Question 1

Find the equation of the normal line to the given surface at the specified point.
$$z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$$

Accepted Answer

The answer of Find the equation of the normal line...

Question 2

Use differentials to estimate the amount of metal in a closed cylindrical can that is $12 \mathrm {~cm}$ high and 8 $\mathrm { cm }$ in diameter if the metal in the top and bottom is $0.09 \mathrm {~cm}$ thick and the metal in the sides is $0.01$ $\mathrm { cm }$ thick. (rounded to the nearest hundredth.)

Accepted Answer

A)  $8.34 \mathrm {~cm} ^ { 3 }$ 
B)  $\quad 6.99 \mathrm {~cm} ^ { 3 }$ 
C)  $6.91 \mathrm {~cm} ^ { 3 }$ 
D)  $6.7 \mathrm {~cm} ^ { 3 }$ 
E)  $12.06 \mathrm {~cm} ^ { 3 }$ 
A)  $8.34 \mathrm {~cm} ^ { 3 }$ 
B)  $\quad 6.99 \mathrm {~cm} ^ { 3 }$ 
C)  $6.91 \mathrm {~cm} ^ { 3 }$ 
D)  $6.7 \mathrm {~cm} ^ { 3 }$ 
E)  $12.06 \mathrm {~cm} ^ { 3 }$

Question 3

Find the differential of the function.
$$y = x ^ { 9 } + 2 x$$

Accepted Answer

The answer of Find the differential of the function.
\[y =...

Question 4

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$, use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$.

Accepted Answer

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

Question 5

$$\text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {. }$$

Accepted Answer

The answer of \[\text { Find the differential of the...

Question 6

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint.
$$f ( x , y , z ) = 14 x + 8 y + 12 z , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 101$$

Accepted Answer

A)  $f ( 14,8,12 ) = 404$ 
B)  $f ( 20,9,18 ) = 568$ 
C)  $f ( 7,4,6 ) = 202$ 
D)  $f ( 18,17,5 ) = 308$ 
E)  $f ( 8,4,12 ) = 212$ 
A)  $f ( 14,8,12 ) = 404$ 
B)  $f ( 20,9,18 ) = 568$ 
C)  $f ( 7,4,6 ) = 202$ 
D)  $f ( 18,17,5 ) = 308$ 
E)  $f ( 8,4,12 ) = 212$

Question 7

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

Accepted Answer

The answer of \[\text { Find the gradient of the...

Question 8

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 7 x - 8$. Find $f ( 2 h , 3 k )$.
Select the correct answer.

Accepted Answer

A)  $9 h ^ { 2 } + 18 h k - 21 h - 8$ 
B)  $4 h ^ { 2 } + 18 h k - 21 h - 8$ 
C)  $4 h ^ { 2 } + 18 h k - 14 h - 8$ 
D)  $6 h ^ { 2 } + 18 h k - 42 h - 48$ 
A)  $9 h ^ { 2 } + 18 h k - 21 h - 8$ 
B)  $4 h ^ { 2 } + 18 h k - 21 h - 8$ 
C)  $4 h ^ { 2 } + 18 h k - 14 h - 8$ 
D)  $6 h ^ { 2 } + 18 h k - 42 h - 48$

Question 9

$$F ( x , y ) = \arctan ( 10 x + 8 \sqrt { y - 4 } )$$

Accepted Answer

The answer of \[F ( x , y ) =...

Question 10

Find the equation of the tangent plane to the given surface at the specified point. $\backslash$
Select the correct answer.
$z + 4 = x e ^ { y } \cos z , ( 4,0,0 )$

Accepted Answer

A)  $x + y - z = 4$ 
B)  $x + 4 y + z = 4$ 
C)  $4 x + y - z = 4$ 
D)  $x + y - 4 z = 4$ 
E)  $x + 4 y - z = 4$ 
A)  $x + y - z = 4$ 
B)  $x + 4 y + z = 4$ 
C)  $4 x + y - z = 4$ 
D)  $x + y - 4 z = 4$ 
E)  $x + 4 y - z = 4$

Question 11

At what point is the following function a local maximum?
$$f ( x , y ) = 3 - 8 x + 12 y - 4 x ^ { 2 } - 6 y ^ { 2 }$$

Accepted Answer

The answer of At what point is the following function...

Question 12

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 13

Find the absolute extrema of the function
$$f ( x , y ) = 2 x + 3 y - 5$$
on the closed triangular region with vertices $( 0,0 ) , ( 5,0 )$, and $( 5,4 )$.

Accepted Answer

The answer of Find the absolute extrema of the function
\[f...

Question 14

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

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 15

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$.

Accepted Answer

The answer of Find equations for the tangent plane and...

Question 16

The radius of a right circular cone is increasing at a rate of $5 \mathrm { in } / \mathrm { s }$ while its height is decreasing at a rate of $3.6 \mathrm { in } / \mathrm { s }$. At what rate is the volume of the cone changing when the radius is $110 \mathrm { in }$. and the height is 145 in.?

