Question 1

Find the critical points of the function. $f ( x , y ) = 72 + 684 x y + 76 x ^ { 2 } + 2160 y + \frac { 9 y ^ { 4 } } { 4 }$

Accepted Answer

A)  $( - 4,4 ) , ( - 6,6 ) , ( 10 , - 10 )$ 
B)  $( - 4,6 ) , ( - 8,6 ) , ( 8 , - 6 )$ 
C)  $( - 4,6 ) , ( - 6 , - 10 ) , ( 10,4 )$ 
D)  $( - 6,6 ) , ( - 8,8 ) , ( 8 , - 8 )$ 
E)  $( 4,4 ) , ( 6,6 ) , ( - 10,0 )$ 
A)  $( - 4,4 ) , ( - 6,6 ) , ( 10 , - 10 )$ 
B)  $( - 4,6 ) , ( - 8,6 ) , ( 8 , - 6 )$ 
C)  $( - 4,6 ) , ( - 6 , - 10 ) , ( 10,4 )$ 
D)  $( - 6,6 ) , ( - 8,8 ) , ( 8 , - 8 )$ 
E)  $( 4,4 ) , ( 6,6 ) , ( - 10,0 )$

Question 2

At what point is the following function a local maximum? $f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }$

Accepted Answer

A)  $( 1,1 )$ 
B)  $( - 1,1 )$ 
C)  $( 3,1 )$ 
D)  $( 0,1 )$ 
E)  $( 1 , - 1 )$ 
A)  $( 1,1 )$ 
B)  $( - 1,1 )$ 
C)  $( 3,1 )$ 
D)  $( 0,1 )$ 
E)  $( 1 , - 1 )$

Question 3

Find the first partial derivatives of the function. $c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)$

Accepted Answer

The answer of Find the first partial derivatives of the...

Question 4

How many nth-order partial derivatives does a function of two variables have?

Accepted Answer

A)  $n ^ { 2 }$ 
B)  $2 ^ { n }$ 
C)  $2 n$ 
D)  $\frac { n } { 2 }$ 
E)  $n 2 ^ { n }$ 
A)  $n ^ { 2 }$ 
B)  $2 ^ { n }$ 
C)  $2 n$ 
D)  $\frac { n } { 2 }$ 
E)  $n 2 ^ { n }$

Question 5

Find the dimensions of the rectangular box with largest volume if the total surface area is given as $294$ $\mathrm { cm } ^ { 2 }$ .

Accepted Answer

A)  $14$ cm, 1.75 cm, 1.75 cm 
B)  $21$ cm, $14$ cm, 1.75 cm 
C)  $294$ cm, $7$ cm, $7$ cm 
D)  $7$ cm, $7$ cm, $7$ cm 
E)  $14$ cm, $14$ cm, 3.5 cm 
A)  $14$ cm, 1.75 cm, 1.75 cm 
B)  $21$ cm, $14$ cm, 1.75 cm 
C)  $294$ cm, $7$ cm, $7$ cm 
D)  $7$ cm, $7$ cm, $7$ cm 
E)  $14$ cm, $14$ cm, 3.5 cm

Question 6

Find the gradient of $f ( x , y , z ) = x ^ { 2 } e ^ { y z }$ at the point $P ( 2,0 , - 2 )$

Accepted Answer

The answer of Find the gradient of \(f ( x...

Question 7

Consider evaluating the limit $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{7 x^{2}+3 y^{2}}$ (a) Evaluate the limit along the path $C _ { 1 } : y = 0$ (b) Evaluate the limit along the path $C _ { 2 } : x = 0$ (c) What can you conclude about the existence of $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } - y ^ { 2 } } { 7 x ^ { 2 } + 3 y ^ { 2 } }$ ?

Accepted Answer

(a)
@#LAT-DLM&
(b)
@#LAT-DLM&

Question 8

Find the equation of the tangent plane to the given surface at the specified point. $f ( x , y ) = 4 x ^ { 3 } y - 7 x ^ { 2 } y , ( 1,1 , - 3 )$

Accepted Answer

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

Question 9

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is $84$

Accepted Answer

A)  $7$ , $84$ , $7$ 
B)  4, 8, 16 
C)  $7$ , $7$ , $7$ 
D)  32, $7$ , 16 
E)  32, 32, 32 
A)  $7$ , $84$ , $7$ 
B)  4, 8, 16 
C)  $7$ , $7$ , $7$ 
D)  32, $7$ , 16 
E)  32, 32, 32

Question 10

Find $f _ { x y }$ for the function $f ( x , y ) = 2 x ^ { 3 } y - 7 x y ^ { 2 }$ .

