Question 1

Given $f ( x , y ) = x e ^ { - 2 y }$ , at the point $( 4,0 )$ find the equation of the tangent plane to the graph of $f$ . Also, find a normal vector to the graph of $f$ .

Accepted Answer

@#IMG-DLM& ; two nor

Question 2

Use a linear approximation to estimate $\frac { 1.02 ^ { 2 } } { 1.98 ^ { 3 } + 1 }$ .

Accepted Answer

The answer of Use a linear approximation to estimate \(\frac...

Question 3

Let $f ( x , y , z ) = ( x - y ) ( y - z ) ( z - x )$ . Compute $\frac { \delta f } { \delta x } + \frac { \delta f } { \delta y } + \frac { \delta f } { \delta z }$ .

Accepted Answer

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

Question 4

Optimize $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 2$ subject to $x y = 4$ .

Accepted Answer

@#IMG-DLM& is a loca

Question 5

The function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x y$ 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 6

Find a normal vector to the surface with parametric equations $x = 2 u v , y = u ^ { 2 } - v ^ { 2 } , z = u ^ { 2 } + v ^ { 2 }$ at the point $( 1,0,1 )$ .

Accepted Answer

The answer of Find a normal vector to the surface...

Question 7

Let $f ( x , y , z ) = x ^ { 2 } y ^ { 3 } - \frac { x } { z } + e ^ { x } \ln y$ . Find $f _ { x }$ , $f _ { y }$ , and $f _ { z }$ .

Accepted Answer

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

Question 8

In which direction does the directional derivative for $f ( x , y ) = x ^ { 2 } y$ at $( 1,1 )$ have value 1? Value $- 2$ ? Is there a direction for which the value is 4?

Accepted Answer

Value 1 in direction

Question 9

Show that there does not exist any function   with continuous second partial derivatives in $\mathbb { R } ^ { 2 }$ such that $\frac { \delta f } { \delta x } = x ^ { 2 } y$ and $\frac { \partial f } { \partial y } = x$ .

Accepted Answer

@#IMG-DLM& for @#IMG-DLM& whic

Question 10

Evaluate $\lim _ { ( x , y , z ) \rightarrow ( 1,0 , - 2 ) } e ^ { x y - z ^ { 2 } }$

Accepted Answer

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

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

G) $\theta$ 
H)Does not exist

Question 11

Let $z = f ( x , y )$ such that $\nabla f ( - 1,2 ) = ( 3 , - 4 )$ . Find an equation of the tangent line to the level curve of $f ( x , y )$ that passes through point $( - 1,2 )$ .

Accepted Answer

The answer of Let \(z = f ( x ,...

Question 12

Evaluate $\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { \tan \left( ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \right) } { ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } }$

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 13

Given $f ( x , y ) = \frac { 1 } { 4 } x ^ { 4 } y ^ { 3 }$ , at the point $( - 1,2 )$ . Find
(a) the maximum value of the directional derivative and
(b) the unit vector in the direction in which the directional derivative takes on its maximum value.

Accepted Answer

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

Question 14

Let $f ( x , y , z ) = z ^ { x y }$ , $z > 0$ . Find the value of the partial derivative $f _ { y } ( 2,1 , e )$ .

Accepted Answer

A) 0 
B) 1 
C)  $e$ 
D)  $2 e$ 
E)  $2 e ^ { 2 }$
F) $\frac { 1 } { 2 }$ 
G) $\ln 2$ 
H)2 
A) 0 
B) 1 
C)  $e$ 
D)  $2 e$ 
E)  $2 e ^ { 2 }$
F) $\frac { 1 } { 2 }$ 
G) $\ln 2$ 
H)2

Given $f ( x , y ) = x e ^ { - 2 y }$ , at the point $( 4,0 )$ find the equation of the tangent plane to the graph of $f$ . Also, find a normal vector to the graph of $f$ .

Use a linear approximation to estimate $\frac { 1.02 ^ { 2 } } { 1.98 ^ { 3 } + 1 }$ .

