Question 1

Let $f ( x , y ) = \sin ( 2 x + y )$ . Find the value of the partial derivative $f _ { x y } \left( \pi , \frac { \pi } { 2 } \right)$ .

Accepted Answer

A)  $\sqrt { 2 }$ 
B)  $- \sqrt { 2 }$ 
C)  $2 \sqrt { 2 }$ 
D)  $- 2 \sqrt { 2 }$ 
E)  $\frac { \sqrt { 2 } } { 2 }$ 
F) $- 2$

G)2

H)0 
A)  $\sqrt { 2 }$ 
B)  $- \sqrt { 2 }$ 
C)  $2 \sqrt { 2 }$ 
D)  $- 2 \sqrt { 2 }$ 
E)  $\frac { \sqrt { 2 } } { 2 }$ 
F) $- 2$

G)2

H)0

Question 2

For the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ , sketch the level curves $z = k$ for $k =$ 0, 1, 2, and 3.

Accepted Answer

The answer of For the function \(z = \sqrt {...

Question 3

Find the directional derivative of $f ( x , y ) = x ^ { 3 } + x y ^ { 2 }$ at the point $( 1 , - 2 )$ in the direction toward the origin.

Accepted Answer

The answer of Find the directional derivative of \(f (...

Question 4

Find an equation of the tangent plant to the surface $z = \frac { x } { y + 1 }$ at the point $\left( 1,4 , \frac { 1 } { 5 } \right)$ .

Accepted Answer

A)  $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y + \frac { 4 } { 25 }$ 
B)  $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$ 
C)  $z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$ 
D) 
$z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$ 
E)  $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$
F) $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y - \frac { 4 } { 25 }$

G) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y - \frac { 4 } { 25 }$

H) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y + \frac { 4 } { 25 }$ 
A)  $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y + \frac { 4 } { 25 }$ 
B)  $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$ 
C)  $z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$ 
D) 
$z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$ 
E)  $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$
F) $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y - \frac { 4 } { 25 }$

G) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y - \frac { 4 } { 25 }$

H) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y + \frac { 4 } { 25 }$

Question 5

Find a normal vector to the surface $x y z = 8$ at the point $( 1,2,4 )$ .

Accepted Answer

A)  $2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }$ 
B)  $\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$ 
C)  $4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }$ 
D)  $\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$ 
E)  $4 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ 
F) $2 \mathbf { k } + \mathbf { j } + 4 \mathbf { i }$ 
 
G) $\mathbf { i } + \mathbf { j } + \mathbf { k }$

H) $\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$ 
A)  $2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }$ 
B)  $\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$ 
C)  $4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }$ 
D)  $\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$ 
E)  $4 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ 
F) $2 \mathbf { k } + \mathbf { j } + 4 \mathbf { i }$ 
 
G) $\mathbf { i } + \mathbf { j } + \mathbf { k }$

H) $\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$

Question 6

Let $x ^ { 3 } + y ^ { 2 } = x ^ { 2 } y ^ { 3 }$ . Use implicit differentiation to find $\delta z / \delta x$ when $( x , y , z ) = ( 1 , - 2,1 )$ .

Accepted Answer

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

Question 7

A rancher intends to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed? Compute the value of $\lambda$ . What does $\lambda$ represent?

Accepted Answer

Minimum fe

Question 8

Find an equation of the tangent plant to the surface $z = \frac { x } { \sqrt { y } }$ at the point $( 4,4,2 )$ .

Accepted Answer

A)  $z = - \frac { 1 } { 2 } x + \frac { 1 } { 4 } y + 1$ 
B)  $z = - \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 1$ 
C)  $z = \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 2$ 
D)  $z = \frac { 1 } { 2 } x + \frac { 1 } { 4 } y + 1$ 
E)  $z = \frac { 1 } { 2 } x - \frac { 1 } { 4 } y - 1$ 
F) $z = \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 1$