Accepted Answer

A)  $f _ { x y } = 6 x ^ { 2 } - 14 y$ 
B)  $f _ { x y } = 6 x ^ { 2 } + 14 y$ 
C)  $f _ { x y } = 14 x ^ { 2 } + 6 y$ 
D)  $f _ { x y } = 14 x ^ { 2 } - 6 y$ 
E)  $f _ { x y } = 24 x ^ { 2 } - 14 y$ 
A)  $f _ { x y } = 6 x ^ { 2 } - 14 y$ 
B)  $f _ { x y } = 6 x ^ { 2 } + 14 y$ 
C)  $f _ { x y } = 14 x ^ { 2 } + 6 y$ 
D)  $f _ { x y } = 14 x ^ { 2 } - 6 y$ 
E)  $f _ { x y } = 24 x ^ { 2 } - 14 y$

Question 11

Let $g ( r , s , t ) = r e ^ { s / t }$ Find $g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)$

Accepted Answer

A)  $\frac { 2 } { 3 }$ 
B)  $4 \sqrt { 3 }$ 
C)  36 
D)  $6$ 
A)  $\frac { 2 } { 3 }$ 
B)  $4 \sqrt { 3 }$ 
C)  36 
D)  $6$

Question 12

Find the limit $\lim _{(x, y) \rightarrow\left(0^{+}, 0^{+}\right)} \frac{e^{\sqrt{x+y}}}{4 x+2 y-5}$

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $- \frac { 1 } { 5 }$ 
C)  $1$ 
D)  $- 1$ 
A)  $\frac { 1 } { 5 }$ 
B)  $- \frac { 1 } { 5 }$ 
C)  $1$ 
D)  $- 1$

Question 13

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 5 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

Accepted Answer

A)  the limit does not exist 
B)  $5$ 
C)  2 
D)  0 
E)  1 
A)  the limit does not exist 
B)  $5$ 
C)  2 
D)  0 
E)  1

Question 14

Determine where the function $f ( x , y ) = \frac { x ^ { 3 } + x y + y ^ { 2 } } { 9 x ^ { 2 } + 8 y ^ { 2 } }$ is continuous.

Accepted Answer

The answer of Determine where the function \(f ( x...

Question 15

Use the Chain Rule to find $\frac { \partial z } { \partial s }$ . $z = e ^ { y } \cos ( \theta ) , r = 8 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }$

Accepted Answer

A)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( 8 t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
B)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( 8 t \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$ 
C)  $\frac { \partial z } { \partial s } = \left( 8 t \cos ( \theta ) + \frac { s e ^ { \gamma } \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
D)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$ 
E)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t } } \right)$ 
A)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( 8 t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
B)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( 8 t \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$ 
C)  $\frac { \partial z } { \partial s } = \left( 8 t \cos ( \theta ) + \frac { s e ^ { \gamma } \sin ( \theta ) } { \sqrt { s ^ { 2 } + t ^ { 2 } } } \right)$ 
D)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( \cos ( \theta ) + \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } - t ^ { 2 } } } \right)$ 
E)  $\frac { \partial z } { \partial s } = e ^ { \gamma } \left( t \cos ( \theta ) - \frac { s \sin ( \theta ) } { \sqrt { s ^ { 2 } + t } } \right)$

Question 16

Find the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 9 x ^ { 2 } + 9 y ^ { 2 } \right) } { 7 x ^ { 2 } + 7 y ^ { 2 } }$ Hint: If $x = r \cos \theta$ and $y = r \sin \theta ,$ then $( x , y ) \rightarrow ( 0,0 )$ if and only if $r \rightarrow 0 ^ { + }$

Accepted Answer

A)  $\frac { 9 } { 7 }$ 
B)  $1$ 
C)  $0$ 
D)  $\frac { 7 } { 9 }$ 
A)  $\frac { 9 } { 7 }$ 
B)  $1$ 
C)  $0$ 
D)  $\frac { 7 } { 9 }$

Question 17

Find all the second partial derivatives of $f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }$

Accepted Answer

\[\begin{array} { l } 
f _ { x

Question 18

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 3 y ^ { 2 } + 9 z ^ { 2 } = 16$ at the point $P ( 2,1,1 )$

Accepted Answer

A)  $x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = y - 3 = z - 9$ 
B)  $2 x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 9 }$ 
C)  $x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 9 }$ 
D)  $2 x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = y - 3 = z - 9$ 
A)  $x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = y - 3 = z - 9$ 
B)  $2 x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 9 }$ 
C)  $x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 9 }$ 
D)  $2 x + 3 y + 9 z = 16$ , $\frac { x - 2 } { 2 } = y - 3 = z - 9$

Question 19

Find and sketch the domain of the function $h ( x , y ) = \sqrt { x y - 2 }$

Accepted Answer

The answer of Find and sketch the domain of the...

