Question 1

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. $$f ( x , y ) = 8 x ^ { 2 } - 4 y ^ { 2 } , 8 x ^ { 2 } + 4 y ^ { 2 } = 9$$

Accepted Answer

A)  $f ( x , y ) = 8$ 
B)  $f ( x , y ) = \frac { 1 } { 4 }$ 
C)  $f ( x , y ) = \frac { 1 } { 9 }$ 
D)  $f ( x , y ) = \frac { 1 } { 8 }$ 
E)  $$f ( x , y ) = 9$$ 
A)  $f ( x , y ) = 8$ 
B)  $f ( x , y ) = \frac { 1 } { 4 }$ 
C)  $f ( x , y ) = \frac { 1 } { 9 }$ 
D)  $f ( x , y ) = \frac { 1 } { 8 }$ 
E)  $$f ( x , y ) = 9$$ E

Question 2

Find the limit $\lim _ { ( x , y , z ) \rightarrow ( 5,1,4 ) } \frac { x y + y z + x z } { x y z - 6 }$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $0$ 
C)  $\frac { 29 } { 14 }$ 
D)  $- \frac { 1 } { 6 }$ 
A)  $\frac { 1 } { 6 }$ 
B)  $0$ 
C)  $\frac { 29 } { 14 }$ 
D)  $- \frac { 1 } { 6 }$ C

Question 3

Find three positive numbers whose sum is $291$ and whose product is a maximum.

Accepted Answer

A)  $x = y = z = 97$ 
B)  $x = 109 , y = 99 , z = 89$ 
C)  $x = 100 , y = 108 , z = 89$ 
D)  $x = 99 , y = z = - 99$ 
E)  $x = 129 , y = 79 , z = 89$ 
A)  $x = y = z = 97$ 
B)  $x = 109 , y = 99 , z = 89$ 
C)  $x = 100 , y = 108 , z = 89$ 
D)  $x = 99 , y = z = - 99$ 
E)  $x = 129 , y = 79 , z = 89$ A

Question 4

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 4,1 )$ (2,1). In what direction does it occur?

Accepted Answer

A)  $\frac { \sqrt { 29 } } { 2 } , \left\langle1 , \frac { 5 } { 2 } \right\rangle$ 
B)  $\frac { 85 } { 2 } , \left\langle2 , \frac { 7 } { 2 } \right\rangle$ 
C)  $\frac { 29 } { 2 } , \left\langle 1 , \frac { 9 } { 2 } \right\rangle$ 
D)  $\frac { \sqrt { 293 } } { 2 } , \left\langle1 , \frac { 17 } { 2 } \right\rangle$ 
E)  none of these 
A)  $\frac { \sqrt { 29 } } { 2 } , \left\langle1 , \frac { 5 } { 2 } \right\rangle$ 
B)  $\frac { 85 } { 2 } , \left\langle2 , \frac { 7 } { 2 } \right\rangle$ 
C)  $\frac { 29 } { 2 } , \left\langle 1 , \frac { 9 } { 2 } \right\rangle$ 
D)  $\frac { \sqrt { 293 } } { 2 } , \left\langle1 , \frac { 17 } { 2 } \right\rangle$ 
E)  none of these

Question 5

Find $h(x, y)=g(f(x, y))$ and determine where h is continuous. $f ( x , y ) = 2 x - y , \quad g ( t ) = \frac { t + 2 } { t - 8 }$

Accepted Answer

The answer of Find $h(x, y)=g(f(x, y))$ and determine where...

Question 6

Find $\frac { \partial z } { \partial z }$ . $x e ^ { 4 y } + 4 y z + z e ^ { 8 x } = 0$

Accepted Answer

The answer of Find \(\frac { \partial z } {...

Question 7

Find $f _ { y k }$ for the function $f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 }$ .

Accepted Answer

The answer of Find $f _ { y k }$...

Question 8

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height $10$ cm if the tin is 0.04 cm thick.

Accepted Answer

The answer of Use differentials to estimate the amount of...

