Question 1

Find the extreme values of the function subject to the given constraint.
-$$f ( x , y ) = x ^ { 2 } + y ^ { 2 } , \quad x y ^ { 2 } = 2000$$

Accepted Answer

A)  Maximum: none; minimum: 300 at $( 10 , \pm 10 \sqrt { 2 } )$ 
B)  Maximum: none; minimum: 0 at $( 0,0 )$ 
C)  Maximum: 300 at $( 10 , \pm 10 \sqrt { 2 } )$; minimum: 0 at $( 0,0 )$ 
D)  Maximum: 300 at $( 10,10 \sqrt { 2 } )$; minimum: $- 300$ at $( 10 , - 10 \sqrt { 2 } )$ 
A)  Maximum: none; minimum: 300 at $( 10 , \pm 10 \sqrt { 2 } )$ 
B)  Maximum: none; minimum: 0 at $( 0,0 )$ 
C)  Maximum: 300 at $( 10 , \pm 10 \sqrt { 2 } )$; minimum: 0 at $( 0,0 )$ 
D)  Maximum: 300 at $( 10,10 \sqrt { 2 } )$; minimum: $- 300$ at $( 10 , - 10 \sqrt { 2 } )$

Question 2

Give an appropriate answer.
-$f ( x , y ) = \ln y ^ { x }$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = 0 ; \frac { \partial f } { \partial y } = - \frac { x } { y }$ 
B)  $\frac { \partial f } { \partial x } = x \ln y ; \frac { \partial f } { \partial y } = - \frac { x } { y }$ 
C)  $\frac { \partial f } { \partial x } = \ln y ; \frac { \partial f } { \partial y } = - x \ln y$ 
D)  $\frac { \partial f } { \partial x } = \ln y ; \frac { \partial f } { \partial y } = \frac { x } { y }$ 
A)  $\frac { \partial f } { \partial x } = 0 ; \frac { \partial f } { \partial y } = - \frac { x } { y }$ 
B)  $\frac { \partial f } { \partial x } = x \ln y ; \frac { \partial f } { \partial y } = - \frac { x } { y }$ 
C)  $\frac { \partial f } { \partial x } = \ln y ; \frac { \partial f } { \partial y } = - x \ln y$ 
D)  $\frac { \partial f } { \partial x } = \ln y ; \frac { \partial f } { \partial y } = \frac { x } { y }$

Question 3

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$
-$f ( x , y , z ) = \sin ( x y ) \cos \left( y z ^ { 2 } \right)$

Accepted Answer

A)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = x \cos ( x y ) \cos \left( y z ^ { 2 } \right) - z ^ { 2 } \sin ( x y ) \sin \left( y z ^ { 2 } \right) ; f _ { z } = - 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$ 
B)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = x \cos ( x y ) \cos \left( y z ^ { 2 } \right) - z ^ { 2 } \sin ( x y ) \sin \left( y z ^ { 2 } \right) ; f _ { z } = 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$ 
C)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = x \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { z } = - 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$ 
D)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = z ^ { 2 } \sin ( x y ) \sin \left( y z ^ { 2 } \right) - x \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { z } = 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$ 
A)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = x \cos ( x y ) \cos \left( y z ^ { 2 } \right) - z ^ { 2 } \sin ( x y ) \sin \left( y z ^ { 2 } \right) ; f _ { z } = - 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$ 
B)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = x \cos ( x y ) \cos \left( y z ^ { 2 } \right) - z ^ { 2 } \sin ( x y ) \sin \left( y z ^ { 2 } \right) ; f _ { z } = 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$ 
C)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = x \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { z } = - 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$ 
D)  $f _ { x } = y \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { y } = z ^ { 2 } \sin ( x y ) \sin \left( y z ^ { 2 } \right) - x \cos ( x y ) \cos \left( y z ^ { 2 } \right) ; f _ { z } = 2 y z \sin ( x y ) \sin \left( y z ^ { 2 } \right)$

Question 4

Sketch the surface z = f(x,y).
-$$f ( x , y ) = 4 x ^ { 2 } + 4 y ^ { 2 } + 2$$

Accepted Answer

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

Question 5

Solve the problem.
-About how much will $f ( x , y , z ) = 2 x - 4 y + 10 z$ change if the point $( x , y , z )$ moves from $( 10 , - 7,10 )$ a distance of $\mathrm { ds } = \frac { 1 } { 10 }$ unit in the direction $2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }$ ?