Accepted Answer

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

Question 17

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 18

Find the indicated partial derivative.
$$u = x ^ { a } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$$

Accepted Answer

A) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = c b ( b - 1 ) c ( a - 1 ) ( a - 2 ) x ^ { c - 1 } y ^ { b - 2 } z ^ { a - 3 }$ 
B)  $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { a - 1 } y ^ { 3 - 2 } z ^ { c - 3 }$ 
C) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a b ( b - 1 ) c ( c - 1 ) ( c - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$ 
D) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a c b ( a - 1 ) ( a - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$ 
E) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { b - 1 } y ^ { c - 2 } z ^ { a - 3 }$ 
A) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = c b ( b - 1 ) c ( a - 1 ) ( a - 2 ) x ^ { c - 1 } y ^ { b - 2 } z ^ { a - 3 }$ 
B)  $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { a - 1 } y ^ { 3 - 2 } z ^ { c - 3 }$ 
C) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a b ( b - 1 ) c ( c - 1 ) ( c - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$ 
D) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a c b ( a - 1 ) ( a - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$ 
E) 
$\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { b - 1 } y ^ { c - 2 } z ^ { a - 3 }$

Question 19

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 }$

Accepted Answer

A)  None 
B)  Relative maximum $f \left( 0 , \frac { \pi } { 8 } \right) = 1$ 
C)  Saddle point $( 0,0,0 )$
$\mathrm { d } .$ Relative minimum $f \left( 0 , \frac { \pi } { 4 } \right) = 0$ 
A)  None 
B)  Relative maximum $f \left( 0 , \frac { \pi } { 8 } \right) = 1$ 
C)  Saddle point $( 0,0,0 )$
$\mathrm { d } .$ Relative minimum $f \left( 0 , \frac { \pi } { 4 } \right) = 0$

Question 20

The height of a hill (in feet) is given by
$$h ( x , y ) = 30 \left( 14 - 5 x ^ { 2 } - 2 y ^ { 2 } + 3 x y + 30 x - 18 y \right)$$
where $x$ is the distance (in miles) east and $y$ is the distance (in miles) north of your cabin. If you are at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the height of the hill (a) in a northerly direction and (b) in an easterly direction?

Accepted Answer

A)  (a) $570 \mathrm { ft } / \mathrm { mi }$, (b) $690 \mathrm { ft } / \mathrm { mi }$ 
B)  (a) $- 570 \mathrm { ft } / \mathrm { mi }$, (b) $690 \mathrm { ft } / \mathrm { mi }$ 
C)  (a) $690 \mathrm { ft } / \mathrm { mi }$, (b) $570 \mathrm { ft } / \mathrm { mi }$ 
D)  (a) $690 \mathrm { ft } / \mathrm { mi }$, (b) $- 570 \mathrm { ft } / \mathrm { mi }$ 
A)  (a) $570 \mathrm { ft } / \mathrm { mi }$, (b) $690 \mathrm { ft } / \mathrm { mi }$ 
B)  (a) $- 570 \mathrm { ft } / \mathrm { mi }$, (b) $690 \mathrm { ft } / \mathrm { mi }$ 
C)  (a) $690 \mathrm { ft } / \mathrm { mi }$, (b) $570 \mathrm { ft } / \mathrm { mi }$ 
D)  (a) $690 \mathrm { ft } / \mathrm { mi }$, (b) $- 570 \mathrm { ft } / \mathrm { mi }$

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

Find the differential of the function. $y = x ^ { 9 } + 2 x$

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$ , use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$ .

$\text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {. }$

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. $f ( x , y , z ) = 14 x + 8 y + 12 z , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 101$

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

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 7 x - 8$ . Find $f ( 2 h , 3 k )$ . Select the correct answer.

$F ( x , y ) = \arctan ( 10 x + 8 \sqrt { y - 4 } )$

Find the equation of the tangent plane to the given surface at the specified point. $\backslash$ Select the correct answer. $z + 4 = x e ^ { y } \cos z , ( 4,0,0 )$

At what point is the following function a local maximum? $f ( x , y ) = 3 - 8 x + 12 y - 4 x ^ { 2 } - 6 y ^ { 2 }$

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

Find the absolute extrema of the function $f ( x , y ) = 2 x + 3 y - 5$ on the closed triangular region with vertices $( 0,0 ) , ( 5,0 )$ , and $( 5,4 )$ .

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

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$ .

The radius of a right circular cone is increasing at a rate of $5 \mathrm { in } / \mathrm { s }$ while its height is decreasing at a rate of $3.6 \mathrm { in } / \mathrm { s }$ . At what rate is the volume of the cone changing when the radius is $110 \mathrm { in }$ . and the height is 145 in.?