Let $f ( x , y , z ) = ( x - y ) ( y - z ) ( z - x )$ . Compute $\frac { \delta f } { \delta x } + \frac { \delta f } { \delta y } + \frac { \delta f } { \delta z }$ .

Optimize $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 2$ subject to $x y = 4$ .

The function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x y$ has one critical point. Determine its location and type.

Find a normal vector to the surface with parametric equations $x = 2 u v , y = u ^ { 2 } - v ^ { 2 } , z = u ^ { 2 } + v ^ { 2 }$ at the point $( 1,0,1 )$ .

Let $f ( x , y , z ) = x ^ { 2 } y ^ { 3 } - \frac { x } { z } + e ^ { x } \ln y$ . Find $f _ { x }$ , $f _ { y }$ , and $f _ { z }$ .

In which direction does the directional derivative for $f ( x , y ) = x ^ { 2 } y$ at $( 1,1 )$ have value 1? Value $- 2$ ? Is there a direction for which the value is 4?

Evaluate $\lim _ { ( x , y , z ) \rightarrow ( 1,0 , - 2 ) } e ^ { x y - z ^ { 2 } }$

Let $z = f ( x , y )$ such that $\nabla f ( - 1,2 ) = ( 3 , - 4 )$ . Find an equation of the tangent line to the level curve of $f ( x , y )$ that passes through point $( - 1,2 )$ .

Evaluate $\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { \tan \left( ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \right) } { ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } }$

Given $f ( x , y ) = \frac { 1 } { 4 } x ^ { 4 } y ^ { 3 }$ , at the point $( - 1,2 )$ . Find (a) the maximum value of the directional derivative and (b) the unit vector in the direction in which the directional derivative takes on its maximum value.

Let $f ( x , y , z ) = z ^ { x y }$ , $z > 0$ . Find the value of the partial derivative $f _ { y } ( 2,1 , e )$ .

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Exam 11: Partial Derivatives

Given f(x,y)=xe−2yf ( x , y ) = x e ^ { - 2 y }f(x,y)=xe−2y , at the point (4,0)( 4,0 )(4,0) find the equation of the tangent plane to the graph of fff . Also, find a normal vector to the graph of fff .

Use a linear approximation to estimate 1.0221.983+1\frac { 1.02 ^ { 2 } } { 1.98 ^ { 3 } + 1 }1.983+11.022​ .

Let f(x,y,z)=(x−y)(y−z)(z−x)f ( x , y , z ) = ( x - y ) ( y - z ) ( z - x )f(x,y,z)=(x−y)(y−z)(z−x) . Compute δfδx+δfδy+δfδz\frac { \delta f } { \delta x } + \frac { \delta f } { \delta y } + \frac { \delta f } { \delta z }δxδf​+δyδf​+δzδf​ .

Optimize f(x,y)=x2+y2+2f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 2f(x,y)=x2+y2+2 subject to xy=4x y = 4xy=4 .

The function f(x,y)=x2+y2+3xyf ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x yf(x,y)=x2+y2+3xy has one critical point. Determine its location and type.

Find a normal vector to the surface with parametric equations x=2uv,y=u2−v2,z=u2+v2x = 2 u v , y = u ^ { 2 } - v ^ { 2 } , z = u ^ { 2 } + v ^ { 2 }x=2uv,y=u2−v2,z=u2+v2 at the point (1,0,1)( 1,0,1 )(1,0,1) .

Let f(x,y,z)=x2y3−xz+exln⁡yf ( x , y , z ) = x ^ { 2 } y ^ { 3 } - \frac { x } { z } + e ^ { x } \ln yf(x,y,z)=x2y3−zx​+exlny . Find fxf _ { x }fx​ , fyf _ { y }fy​ , and fzf _ { z }fz​ .

In which direction does the directional derivative for f(x,y)=x2yf ( x , y ) = x ^ { 2 } yf(x,y)=x2y at (1,1)( 1,1 )(1,1) have value 1? Value −2- 2−2 ? Is there a direction for which the value is 4?

