Question 1

Prove that the following limit does not exist: $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } f ( x , y )$ where $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 4 } }$ .

Accepted Answer

@#IMG-DLM& is defined everywhere in @#IMG-DLM& except at @#IMG-DLM& .

Question 2

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = x ^ { 3 } - 3 x y - y ^ { 3 }$ .

Accepted Answer

@#IMG-DLM& is a sadd

Question 3

Find the differential of $z = 3 x + 2 y$

Accepted Answer

A)  $3 x d x + 2 y d y$ 
B)  $3 x ^ { 2 } d x + 2 y ^ { 2 } d y$ 
C)  $2 x d x + 3 y d y$ 
D)  $2 x ^ { 2 } d x + 3 y ^ { 2 } d y$ 
E)  $3 y d x + 2 x d y$  
F) $2 y d x + 3 x d y$

G) $2 d x + 3 d y$

H) $3 d x + 2 d y$ 
A)  $3 x d x + 2 y d y$ 
B)  $3 x ^ { 2 } d x + 2 y ^ { 2 } d y$ 
C)  $2 x d x + 3 y d y$ 
D)  $2 x ^ { 2 } d x + 3 y ^ { 2 } d y$ 
E)  $3 y d x + 2 x d y$  
F) $2 y d x + 3 x d y$

G) $2 d x + 3 d y$

H) $3 d x + 2 d y$

Question 4

Find $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { x - y ^ { 4 } } { x ^ { 3 } - y ^ { 4 } }$ if it exists, or show that the limit does not exist.

Accepted Answer

Along @#IMG-DLM& , @#IMG-DLM& (if the limit exists

Question 5

Let $f ( x , y ) = 3 x ^ { 3 } + y ^ { 2 } - 9 x + 4 y$ . Find the direction at the origin in which $f ( x , y )$ is decreasing the fastest.

Accepted Answer

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

Question 6

The graph of level curves of $f ( x , y )$ is given. Find a possible formula for $f ( x , y )$ and sketch the surface $z = f ( x , y )$ .

Accepted Answer

@#IMG-DLM& . There a

Question 7

Find $\frac { \delta z } { \delta y }$ for $x ^ { 3 } - y ^ { 2 } z + \sin ( x y z ) = 0$

Accepted Answer

The answer of Find \(\frac { \delta z } {...

Question 8

Let $f ( x , y , z ) = ( x y z ) ^ { 2 }$ . Find all second-order partial derivatives of $f$ .

Accepted Answer

@#IMG-DLM& , @#IMG-DLM& , @#IMG-DLM& , @#IMG-DLM&

Question 9

Show that the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 2 x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + 2 y ^ { 2 } }$ does not exist.

Accepted Answer

Approaching @#IMG-DLM& along the @#IMG-DLM& -axis, we have @#IMG-DLM&

Question 10

Find an equation of the tangent plant to the surface $z = \ln ( 2 x + y )$ at the point $( - 1,3,0 )$ .

Accepted Answer

A)  $2 x + y - z = 1$ 
B)  $x + 2 y + z = 5$ 
C)  $x - 2 y + z = 7$ 
D)  $3 x + 2 y + z = 3$ 
E)  $2 x + 3 y + z = 7$ 
F) $x + 2 y + 3 z = 5$

G) $3 x + y - z = 0$

H) $x + 3 y + 2 z = 8$ 
A)  $2 x + y - z = 1$ 
B)  $x + 2 y + z = 5$ 
C)  $x - 2 y + z = 7$ 
D)  $3 x + 2 y + z = 3$ 
E)  $2 x + 3 y + z = 7$ 
F) $x + 2 y + 3 z = 5$

G) $3 x + y - z = 0$

H) $x + 3 y + 2 z = 8$

Question 11

Find two positive numbers who sum is eighteen and whose product is a maximum, using the method of Lagrange multipliers

Accepted Answer

The answer of Find two positive numbers who sum is...