G) $z = - \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 2$

H) $z = \frac { 1 } { 2 } x + \frac { 1 } { 4 } y + 2$ 
A)  $z = - \frac { 1 } { 2 } x + \frac { 1 } { 4 } y + 1$ 
B)  $z = - \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 1$ 
C)  $z = \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 2$ 
D)  $z = \frac { 1 } { 2 } x + \frac { 1 } { 4 } y + 1$ 
E)  $z = \frac { 1 } { 2 } x - \frac { 1 } { 4 } y - 1$ 
F) $z = \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 1$

G) $z = - \frac { 1 } { 2 } x - \frac { 1 } { 4 } y + 2$

H) $z = \frac { 1 } { 2 } x + \frac { 1 } { 4 } y + 2$

Question 9

Let $z = e ^ { x ^ { 2 } } \sin y$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 0 , y ( 1 ) = 0 , x ^ { \prime } ( 1 ) = 3$ , and $y ^ { \prime } ( 1 ) = 4$ . 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)7 
A) 0 
B) 1 
C) 2 
D) 3 
E) 4
F)5
G)6
H)7

Question 10

Let $f ( x , y ) = \frac { 1 } { x + y ^ { 2 } }$ . Find the gradient vector $\nabla f ( 1,1 )$ at the point $( x , y ) = ( 1,1 )$ .

Accepted Answer

A)  $- i - j$ 
B)  $- \frac { 1 } { 4 } \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$ 
C)  $- \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$ 
D)  $- \frac { 1 } { 2 } \mathbf { i } - \mathbf { j }$ 
E)  $\mathbf { i } + \mathbf { j }$
F) $\frac { 1 } { 4 } \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$

G) $\mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$

H) $\frac { 1 } { 2 } \mathbf { i } + \mathbf { j }$ 
A)  $- i - j$ 
B)  $- \frac { 1 } { 4 } \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$ 
C)  $- \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$ 
D)  $- \frac { 1 } { 2 } \mathbf { i } - \mathbf { j }$ 
E)  $\mathbf { i } + \mathbf { j }$
F) $\frac { 1 } { 4 } \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$

G) $\mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$

H) $\frac { 1 } { 2 } \mathbf { i } + \mathbf { j }$

Question 11

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

Accepted Answer

A)  $( 2,1 )$ , saddle point 
B)  $( 2,1 )$ , minimum point 
C)  $( 2,1 )$ , maximum point 
D)  $( \sqrt { 2 } , 1 )$ , saddle point 
E)  $( \sqrt { 2 } , 1 )$ , minimum point
F) $( \sqrt { 2 } , 1 )$ , maximum point
G) $( 0,0 )$ , saddle point
H) $( 0,0 )$ , minimum point 
A)  $( 2,1 )$ , saddle point 
B)  $( 2,1 )$ , minimum point 
C)  $( 2,1 )$ , maximum point 
D)  $( \sqrt { 2 } , 1 )$ , saddle point 
E)  $( \sqrt { 2 } , 1 )$ , minimum point
F) $( \sqrt { 2 } , 1 )$ , maximum point
G) $( 0,0 )$ , saddle point
H) $( 0,0 )$ , minimum point

Question 12

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y } { x ^ { 2 } + y ^ { 2 } }$

Accepted Answer

A) 1 
B) 0 
C)  $\frac { 1 } { 2 }$ 
D)  $\infty$ 
E) 2
F)4
G)8
H)Does not exist 
A) 1 
B) 0 
C)  $\frac { 1 } { 2 }$ 
D)  $\infty$ 
E) 2
F)4
G)8
H)Does not exist

Question 13

Evaluate $\lim _ { ( x , y , z ) \rightarrow ( 0,0,0 ) } \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { \sin \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }$

Accepted Answer

A)  $e ^ { - 4 }$ 
B)  $e ^ { - 2 }$ 
C)  $e ^ { - 1 }$ 
D) 1 
E)  $e ^ { 4 }$ 
F) $e ^ { 2 }$

G) $e$ 
H)Does not exist 
A)  $e ^ { - 4 }$ 
B)  $e ^ { - 2 }$ 
C)  $e ^ { - 1 }$ 
D) 1 
E)  $e ^ { 4 }$ 
F) $e ^ { 2 }$

G) $e$ 
H)Does not exist

Question 14

If $z ^ { 3 } + x z - y = 0$ , find $\frac { \delta ^ { 2 } z } { \delta x \delta y }$ in terms of $x$ , $y$ , and $Z$ .