Accepted Answer

A)  $\frac { 36 } { 35 }$ 
B)  $\frac { 38 } { 35 }$ 
C)  $\frac { 8 } { 7 }$ 
D)  $\frac { 6 } { 5 }$ 
A)  $\frac { 36 } { 35 }$ 
B)  $\frac { 38 } { 35 }$ 
C)  $\frac { 8 } { 7 }$ 
D)  $\frac { 6 } { 5 }$

Question 6

Provide an appropriate response.
-If $\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } )$ is differentiable, $\mathrm { x } = \mathrm { r } - \mathrm { s } , \mathrm { y } = \mathrm { s } - \mathrm { t }$, and $\mathrm { z } = \mathrm { t } - \mathrm { r }$, show that $\frac { \partial f } { \partial r } + \frac { \partial f } { \partial s } + \frac { \partial f } { \partial t } = 0$

Accepted Answer

This follo

Question 7

Write a chain rule formula for the following derivative.
-$$\frac { \partial \mathrm { z } } { \partial \mathrm { t } } \text { for } \mathrm { z } = \mathrm { f } ( \mathrm { r } , \mathrm { s } ) ; \mathrm { r } = \mathrm { g } ( \mathrm { t } ) , \mathrm { s } = \mathrm { h } ( \mathrm { t } )$$

Accepted Answer

A)  $\frac { \partial z } { \partial t } = \frac { \partial z } { \partial r } \frac { d r } { d s }$ 
B)  $\frac { \partial \mathrm { z } } { \partial \mathrm { t } } = \frac { \partial \mathrm { z } } { \partial \mathrm { r } } \frac { \mathrm { dr } } { \mathrm { dt } } + \frac { \partial \mathrm { z } } { \partial \mathrm { s } } \frac { \mathrm { ds } } { \mathrm { dt } }$ 
C)  $\frac { \partial \mathrm { z } } { \partial \mathrm { t } } = \frac { \mathrm { dr } } { \mathrm { dt } } + \frac { \mathrm { ds } } { \mathrm { dt } }$ 
D)  $\frac { \partial \mathrm { z } } { \partial \mathrm { t } } = \frac { \partial \mathrm { z } } { \partial \mathrm { r } } \frac { \mathrm { dt } } { \mathrm { dr } } + \frac { \partial \mathrm { z } } { \partial \mathrm { s } } \frac { \mathrm { dt } } { \mathrm { ds } }$ 
A)  $\frac { \partial z } { \partial t } = \frac { \partial z } { \partial r } \frac { d r } { d s }$ 
B)  $\frac { \partial \mathrm { z } } { \partial \mathrm { t } } = \frac { \partial \mathrm { z } } { \partial \mathrm { r } } \frac { \mathrm { dr } } { \mathrm { dt } } + \frac { \partial \mathrm { z } } { \partial \mathrm { s } } \frac { \mathrm { ds } } { \mathrm { dt } }$ 
C)  $\frac { \partial \mathrm { z } } { \partial \mathrm { t } } = \frac { \mathrm { dr } } { \mathrm { dt } } + \frac { \mathrm { ds } } { \mathrm { dt } }$ 
D)  $\frac { \partial \mathrm { z } } { \partial \mathrm { t } } = \frac { \partial \mathrm { z } } { \partial \mathrm { r } } \frac { \mathrm { dt } } { \mathrm { dr } } + \frac { \partial \mathrm { z } } { \partial \mathrm { s } } \frac { \mathrm { dt } } { \mathrm { ds } }$

Question 8

Sketch the surface z = f(x,y).
-$f ( x , y ) = - x ^ { 2 } - y ^ { 2 }$

Accepted Answer

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

Question 9

Provide an appropriate response.
-Find $F ^ { \prime } ( x )$ if $F ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t ^ { 2 } + x ^ { 2 } } d t$.

