Question 1

Sketch the domain of the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ .

Accepted Answer

The answer of Sketch the domain of the function \(z...

Question 2

Let $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .(a) Sketch the intersection of $z = f ( x , y )$ and $z = 1$ in the $x y$ -plane.(b) Sketch the intersection of $z = f ( x , y )$ and $x = 0$ in the $y z$ -plane.(c) Sketch the intersection of $z = f ( x , y )$ and $y = 1$ in the $x z$ -plane.(d) Sketch the graph of $z = f ( x , y )$ in $\mathbb { R } ^ { 3 }$ .

Accepted Answer

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

Question 3

What is the shortest distance from the origin to the surface $x y z ^ { 2 } = 2$ ?

Accepted Answer

The answer of What is the shortest distance from the...

Question 4

Let $f ( x , y ) = \left\{ \begin{array} { l l } 
\frac { x ^ { 3 } y ^ { 2 } } { x ^ { 4 } + y ^ { 4 } } & \text { if } ( x , y ) \neq ( 0,0 ) \
0 & \text { if } ( x , y ) = ( 0,0 )
\end{array} \right.$ Where is $f$ continuous?

Accepted Answer

Continuous

Question 5

Find the shortest distance from the origin to the surface $z ^ { 2 } = 2 x y + 2$ .

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 6

Let $e ^ { xy ^ { 2 } } + \cos ( x z ) + y ^ { 2 } = 10$ , find $\frac { \delta z } { \delta x }$ .

Accepted Answer

A)  $\frac { 2 x y e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$ 
B)  $- \frac { y ^ { 2 } e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$ 
C)  $\frac { y ^ { 2 } e ^ { xy ^ { 2 } } - z \sin ( x z ) } { x \sin ( x z ) }$ 
D)  $- \frac { x \sin ( x z ) } { y ^ { 2 } e ^ { x y ^ { 2 } } - z \sin ( x z ) }$ 
E)  $- \frac { 2 x y e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$ 
F) $\frac { y ^ { 2 } e ^ { x ^ { 2 } } + 2 y } { x \sin ( x z ) }$

G) $- \frac { y ^ { 2 } e ^ { x ^ { 2 } } - z \sin ( x z ) } { x \sin ( x z ) }$

H) $\frac { x \sin ( x z ) } { y ^ { 2 } e ^ { x y ^ { 2 } } - z \sin ( x z ) }$ 
A)  $\frac { 2 x y e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$ 
B)  $- \frac { y ^ { 2 } e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$ 
C)  $\frac { y ^ { 2 } e ^ { xy ^ { 2 } } - z \sin ( x z ) } { x \sin ( x z ) }$ 
D)  $- \frac { x \sin ( x z ) } { y ^ { 2 } e ^ { x y ^ { 2 } } - z \sin ( x z ) }$ 
E)  $- \frac { 2 x y e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$ 
F) $\frac { y ^ { 2 } e ^ { x ^ { 2 } } + 2 y } { x \sin ( x z ) }$

G) $- \frac { y ^ { 2 } e ^ { x ^ { 2 } } - z \sin ( x z ) } { x \sin ( x z ) }$

H) $\frac { x \sin ( x z ) } { y ^ { 2 } e ^ { x y ^ { 2 } } - z \sin ( x z ) }$

Question 7

Let $f ( x , y , z ) = 2 z \ln \left( x ^ { 2 } y z \right)$ . Find $f _ { z } \left( \frac { 1 } { 2 } , 1,4 \right)$ .

Accepted Answer

A) 0 
B)  $2 \ln 2$ 
C) 12 
D) 2 
E) 4
F) $\ln 2$ 
G)16
H)8 
A) 0 
B)  $2 \ln 2$ 
C) 12 
D) 2 
E) 4
F) $\ln 2$ 
G)16
H)8

Question 8

Use differentials to approximate $\sqrt { 24 } + \sqrt { 5 }$ .

Accepted Answer

A) 6.85 
B) 6.90 
C) 6.95 
D) 7.00 
E) 7.05
F)7.10
G)7.15
H)7.20 
A) 6.85 
B) 6.90 
C) 6.95 
D) 7.00 
E) 7.05
F)7.10
G)7.15
H)7.20

Question 9

Let $f ( x , y ) = x ^ { 2 } y ^ { 3 }$ . Find the value of the partial derivative $f _ { x } ( 1,1 )$ .

Accepted Answer

A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)7 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)7

Question 10

Find an equation of the tangent plane to the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4$ at the point $( 1,0,1 )$ .

