Question 1

Compute the minimum value of $f ( x , y , z ) = x ^ { 2 } + y + z ^ { 2 }$ subject to the condition that $g ( x ) = 2 x + y + 4 z = 6$ .

Accepted Answer

The answer of Compute the minimum value of \(f (...

Question 2

Let $f ( x , y , z ) = x ^ { y z }$ , $x > 0$ . Find the value of the partial derivative $f _ { z } ( 2,3,0 )$ .

Accepted Answer

A) 0 
B) 2 
C) 3 
D) 8 
E)  $\ln 2$ 
F) $\ln 3$ 
G) $2 \ln 2$
H) $3 \ln 2$ 
A) 0 
B) 2 
C) 3 
D) 8 
E)  $\ln 2$ 
F) $\ln 3$ 
G) $2 \ln 2$
H) $3 \ln 2$

Question 3

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = 3 x ^ { 3 } + y ^ { 2 } - 9 x + 4 y$ .

Accepted Answer

@#IMG-DLM& is a loca

Question 4

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = - x ^ { 4 } + 4 x y - 2 y ^ { 2 } + 1$ .

Accepted Answer

@#IMG-DLM& is a local maximum;

Question 5

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

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

Question 6

Suppose that the equation $F ( x , y , z ) = 0$ defines $Z$ implicitly as a function of $x$ and $y$ . Let $( a , b , c )$ be a point such that $F ( a , b , c ) = 0$ and $\nabla F ( a , b , c ) = ( 2 , - 3,4 )$ . Find $\frac { \delta z } { \delta x } ( a , b )$ and $\frac { \delta z } { \delta y } ( a , b )$ .

Accepted Answer

The answer of Suppose that the equation \(F ( x...

Question 7

Find an equation of the tangent plant to the parametric surface $x = u , y = v , z = u ^ { 2 } + v$ at the point $( 1,2,3 )$ .

Accepted Answer

A)  $x + y + z = 6$ 
B)  $x + y - z = 0$ 
C)  $x + y + 2 z = 9$ 
D)  $x + y - 2 z = - 3$ 
E)  $x + 2 y + z = 8$ 
F) $x - 2 y + z = 0$

G) $2 x + y + z = 7$

H) $2 x + y - z = 1$ 
A)  $x + y + z = 6$ 
B)  $x + y - z = 0$ 
C)  $x + y + 2 z = 9$ 
D)  $x + y - 2 z = - 3$ 
E)  $x + 2 y + z = 8$ 
F) $x - 2 y + z = 0$

G) $2 x + y + z = 7$

H) $2 x + y - z = 1$

Question 8

The radius of a right circular cylinder is increasing at a rate of 2 cm/min and its height is decreasing at 4cm/min. At what rate is the volume changing at the instant when the radius is 4 cm and the height is 10 cm?

Accepted Answer

The answer of The radius of a right circular cylinder...

Question 9

Suppose the point $( 2,3 )$ is on a curve $C$ which is a level curve of the surface $z = x ^ { 2 } + 2 y$ . Can it be concluded that the point $( 4 , - 3 )$ is also on $C$ ? Explain.

Accepted Answer

Yes, given that @#IMG-DLM& is a level cu

Question 10

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

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $- 2$ 
C) 0 
D) 1 
E)  $- \frac { 1 } { 2 }$ 
F)2

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

G) $- 1$ 
H)Does not exist

Question 11

Let $f ( x , y ) = \frac { 1 } { x ^ { 2 } - y }$ .(a) Evaluate $f ( 3 , - 1 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

Accepted Answer

(a) @#IMG-DLM& (b) All the poi

Question 12

If $f ( x , y ) = 6 - 2 x ^ { 2 } - y ^ { 2 }$ , find $f _ { x } ( 1,1 )$ and $f _ { y } ( 1,1 )$ and interpret these numbers as slopes. Illustrate with sketches.

Accepted Answer

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

Question 13

Use a linear approximation to estimate $\sqrt { 2.9 ^ { 2 } + 4.1 ^ { 2 } }$ .

Accepted Answer

A) 4.996 
B) 5.02 
C) 4.99 
D) 5.021 
E) 5.022
F)5.01
G)5.11
H)5.002 
A) 4.996 
B) 5.02 
C) 4.99 
D) 5.021 
E) 5.022
F)5.01
G)5.11
H)5.002

Question 14

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

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 15

Let $x + y + z - \sin ( x y z ) = 3$ . Use implicit differentiation to find $\delta z/\delta y$ when $( x , y , z ) = ( 1,0,2 )$ .

