+9y),P0​(9,−9),u=6i+8j