Accepted Answer

A)  $2 x \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$ 
B)  $2 x ^ { 2 } \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x ^ { 2 } } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$ 
C)  $2 x ^ { 2 } \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$ 
D)  $x \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$ 
A)  $2 x \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$ 
B)  $2 x ^ { 2 } \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x ^ { 2 } } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$ 
C)  $2 x ^ { 2 } \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$ 
D)  $x \sqrt { x ^ { 2 } + 1 } + \int _ { 0 } ^ { x ^ { 2 } } \frac { x } { \sqrt { t ^ { 2 } + x ^ { 2 } } } d t$

Question 10

Find the linearization of the function at the given point.
-$\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { e } ^ { 5 \mathrm { x } - 8 \mathrm { y } + 3 \mathrm { z } }$ at $( 0,0,0 )$

Accepted Answer

A)  $L ( x , y , z ) = \frac { 5 } { 2 } x - 4 y + \frac { 3 } { 2 } z + 1$ 
B)  $L ( x , y , z ) = 5 x - 8 y + 3 z$ 
C)  $L ( x , y , z ) = 5 x - 8 y + 3 z + 1$ 
D)  $L ( x , y , z ) = \frac { 5 } { 2 } x - 4 y + \frac { 3 } { 2 } z$ 
A)  $L ( x , y , z ) = \frac { 5 } { 2 } x - 4 y + \frac { 3 } { 2 } z + 1$ 
B)  $L ( x , y , z ) = 5 x - 8 y + 3 z$ 
C)  $L ( x , y , z ) = 5 x - 8 y + 3 z + 1$ 
D)  $L ( x , y , z ) = \frac { 5 } { 2 } x - 4 y + \frac { 3 } { 2 } z$

Question 11

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

Accepted Answer

A)  $- 1$ 
B)  $- \frac { 3 } { 10 }$ 
C)  1 
D)  $\frac { 3 } { 10 }$ 
A)  $- 1$ 
B)  $- \frac { 3 } { 10 }$ 
C)  1 
D)  $\frac { 3 } { 10 }$

Question 12

Find the domain and range and describe the level curves for the function f(x,y).
-f(x, y) = -6x + 8y

Accepted Answer

A)  Domain: all points in the $x y$-plane; range: real numbers $z \geq 0$; level curves: lines $- 6 x + 8 y = c , c \leq 0$ 
B)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $- 6 x + 8 y = c$ 
C)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $- 6 x + 8 y = c , c > 0$ 
D)  Domain: all points in the $x y$-plane; range: real numbers $z \geq 0$; level curves: lines $- 6 x + 8 y = c , c \geq 0$ 
A)  Domain: all points in the $x y$-plane; range: real numbers $z \geq 0$; level curves: lines $- 6 x + 8 y = c , c \leq 0$ 
B)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $- 6 x + 8 y = c$ 
C)  Domain: all points in the $x y$-plane; range: all real numbers; level curves: lines $- 6 x + 8 y = c , c > 0$ 
D)  Domain: all points in the $x y$-plane; range: real numbers $z \geq 0$; level curves: lines $- 6 x + 8 y = c , c \geq 0$

Question 13

Show that the function is a solution of the wave equation.
-$$w ( x , t ) = \cos ( 9 t - 9 c x ) , c \neq 1$$

Accepted Answer

A) True 
 B)False

Question 14

Determine whether the given function satisfies a Laplace equation.
-$$f ( x , y , z ) = 5 x + 2 y ^ { 3 } z ^ { 2 }$$

Accepted Answer

A) True 
 B)False

Question 15

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-Does knowing that $\left| \sin \left( \frac { 1 } { y } \right) \right| \leq 1$ tell you anything about $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \sin ( x ) \sin \left( \frac { 1 } { y } \right)$ ? Give reasons for your answer.