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

Find the indicated partial derivative. $u = x ^ { a } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 }$

Functions and Models

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Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

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Exam 14: Partial Derivatives

Find the equation of the normal line to the given surface at the specified point. z+2=xeycos⁡z,(4,0,0)z + 2 = x e ^ { y } \cos z , ( 4,0,0 )z+2=xeycosz,(4,0,0)

Find the differential of the function. y=x9+2xy = x ^ { 9 } + 2 xy=x9+2x

If f(x,y)=x2+6y2f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }f(x,y)=x2+6y2 , use the gradient vector ∇f(10,2)\nabla f ( 10,2 )∇f(10,2) to find the tangent line to the level curve f(x,y)=124f ( x , y ) = 124f(x,y)=124 at the point (10,2)( 10,2 )(10,2) .

Find the differential of the function z=e3xsin⁡5y. \text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {. } Find the differential of the function z=e3xsin5y.

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. f(x,y,z)=14x+8y+12z,x2+y2+z2=101f ( x , y , z ) = 14 x + 8 y + 12 z , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 101f(x,y,z)=14x+8y+12z,x2+y2+z2=101

Find the gradient of the function f(x,y,z)=zγe2xy. \text { Find the gradient of the function } f ( x , y , z ) = z ^ { \gamma } e ^ { 2 x \sqrt { y } } \text {. } Find the gradient of the function f(x,y,z)=zγe2xy​.

Let f(x,y)=x2+3xy−7x−8f ( x , y ) = x ^ { 2 } + 3 x y - 7 x - 8f(x,y)=x2+3xy−7x−8 . Find f(2h,3k)f ( 2 h , 3 k )f(2h,3k) . Select the correct answer.

F(x,y)=arctan⁡(10x+8y−4)F ( x , y ) = \arctan ( 10 x + 8 \sqrt { y - 4 } )F(x,y)=arctan(10x+8y−4​)

Find the equation of the tangent plane to the given surface at the specified point. \\backslash\ Select the correct answer. z+4=xeycos⁡z,(4,0,0)z + 4 = x e ^ { y } \cos z , ( 4,0,0 )z+4=xeycosz,(4,0,0)

At what point is the following function a local maximum? f(x,y)=3−8x+12y−4x2−6y2f ( x , y ) = 3 - 8 x + 12 y - 4 x ^ { 2 } - 6 y ^ { 2 }f(x,y)=3−8x+12y−4x2−6y2

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​

Find the absolute extrema of the function f(x,y)=2x+3y−5f ( x , y ) = 2 x + 3 y - 5f(x,y)=2x+3y−5 on the closed triangular region with vertices (0,0),(5,0)( 0,0 ) , ( 5,0 )(0,0),(5,0) , and (5,4)( 5,4 )(5,4) .

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

Find equations for the tangent plane and the normal line to the surface with equation x2+9y2+9z2=22x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22x2+9y2+9z2=22 at the point P(2,1,1)P ( 2,1,1 )P(2,1,1) .

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​)

Find the indicated partial derivative. u=xaybzc;∂6u∂x∂y2∂z3,a>1,b>2,c>3u = x ^ { a } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3u=xaybzc;∂x∂y2∂z3∂6u​,a>1,b>2,c>3

Find and classify the relative extrema and saddle points of the function f(x,y)=e−2xsin⁡4yf ( x , y ) = e ^ { - 2 x } \sin 4 yf(x,y)=e−2xsin4y for x≥0x \geq 0x≥0 and 0≤y≤π20 \leq y \leq \frac { \pi } { 2 }0≤y≤2π​

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

Filters

Find the equation of the normal line to the given surface at the specified point. $z + 2 = x e ^ { y } \cos z , ( 4,0,0 )$

Find the differential of the function. $y = x ^ { 9 } + 2 x$

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$ , use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$ .

$\text { Find the differential of the function } z = e ^ { 3 x } \sin 5 y \text {. }$

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. $f ( x , y , z ) = 14 x + 8 y + 12 z , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 101$

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

Let $f ( x , y ) = x ^ { 2 } + 3 x y - 7 x - 8$ . Find $f ( 2 h , 3 k )$ . Select the correct answer.

$F ( x , y ) = \arctan ( 10 x + 8 \sqrt { y - 4 } )$

Find the equation of the tangent plane to the given surface at the specified point. $\backslash$ Select the correct answer. $z + 4 = x e ^ { y } \cos z , ( 4,0,0 )$

At what point is the following function a local maximum? $f ( x , y ) = 3 - 8 x + 12 y - 4 x ^ { 2 } - 6 y ^ { 2 }$

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

Find the absolute extrema of the function $f ( x , y ) = 2 x + 3 y - 5$ on the closed triangular region with vertices $( 0,0 ) , ( 5,0 )$ , and $( 5,4 )$ .

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

Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$ .

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

Find the indicated partial derivative. $u = x ^ { a } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = e ^ { - 2 x } \sin 4 y$ for $x \geq 0$ and $0 \leq y \leq \frac { \pi } { 2 }$