Accepted Answer

A)  $- 3$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)3
H)4 
A)  $- 3$ 
B)  $- 2$ 
C)  $- 1$ 
D) 0 
E) 1
F)2
G)3
H)4

Question 16

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x y = 2$ , what is the value of the multiplier $\lambda$ ?

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 17

Find the greatest product three numbers can have if the sum of their squares must be 48.

Accepted Answer

The answer of Find the greatest product three numbers can...

Question 18

Find the absolute maximum and minimum value of $f ( x , y ) = 2 x + y$ on the square region $D$ with vertices $( 0,0 )$ , $( 0,2 )$ , $( 1,2 )$ , and $( 1,0 )$ .

Accepted Answer

Absolute maximum val

Question 19

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

Accepted Answer

@#IMG-DLM& . There a

Question 20

Find three positive numbers $x$ , $y$ , and $Z$ whose product is 343 and sum is minimum.

Accepted Answer

The answer of Find three positive numbers $x$ , $y$...

Compute the minimum value of $f ( x , y , z ) = x ^ { 2 } + y + z ^ { 2 }$ subject to the condition that $g ( x ) = 2 x + y + 4 z = 6$ .

Let $f ( x , y , z ) = x ^ { y z }$ , $x > 0$ . Find the value of the partial derivative $f _ { z } ( 2,3,0 )$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = 3 x ^ { 3 } + y ^ { 2 } - 9 x + 4 y$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = - x ^ { 4 } + 4 x y - 2 y ^ { 2 } + 1$ .

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

Find an equation of the tangent plant to the parametric surface $x = u , y = v , z = u ^ { 2 } + v$ at the point $( 1,2,3 )$ .

The radius of a right circular cylinder is increasing at a rate of 2 cm/min and its height is decreasing at 4cm/min. At what rate is the volume changing at the instant when the radius is 4 cm and the height is 10 cm?

Suppose the point $( 2,3 )$ is on a curve $C$ which is a level curve of the surface $z = x ^ { 2 } + 2 y$ . Can it be concluded that the point $( 4 , - 3 )$ is also on $C$ ? Explain.

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

Let $f ( x , y ) = \frac { 1 } { x ^ { 2 } - y }$ .(a) Evaluate $f ( 3 , - 1 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

If $f ( x , y ) = 6 - 2 x ^ { 2 } - y ^ { 2 }$ , find $f _ { x } ( 1,1 )$ and $f _ { y } ( 1,1 )$ and interpret these numbers as slopes. Illustrate with sketches.

Use a linear approximation to estimate $\sqrt { 2.9 ^ { 2 } + 4.1 ^ { 2 } }$ .

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

Let $x + y + z - \sin ( x y z ) = 3$ . Use implicit differentiation to find $\delta z/\delta y$ when $( x , y , z ) = ( 1,0,2 )$ .

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x y = 2$ , what is the value of the multiplier $\lambda$ ?

Find the greatest product three numbers can have if the sum of their squares must be 48.

Find the absolute maximum and minimum value of $f ( x , y ) = 2 x + y$ on the square region $D$ with vertices $( 0,0 )$ , $( 0,2 )$ , $( 1,2 )$ , and $( 1,0 )$ .

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

Find three positive numbers $x$ , $y$ , and $Z$ whose product is 343 and sum is minimum.

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Exam 11: Partial Derivatives

Compute the minimum value of f(x,y,z)=x2+y+z2f ( x , y , z ) = x ^ { 2 } + y + z ^ { 2 }f(x,y,z)=x2+y+z2 subject to the condition that g(x)=2x+y+4z=6g ( x ) = 2 x + y + 4 z = 6g(x)=2x+y+4z=6 .

Let f(x,y,z)=xyzf ( x , y , z ) = x ^ { y z }f(x,y,z)=xyz , x>0x > 0x>0 . Find the value of the partial derivative fz(2,3,0)f _ { z } ( 2,3,0 )fz​(2,3,0) .

Find the local maximum and minimum values and saddle points of the function 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 local maximum and minimum values and saddle points of the function f(x,y)=−x4+4xy−2y2+1f ( x , y ) = - x ^ { 4 } + 4 x y - 2 y ^ { 2 } + 1f(x,y)=−x4+4xy−2y2+1 .

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

Find an equation of the tangent plant to the parametric surface x=u,y=v,z=u2+vx = u , y = v , z = u ^ { 2 } + vx=u,y=v,z=u2+v at the point (1,2,3)( 1,2,3 )(1,2,3) .

The radius of a right circular cylinder is increasing at a rate of 2 cm/min and its height is decreasing at 4cm/min. At what rate is the volume changing at the instant when the radius is 4 cm and the height is 10 cm?

