Deck 14: Partial Derivatives

Question 1

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

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) $x = 129 , y = 79 , z = 89$

Answer 1

$x = y = z = 97$

Question 2

Find the direction in which the maximum rate of change of f at the given point occurs. $f ( x , y ) = 2 \sin ( x y ) , ( 1,0 )$

A) $\langle2,0 \rangle$
B) $\langle1 , - \sqrt { 2 } \rangle$
C) $\langle \sqrt { 2 } , 0 \rangle$
D) $\langle0,2 \rangle$
E) $\left\langle\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right\rangle$

Answer 2

$\langle0,2 \rangle$

Question 3

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

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) $$f ( x , y ) = 9$$

Answer 3

$$f ( x , y ) = 9$$

Question 4

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

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

Question 5

Use Lagrange multipliers to find the maximum value of the function subject to the given constraints.

Question 6

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

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 7

Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In the case of $f _ { x } ( 1,1 ) = 8$ , $f _ { x y } ( 1,1 ) = 8$ , $f _ { y y } ( 1,1 ) = 10$ what can you say about f ?

A) f has a local maximum at (1,1)
B) f has a saddle point at (1,1)
C) f has a local minimum at (1,1)

Question 8

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 $0 \leq y \leq \frac { \pi } { 2 }$ .

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

Question 9

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the constraints and .

Question 10

Find all the saddle points of the function. $f ( x , y ) = x \sin \frac { y } { 2 }$

A) $( 3 \pi n , 0 )$
B) $\left( 0 , \frac { \pi n } { 2 } \right)$
C) $( 2 \pi n , 1 )$
D) $$( 0,2 \pi n )$$
E) $\left( \frac { 2 n } { \pi } , 0 \right)$

Question 11

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

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 12

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

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 13

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s).

Question 14

At what point is the following function a local minimum? $f ( x , y ) = 8 x ^ { 2 } + 5 y ^ { 2 }$

A) $( 5,0 )$
B) $( 8 , - 5 )$
C) $( 8,0 )$
D) $( 8,5 )$
E) $( 0,0 )$

Question 15

Use Lagrange multipliers to find the minimum value of the function subject to the given constraints.

Question 16

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

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 17

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

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

Question 18

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

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

Question 19

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 $( 5,4 )$ .

A) Absolute minimum 0, Absolute maximum 5
B) Absolute minimum -5, Absolute maximum 5
C) Absolute minimum -5, Absolute maximum 17
D) Absolute minimum 5, Absolute maximum 17

Question 20

Find the absolute minimum value of the function $f ( x , y ) = 6 + 3 x y - 2 x - 4 y$ on the set D. D is the region bounded by the parabola $y = x ^ { 2 }$ and the line $y = 4$

A) $- 31$
B) $- 32$
C) $- 30$
D) 30
E) 0

Question 21

Find the equation of the tangent plane to the given surface at the specified point.

Question 22

Find the directional derivative of the function $f ( x , y ) = ( x + 5 ) e ^ { y }$ at the point $P ( 6,0 )$ in the direction of the unit vector that makes the angle $\theta = \frac { \pi } { 2 }$ with the positive x-axis.

A) $e ^ { 6 }$
B) 1
C) $5 e ^ { 6 }$
D) 11

Question 23

Find the directional derivative of $f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }$ at the point (1, 3) in the direction toward the point (3, 1). Select the correct answer.

A) $7 \sqrt { 2 }$
B) $14 \sqrt { 2 }$
C) $\sqrt { 2 }$
D) 28
E) none of these

Question 24

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

Question 25

Suppose that over a certain region of space the electrical potential V is given by $V ( x , y , z ) = 8 x ^ { 2 } - 7 x y + 7 x y z$ . Find the rate of change of the potential at $( - 1,1 , - 1 )$ in the direction of the vector $\mathbf { v } = 8 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }$ .

