Question 1

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

Find the maximum and minimum values of the function $f ( x , y ) = x y$ on the ellipse given by the equation $x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ .

Accepted Answer

Maximum va

Question 3

Find the point on the plane $x - 2 y + z = 3$ where $x ^ { 2 } + 4 y ^ { 2 } + 2 z ^ { 2 }$ is minimum.

Accepted Answer

The answer of Find the point on the plane \(x...

Question 4

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

Accepted Answer

A) 0 
B) 1 
C)  $- 1$ 
D)  $\frac { 1 } { 2 }$ 
E)  $- \frac { 1 } { 2 }$
F) $\frac { 1 } { 3 }$ 
G) $- \frac { 1 } { 3 }$ 
H)Does not exist 
A) 0 
B) 1 
C)  $- 1$ 
D)  $\frac { 1 } { 2 }$ 
E)  $- \frac { 1 } { 2 }$
F) $\frac { 1 } { 3 }$ 
G) $- \frac { 1 } { 3 }$ 
H)Does not exist

Question 5

Let $f ( x , y , z ) = x ^ { 3 } y ^ { 2 } z + 1$ . If $x = 1$ , find $f \left( x , x ^ { 2 } , - x \right)$ .

Accepted Answer

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

Question 6

The function $f ( x , y ) = x ^ { 2 } - 2 y ^ { 2 }$ 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 7

Find all solutions to the partial differential equation $\frac { \delta ^ { 2 } f } { \delta x \delta x } = 0$ .

Accepted Answer

@#IMG-DLM& where @#IMG-DLM& an

Question 8

Find both partial derivatives if $f ( x , y ) = \frac { 2 x - 3 y } { 3 x - 2 y }$

Accepted Answer

The answer of Find both partial derivatives if \(f (...

Question 9

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x + y = 3$ , 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 10

Let $f ( x , y ) = x \sin y$ . Find $f \left( 2 , \frac { \pi } { 3 } \right)$ .

Accepted Answer

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

G)1

H)0 
A)  $\sqrt { 3 }$ 
B)  $\sqrt { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  $\frac { 1 } { 2 }$
F) $\frac { 1 } { 3 }$

G)1

H)0

Question 11

The dimensions of a closed rectangular box are measured to be 60 cm, 40 cm, and 30 cm. The ruler that is used has a possible error in measurement of at most 0.1 cm. Use differentials to estimate the maximum error in the calculated volume of the box.

Accepted Answer

The answer of The dimensions of a closed rectangular box...

Question 12

Use the level curves of $f ( x , y )$ shown below to estimate the critical points of $f$ . Indicate whether $f$ has a saddle point or a local maximum or minimum at each of those points.

Accepted Answer

@#IMG-DLM& has a saddle point

Question 13

Find the directions in which the directional derivative of $f ( x , y ) = x ^ { 2 } y + x ^ { 3 }$ at the point $( - 1,2 )$ has the value 1.

Accepted Answer

The answer of Find the directions in which the directional...

Question 14

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

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 15

Describe the level surfaces of the function $f ( x , y , z ) = z - x ^ { 2 } - y ^ { 2 }$ .

Accepted Answer

Paraboloid

Question 16

Given $f ( x , y ) = 2 x ^ { 3 } y ^ { 2 } - 3 x ^ { 3 } + 8 y ^ { 4 }$ , find $f _ { x } \text { and } f _ { y }$ and evaluate each at $( 1,2 )$ .

Accepted Answer

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

Question 17

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

The answer of The graph of level curves of \(f...

Question 18

Show that $f ( x , y ) = e ^ { y } \cos x - 2 x y$ satisfies Laplace Equation $f _ { xx } + f _ { y y } = 0$ .

Accepted Answer

The answer of Show that \(f ( x , y...

Question 19

Find the absolute maximum and minimum value of $f ( x , y ) = x ^ { 2 } - 3 y ^ { 2 } - 2 x + 6 y$ on the square region $D$ with vertices $( 0,0 )$ , $( 0,2 )$ , $( 2,2 )$ , and $( 2,0 )$ .

Accepted Answer

Absolute maximum val

Question 20

Find an equation of the plane tangent to the surface $x y z = 2$ at $( 1 , - 1 , - 2 )$ .

Accepted Answer

The answer of Find an equation of the plane tangent...