Accepted Answer

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

G) $x + y - z = 0$

H) $5 x + 3 y - 2 z = 3$ 
A)  $3 x - y + z = 4$ 
B)  $2 x - 4 y + z = 3$ 
C)  $x + 2 y + 3 z = 4$ 
D)  $2 x + y - z = 1$ 
E)  $3 x + 2 y - z = 2$ 
F) $4 x - y + 2 z = 6$

G) $x + y - z = 0$

H) $5 x + 3 y - 2 z = 3$

Question 11

Find the direction $\theta$ in which the directional derivative of the function $f ( x , y ) = x y + y ^ { 2 }$ at the point $( 1,1 )$ is maximum.

Accepted Answer

A)  $\cot ^ { - 1 } 2$ 
B) 
$\cot ^ { - 1 } 3$ 
C)  $\cos ^ { - 1 } \frac { 1 } { 2 }$ 
D)  $\cos ^ { - 1 } \frac { 1 } { 3 }$ 
E)  $\sin ^ { - 1 } \frac { 1 } { 2 }$
F)
$\sin ^ { - 1 } \frac { 1 } { 3 }$

G)
$\tan ^ { - 1 } 2$

H) $\tan ^ { - 1 } 3$ 
A)  $\cot ^ { - 1 } 2$ 
B) 
$\cot ^ { - 1 } 3$ 
C)  $\cos ^ { - 1 } \frac { 1 } { 2 }$ 
D)  $\cos ^ { - 1 } \frac { 1 } { 3 }$ 
E)  $\sin ^ { - 1 } \frac { 1 } { 2 }$
F)
$\sin ^ { - 1 } \frac { 1 } { 3 }$

G)
$\tan ^ { - 1 } 2$

H) $\tan ^ { - 1 } 3$

Question 12

Identify the graph of the function $f ( x , y ) = \sqrt { 3 - x ^ { 2 } + y ^ { 2 } }$ .

Accepted Answer

A) Cone 
B) Paraboloid 
C) Ellipsoid 
D) Hyperboloid of one sheet 
E) Hyperboloid of two sheets
F)Hyperbolic cylinder
G)Elliptic cylinder
H)Parabolic cylinder 
A) Cone 
B) Paraboloid 
C) Ellipsoid 
D) Hyperboloid of one sheet 
E) Hyperboloid of two sheets
F)Hyperbolic cylinder
G)Elliptic cylinder
H)Parabolic cylinder

Question 13

What point on the surface $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = 1$ , $x > 0$ , $y > 0$ , $z > 0$ is closest to the origin?

Accepted Answer

The answer of What point on the surface \(\frac {...

Question 14

Find the point at which the function $f ( x , y ) = x y - x ^ { 2 } y - x y ^ { 2 }$ has a local maximum.

Accepted Answer

A)  $( 0,0 )$ 
B)  $( 1,1 )$ 
C)  $( 0,2 )$ 
D)  $( 2,0 )$ 
E)  $( - 1,0 )$
F) $( 0 , - 1 )$

G) $\left( \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$ 
 
H) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ 
A)  $( 0,0 )$ 
B)  $( 1,1 )$ 
C)  $( 0,2 )$ 
D)  $( 2,0 )$ 
E)  $( - 1,0 )$
F) $( 0 , - 1 )$

G) $\left( \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$ 
 
H) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

Question 15

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x } { \sin y }$

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 16

Describe how the graph of $g ( x , y ) = 2 f ( x - 1 , y - 1 )$ is obtained from the graph of $z = f ( x , y )$ .

Accepted Answer

The graph of @#IMG-DLM& is the graph of

Question 17

If $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ then $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } f ( x , y )$

Accepted Answer

A) exists 
B) does not exist 
C) is equal to 0 
D) is equal to $\frac { 1 } { 2 }$ 
E) is equal to 1 
A) exists 
B) does not exist 
C) is equal to 0 
D) is equal to $\frac { 1 } { 2 }$ 
E) is equal to 1

Question 18

Let $w = x y z$ , $x = s + t$ , $y = s ^ { 2 } - t ^ { 2 }$ and $z = s t$ . Use the chain rule to show that $s \left( \frac { \delta w } { \delta s } \right) + t \left( \frac { \delta w } { \delta t } \right) = 5 x y z$ .

Accepted Answer

To show that s \left( \frac { \delta w }

Question 19

Show that there does not exist any function $f ( x , y )$ with continuous second partial derivatives such that $\frac { \delta f } { \delta x } = 2 y$ and $\frac { \partial f } { \partial y } = - 2 x$

Accepted Answer

@#IMG-DLM& which con

Question 20

Let $f ( x , y ) = \frac { x } { y } + \frac { y } { x }$ . Find the gradient vector $\nabla f$ .

