Question 1

Find the domain and range of the function  f(x,y,z) =4−x2−y2−z2f ( x , y , z )  = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }f(x,y,z) =4−x2−y2−z2​  .

Accepted Answer

A)    D={(x,y,z) ∣x2+y2+z2≤4}R={w∣0≤w≤2}\begin{array} { l } D = \left\{ ( x , y , z )  \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 4 ight\} \R = \{ w \mid 0 \leq w \leq 2 \}\end{array}D={(x,y,z) ∣x2+y2+z2≤4}R={w∣0≤w≤2}​ 
B)    D={(x,y,z) ∣x2+y2+z2≤4}R={w∣0≤w}\begin{array} { l } D = \left\{ ( x , y , z )  \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 4 ight\} \R = \{ w \mid 0 \leq w \}\end{array}D={(x,y,z) ∣x2+y2+z2≤4}R={w∣0≤w}​ 
C)    D={(x,y,z) ∣x2+y2+z2≥4}R={w∣0≤w}\begin{array} { l } D = \left\{ ( x , y , z )  \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geq 4 ight\} \R = \{ w \mid 0 \leq w \}\end{array}D={(x,y,z) ∣x2+y2+z2≥4}R={w∣0≤w}​ 
D)    D={(x,y,z) ∣x2+y2+z2≥4}R={w∣0≤w≤2}\begin{array} { l } D = \left\{ ( x , y , z )  \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geq 4 ight\} \R = \{ w \mid 0 \leq w \leq 2 \}\end{array}D={(x,y,z) ∣x2+y2+z2≥4}R={w∣0≤w≤2}​

Question 2

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

Accepted Answer

The answer of Find the differential of the function \(z...

Question 3

Evaluate the gradient of f at the point P. $f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } , \quad P ( - 1,3 , - 1 )$

Accepted Answer

The answer of Evaluate the gradient of f at the...

Question 4

Find the absolute minimum value of the function  f(x,y) =6+3xy−2x−4yf ( x , y )  = 6 + 3 x y - 2 x - 4 yf(x,y) =6+3xy−2x−4y  on the set D. D is the region bounded by the parabola  y=x2y = x ^ { 2 }y=x2  and the line  y=4y = 4y=4

Accepted Answer

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

Question 5

Find the linearization L(x, y) of the function at the given point. $f ( x , y ) = x \sqrt { y } , ( - 5,25 )$ Round the answers to the nearest hundredth.

Accepted Answer

The answer of Find the linearization L(x, y) of the...

Question 6

Use partial derivatives to find the implicit derivative $\frac { d y } { d x }$ $8 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 5$

Accepted Answer

The answer of Use partial derivatives to find the implicit...

Question 7

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$ .

Accepted Answer

The answer of Use Lagrange multipliers to find the maximum...

Question 8

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

Accepted Answer

A)    121212 
B)    −4- 4−4 
C)    −12- 12−12 
D)    444 
E)   0

Question 9

Find the equation of the tangent plane to the given surface at the specified point. $3 x ^ { 2 } + 3 y ^ { 2 } + 8 z ^ { 2 } = 353 , \quad ( 3,6,5 )$

Accepted Answer

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

Question 10

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

Accepted Answer

A)    ∂6u∂x∂y2∂z3=cb(b−1) c(a−1) (a−2) xc−1yb−2za−3\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 }∂x∂y2∂z3∂6u​=cb(b−1) c(a−1) (a−2) xc−1yb−2za−3 
B)    ∂6u∂x∂y2∂z3=xa−1yb−2zc−3\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }∂x∂y2∂z3∂6u​=xa−1yb−2zc−3 
C)    ∂6u∂x∂y2∂z3=ab(b−1) c(c−1) (c−2) xa−1yb−2zc−3\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 }∂x∂y2∂z3∂6u​=ab(b−1) c(c−1) (c−2) xa−1yb−2zc−3 
D)    ∂6u∂x∂y2∂z3=acb(a−1) (a−2) xa−1yb−2zc−3\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 }∂x∂y2∂z3∂6u​=acb(a−1) (a−2) xa−1yb−2zc−3 
E)    ∂6u∂x∂y2∂z3=xb−1yc−2za−3\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { b - 1 } y ^ { c - 2 } z ^ { a - 3 }∂x∂y2∂z3∂6u​=xb−1yc−2za−3

Question 11

Suppose that over a certain region of space the electrical potential V is given by  V(x,y,z) =8x2−7xy+7xyzV ( x , y , z )  = 8 x ^ { 2 } - 7 x y + 7 x y zV(x,y,z) =8x2−7xy+7xyz  . Find the rate of change of the potential at  (−1,1,−1) ( - 1,1 , - 1 ) (−1,1,−1)   in the direction of the vector  v=8i+10j−8k\mathbf { v } = 8 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }v=8i+10j−8k  .