Question 12

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

Accepted Answer

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

F)3
G) $\frac { 7 } { 2 }$ 
H)4 
A)  $\frac { 1 } { 2 }$ 
B) 1 
C)  $\frac { 3 } { 2 }$ 
D) 2 
E)  $\frac { 5 } { 2 }$

F)3
G) $\frac { 7 } { 2 }$ 
H)4

Question 13

The function $f ( x , y ) = 4 - x ^ { 2 } - y ^ { 2 } - x y - x$ has one critical point. Determine its location and type.

Accepted Answer

A)  $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , saddle point 
B)  $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , minimum point 
C)  $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , maximum point 
D)  $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , saddle point 
E)  $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , minimum point
F) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , maximum point
G) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , saddle point
H) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , minimum point 
A)  $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , saddle point 
B)  $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , minimum point 
C)  $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , maximum point 
D)  $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , saddle point 
E)  $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , minimum point
F) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , maximum point
G) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , saddle point
H) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , minimum point

Question 14

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + \frac { 9 } { 2 } y ^ { 2 } - 9 y - 4 x y + 50$

Accepted Answer

A) 8.5 
B) 18 
C) 19 
D) 10 
E) 10.5
F)9
G)9.5
H)50 
A) 8.5 
B) 18 
C) 19 
D) 10 
E) 10.5
F)9
G)9.5
H)50

Question 15

Describe the difference between the horizontal trace in $z = k$ for the function $z = f ( x , y )$ and the contour curve $f ( x , y ) = k$

Accepted Answer

The contour curve @#IMG-DLM& i

Question 16

The pressure $P$ (in kilopascals), volume $V$ (in liters), and temperature $T$ (in $^\circ$ K) of a mole of an ideal gas are related by the equation $P = \frac { 8.31 T } { V }$ . Find the rate at which the pressure is changing when the temperature is 300 $^\circ$ K and increasing at a rate of 0.1 $^\circ$ K/s and the volume is 100 L and increasing at a rate of 0.2 L/s.

Accepted Answer

The answer of The pressure $P$ (in kilopascals), volume $V$...

Question 17

A rectangular area of 3200 ft $2$ is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of $\lambda$ . What does $\lambda$ represent?

Accepted Answer

80 ft @#IMG-DLM& 40

Question 18

Let $f ( x , y ) = \left\{ \begin{array} { l l } 
\frac { x ^ { 2 } } { x + y } & \text { if } ( x , y ) \neq ( 0,0 ) \
0 & \text { if } ( x , y ) = ( 0,0 )
\end{array} \right.$ (a) Find $\frac { \delta f } { \delta x }$ and $\frac { \delta f } { \delta y }$ .(b) Find the value of the above derivatives at $( 0,0 )$ , if they exist.

Accepted Answer

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

Question 19

Let $z = \sin ( x y )$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 0 , y ( 1 ) = 1 , x ^ { \prime } ( 1 ) = 2 , \text { and } y ^ { \prime } ( 1 ) = 3$ . Find $d z / d t$ when $t = 1$ .

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)12 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)12

Question 20

Find the domain of the function $f ( x , y ) = e ^ { x - y ^ { 2 } }$ .

Accepted Answer

A) All points on or to the left of $x = y ^ { 2 }$ 
B) All points on or to the right of $x = y ^ { 2 }$ 
C) All points to the left of $x = y ^ { 2 }$ 
D) All points to the right of $x = y ^ { 2 }$ 
E) All points on or to the left of $x = 0$
F)All points on or to the right of $x = 0$

G)All points to the left of $x = 0$

H)All points in the $x y$ -plane 
A) All points on or to the left of $x = y ^ { 2 }$ 
B) All points on or to the right of $x = y ^ { 2 }$ 
C) All points to the left of $x = y ^ { 2 }$ 
D) All points to the right of $x = y ^ { 2 }$ 
E) All points on or to the left of $x = 0$
F)All points on or to the right of $x = 0$

G)All points to the left of $x = 0$

H)All points in the $x y$ -plane

Prove that the following limit does not exist: $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } f ( x , y )$ where $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 4 } }$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = x ^ { 3 } - 3 x y - y ^ { 3 }$ .