Accepted Answer

The answer of If \(z ^ { 3 } +...

Question 15

Determine how many critical points the function $f ( x , y ) = x ^ { 4 } - 2 x ^ { 2 } + 3 y - y ^ { 3 }$ has.

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 16

Determine how many critical points the function $f ( x , y ) = x y - x ^ { 2 } y + x y ^ { 2 }$ has.

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 17

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

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) $\frac { y ^ { 2 } e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$

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

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) $\frac { y ^ { 2 } e ^ { xy ^ { 2 } } + 2 y } { x \sin ( x z ) }$

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

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

Question 18

Let $z = f ( x , y )$ , $x = r \cos \theta$ , and $y = r \sin \theta$ .(a) Show that $| \nabla f | ^ { 2 } = \left( \frac { \delta f } { \delta r } \right) ^ { 2 } + \frac { 1 } { r ^ { 2 } } \left( \frac { \delta f } { \delta \theta } \right) ^ { 2 }$ (b) Let $z = \ln \left( x ^ { 2 } + y ^ { 2 } \right)$ compute $| \nabla f |$ by using the formula in part (a).

Accepted Answer

(a) Answer

Question 19

Let $f ( x , y ) = \frac { 3 x - y } { x + y }$ . Find the value of the partial derivative $f _ { y } ( 1,1 )$ .

Accepted Answer

A)  $- 4$ 
B) 1 
C) 2 
D) 3 
E) 4
F) $- 1$ 
G) $- 2$ 
H) $- 3$ 
A)  $- 4$ 
B) 1 
C) 2 
D) 3 
E) 4
F) $- 1$ 
G) $- 2$ 
H) $- 3$

Question 20

Describe the level curves of the function $f ( x , y ) = \sqrt { x ^ { 2 } + ( y - 1 ) ^ { 2 } } - y$ .

Accepted Answer

A) Concentric circles 
B) Non-concentric circles 
C) Concentric ellipses (not circles) 
D) Non-concentric ellipses (not circles) 
E) Parabolas with the same vertex
F)Parabolas with the same focus
G)Hyperbolas with the same vertices
H)Hyperbolas with the same foci 
A) Concentric circles 
B) Non-concentric circles 
C) Concentric ellipses (not circles) 
D) Non-concentric ellipses (not circles) 
E) Parabolas with the same vertex
F)Parabolas with the same focus
G)Hyperbolas with the same vertices
H)Hyperbolas with the same foci

Let $f ( x , y ) = \sin ( 2 x + y )$ . Find the value of the partial derivative $f _ { x y } \left( \pi , \frac { \pi } { 2 } \right)$ .

For the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ , sketch the level curves $z = k$ for $k =$ 0, 1, 2, and 3.

Find the directional derivative of $f ( x , y ) = x ^ { 3 } + x y ^ { 2 }$ at the point $( 1 , - 2 )$ in the direction toward the origin.

Find an equation of the tangent plant to the surface $z = \frac { x } { y + 1 }$ at the point $\left( 1,4 , \frac { 1 } { 5 } \right)$ .

Find a normal vector to the surface $x y z = 8$ at the point $( 1,2,4 )$ .

Let $x ^ { 3 } + y ^ { 2 } = x ^ { 2 } y ^ { 3 }$ . Use implicit differentiation to find $\delta z / \delta x$ when $( x , y , z ) = ( 1 , - 2,1 )$ .

A rancher intends to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed? Compute the value of $\lambda$ . What does $\lambda$ represent?

Find an equation of the tangent plant to the surface $z = \frac { x } { \sqrt { y } }$ at the point $( 4,4,2 )$ .

Let $z = e ^ { x ^ { 2 } } \sin y$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 0 , y ( 1 ) = 0 , x ^ { \prime } ( 1 ) = 3$ , and $y ^ { \prime } ( 1 ) = 4$ . Find $d z / d t$ when $t = 1$ .