A) $- 15.099$
B) 44
C) -2.91
D) 20
E) $\text { 14. } 569856$

Question 26

Find the equation of the normal line to the given surface at the specified point. $2 x ^ { 2 } + 8 y ^ { 2 } + 3 z ^ { 2 } = 235 , ( 4,4,5 )$

A) $2 x + 8 y + 3 z = 1$
B) $16 x + 64 y + 30 z = 235$
C) $\frac { x - 4 } { 55 } = \frac { y - 4 } { 55 } = \frac { z - 5 } { 55 }$
D) $\frac { x - 4 } { 16 } = \frac { y - 4 } { 64 } = \frac { z - 5 } { 30 }$
E) $\frac { x + 4 } { 16 } = \frac { y + 4 } { 64 } = \frac { z + 5 } { 30 }$

Question 27

Find and classify the relative extrema and saddle points of the function .

Question 28

Which of the given points are the points on the hyperboloid $x ^ { 2 } - y ^ { 2 } + 4 z ^ { 2 } = 4$ where the normal line is parallel to the line that joins the points $( - 1,1,3 )$ and $( 0,2,5 )$ .
Select all that apply.

A) $( - 2,2,1 )$
B) $( - 2,2 , - 1 )$
C) $( 2 , - 2,1 )$
D) $( 2 , - 2 , - 1 )$
E) $( 2,2,1 )$

Question 29

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?

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 30

Find the maximum rate of change of f at the given point.

Question 31

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 $P ( 2,1,1 )$

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 32

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

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 33

A cardboard box without a lid is to have a volume of cm . Find the dimensions that minimize the amount of cardboard used.

Question 34

Evaluate the gradient of f at the point P.

Question 35

Find equations for the tangent plane and the normal line to the surface with equation $x y + y z + x z = 38$ at the point $P ( 2,4,5 )$

A) $6 x + 7 y + 9 z = 76$ , $\frac { x - 2 } { 6 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 9 }$
B) $9 x + 7 y + 6 z = 76$ , $\frac { x - 2 } { 9 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 6 }$
C) $2 x + 4 y + 5 z = 76$ , $\frac { x - 6 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 9 } { 5 }$
D) $2 x + 4 y + 5 z = 76$ , $\frac { x - 9 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 6 } { 5 }$

Question 36

Find the absolute extrema of the function on the region bounded by the disk defined by .

Question 37

Find three positive real numbers whose sum is 388 and whose product is as large as possible.

Question 38

Find the direction in which the function decreases fastest at the point .

Question 39

Find the local maximum, and minimum value and saddle points of the function.

Question 40

If $f ( x , y ) = x ^ { 2 } + 7 y ^ { 2 }$ use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 136$ at the point $( 10,2 )$ .

A) $10 x + 14 y = 12$
B) $10 x + 14 y = 136$
C) $100 x - 49 y = 12$
D) $100 x + 49 y = 136$
E) $100 x - 14 y = 12$

Question 41

Use the Chain Rule to find $\frac { \partial u } { \partial p }$ . $u = \frac { x + y } { y + z }$ $x = p + 5 r + 7 t , y = p - 5 r + 7 t , z = p + 5 r - 7 t$

A) $\frac { \partial u } { \partial p } = - \frac { 7 t } { p ^ { 2 } }$
B) $\frac { \partial u } { \partial p } = \frac { 7 t } { p }$
C) $\frac { \partial u } { \partial p } = - \frac { 5 t } { p ^ { 3 } }$
D) $\frac { \partial u } { \partial p } = - \frac { 35 t } { p ^ { 2 } }$
E) $\frac { \partial u } { \partial p } = \frac { 35 t } { p ^ { 2 } } + t$

Question 42

Find the limit if .

Question 43

The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is $108$ in. and the height is $132$ in.?

A) $108316.24 \mathrm { in } ^ { 3 } / \mathrm { s }$
B) $105316.24 \mathrm { in } ^ { 3 } / \mathrm { s }$
C) $111316.24 \mathrm { in } ^ { 3 } / \mathrm { s }$
D) $102316.24 \mathrm { in } ^ { 3 } / \mathrm { s }$
E) $99316.24 \mathrm { in } ^ { 3 } / \mathrm { s }$

Question 44

Find the differential of the function $z = 3 x ^ { 3 } y ^ { 6 }$

A) $d z = 9 x ^ { 2 } y ^ { 5 } d x + 18 x ^ { 2 } y ^ { 5 } d y$
B) $d z = 9 x ^ { 2 } y ^ { 6 } d x + 18 x ^ { 3 } y ^ { 5 } d y$
C) $d z = 18 x ^ { 2 } y ^ { 5 } d x + 9 x ^ { 2 } y ^ { 5 } d y$
D) $d z = 18 x ^ { 3 } y ^ { 5 } d x + 9 x ^ { 2 } y ^ { 6 } d y$

Question 45

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

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 46

Use the Chain Rule to find where .