Accepted Answer

A)    −15.099- 15.099−15.099 
B)   44
C)   -2.91
D)   20
E)     14. 569856	ext { 14. } 569856 14. 569856

Question 12

Let  g(r,s,t) =res/tg ( r , s , t )  = r e ^ { s / t }g(r,s,t) =res/t  Find  g(8,ln⁡3,12) g \left( 8 , \ln 3 , \frac { 1 } { 2 } \right) g(8,ln3,21​)

Accepted Answer

A)    43\frac { 4 } { 3 }34​ 
B)    121212 
C)    838 \sqrt { 3 }83c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)   72

Exam 14: Partial Derivatives

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

Correct Answer
verified

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 $\mathbf { v } = 8 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }$ .

Correct Answer
verified

Use partial derivatives to find the implicit derivative $\frac { d y } { d x }$ $8 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 5$

Correct Answer
verified

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$ .

Correct Answer
verified

Find the equation of the tangent plane to the given surface at the specified point. $3 x ^ { 2 } + 3 y ^ { 2 } + 8 z ^ { 2 } = 353 , \quad ( 3,6,5 )$

Correct Answer
verified

Find the linearization L(x, y) of the function at the given point. $f ( x , y ) = x \sqrt { y } , ( - 5,25 )$ Round the answers to the nearest hundredth.

Correct Answer
verified

Let $g ( r , s , t ) = r e ^ { s / t }$ Find $g \left( 8 , \ln 3 , \frac { 1 } { 2 } \right)$

Correct Answer
verified

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

Correct Answer
verified

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 $y = 4$

Correct Answer
verified

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

Correct Answer
verified

Evaluate the gradient of f at the point P. $f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } , \quad P ( - 1,3 , - 1 )$

Correct Answer
verified

Find the domain and range of the function $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }$ .

Correct Answer
verified

Exam 14: Partial Derivatives

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Use partial derivatives to find the implicit derivative dydx\frac { d y } { d x }dxdy​ 8x2+9xy−7y=58 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 58x2+9xy​−7y=5

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Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y,z)=5x−y−5zf ( x , y , z ) = 5 x - y - 5 zf(x,y,z)=5x−y−5z subject to the constraints x+2y−z=0x + 2 y - z = 0x+2y−z=0 and x2+4y2=1x ^ { 2 } + 4 y ^ { 2 } = 1x2+4y2=1 .

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Find the equation of the tangent plane to the given surface at the specified point. 3x2+3y2+8z2=353,(3,6,5)3 x ^ { 2 } + 3 y ^ { 2 } + 8 z ^ { 2 } = 353 , \quad ( 3,6,5 )3x2+3y2+8z2=353,(3,6,5)

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Find the linearization L(x, y) of the function at the given point. f(x,y)=xy,(−5,25)f ( x , y ) = x \sqrt { y } , ( - 5,25 )f(x,y)=xy​,(−5,25) Round the answers to the nearest hundredth.

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Let g(r,s,t) =res/tg ( r , s , t ) = r e ^ { s / t }g(r,s,t) =res/t Find g(8,ln⁡3,12) g \left( 8 , \ln 3 , \frac { 1 } { 2 } \right) g(8,ln3,21​)

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

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Find the absolute minimum value of the function f(x,y) =6+3xy−2x−4yf ( x , y ) = 6 + 3 x y - 2 x - 4 yf(x,y) =6+3xy−2x−4y on the set D. D is the region bounded by the parabola y=x2y = x ^ { 2 }y=x2 and the line y=4y = 4y=4

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

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Evaluate the gradient of f at the point P. f(x,y,z)=xy2z3,P(−1,3,−1)f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } , \quad P ( - 1,3 , - 1 )f(x,y,z)=xy2z3,P(−1,3,−1)

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Find the domain and range of the function f(x,y,z) =4−x2−y2−z2f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }f(x,y,z) =4−x2−y2−z2​ .

Correct AnswerverifiedShow Answer

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

Correct Answer
verified

Correct Answer
verified

Use partial derivatives to find the implicit derivative $\frac { d y } { d x }$ $8 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 5$

Correct Answer
verified

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$ .

Correct Answer
verified

Find the equation of the tangent plane to the given surface at the specified point. $3 x ^ { 2 } + 3 y ^ { 2 } + 8 z ^ { 2 } = 353 , \quad ( 3,6,5 )$

Correct Answer
verified

Find the linearization L(x, y) of the function at the given point. $f ( x , y ) = x \sqrt { y } , ( - 5,25 )$ Round the answers to the nearest hundredth.

Correct Answer
verified

Let $g ( r , s , t ) = r e ^ { s / t }$ Find $g \left( 8 , \ln 3 , \frac { 1 } { 2 } \right)$

Correct Answer
verified

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

Correct Answer
verified

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 $y = 4$

Correct Answer
verified

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

Correct Answer
verified

Evaluate the gradient of f at the point P. $f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } , \quad P ( - 1,3 , - 1 )$

Correct Answer
verified

Find the domain and range of the function $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }$ .

Correct Answer
verified