Question 9

Find the gradient of the function $f ( x , y , z ) = z ^ { 6 } e ^ { 2 x \sqrt { y } }$ .

Accepted Answer

A)  $z e ^ { 2 x \sqrt { y } } \left\langle 2 z ^ { 5 } \sqrt { y } , \frac { x z ^ { 5 } } { \sqrt { y } } , 6 z ^ { 4 } \right\rangle$ 
B)  $e ^ { x ^ { 2 } \sqrt { y } } \left\langle 6 x z \sqrt { y } , \frac { x ^ { 2 } z } { 2 \sqrt { y } } , 2 \right\rangle$ 
C)  $z e ^ { x \sqrt { y } } \left\langle z \sqrt { y } , \frac { x z } { 2 \sqrt { y } } , 6 \right\rangle$ 
D)  $e ^ { x ^ { 2 } \sqrt { y } } \left\langle 2 x z \sqrt { y } , \frac { x ^ { 2 } z } { 6 \sqrt { y } } , 1 \right\rangle$ 
E)  $z e ^ { 2 x \sqrt { y } } \left\langle 2 z \sqrt { y } , \frac { x z } { \sqrt { y } } , 2 \right\rangle$ 
A)  $z e ^ { 2 x \sqrt { y } } \left\langle 2 z ^ { 5 } \sqrt { y } , \frac { x z ^ { 5 } } { \sqrt { y } } , 6 z ^ { 4 } \right\rangle$ 
B)  $e ^ { x ^ { 2 } \sqrt { y } } \left\langle 6 x z \sqrt { y } , \frac { x ^ { 2 } z } { 2 \sqrt { y } } , 2 \right\rangle$ 
C)  $z e ^ { x \sqrt { y } } \left\langle z \sqrt { y } , \frac { x z } { 2 \sqrt { y } } , 6 \right\rangle$ 
D)  $e ^ { x ^ { 2 } \sqrt { y } } \left\langle 2 x z \sqrt { y } , \frac { x ^ { 2 } z } { 6 \sqrt { y } } , 1 \right\rangle$ 
E)  $z e ^ { 2 x \sqrt { y } } \left\langle 2 z \sqrt { y } , \frac { x z } { \sqrt { y } } , 2 \right\rangle$

Question 10

Use polar coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 6 } + y ^ { 6 } } { x ^ { 5 } + y ^ { 5 } }$

Accepted Answer

The answer of Use polar coordinates to find the limit....

Question 11

Find the limit $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 8 x ^ { 2 } + 2 y ^ { 2 } \right)$

Accepted Answer

A)  34 
B)  80 
C)  260 
D)  16 
A)  34 
B)  80 
C)  260 
D)  16

Question 12

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 9 , - 5 ) } x ^ { 5 } + 8 x ^ { 3 } y - 4 x y ^ { 2 }$

Accepted Answer

The answer of Find the limit. \(\lim _ { (...

Question 13

Find the indicated partial derivative. $f ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y ; f _ { m x }$

Accepted Answer

A)  $f _ { m x } = 12 x ^ { 2 } y$ 
B)  $f _ { m x } = - 36 x y$ 
C)  $f _ { m x } = 3 x y$ 
D)  $f _ { m x } = - 72 x y$ 
E)  $f _ { m x } = - 3 x y$ 
A)  $f _ { m x } = 12 x ^ { 2 } y$ 
B)  $f _ { m x } = - 36 x y$ 
C)  $f _ { m x } = 3 x y$ 
D)  $f _ { m x } = - 72 x y$ 
E)  $f _ { m x } = - 3 x y$

Question 14

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

Accepted Answer

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

Question 15

Use the Chain Rule to find $\frac { \partial w } { \partial s }$ where $s = 4 , t = 0$ . $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin t$

Accepted Answer

The answer of Use the Chain Rule to find \(\frac...

Question 16

A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft. Use differentials to approximate the number of square feet of paint in the stripe.