Find the differential of $z = 3 x + 2 y$

Find $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { x - y ^ { 4 } } { x ^ { 3 } - y ^ { 4 } }$ if it exists, or show that the limit does not exist.

Let $f ( x , y ) = 3 x ^ { 3 } + y ^ { 2 } - 9 x + 4 y$ . Find the direction at the origin in which $f ( x , y )$ is decreasing the fastest.

The graph of level curves of $f ( x , y )$ is given. Find a possible formula for $f ( x , y )$ and sketch the surface $z = f ( x , y )$ .

Find $\frac { \delta z } { \delta y }$ for $x ^ { 3 } - y ^ { 2 } z + \sin ( x y z ) = 0$

Let $f ( x , y , z ) = ( x y z ) ^ { 2 }$ . Find all second-order partial derivatives of $f$ .

Show that the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 2 x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + 2 y ^ { 2 } }$ does not exist.

Find an equation of the tangent plant to the surface $z = \ln ( 2 x + y )$ at the point $( - 1,3,0 )$ .

Find two positive numbers who sum is eighteen and whose product is a maximum, using the method of Lagrange multipliers

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

The function $f ( x , y ) = 4 - x ^ { 2 } - y ^ { 2 } - x y - x$ has one critical point. Determine its location and type.

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + \frac { 9 } { 2 } y ^ { 2 } - 9 y - 4 x y + 50$

Describe the difference between the horizontal trace in $z = k$ for the function $z = f ( x , y )$ and the contour curve $f ( x , y ) = k$

A rectangular area of 3200 ft $2$ is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of $\lambda$ . What does $\lambda$ represent?

Let $z = \sin ( x y )$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 0 , y ( 1 ) = 1 , x ^ { \prime } ( 1 ) = 2 , \text { and } y ^ { \prime } ( 1 ) = 3$ . Find $d z / d t$ when $t = 1$ .

Find the domain of the function $f ( x , y ) = e ^ { x - y ^ { 2 } }$ .

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Exam 11: Partial Derivatives

Prove that the following limit does not exist: lim⁡(x,y)→(0,0)f(x,y)\lim _ { ( x , y ) \rightarrow ( 0,0 ) } f ( x , y )lim(x,y)→(0,0)​f(x,y) where f(x,y)=xyx2+y4f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 4 } }f(x,y)=x2+y4xy​ .

Find the local maximum and minimum values and saddle points of the function f(x,y)=x3−3xy−y3f ( x , y ) = x ^ { 3 } - 3 x y - y ^ { 3 }f(x,y)=x3−3xy−y3 .

Find the differential of z=3x+2yz = 3 x + 2 yz=3x+2y

Find lim⁡(x,y)→(1,1)x−y4x3−y4\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { x - y ^ { 4 } } { x ^ { 3 } - y ^ { 4 } }lim(x,y)→(1,1)​x3−y4x−y4​ if it exists, or show that the limit does not exist.

Let f(x,y)=3x3+y2−9x+4yf ( x , y ) = 3 x ^ { 3 } + y ^ { 2 } - 9 x + 4 yf(x,y)=3x3+y2−9x+4y . Find the direction at the origin in which f(x,y)f ( x , y )f(x,y) is decreasing the fastest.

The graph of level curves of f(x,y)f ( x , y )f(x,y) is given. Find a possible formula for f(x,y)f ( x , y )f(x,y) and sketch the surface z=f(x,y)z = f ( x , y )z=f(x,y) .