Accepted Answer

A)  $2 x \mathbf { i } + 2 y \mathbf { j }$ 
B)  $2 y \mathbf { i } + 2 x \mathbf { j }$ 
C)  $- \frac { y } { x ^ { 2 } } \mathbf { i } - \frac { x } { y ^ { 2 } } \mathbf { j }$ 
D)  $\frac { 1 } { y } \mathbf { i } + \frac { 1 } { x } \mathbf { j }$ 
E)  $x \mathbf { i } + y \mathbf { j }$  
F) $y \mathbf { i } + x \mathbf { j }$

G) $\frac { y } { x ^ { 2 } } \mathbf { i } + \frac { x } { y ^ { 2 } } \mathbf { j }$

H) $\left( \frac { 1 } { y } - \frac { y } { x ^ { 2 } } \right) \mathbf { i } + \left( \frac { 1 } { x } - \frac { x } { y ^ { 2 } } \right) \mathbf { j }$ 
A)  $2 x \mathbf { i } + 2 y \mathbf { j }$ 
B)  $2 y \mathbf { i } + 2 x \mathbf { j }$ 
C)  $- \frac { y } { x ^ { 2 } } \mathbf { i } - \frac { x } { y ^ { 2 } } \mathbf { j }$ 
D)  $\frac { 1 } { y } \mathbf { i } + \frac { 1 } { x } \mathbf { j }$ 
E)  $x \mathbf { i } + y \mathbf { j }$  
F) $y \mathbf { i } + x \mathbf { j }$

G) $\frac { y } { x ^ { 2 } } \mathbf { i } + \frac { x } { y ^ { 2 } } \mathbf { j }$

H) $\left( \frac { 1 } { y } - \frac { y } { x ^ { 2 } } \right) \mathbf { i } + \left( \frac { 1 } { x } - \frac { x } { y ^ { 2 } } \right) \mathbf { j }$

Sketch the domain of the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ .

What is the shortest distance from the origin to the surface $x y z ^ { 2 } = 2$ ?

Let $f ( x , y ) = \left\{ \begin{array} { l l } \frac { x ^ { 3 } y ^ { 2 } } { x ^ { 4 } + y ^ { 4 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\0 & \text { if } ( x , y ) = ( 0,0 )\end{array} \right.$ Where is $f$ continuous?

Find the shortest distance from the origin to the surface $z ^ { 2 } = 2 x y + 2$ .

Let $e ^ { xy ^ { 2 } } + \cos ( x z ) + y ^ { 2 } = 10$ , find $\frac { \delta z } { \delta x }$ .

Let $f ( x , y , z ) = 2 z \ln \left( x ^ { 2 } y z \right)$ . Find $f _ { z } \left( \frac { 1 } { 2 } , 1,4 \right)$ .

Use differentials to approximate $\sqrt { 24 } + \sqrt { 5 }$ .

Let $f ( x , y ) = x ^ { 2 } y ^ { 3 }$ . Find the value of the partial derivative $f _ { x } ( 1,1 )$ .

Find an equation of the tangent plane to the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4$ at the point $( 1,0,1 )$ .

Find the direction $\theta$ in which the directional derivative of the function $f ( x , y ) = x y + y ^ { 2 }$ at the point $( 1,1 )$ is maximum.

Identify the graph of the function $f ( x , y ) = \sqrt { 3 - x ^ { 2 } + y ^ { 2 } }$ .

What point on the surface $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = 1$ , $x > 0$ , $y > 0$ , $z > 0$ is closest to the origin?

Find the point at which the function $f ( x , y ) = x y - x ^ { 2 } y - x y ^ { 2 }$ has a local maximum.

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x } { \sin y }$

Describe how the graph of $g ( x , y ) = 2 f ( x - 1 , y - 1 )$ is obtained from the graph of $z = f ( x , y )$ .

If $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ then $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } f ( x , y )$

Let $w = x y z$ , $x = s + t$ , $y = s ^ { 2 } - t ^ { 2 }$ and $z = s t$ . Use the chain rule to show that $s \left( \frac { \delta w } { \delta s } \right) + t \left( \frac { \delta w } { \delta t } \right) = 5 x y z$ .

Show that there does not exist any function $f ( x , y )$ with continuous second partial derivatives such that $\frac { \delta f } { \delta x } = 2 y$ and $\frac { \partial f } { \partial y } = - 2 x$

Let $f ( x , y ) = \frac { x } { y } + \frac { y } { x }$ . Find the gradient vector $\nabla f$ .

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Exam 11: Partial Derivatives

Sketch the domain of the function z=x2+y2−1z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }z=x2+y2−1​ .

What is the shortest distance from the origin to the surface xyz2=2x y z ^ { 2 } = 2xyz2=2 ?

Find the shortest distance from the origin to the surface z2=2xy+2z ^ { 2 } = 2 x y + 2z2=2xy+2 .

Let exy2+cos⁡(xz)+y2=10e ^ { xy ^ { 2 } } + \cos ( x z ) + y ^ { 2 } = 10exy2+cos(xz)+y2=10 , find δzδx\frac { \delta z } { \delta x }δxδz​ .