Question 20

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7$ Find $f ( 3 h , 4 k )$

Accepted Answer

A)  $12 h ^ { 2 } + 36 h k - 60 h - 84$ 
B)  $9 h ^ { 2 } + 36 h k - 20 h - 7$ 
C)  $16 h ^ { 2 } + 36 h k - 20 h - 7$ 
D)  $9 h ^ { 2 } + 36 h k - 15 h - 7$ 
A)  $12 h ^ { 2 } + 36 h k - 60 h - 84$ 
B)  $9 h ^ { 2 } + 36 h k - 20 h - 7$ 
C)  $16 h ^ { 2 } + 36 h k - 20 h - 7$ 
D)  $9 h ^ { 2 } + 36 h k - 15 h - 7$

Find the critical points of the function. $f ( x , y ) = 72 + 684 x y + 76 x ^ { 2 } + 2160 y + \frac { 9 y ^ { 4 } } { 4 }$

At what point is the following function a local maximum? $f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }$

Find the first partial derivatives of the function. $c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)$

How many nth-order partial derivatives does a function of two variables have?

Find the dimensions of the rectangular box with largest volume if the total surface area is given as $294$ $\mathrm { cm } ^ { 2 }$ .

Find the gradient of $f ( x , y , z ) = x ^ { 2 } e ^ { y z }$ at the point $P ( 2,0 , - 2 )$

Find the equation of the tangent plane to the given surface at the specified point. $f ( x , y ) = 4 x ^ { 3 } y - 7 x ^ { 2 } y , ( 1,1 , - 3 )$

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is $84$

Find $f _ { x y }$ for the function $f ( x , y ) = 2 x ^ { 3 } y - 7 x y ^ { 2 }$ .

Let $g ( r , s , t ) = r e ^ { s / t }$ Find $g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)$

Find the limit $\lim _{(x, y) \rightarrow\left(0^{+}, 0^{+}\right)} \frac{e^{\sqrt{x+y}}}{4 x+2 y-5}$

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 5 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

Determine where the function $f ( x , y ) = \frac { x ^ { 3 } + x y + y ^ { 2 } } { 9 x ^ { 2 } + 8 y ^ { 2 } }$ is continuous.

Use the Chain Rule to find $\frac { \partial z } { \partial s }$ . $z = e ^ { y } \cos ( \theta ) , r = 8 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }$

Find the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 9 x ^ { 2 } + 9 y ^ { 2 } \right) } { 7 x ^ { 2 } + 7 y ^ { 2 } }$ Hint: If $x = r \cos \theta$ and $y = r \sin \theta ,$ then $( x , y ) \rightarrow ( 0,0 )$ if and only if $r \rightarrow 0 ^ { + }$

Find all the second partial derivatives of $f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }$

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 3 y ^ { 2 } + 9 z ^ { 2 } = 16$ at the point $P ( 2,1,1 )$

Find and sketch the domain of the function $h ( x , y ) = \sqrt { x y - 2 }$

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7$ Find $f ( 3 h , 4 k )$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

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Exam 14: Partial Derivatives

Find the critical points of the function. f(x,y)=72+684xy+76x2+2160y+9y44f ( x , y ) = 72 + 684 x y + 76 x ^ { 2 } + 2160 y + \frac { 9 y ^ { 4 } } { 4 }f(x,y)=72+684xy+76x2+2160y+49y4​

At what point is the following function a local maximum? f(x,y)=3−10x+12y−5x2−6y2f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }f(x,y)=3−10x+12y−5x2−6y2

Find the first partial derivatives of the function. c=ln⁡(a+a4+b4)c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)c=ln(a+a4+b4​)

How many nth-order partial derivatives does a function of two variables have?

Find the dimensions of the rectangular box with largest volume if the total surface area is given as 294294294 cm2\mathrm { cm } ^ { 2 }cm2 .

Find the gradient of f(x,y,z)=x2eyzf ( x , y , z ) = x ^ { 2 } e ^ { y z }f(x,y,z)=x2eyz at the point P(2,0,−2)P ( 2,0 , - 2 )P(2,0,−2)

Find the equation of the tangent plane to the given surface at the specified point. f(x,y)=4x3y−7x2y,(1,1,−3)f ( x , y ) = 4 x ^ { 3 } y - 7 x ^ { 2 } y , ( 1,1 , - 3 )f(x,y)=4x3y−7x2y,(1,1,−3)

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is 848484

Find fxyf _ { x y }fxy​ for the function f(x,y)=2x3y−7xy2f ( x , y ) = 2 x ^ { 3 } y - 7 x y ^ { 2 }f(x,y)=2x3y−7xy2 .