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

Find the maximum and minimum values of the function $f ( x , y ) = x y$ on the ellipse given by the equation $x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ .

Find the point on the plane $x - 2 y + z = 3$ where $x ^ { 2 } + 4 y ^ { 2 } + 2 z ^ { 2 }$ is minimum.

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

Let $f ( x , y , z ) = x ^ { 3 } y ^ { 2 } z + 1$ . If $x = 1$ , find $f \left( x , x ^ { 2 } , - x \right)$ .

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

Find all solutions to the partial differential equation $\frac { \delta ^ { 2 } f } { \delta x \delta x } = 0$ .

Find both partial derivatives if $f ( x , y ) = \frac { 2 x - 3 y } { 3 x - 2 y }$

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x + y = 3$ , what is the value of the multiplier $\lambda$ ?

Let $f ( x , y ) = x \sin y$ . Find $f \left( 2 , \frac { \pi } { 3 } \right)$ .

The dimensions of a closed rectangular box are measured to be 60 cm, 40 cm, and 30 cm. The ruler that is used has a possible error in measurement of at most 0.1 cm. Use differentials to estimate the maximum error in the calculated volume of the box.

Use the level curves of $f ( x , y )$ shown below to estimate the critical points of $f$ . Indicate whether $f$ has a saddle point or a local maximum or minimum at each of those points.

Find the directions in which the directional derivative of $f ( x , y ) = x ^ { 2 } y + x ^ { 3 }$ at the point $( - 1,2 )$ has the value 1.

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

Describe the level surfaces of the function $f ( x , y , z ) = z - x ^ { 2 } - y ^ { 2 }$ .

Given $f ( x , y ) = 2 x ^ { 3 } y ^ { 2 } - 3 x ^ { 3 } + 8 y ^ { 4 }$ , find $f _ { x } \text { and } f _ { y }$ and evaluate each at $( 1,2 )$ .

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

Show that $f ( x , y ) = e ^ { y } \cos x - 2 x y$ satisfies Laplace Equation $f _ { xx } + f _ { y y } = 0$ .

Find the absolute maximum and minimum value of $f ( x , y ) = x ^ { 2 } - 3 y ^ { 2 } - 2 x + 6 y$ on the square region $D$ with vertices $( 0,0 )$ , $( 0,2 )$ , $( 2,2 )$ , and $( 2,0 )$ .

Find an equation of the plane tangent to the surface $x y z = 2$ at $( 1 , - 1 , - 2 )$ .

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Exam 11: Partial Derivatives

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

Find the maximum and minimum values of the function f(x,y)=xyf ( x , y ) = x yf(x,y)=xy on the ellipse given by the equation x2+y24=1x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1x2+4y2​=1 .

Find the point on the plane x−2y+z=3x - 2 y + z = 3x−2y+z=3 where x2+4y2+2z2x ^ { 2 } + 4 y ^ { 2 } + 2 z ^ { 2 }x2+4y2+2z2 is minimum.

Evaluate lim⁡(x,y)→(0,0)x2−y2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2x2−y2​

Let f(x,y,z)=x3y2z+1f ( x , y , z ) = x ^ { 3 } y ^ { 2 } z + 1f(x,y,z)=x3y2z+1 . If x=1x = 1x=1 , find f(x,x2,−x)f \left( x , x ^ { 2 } , - x \right)f(x,x2,−x) .

The function f(x,y)=x2−2y2f ( x , y ) = x ^ { 2 } - 2 y ^ { 2 }f(x,y)=x2−2y2 has one critical point. Determine its location and type.

Find all solutions to the partial differential equation δ2fδxδx=0\frac { \delta ^ { 2 } f } { \delta x \delta x } = 0δxδxδ2f​=0 .

Find both partial derivatives if f(x,y)=2x−3y3x−2yf ( x , y ) = \frac { 2 x - 3 y } { 3 x - 2 y }f(x,y)=3x−2y2x−3y​

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 x+y=3x + y = 3x+y=3 , what is the value of the multiplier λ\lambdaλ ?

Let f(x,y)=xsin⁡yf ( x , y ) = x \sin yf(x,y)=xsiny . Find f(2,π3)f \left( 2 , \frac { \pi } { 3 } \right)f(2,3π​) .

The dimensions of a closed rectangular box are measured to be 60 cm, 40 cm, and 30 cm. The ruler that is used has a possible error in measurement of at most 0.1 cm. Use differentials to estimate the maximum error in the calculated volume of the box.