Find δzδy\frac { \delta z } { \delta y }δyδz​ for x3−y2z+sin⁡(xyz)=0x ^ { 3 } - y ^ { 2 } z + \sin ( x y z ) = 0x3−y2z+sin(xyz)=0

Let f(x,y,z)=(xyz)2f ( x , y , z ) = ( x y z ) ^ { 2 }f(x,y,z)=(xyz)2 . Find all second-order partial derivatives of fff .

Show that the limit lim⁡(x,y)→(0,0)2x2−y2x2+2y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 2 x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + 2 y ^ { 2 } }lim(x,y)→(0,0)​x2+2y22x2−y2​ does not exist.

Find an equation of the tangent plant to the surface z=ln⁡(2x+y)z = \ln ( 2 x + y )z=ln(2x+y) at the point (−1,3,0)( - 1,3,0 )(−1,3,0) .

Find two positive numbers who sum is eighteen and whose product is a maximum, using the method of Lagrange multipliers

Find the minimum value of the function f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2 subject to the constraint that xy=2x y = 2xy=2 .

The function f(x,y)=4−x2−y2−xy−xf ( x , y ) = 4 - x ^ { 2 } - y ^ { 2 } - x y - xf(x,y)=4−x2−y2−xy−x has one critical point. Determine its location and type.

Find the minimum value of the function f(x,y)=x2+92y2−9y−4xy+50f ( x , y ) = x ^ { 2 } + \frac { 9 } { 2 } y ^ { 2 } - 9 y - 4 x y + 50f(x,y)=x2+29​y2−9y−4xy+50

Describe the difference between the horizontal trace in z=kz = kz=k for the function z=f(x,y)z = f ( x , y )z=f(x,y) and the contour curve f(x,y)=kf ( x , y ) = kf(x,y)=k

A rectangular area of 3200 ft 222 is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of λ\lambdaλ . What does λ\lambdaλ represent?

Find the domain of the function f(x,y)=ex−y2f ( x , y ) = e ^ { x - y ^ { 2 } }f(x,y)=ex−y2 .

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Prove that the following limit does not exist: $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } f ( x , y )$ where $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 4 } }$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = x ^ { 3 } - 3 x y - y ^ { 3 }$ .

Find the differential of $z = 3 x + 2 y$

Find $\lim _ { ( x , y ) \rightarrow ( 1,1 ) } \frac { x - y ^ { 4 } } { x ^ { 3 } - y ^ { 4 } }$ if it exists, or show that the limit does not exist.

Let $f ( x , y ) = 3 x ^ { 3 } + y ^ { 2 } - 9 x + 4 y$ . Find the direction at the origin in which $f ( x , y )$ is decreasing the fastest.

The graph of level curves of $f ( x , y )$ is given. Find a possible formula for $f ( x , y )$ and sketch the surface $z = f ( x , y )$ .

Find $\frac { \delta z } { \delta y }$ for $x ^ { 3 } - y ^ { 2 } z + \sin ( x y z ) = 0$

Let $f ( x , y , z ) = ( x y z ) ^ { 2 }$ . Find all second-order partial derivatives of $f$ .

Show that the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 2 x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + 2 y ^ { 2 } }$ does not exist.

Find an equation of the tangent plant to the surface $z = \ln ( 2 x + y )$ at the point $( - 1,3,0 )$ .

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

The function $f ( x , y ) = 4 - x ^ { 2 } - y ^ { 2 } - x y - x$ has one critical point. Determine its location and type.

Find the minimum value of the function $f ( x , y ) = x ^ { 2 } + \frac { 9 } { 2 } y ^ { 2 } - 9 y - 4 x y + 50$

Describe the difference between the horizontal trace in $z = k$ for the function $z = f ( x , y )$ and the contour curve $f ( x , y ) = k$

A rectangular area of 3200 ft $2$ is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of $\lambda$ . What does $\lambda$ represent?

Find the domain of the function $f ( x , y ) = e ^ { x - y ^ { 2 } }$ .