Let g(r,s,t)=res/tg ( r , s , t ) = r e ^ { s / t }g(r,s,t)=res/t Find g(4,ln⁡3,12)g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)g(4,ln3,21​)

Find the limit lim⁡(x,y)→(0+,0+)ex+y4x+2y−5\lim _{(x, y) \rightarrow\left(0^{+}, 0^{+}\right)} \frac{e^{\sqrt{x+y}}}{4 x+2 y-5}lim(x,y)→(0+,0+)​4x+2y−5ex+y​​

Evaluate the limit. lim⁡(x,y)→(0,0)5xy2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 5 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y25xy2​

Determine where the function f(x,y)=x3+xy+y29x2+8y2f ( x , y ) = \frac { x ^ { 3 } + x y + y ^ { 2 } } { 9 x ^ { 2 } + 8 y ^ { 2 } }f(x,y)=9x2+8y2x3+xy+y2​ is continuous.

Use the Chain Rule to find ∂z∂s\frac { \partial z } { \partial s }∂s∂z​ . z=eycos⁡(θ),r=8st,θ=s2+t2z = e ^ { y } \cos ( \theta ) , r = 8 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }z=eycos(θ),r=8st,θ=s2+t2​

Find all the second partial derivatives of f(x,y)=2x3y−6xy2f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }f(x,y)=2x3y−6xy2

Find equations for the tangent plane and the normal line to the surface with equation x2+3y2+9z2=16x ^ { 2 } + 3 y ^ { 2 } + 9 z ^ { 2 } = 16x2+3y2+9z2=16 at the point P(2,1,1)P ( 2,1,1 )P(2,1,1)

Find and sketch the domain of the function h(x,y)=xy−2h ( x , y ) = \sqrt { x y - 2 }h(x,y)=xy−2​

Let f(x,y)=x2+3xy−5x−7f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7f(x,y)=x2+3xy−5x−7 Find f(3h,4k)f ( 3 h , 4 k )f(3h,4k)

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the critical points of the function. $f ( x , y ) = 72 + 684 x y + 76 x ^ { 2 } + 2160 y + \frac { 9 y ^ { 4 } } { 4 }$

At what point is the following function a local maximum? $f ( x , y ) = 3 - 10 x + 12 y - 5 x ^ { 2 } - 6 y ^ { 2 }$

Find the first partial derivatives of the function. $c = \ln \left( a + \sqrt { a ^ { 4 } + b ^ { 4 } } \right)$

Find the dimensions of the rectangular box with largest volume if the total surface area is given as $294$ $\mathrm { cm } ^ { 2 }$ .

Find the gradient of $f ( x , y , z ) = x ^ { 2 } e ^ { y z }$ at the point $P ( 2,0 , - 2 )$

Find the equation of the tangent plane to the given surface at the specified point. $f ( x , y ) = 4 x ^ { 3 } y - 7 x ^ { 2 } y , ( 1,1 , - 3 )$

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is $84$

Find $f _ { x y }$ for the function $f ( x , y ) = 2 x ^ { 3 } y - 7 x y ^ { 2 }$ .

Let $g ( r , s , t ) = r e ^ { s / t }$ Find $g \left( 4 , \ln 3 , \frac { 1 } { 2 } \right)$

Find the limit $\lim _{(x, y) \rightarrow\left(0^{+}, 0^{+}\right)} \frac{e^{\sqrt{x+y}}}{4 x+2 y-5}$

Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 5 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$

Determine where the function $f ( x , y ) = \frac { x ^ { 3 } + x y + y ^ { 2 } } { 9 x ^ { 2 } + 8 y ^ { 2 } }$ is continuous.

Use the Chain Rule to find $\frac { \partial z } { \partial s }$ . $z = e ^ { y } \cos ( \theta ) , r = 8 s t , \theta = \sqrt { s ^ { 2 } + t ^ { 2 } }$

Find all the second partial derivatives of $f ( x , y ) = 2 x ^ { 3 } y - 6 x y ^ { 2 }$

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 3 y ^ { 2 } + 9 z ^ { 2 } = 16$ at the point $P ( 2,1,1 )$

Find and sketch the domain of the function $h ( x , y ) = \sqrt { x y - 2 }$

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 5 x - 7$ Find $f ( 3 h , 4 k )$