Let $f ( x , y ) = \frac { 1 } { x + y ^ { 2 } }$ . Find the gradient vector $\nabla f ( 1,1 )$ at the point $( x , y ) = ( 1,1 )$ .

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

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y } { x ^ { 2 } + y ^ { 2 } }$

Evaluate $\lim _ { ( x , y , z ) \rightarrow ( 0,0,0 ) } \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { \sin \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }$

If $z ^ { 3 } + x z - y = 0$ , find $\frac { \delta ^ { 2 } z } { \delta x \delta y }$ in terms of $x$ , $y$ , and $Z$ .

Determine how many critical points the function $f ( x , y ) = x ^ { 4 } - 2 x ^ { 2 } + 3 y - y ^ { 3 }$ has.

Determine how many critical points the function $f ( x , y ) = x y - x ^ { 2 } y + x y ^ { 2 }$ has.

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

Let $f ( x , y ) = \frac { 3 x - y } { x + y }$ . Find the value of the partial derivative $f _ { y } ( 1,1 )$ .

Describe the level curves of the function $f ( x , y ) = \sqrt { x ^ { 2 } + ( y - 1 ) ^ { 2 } } - y$ .

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Exam 11: Partial Derivatives

Let f(x,y)=sin⁡(2x+y)f ( x , y ) = \sin ( 2 x + y )f(x,y)=sin(2x+y) . Find the value of the partial derivative fxy(π,π2)f _ { x y } \left( \pi , \frac { \pi } { 2 } \right)fxy​(π,2π​) .

For the function z=x2+y2−1z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }z=x2+y2−1​ , sketch the level curves z=kz = kz=k for k=k =k= 0, 1, 2, and 3.

Find the directional derivative of f(x,y)=x3+xy2f ( x , y ) = x ^ { 3 } + x y ^ { 2 }f(x,y)=x3+xy2 at the point (1,−2)( 1 , - 2 )(1,−2) in the direction toward the origin.

Find an equation of the tangent plant to the surface z=xy+1z = \frac { x } { y + 1 }z=y+1x​ at the point (1,4,15)\left( 1,4 , \frac { 1 } { 5 } \right)(1,4,51​) .

Find a normal vector to the surface xyz=8x y z = 8xyz=8 at the point (1,2,4)( 1,2,4 )(1,2,4) .

Let x3+y2=x2y3x ^ { 3 } + y ^ { 2 } = x ^ { 2 } y ^ { 3 }x3+y2=x2y3 . Use implicit differentiation to find δz/δx\delta z / \delta xδz/δx when (x,y,z)=(1,−2,1)( x , y , z ) = ( 1 , - 2,1 )(x,y,z)=(1,−2,1) .

A rancher intends to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed? Compute the value of λ\lambdaλ . What does λ\lambdaλ represent?

Find an equation of the tangent plant to the surface z=xyz = \frac { x } { \sqrt { y } }z=y​x​ at the point (4,4,2)( 4,4,2 )(4,4,2) .

Let z=ex2sin⁡yz = e ^ { x ^ { 2 } } \sin yz=ex2siny , and let xxx and yyy be functions of ttt with x(1)=0,y(1)=0,x′(1)=3x ( 1 ) = 0 , y ( 1 ) = 0 , x ^ { \prime } ( 1 ) = 3x(1)=0,y(1)=0,x′(1)=3 , and y′(1)=4y ^ { \prime } ( 1 ) = 4y′(1)=4 . Find dz/dtd z / d tdz/dt when t=1t = 1t=1 .

Let f(x,y)=1x+y2f ( x , y ) = \frac { 1 } { x + y ^ { 2 } }f(x,y)=x+y21​ . Find the gradient vector ∇f(1,1)\nabla f ( 1,1 )∇f(1,1) at the point (x,y)=(1,1)( x , y ) = ( 1,1 )(x,y)=(1,1) .

The function f(x,y)=x2+y2+xyf ( x , y ) = x ^ { 2 } + y ^ { 2 } + x yf(x,y)=x2+y2+xy has one critical point. Determine its location and type.