Use the level curves of f(x,y)f ( x , y )f(x,y) shown below to estimate the critical points of fff . Indicate whether fff has a saddle point or a local maximum or minimum at each of those points.

Find the directions in which the directional derivative of f(x,y)=x2y+x3f ( x , y ) = x ^ { 2 } y + x ^ { 3 }f(x,y)=x2y+x3 at the point (−1,2)( - 1,2 )(−1,2) has the value 1.

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

Describe the level surfaces of the function f(x,y,z)=z−x2−y2f ( x , y , z ) = z - x ^ { 2 } - y ^ { 2 }f(x,y,z)=z−x2−y2 .

Given f(x,y)=2x3y2−3x3+8y4f ( x , y ) = 2 x ^ { 3 } y ^ { 2 } - 3 x ^ { 3 } + 8 y ^ { 4 }f(x,y)=2x3y2−3x3+8y4 , find fx and fyf _ { x } \text { and } f _ { y }fx​ and fy​ and evaluate each at (1,2)( 1,2 )(1,2) .

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) .

Show that f(x,y)=eycos⁡x−2xyf ( x , y ) = e ^ { y } \cos x - 2 x yf(x,y)=eycosx−2xy satisfies Laplace Equation fxx+fyy=0f _ { xx } + f _ { y y } = 0fxx​+fyy​=0 .

Find the absolute maximum and minimum value of f(x,y)=x2−3y2−2x+6yf ( x , y ) = x ^ { 2 } - 3 y ^ { 2 } - 2 x + 6 yf(x,y)=x2−3y2−2x+6y on the square region DDD with vertices (0,0)( 0,0 )(0,0) , (0,2)( 0,2 )(0,2) , (2,2)( 2,2 )(2,2) , and (2,0)( 2,0 )(2,0) .

Find an equation of the plane tangent to the surface xyz=2x y z = 2xyz=2 at (1,−1,−2)( 1 , - 1 , - 2 )(1,−1,−2) .

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Limits and Derivatives

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

Find the maximum and minimum values of the function $f ( x , y ) = x y$ on the ellipse given by the equation $x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ .

Find the point on the plane $x - 2 y + z = 3$ where $x ^ { 2 } + 4 y ^ { 2 } + 2 z ^ { 2 }$ is minimum.

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

Let $f ( x , y , z ) = x ^ { 3 } y ^ { 2 } z + 1$ . If $x = 1$ , find $f \left( x , x ^ { 2 } , - x \right)$ .

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

Find all solutions to the partial differential equation $\frac { \delta ^ { 2 } f } { \delta x \delta x } = 0$ .

Find both partial derivatives if $f ( x , y ) = \frac { 2 x - 3 y } { 3 x - 2 y }$

In using Lagrange multipliers to minimize the function $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ subject to the constraint that $x + y = 3$ , what is the value of the multiplier $\lambda$ ?

Let $f ( x , y ) = x \sin y$ . Find $f \left( 2 , \frac { \pi } { 3 } \right)$ .

Use the level curves of $f ( x , y )$ shown below to estimate the critical points of $f$ . Indicate whether $f$ has a saddle point or a local maximum or minimum at each of those points.

Find the directions in which the directional derivative of $f ( x , y ) = x ^ { 2 } y + x ^ { 3 }$ at the point $( - 1,2 )$ has the value 1.

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

Describe the level surfaces of the function $f ( x , y , z ) = z - x ^ { 2 } - y ^ { 2 }$ .

Given $f ( x , y ) = 2 x ^ { 3 } y ^ { 2 } - 3 x ^ { 3 } + 8 y ^ { 4 }$ , find $f _ { x } \text { and } f _ { y }$ and evaluate each at $( 1,2 )$ .

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

Show that $f ( x , y ) = e ^ { y } \cos x - 2 x y$ satisfies Laplace Equation $f _ { xx } + f _ { y y } = 0$ .

Find the absolute maximum and minimum value of $f ( x , y ) = x ^ { 2 } - 3 y ^ { 2 } - 2 x + 6 y$ on the square region $D$ with vertices $( 0,0 )$ , $( 0,2 )$ , $( 2,2 )$ , and $( 2,0 )$ .

Find an equation of the plane tangent to the surface $x y z = 2$ at $( 1 , - 1 , - 2 )$ .