Question 47

Find an equation of the tangent plane to the given surface at the specified point.

Question 48

Use the Chain Rule to find $\frac { \partial w } { \partial r }$ and $\frac { \partial w } { \partial t }$ if $r = 5$ $s = 2$ and $t = 0$ $w = \frac { x ^ { 2 } y } { z ^ { 2 } } , \quad x = r e ^ { s t } , \quad y = s e ^ { n t } , \quad z = e ^ { r s t }$

A) $\frac { \partial w } { \partial r } = 20 , \quad \frac { \partial w } { \partial t } = - 550$
B) $\frac { \partial w } { \partial r } = 100 , \quad \frac { \partial w } { \partial t } = 200$
C) $\frac { \partial w } { \partial r } = 10 , \quad \frac { \partial w } { \partial t } = 0$
D) $\frac { \partial w } { \partial r } = 7 , \quad \frac { \partial w } { \partial t } = - 550$

Question 49

Find the equation of the tangent plane to the given surface at the specified point.

Question 50

Use the equation $\frac { d y } { d x } = - \frac { \frac { \partial F } { \partial x } } { \frac { \partial F } { \partial y } } = - \frac { F _ { x } } { F _ { y } }$ to find $\frac { d y } { d x }$ . $\cos ( x - 7 y ) = x e ^ { 4 y }$

A) $\frac { d y } { d x } = \frac { \sin ( x - y ) + e ^ { 4 y } } { 7 \sin ( x - 7 y ) - x e ^ { 4 y } }$
B) $\frac { d y } { d x } = \frac { \sin ( x - y ) + e ^ { 4 y } } { \sin ( x - y ) - x e ^ { y } }$
C) $\frac { d y } { d x } = \frac { \sin ( x - 7 y ) + e ^ { 4 y } } { 7 \sin ( x - 7 y ) - 4 x e ^ { 4 y } }$
D) $\frac { d y } { d x } = \frac { 7 \sin ( x - y ) + e ^ { 4 y } } { \sin ( x - 7 y ) - 7 x e ^ { y } }$
E) $\frac { d y } { d x } = \frac { \sin ( x - 7 y ) + e ^ { 4 y } } { \sin ( x - 7 y ) - x e ^ { 4 y } }$

Question 51

The length l, width w and height h of a box change with time. At a certain instant the dimensions are and , and l and w are increasing at a rate of 10 m/s while h is decreasing at a rate of 1 m/s. At that instant find the rates at which the surface area is changing.

Question 52

Use implicit differentiation to find .

Question 53

Find the equation of the tangent plane to the given surface at the specified point. $z + 7 = x e ^ { y } \cos z , ( 7,0,0 )$

A) $x + 7 y + z = 7$
B) $x + 7 y - z = 7$
C) $x + y - z = 7$
D) $7 x + y - z = 7$
E) $x + y - 7 z = 7$

Question 54

Use the Chain Rule to find $\frac { d w } { d t }$ $w = 9 x ^ { 4 } y ^ { 3 } z , \quad x = 6 t , \quad y = \cos 7 t , \quad z = t \sin t$

A) $4,536 x ^ { 4 } y ^ { 3 } z \sin 7 t ( \sin t + t \cos t )$
B) $9 x ^ { 4 } y ^ { 3 } ( 24 y z - 21 x z \sin 7 t + x y ( \sin t + t \cos t ) )$
C) $4,536 x ^ { 3 } y ^ { 2 } \sin 7 t ( \sin t + t \cos t )$
D) $9 x ^ { 3 } y ^ { 2 } ( 24 y z - 21 x z \sin 7 t + x y ( \sin t + t \cos t ) )$

Question 55

Find the gradient of the function .

Question 56

Find the equation of the normal line to the given surface at the specified point.

Question 57

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.