Evaluate lim⁡(x,y)→(0,0)x2yx2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2x2y​

Evaluate lim⁡(x,y,z)→(0,0,0)x2+y2+z2sin⁡(x2+y2+z2)\lim _ { ( x , y , z ) \rightarrow ( 0,0,0 ) } \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { \sin \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }lim(x,y,z)→(0,0,0)​sin(x2+y2+z2)x2+y2+z2​

If z3+xz−y=0z ^ { 3 } + x z - y = 0z3+xz−y=0 , find δ2zδxδy\frac { \delta ^ { 2 } z } { \delta x \delta y }δxδyδ2z​ in terms of xxx , yyy , and ZZZ .

Determine how many critical points the function f(x,y)=x4−2x2+3y−y3f ( x , y ) = x ^ { 4 } - 2 x ^ { 2 } + 3 y - y ^ { 3 }f(x,y)=x4−2x2+3y−y3 has.

Determine how many critical points 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.

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

Let f(x,y)=3x−yx+yf ( x , y ) = \frac { 3 x - y } { x + y }f(x,y)=x+y3x−y​ . Find the value of the partial derivative fy(1,1)f _ { y } ( 1,1 )fy​(1,1) .

Describe the level curves of the function f(x,y)=x2+(y−1)2−yf ( x , y ) = \sqrt { x ^ { 2 } + ( y - 1 ) ^ { 2 } } - yf(x,y)=x2+(y−1)2​−y .

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Let $f ( x , y ) = \sin ( 2 x + y )$ . Find the value of the partial derivative $f _ { x y } \left( \pi , \frac { \pi } { 2 } \right)$ .

For the function $z = \sqrt { x ^ { 2 } + y ^ { 2 } - 1 }$ , sketch the level curves $z = k$ for $k =$ 0, 1, 2, and 3.

Find the directional derivative of $f ( x , y ) = x ^ { 3 } + x y ^ { 2 }$ at the point $( 1 , - 2 )$ in the direction toward the origin.

Find an equation of the tangent plant to the surface $z = \frac { x } { y + 1 }$ at the point $\left( 1,4 , \frac { 1 } { 5 } \right)$ .

Find a normal vector to the surface $x y z = 8$ at the point $( 1,2,4 )$ .

Let $x ^ { 3 } + y ^ { 2 } = x ^ { 2 } y ^ { 3 }$ . Use implicit differentiation to find $\delta z / \delta x$ when $( x , y , z ) = ( 1 , - 2,1 )$ .

A rancher intends to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed? Compute the value of $\lambda$ . What does $\lambda$ represent?

Find an equation of the tangent plant to the surface $z = \frac { x } { \sqrt { y } }$ at the point $( 4,4,2 )$ .

Let $z = e ^ { x ^ { 2 } } \sin y$ , and let $x$ and $y$ be functions of $t$ with $x ( 1 ) = 0 , y ( 1 ) = 0 , x ^ { \prime } ( 1 ) = 3$ , and $y ^ { \prime } ( 1 ) = 4$ . Find $d z / d t$ when $t = 1$ .

Let $f ( x , y ) = \frac { 1 } { x + y ^ { 2 } }$ . Find the gradient vector $\nabla f ( 1,1 )$ at the point $( x , y ) = ( 1,1 )$ .

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

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } y } { x ^ { 2 } + y ^ { 2 } }$

Evaluate $\lim _ { ( x , y , z ) \rightarrow ( 0,0,0 ) } \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { \sin \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }$

If $z ^ { 3 } + x z - y = 0$ , find $\frac { \delta ^ { 2 } z } { \delta x \delta y }$ in terms of $x$ , $y$ , and $Z$ .

Determine how many critical points the function $f ( x , y ) = x ^ { 4 } - 2 x ^ { 2 } + 3 y - y ^ { 3 }$ has.

Determine how many critical points the function $f ( x , y ) = x y - x ^ { 2 } y + x y ^ { 2 }$ has.

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

Let $f ( x , y ) = \frac { 3 x - y } { x + y }$ . Find the value of the partial derivative $f _ { y } ( 1,1 )$ .

Describe the level curves of the function $f ( x , y ) = \sqrt { x ^ { 2 } + ( y - 1 ) ^ { 2 } } - y$ .