Show that there does not exist any function with continuous second partial derivatives in R2\mathbb { R } ^ { 2 }R2 such that δfδx=x2y\frac { \delta f } { \delta x } = x ^ { 2 } yδxδf​=x2y and ∂f∂y=x\frac { \partial f } { \partial y } = x∂y∂f​=x .

Evaluate lim⁡(x,y,z)→(1,0,−2)exy−z2\lim _ { ( x , y , z ) \rightarrow ( 1,0 , - 2 ) } e ^ { x y - z ^ { 2 } }lim(x,y,z)→(1,0,−2)​exy−z2

Let z=f(x,y)z = f ( x , y )z=f(x,y) such that ∇f(−1,2)=(3,−4)\nabla f ( - 1,2 ) = ( 3 , - 4 )∇f(−1,2)=(3,−4) . Find an equation of the tangent line to the level curve of f(x,y)f ( x , y )f(x,y) that passes through point (−1,2)( - 1,2 )(−1,2) .

Evaluate lim⁡(x,y)→(2,2)tan⁡((x−2)2+(y−2)2)(x−2)2+(y−2)2\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { \tan \left( ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \right) } { ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } }lim(x,y)→(2,2)​(x−2)2+(y−2)2tan((x−2)2+(y−2)2)​

Given f(x,y)=14x4y3f ( x , y ) = \frac { 1 } { 4 } x ^ { 4 } y ^ { 3 }f(x,y)=41​x4y3 , at the point (−1,2)( - 1,2 )(−1,2) . Find (a) the maximum value of the directional derivative and (b) the unit vector in the direction in which the directional derivative takes on its maximum value.

Let f(x,y,z)=zxyf ( x , y , z ) = z ^ { x y }f(x,y,z)=zxy , z>0z > 0z>0 . Find the value of the partial derivative fy(2,1,e)f _ { y } ( 2,1 , e )fy​(2,1,e) .

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Given $f ( x , y ) = x e ^ { - 2 y }$ , at the point $( 4,0 )$ find the equation of the tangent plane to the graph of $f$ . Also, find a normal vector to the graph of $f$ .

Use a linear approximation to estimate $\frac { 1.02 ^ { 2 } } { 1.98 ^ { 3 } + 1 }$ .

Let $f ( x , y , z ) = ( x - y ) ( y - z ) ( z - x )$ . Compute $\frac { \delta f } { \delta x } + \frac { \delta f } { \delta y } + \frac { \delta f } { \delta z }$ .

Optimize $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 2$ subject to $x y = 4$ .

The function $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + 3 x y$ has one critical point. Determine its location and type.

Find a normal vector to the surface with parametric equations $x = 2 u v , y = u ^ { 2 } - v ^ { 2 } , z = u ^ { 2 } + v ^ { 2 }$ at the point $( 1,0,1 )$ .

Let $f ( x , y , z ) = x ^ { 2 } y ^ { 3 } - \frac { x } { z } + e ^ { x } \ln y$ . Find $f _ { x }$ , $f _ { y }$ , and $f _ { z }$ .

In which direction does the directional derivative for $f ( x , y ) = x ^ { 2 } y$ at $( 1,1 )$ have value 1? Value $- 2$ ? Is there a direction for which the value is 4?

Evaluate $\lim _ { ( x , y , z ) \rightarrow ( 1,0 , - 2 ) } e ^ { x y - z ^ { 2 } }$

Let $z = f ( x , y )$ such that $\nabla f ( - 1,2 ) = ( 3 , - 4 )$ . Find an equation of the tangent line to the level curve of $f ( x , y )$ that passes through point $( - 1,2 )$ .

Evaluate $\lim _ { ( x , y ) \rightarrow ( 2,2 ) } \frac { \tan \left( ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \right) } { ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } }$

Given $f ( x , y ) = \frac { 1 } { 4 } x ^ { 4 } y ^ { 3 }$ , at the point $( - 1,2 )$ . Find (a) the maximum value of the directional derivative and (b) the unit vector in the direction in which the directional derivative takes on its maximum value.

Let $f ( x , y , z ) = z ^ { x y }$ , $z > 0$ . Find the value of the partial derivative $f _ { y } ( 2,1 , e )$ .