Accepted Answer

A)  113 $\mathrm { ft } ^ { 2 }$ 
B)  113.81 $\mathrm { ft } ^ { 2 }$ 
C)  113.23 $\mathrm { ft } ^ { 2 }$ 
D)  113.89 $\mathrm { ft } ^ { 2 }$ 
E)  113.33 $\mathrm { ft } ^ { 2 }$ 
A)  113 $\mathrm { ft } ^ { 2 }$ 
B)  113.81 $\mathrm { ft } ^ { 2 }$ 
C)  113.23 $\mathrm { ft } ^ { 2 }$ 
D)  113.89 $\mathrm { ft } ^ { 2 }$ 
E)  113.33 $\mathrm { ft } ^ { 2 }$

Question 17

Sketch the level curves $f ( x , y ) = k$ of the function for the indicated values of k. $f ( x , y ) = x - 4 y ^ { 2 } ; \quad k = - 2 , - 1,0,1,2$

Accepted Answer

The answer of Sketch the level curves \(f ( x...

Question 18

Find the points on the surface $z ^ { 2 } = x y + 49$ that are closest to the origin.

Accepted Answer

A)  $( 0,0 , - 49 )$ 
B)  $( 0,0,49 ) ( 0,0 , - 49 )$ 
C)  $( 0,7,0 )$ 
D)  $( 0,0,7 ) ( 0,0 , - 7 )$ 
E)  $( 0,0,7 )$ 
A)  $( 0,0 , - 49 )$ 
B)  $( 0,0,49 ) ( 0,0 , - 49 )$ 
C)  $( 0,7,0 )$ 
D)  $( 0,0,7 ) ( 0,0 , - 7 )$ 
E)  $( 0,0,7 )$

Question 19

Find the limit $\lim _ { ( x , y , z ) \rightarrow ( 3,2,4 ) } \frac { x y + y z + x z } { x y z - 6 }$

Accepted Answer

A)  $\frac { 13 } { 9 }$ 
B)  $0$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
A)  $\frac { 13 } { 9 }$ 
B)  $0$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$

Question 20

Find the shortest distance from the point $( 3,9,8 )$ to the plane $3 x + 9 y + 4 z = 16$ .

Accepted Answer

A)  $D = 61$ 
B)  $D = 90$ 
C)  $D = 10.29$ 
D)  $D = \sqrt { 61 }$ 
E)  $D = 106$ 
A)  $D = 61$ 
B)  $D = 90$ 
C)  $D = 10.29$ 
D)  $D = \sqrt { 61 }$ 
E)  $D = 106$

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. $f ( x , y ) = 8 x ^ { 2 } - 4 y ^ { 2 } , 8 x ^ { 2 } + 4 y ^ { 2 } = 9$

Find the limit $\lim _ { ( x , y , z ) \rightarrow ( 5,1,4 ) } \frac { x y + y z + x z } { x y z - 6 }$

Find three positive numbers whose sum is $291$ and whose product is a maximum.

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 4,1 )$ (2,1). In what direction does it occur?

Find $h(x, y)=g(f(x, y))$ and determine where h is continuous. $f ( x , y ) = 2 x - y , \quad g ( t ) = \frac { t + 2 } { t - 8 }$

Find $\frac { \partial z } { \partial z }$ . $x e ^ { 4 y } + 4 y z + z e ^ { 8 x } = 0$

Find $f _ { y k }$ for the function $f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 }$ .

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height $10$ cm if the tin is 0.04 cm thick.

Find the gradient of the function $f ( x , y , z ) = z ^ { 6 } e ^ { 2 x \sqrt { y } }$ .

Use polar coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 6 } + y ^ { 6 } } { x ^ { 5 } + y ^ { 5 } }$

Find the limit $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 8 x ^ { 2 } + 2 y ^ { 2 } \right)$

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 9 , - 5 ) } x ^ { 5 } + 8 x ^ { 3 } y - 4 x y ^ { 2 }$

Find the indicated partial derivative. $f ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y ; f _ { m x }$

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

Use the Chain Rule to find $\frac { \partial w } { \partial s }$ where $s = 4 , t = 0$ . $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin t$

A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft. Use differentials to approximate the number of square feet of paint in the stripe.