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 58

Use partial derivatives to find the implicit partial derivatives and

Question 59

Find the gradient of at the point

Question 60

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

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

Question 61

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

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 62

If and changes from (2, 1) to find dz.

Question 63

Find the differential of the function.

Question 64

Use the linearization L(x, y) of the function. at to approximate .

Question 65

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

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

Question 66

Find the differential of the function

Question 67

Find $f _ { m w } ( x , y )$ for the function $f ( x , y ) = x ^ { 4 } - 9 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } + 6 y ^ { 4 }$

A) $24 x$
B) $92 x ^ { 2 }$
C) $- 15 y ^ { 2 }$
D) $- 36 y$

Question 68

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

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 69

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ $\ln \left( x ^ { 2 } + z ^ { 2 } \right) + y z ^ { 3 } + 4 x ^ { 2 } = 6$

A) $\frac { \partial z } { \partial x } = - \frac { 2 x \left( x ^ { 2 } + z ^ { 2 } + 6 \right) } { z \left( 3 y z ^ { 3 } + 3 x ^ { 2 } y z + 6 \right) }$
B) $\frac { \partial z } { \partial x } = - \frac { 2 x \left( 4 x ^ { 2 } + 4 z ^ { 2 } + 1 \right) } { z \left( 3 y z ^ { 3 } + 3 x ^ { 2 } y z + 2 \right) }$
C) $\frac { \partial z } { \partial x } = - \frac { 2 x \left( 4 x ^ { 2 } + 4 z ^ { 2 } + 6 \right) } { z \left( 3 y z ^ { 3 } + 3 x ^ { 2 } y z + 6 \right) }$
D) $\frac { \partial z } { \partial x } = - \frac { 2 x \left( 6 x ^ { 2 } + 6 z ^ { 2 } + 1 \right) } { z \left( 3 y z ^ { 3 } + 3 x ^ { 2 } y z + 2 \right) }$

Question 70

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

Question 71

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

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) $\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }$
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 72

Find the differential of the function

Question 73

Let and suppose that changes from to (a) Compute (b) Compute

Question 74

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?

A) (a) 570 ft/mi, (b) 690 ft/mi
B) (a) -570 ft/mi, (b) 690 ft/mi
C) (a) 690 ft/mi, (b) 570 ft/mi
D) (a) 690 ft/mi, (b) -570 ft/mi

Question 75

Find $h_{m y}(x, y, z)$ for the function $h ( x , y , z ) = e ^ { 9 x } \cos ( y + 7 z )$

A) $63 e ^ { 9 x } \cos ( y + 7 z )$
B) $49 e ^ { 9 x } \sin ( y + 7 z )$
C) $- 63 e ^ { 9 x } \sin ( y + 7 z )$
D) $- 49 e ^ { 9 x } \cos ( y + 7 z )$

Question 76

Use the definition of partial derivatives as limits to find $f _ { x } ( x , y )$ if $f ( x , y ) = 5 x ^ { 2 } - 9 x y + 2 y ^ { 2 }$ .

A) $5 y - 9 x$
B) $10 x - 9 y$
C) $10 x - 9$
D) $10 x - 9 x y$
E) $5 x - 9 y$

Question 77

The wind-chill index I is the perceived temperature when the actual temperature is T and the wind speed is v so we can write . The following table of values is an excerpt from a table compiled by the National Atmospheric and Oceanic Administration. Use the table to find a linear approximation to the wind chill index function when T is near and v is near 30 kmh.

Question 78

Find the linearization L(x, y) of the function at the given point. Round the answers to the nearest hundredth.

Question 79

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ $x ^ { 4 } y + x z + y z ^ { 2 } = 7$

A) $\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } y } { 1 + 2 y }$
B) $\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } y + z } { x + 2 y z }$
C) $\frac { \partial z } { \partial x } = - \frac { 4 x ^ { 3 } } { 1 + 2 y }$
D) $\frac { \partial z } { \partial x } = \frac { 1 } { 1 + 2 y }$

Question 80

Find $f _ { y } ( - 24,8 )$ for $f ( x , y ) = \sin ( 4 x + 12 y )$ .

A) $12$
B) $- 4$
C) $- 12$
D) $4$
E) 0