Let f(x,y,z)=2zln⁡(x2yz)f ( x , y , z ) = 2 z \ln \left( x ^ { 2 } y z \right)f(x,y,z)=2zln(x2yz) . Find fz(12,1,4)f _ { z } \left( \frac { 1 } { 2 } , 1,4 \right)fz​(21​,1,4) .

Use differentials to approximate 24+5\sqrt { 24 } + \sqrt { 5 }24​+5​ .

Let f(x,y)=x2y3f ( x , y ) = x ^ { 2 } y ^ { 3 }f(x,y)=x2y3 . Find the value of the partial derivative fx(1,1)f _ { x } ( 1,1 )fx​(1,1) .

Find an equation of the tangent plane to the hyperboloid x2+y2−z2−2xy+4xz=4x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4x2+y2−z2−2xy+4xz=4 at the point (1,0,1)( 1,0,1 )(1,0,1) .

Find the direction θ\thetaθ in which the directional derivative of the function f(x,y)=xy+y2f ( x , y ) = x y + y ^ { 2 }f(x,y)=xy+y2 at the point (1,1)( 1,1 )(1,1) is maximum.

Identify the graph of the function f(x,y)=3−x2+y2f ( x , y ) = \sqrt { 3 - x ^ { 2 } + y ^ { 2 } }f(x,y)=3−x2+y2​ .

What point on the surface 1x+1y+1z=1\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = 1x1​+y1​+z1​=1 , x>0x > 0x>0 , y>0y > 0y>0 , z>0z > 0z>0 is closest to the origin?

Find the point at which the function f(x,y)=xy−x2y−xy2f ( x , y ) = x y - x ^ { 2 } y - x y ^ { 2 }f(x,y)=xy−x2y−xy2 has a local maximum.

Evaluate lim⁡(x,y)→(0,0)xsin⁡y\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x } { \sin y }lim(x,y)→(0,0)​sinyx​

Describe how the graph of g(x,y)=2f(x−1,y−1)g ( x , y ) = 2 f ( x - 1 , y - 1 )g(x,y)=2f(x−1,y−1) is obtained from the graph of z=f(x,y)z = f ( x , y )z=f(x,y) .

If f(x,y)=xyx2+y2f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }f(x,y)=x2+y2xy​ then 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)

Show that there does not exist any function f(x,y)f ( x , y )f(x,y) with continuous second partial derivatives such that δfδx=2y\frac { \delta f } { \delta x } = 2 yδxδf​=2y and ∂f∂y=−2x\frac { \partial f } { \partial y } = - 2 x∂y∂f​=−2x

Let f(x,y)=xy+yxf ( x , y ) = \frac { x } { y } + \frac { y } { x }f(x,y)=yx​+xy​ . Find the gradient vector ∇f\nabla f∇f .

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Sketch the domain of the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ .

What is the shortest distance from the origin to the surface $x y z ^ { 2 } = 2$ ?

Find the shortest distance from the origin to the surface $z ^ { 2 } = 2 x y + 2$ .

Let $e ^ { xy ^ { 2 } } + \cos ( x z ) + y ^ { 2 } = 10$ , find $\frac { \delta z } { \delta x }$ .

Let $f ( x , y , z ) = 2 z \ln \left( x ^ { 2 } y z \right)$ . Find $f _ { z } \left( \frac { 1 } { 2 } , 1,4 \right)$ .

Use differentials to approximate $\sqrt { 24 } + \sqrt { 5 }$ .

Let $f ( x , y ) = x ^ { 2 } y ^ { 3 }$ . Find the value of the partial derivative $f _ { x } ( 1,1 )$ .

Find an equation of the tangent plane to the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4$ at the point $( 1,0,1 )$ .

Find the direction $\theta$ in which the directional derivative of the function $f ( x , y ) = x y + y ^ { 2 }$ at the point $( 1,1 )$ is maximum.

Identify the graph of the function $f ( x , y ) = \sqrt { 3 - x ^ { 2 } + y ^ { 2 } }$ .

What point on the surface $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = 1$ , $x > 0$ , $y > 0$ , $z > 0$ is closest to the origin?

Find the point at which the function $f ( x , y ) = x y - x ^ { 2 } y - x y ^ { 2 }$ has a local maximum.

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x } { \sin y }$

Describe how the graph of $g ( x , y ) = 2 f ( x - 1 , y - 1 )$ is obtained from the graph of $z = f ( x , y )$ .

If $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ then $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } f ( x , y )$

Show that there does not exist any function $f ( x , y )$ with continuous second partial derivatives such that $\frac { \delta f } { \delta x } = 2 y$ and $\frac { \partial f } { \partial y } = - 2 x$

Let $f ( x , y ) = \frac { x } { y } + \frac { y } { x }$ . Find the gradient vector $\nabla f$ .