Sketch the level curves $f ( x , y ) = k$ of the function for the indicated values of k. $f ( x , y ) = x - 4 y ^ { 2 } ; \quad k = - 2 , - 1,0,1,2$

Find the points on the surface $z ^ { 2 } = x y + 49$ that are closest to the origin.

Find the limit $\lim _ { ( x , y , z ) \rightarrow ( 3,2,4 ) } \frac { x y + y z + x z } { x y z - 6 }$

Find the shortest distance from the point $( 3,9,8 )$ to the plane $3 x + 9 y + 4 z = 16$ .

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

Exam 14: Partial Derivatives

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. f(x,y)=8x2−4y2,8x2+4y2=9f ( x , y ) = 8 x ^ { 2 } - 4 y ^ { 2 } , 8 x ^ { 2 } + 4 y ^ { 2 } = 9f(x,y)=8x2−4y2,8x2+4y2=9

Find the limit lim⁡(x,y,z)→(5,1,4)xy+yz+xzxyz−6\lim _ { ( x , y , z ) \rightarrow ( 5,1,4 ) } \frac { x y + y z + x z } { x y z - 6 }lim(x,y,z)→(5,1,4)​xyz−6xy+yz+xz​

Find three positive numbers whose sum is 291291291 and whose product is a maximum.

Find the maximum rate of change of f(x,y)=xy2+yf ( x , y ) = x y ^ { 2 } + \sqrt { y }f(x,y)=xy2+y​ at the point (4,1)( 4,1 )(4,1) (2,1). In what direction does it occur?

Find h(x,y)=g(f(x,y))h(x, y)=g(f(x, y))h(x,y)=g(f(x,y)) and determine where h is continuous. f(x,y)=2x−y,g(t)=t+2t−8f ( x , y ) = 2 x - y , \quad g ( t ) = \frac { t + 2 } { t - 8 }f(x,y)=2x−y,g(t)=t−8t+2​

Find ∂z∂z\frac { \partial z } { \partial z }∂z∂z​ . xe4y+4yz+ze8x=0x e ^ { 4 y } + 4 y z + z e ^ { 8 x } = 0xe4y+4yz+ze8x=0

Find fykf _ { y k }fyk​ for the function f(x,y)=4x3y−7xy2f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 }f(x,y)=4x3y−7xy2 .

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 101010 cm if the tin is 0.04 cm thick.

Find the gradient of the function f(x,y,z)=z6e2xyf ( x , y , z ) = z ^ { 6 } e ^ { 2 x \sqrt { y } }f(x,y,z)=z6e2xy​ .

Use polar coordinates to find the limit. lim⁡(x,y)→(0,0)x6+y6x5+y5\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 6 } + y ^ { 6 } } { x ^ { 5 } + y ^ { 5 } }lim(x,y)→(0,0)​x5+y5x6+y6​

Find the limit lim⁡(x,y)→(1,2)(8x2+2y2)\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 8 x ^ { 2 } + 2 y ^ { 2 } \right)lim(x,y)→(1,2)​(8x2+2y2)

Find the limit. lim⁡(x,y)→(9,−5)x5+8x3y−4xy2\lim _ { ( x , y ) \rightarrow ( 9 , - 5 ) } x ^ { 5 } + 8 x ^ { 3 } y - 4 x y ^ { 2 }lim(x,y)→(9,−5)​x5+8x3y−4xy2

Find the indicated partial derivative. f(x,y)=x2y4−3x4y;fmxf ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y ; f _ { m x }f(x,y)=x2y4−3x4y;fmx​

Determine where the function f(x,y,z)=xyzx2+y2+z2−2f ( x , y , z ) = \frac { x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 2 }f(x,y,z)=x2+y2+z2−2xyz​ is continuous.