Accepted Answer

@#LAT-DLM& implies @#LAT-DLM&, which all

Question 16

Find the specific function value.
-Find $f ( 3,4 )$ when $f ( x , y ) = \frac { x - 6 y } { x ^ { 2 } + y ^ { 2 } }$

Accepted Answer

A)  $ = \frac { 25 } { 21 }$ 
B)  $- \frac { 5 } { 21 }$ 
C)  $ = \frac { 21 } { 25 }$ 
D)  $- \frac { 21 } { 5 }$ 
A)  $ = \frac { 25 } { 21 }$ 
B)  $- \frac { 5 } { 21 }$ 
C)  $ = \frac { 21 } { 25 }$ 
D)  $- \frac { 21 } { 5 }$

Question 17

Solve the problem.
-The resistance $R$ produced by wiring resistors of $R _ { 1 }$ and $R _ { 2 }$ ohms in parallel can be calculated from the formul $\frac { 1 } { R } = \frac { 1 } { R _ { 1 } } + \frac { 1 } { R _ { 2 } } .$
If $R _ { 1 }$ and $R _ { 2 }$ are measured to be 10 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)  0.032 
B)  0.016 
C)  0.021 
D)  0.027 
A)  0.032 
B)  0.016 
C)  0.021 
D)  0.027

Question 18

Solve the problem.
-Write parametric equations for the tangent line to the curve of intersection of the surfaces $z = 10 x ^ { 2 } + 2 y ^ { 2 }$ and $z = x + y + 10$ at the point $( 1,1,12 )$.

Accepted Answer

A)  x = -19t + 1, y = 21t + 1, z = 16t + 12 
B)  x = -3t + 1, y = 21t + 1, z = 16t + 12 
C)  x = -19t + 1, y = 19t + 1, z = 16t + 12 
D)  x = -3t + 1, y = 19t + 1, z = 16t + 12 
A)  x = -19t + 1, y = 21t + 1, z = 16t + 12 
B)  x = -3t + 1, y = 21t + 1, z = 16t + 12 
C)  x = -19t + 1, y = 19t + 1, z = 16t + 12 
D)  x = -3t + 1, y = 19t + 1, z = 16t + 12

Question 19

Solve the problem.
-Find an equation for the level surface of the function $f ( x , y , z ) = x + e y ^ { + } z$ that passes through the point $( 9 , \ln ( 2 ) , \ln ( 9 ) )$.

Accepted Answer

A)  $\ln ( x ) + y + z = 27$ 
B)  $x + e ^ { y + z } = 27$ 
C)  $x + \mathrm { e } ^ { + } \mathrm { z } = 20$ 
D)  $x + e ^ { y + z } = 18$ 
A)  $\ln ( x ) + y + z = 27$ 
B)  $x + e ^ { y + z } = 27$ 
C)  $x + \mathrm { e } ^ { + } \mathrm { z } = 20$ 
D)  $x + e ^ { y + z } = 18$

Question 20

Solve the problem.
-Find the derivative of the function $f ( x , y ) = e ^ { x y }$ at the point $( 0 , - 6 )$ in the direction in which the function decreases most rapidly.

Accepted Answer

A)  -12 
B)  -18 
C)  -7 
D)  -6 
A)  -12 
B)  -18 
C)  -7 
D)  -6

Find the extreme values of the function subject to the given constraint. - $f ( x , y ) = x ^ { 2 } + y ^ { 2 } , \quad x y ^ { 2 } = 2000$

Give an appropriate answer. - $f ( x , y ) = \ln y ^ { x }$

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = \sin ( x y ) \cos \left( y z ^ { 2 } \right)$

Sketch the surface z = f(x,y). - $f ( x , y ) = 4 x ^ { 2 } + 4 y ^ { 2 } + 2$

Solve the problem. -About how much will $f ( x , y , z ) = 2 x - 4 y + 10 z$ change if the point $( x , y , z )$ moves from $( 10 , - 7,10 )$ a distance of $\mathrm { ds } = \frac { 1 } { 10 }$ unit in the direction $2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }$ ?