Suppose the point (2,3)( 2,3 )(2,3) is on a curve CCC which is a level curve of the surface z=x2+2yz = x ^ { 2 } + 2 yz=x2+2y . Can it be concluded that the point (4,−3)( 4 , - 3 )(4,−3) is also on CCC ? Explain.

Evaluate lim⁡(x,yz)→(1,0,−2)xy+z\lim _ { ( x , y z ) \rightarrow ( 1,0 , - 2 ) } \frac { x } { y + z }lim(x,yz)→(1,0,−2)​y+zx​

Let f(x,y)=1x2−yf ( x , y ) = \frac { 1 } { x ^ { 2 } - y }f(x,y)=x2−y1​ .(a) Evaluate f(3,−1)f ( 3 , - 1 )f(3,−1) .(b) Sketch the domain of fff .(c) What is the range of the function fff ?

If f(x,y)=6−2x2−y2f ( x , y ) = 6 - 2 x ^ { 2 } - y ^ { 2 }f(x,y)=6−2x2−y2 , find fx(1,1)f _ { x } ( 1,1 )fx​(1,1) and fy(1,1)f _ { y } ( 1,1 )fy​(1,1) and interpret these numbers as slopes. Illustrate with sketches.

Use a linear approximation to estimate 2.92+4.12\sqrt { 2.9 ^ { 2 } + 4.1 ^ { 2 } }2.92+4.12​ .

Find the directional derivative of the function f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2 at the point (1,2)( 1,2 )(1,2) in the direction θ=π2\theta = \frac { \pi } { 2 }θ=2π​ .

Let x+y+z−sin⁡(xyz)=3x + y + z - \sin ( x y z ) = 3x+y+z−sin(xyz)=3 . Use implicit differentiation to find δz/δy\delta z/\delta yδz/δy when (x,y,z)=(1,0,2)( x , y , z ) = ( 1,0,2 )(x,y,z)=(1,0,2) .

In using Lagrange multipliers to minimize 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 , what is the value of the multiplier λ\lambdaλ ?

Find the greatest product three numbers can have if the sum of their squares must be 48.

Find the absolute maximum and minimum value of f(x,y)=2x+yf ( x , y ) = 2 x + yf(x,y)=2x+y on the square region DDD with vertices (0,0)( 0,0 )(0,0) , (0,2)( 0,2 )(0,2) , (1,2)( 1,2 )(1,2) , and (1,0)( 1,0 )(1,0) .

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 stretch the surface z=f(x,y)z = f ( x , y )z=f(x,y) .

Find three positive numbers xxx , yyy , and ZZZ whose product is 343 and sum is minimum.

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Differential Equations

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Vector Calculus

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Compute the minimum value of $f ( x , y , z ) = x ^ { 2 } + y + z ^ { 2 }$ subject to the condition that $g ( x ) = 2 x + y + 4 z = 6$ .

Let $f ( x , y , z ) = x ^ { y z }$ , $x > 0$ . Find the value of the partial derivative $f _ { z } ( 2,3,0 )$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = 3 x ^ { 3 } + y ^ { 2 } - 9 x + 4 y$ .

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = - x ^ { 4 } + 4 x y - 2 y ^ { 2 } + 1$ .

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

Find an equation of the tangent plant to the parametric surface $x = u , y = v , z = u ^ { 2 } + v$ at the point $( 1,2,3 )$ .

Suppose the point $( 2,3 )$ is on a curve $C$ which is a level curve of the surface $z = x ^ { 2 } + 2 y$ . Can it be concluded that the point $( 4 , - 3 )$ is also on $C$ ? Explain.

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

Let $f ( x , y ) = \frac { 1 } { x ^ { 2 } - y }$ .(a) Evaluate $f ( 3 , - 1 )$ .(b) Sketch the domain of $f$ .(c) What is the range of the function $f$ ?

If $f ( x , y ) = 6 - 2 x ^ { 2 } - y ^ { 2 }$ , find $f _ { x } ( 1,1 )$ and $f _ { y } ( 1,1 )$ and interpret these numbers as slopes. Illustrate with sketches.

Use a linear approximation to estimate $\sqrt { 2.9 ^ { 2 } + 4.1 ^ { 2 } }$ .

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

Let $x + y + z - \sin ( x y z ) = 3$ . Use implicit differentiation to find $\delta z/\delta y$ when $( x , y , z ) = ( 1,0,2 )$ .

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x y = 2$ , what is the value of the multiplier $\lambda$ ?

Find the absolute maximum and minimum value of $f ( x , y ) = 2 x + y$ on the square region $D$ with vertices $( 0,0 )$ , $( 0,2 )$ , $( 1,2 )$ , and $( 1,0 )$ .

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

Find three positive numbers $x$ , $y$ , and $Z$ whose product is 343 and sum is minimum.