Use the Chain Rule to find ∂w∂s\frac { \partial w } { \partial s }∂s∂w​ where s=4,t=0s = 4 , t = 0s=4,t=0 . w=x2+y2+z2,x=st,y=scos⁡t,z=ssin⁡tw = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin tw=x2+y2+z2,x=st,y=scost,z=ssint

A boundary stripe 2 in. wide is painted around a rectangle whose dimensions are 100 ft by 240 ft. Use differentials to approximate the number of square feet of paint in the stripe.

Sketch the level curves f(x,y)=kf ( x , y ) = kf(x,y)=k of the function for the indicated values of k. f(x,y)=x−4y2;k=−2,−1,0,1,2f ( x , y ) = x - 4 y ^ { 2 } ; \quad k = - 2 , - 1,0,1,2f(x,y)=x−4y2;k=−2,−1,0,1,2

Find the points on the surface z2=xy+49z ^ { 2 } = x y + 49z2=xy+49 that are closest to the origin.

Find the limit lim⁡(x,y,z)→(3,2,4)xy+yz+xzxyz−6\lim _ { ( x , y , z ) \rightarrow ( 3,2,4 ) } \frac { x y + y z + x z } { x y z - 6 }lim(x,y,z)→(3,2,4)​xyz−6xy+yz+xz​

Find the shortest distance from the point (3,9,8)( 3,9,8 )(3,9,8) to the plane 3x+9y+4z=163 x + 9 y + 4 z = 163x+9y+4z=16 .

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

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. $f ( x , y ) = 8 x ^ { 2 } - 4 y ^ { 2 } , 8 x ^ { 2 } + 4 y ^ { 2 } = 9$

Find the limit $\lim _ { ( x , y , z ) \rightarrow ( 5,1,4 ) } \frac { x y + y z + x z } { x y z - 6 }$

Find three positive numbers whose sum is $291$ and whose product is a maximum.

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 4,1 )$ (2,1). In what direction does it occur?

Find $h(x, y)=g(f(x, y))$ and determine where h is continuous. $f ( x , y ) = 2 x - y , \quad g ( t ) = \frac { t + 2 } { t - 8 }$

Find $\frac { \partial z } { \partial z }$ . $x e ^ { 4 y } + 4 y z + z e ^ { 8 x } = 0$

Find $f _ { y k }$ for the function $f ( x , y ) = 4 x ^ { 3 } y - 7 x y ^ { 2 }$ .

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height $10$ cm if the tin is 0.04 cm thick.

Find the gradient of the function $f ( x , y , z ) = z ^ { 6 } e ^ { 2 x \sqrt { y } }$ .

Use polar coordinates to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 6 } + y ^ { 6 } } { x ^ { 5 } + y ^ { 5 } }$

Find the limit $\lim _ { ( x , y ) \rightarrow ( 1,2 ) } \left( 8 x ^ { 2 } + 2 y ^ { 2 } \right)$

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 9 , - 5 ) } x ^ { 5 } + 8 x ^ { 3 } y - 4 x y ^ { 2 }$

Find the indicated partial derivative. $f ( x , y ) = x ^ { 2 } y ^ { 4 } - 3 x ^ { 4 } y ; f _ { m x }$

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

Use the Chain Rule to find $\frac { \partial w } { \partial s }$ where $s = 4 , t = 0$ . $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , \quad x = s t , \quad y = s \cos t , \quad z = s \sin t$

Sketch the level curves $f ( x , y ) = k$ of the function for the indicated values of k. $f ( x , y ) = x - 4 y ^ { 2 } ; \quad k = - 2 , - 1,0,1,2$

Find the points on the surface $z ^ { 2 } = x y + 49$ that are closest to the origin.

Find the limit $\lim _ { ( x , y , z ) \rightarrow ( 3,2,4 ) } \frac { x y + y z + x z } { x y z - 6 }$

Find the shortest distance from the point $( 3,9,8 )$ to the plane $3 x + 9 y + 4 z = 16$ .