Write a chain rule formula for the following derivative. - $\frac { \partial \mathrm { z } } { \partial \mathrm { t } } \text { for } \mathrm { z } = \mathrm { f } ( \mathrm { r } , \mathrm { s } ) ; \mathrm { r } = \mathrm { g } ( \mathrm { t } ) , \mathrm { s } = \mathrm { h } ( \mathrm { t } )$

Sketch the surface z = f(x,y). - $f ( x , y ) = - x ^ { 2 } - y ^ { 2 }$

Provide an appropriate response. -Find $F ^ { \prime } ( x )$ if $F ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t ^ { 2 } + x ^ { 2 } } d t$ .

Find the linearization of the function at the given point. - $\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { e } ^ { 5 \mathrm { x } - 8 \mathrm { y } + 3 \mathrm { z } }$ at $( 0,0,0 )$

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

Find the domain and range and describe the level curves for the function f(x,y). -f(x, y) = -6x + 8y

Show that the function is a solution of the wave equation. - $w ( x , t ) = \cos ( 9 t - 9 c x ) , c \neq 1$

Determine whether the given function satisfies a Laplace equation. - $f ( x , y , z ) = 5 x + 2 y ^ { 3 } z ^ { 2 }$

Find the specific function value. -Find $f ( 3,4 )$ when $f ( x , y ) = \frac { x - 6 y } { x ^ { 2 } + y ^ { 2 } }$

Solve the problem. -Write parametric equations for the tangent line to the curve of intersection of the surfaces $z = 10 x ^ { 2 } + 2 y ^ { 2 }$ and $z = x + y + 10$ at the point $( 1,1,12 )$ .

Solve the problem. -Find an equation for the level surface of the function $f ( x , y , z ) = x + e y ^ { + } z$ that passes through the point $( 9 , \ln ( 2 ) , \ln ( 9 ) )$ .

Solve the problem. -Find the derivative of the function $f ( x , y ) = e ^ { x y }$ at the point $( 0 , - 6 )$ in the direction in which the function decreases most rapidly.

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Exam 15: Partial Derivatives

Find the extreme values of the function subject to the given constraint. - f(x,y)=x2+y2,xy2=2000f ( x , y ) = x ^ { 2 } + y ^ { 2 } , \quad x y ^ { 2 } = 2000f(x,y)=x2+y2,xy2=2000

Give an appropriate answer. - f(x,y)=ln⁡yxf ( x , y ) = \ln y ^ { x }f(x,y)=lnyx

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)=sin⁡(xy)cos⁡(yz2)f ( x , y , z ) = \sin ( x y ) \cos \left( y z ^ { 2 } \right)f(x,y,z)=sin(xy)cos(yz2)

Sketch the surface z = f(x,y). - f(x,y)=4x2+4y2+2f ( x , y ) = 4 x ^ { 2 } + 4 y ^ { 2 } + 2f(x,y)=4x2+4y2+2

Sketch the surface z = f(x,y). - f(x,y)=−x2−y2f ( x , y ) = - x ^ { 2 } - y ^ { 2 }f(x,y)=−x2−y2

Provide an appropriate response. -Find F′(x)F ^ { \prime } ( x )F′(x) if F(x)=∫0x2t2+x2dtF ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t ^ { 2 } + x ^ { 2 } } d tF(x)=∫0x2​t2+x2​dt .

Find the linearization of the function at the given point. - f(x,y,z)=e5x−8y+3z\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { e } ^ { 5 \mathrm { x } - 8 \mathrm { y } + 3 \mathrm { z } }f(x,y,z)=e5x−8y+3z at (0,0,0)( 0,0,0 )(0,0,0)

Use implicit differentiation to find the specified derivative at the given point. -Find dydx\frac { d y } { d x }dxdy​ at the point (2,1)( 2,1 )(2,1) for ln⁡x+xy2+ln⁡y=0\ln x + x y ^ { 2 } + \ln y = 0lnx+xy2+lny=0 .