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Find the second-order partial derivative. -Find 2zx2\frac{\partial^{2} z}{\partial x^{2}} when z=exlnyz=e^{x} \ln |y| .


A) exlnye^{x} \ln |y|
B) exy2-\frac{e^{x}}{y^{2}}
C) exy2\frac{e^{x}}{y^{2}}
D) exlny-e^{x} \ln |y|

E) None of the above
F) A) and D)

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A

Sketch the indicated plane. - f(x,y) =1+xyf(x, y) =1+x-y  Sketch the indicated plane. - f(x, y) =1+x-y     A)     B)    C)     D)


A)  Sketch the indicated plane. - f(x, y) =1+x-y     A)     B)    C)     D)
B)  Sketch the indicated plane. - f(x, y) =1+x-y     A)     B)    C)     D)
C)  Sketch the indicated plane. - f(x, y) =1+x-y     A)     B)    C)     D)
D)  Sketch the indicated plane. - f(x, y) =1+x-y     A)     B)    C)     D)

E) A) and B)
F) A) and C)

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Find the second-order partial derivative. -Find 2zy2\frac{\partial^{2} z}{\partial y^{2}} when z=8xexyz=8 x e^{x y} .


A) 8x3ex8 x^{3} e^{x}
B) 8x3exy8 x^{3} e^{x y}
C) 16x2exy16 x^{2} e^{x y}
D) 16x2ex16 x^{2} e^{x}

E) None of the above
F) B) and C)

Correct Answer

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Find the partial derivative. -Find fx(9,9) f_{x}(9,-9) when f(x,y) =7x29xyf(x, y) =7 x^{2}-9 x y .


A) 45
B) -45
C) -162
D) 207

E) A) and D)
F) None of the above

Correct Answer

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Find the indicated maxima or minima. -Minimum of f(x,y) =x2+4y2+6f(x, y) =x^{2}+4 y^{2}+6 , subject to 2x8y=202 x-8 y=20


A) 26 when x=2,y=2x=2, y=2
B) 26 when x=2,y=2x=-2, y=2
C) 26 when x=2,y=2x=2, y=-2
D) 26 when x=2,y=2x=-2, y=-2

E) B) and C)
F) A) and D)

Correct Answer

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Find the second-order partial derivative. -Find fyyf_{y y} when f(x,y) =8x3y7y2+2xf(x, y) =8 x^{3} y-7 y^{2}+2 x .


A) 48xy
B) -28
C) -14
D) 24x224 x^{2}

E) All of the above
F) C) and D)

Correct Answer

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Evaluate the required second order partial derivative. -Find fyx(2,4) f_{y x}(2,-4) , if f(x,y) =4x3y2+4x4+2xy3f(x, y) =4 x^{3} y^{2}+4 x^{4}+2 x y^{3} .


A) -96
B) 752
C) -192
D) -288

E) All of the above
F) B) and D)

Correct Answer

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Evaluate the required second order partial derivative. -Find fxy(5,2) f_{x y}(5,-2) , if f(x,y) =5x3y2+4x43xy3f(x, y) =-5 x^{3} y^{2}+4 x^{4}-3 x y^{3} .


A) 542
B) -633
C) 1464
D) -825

E) All of the above
F) A) and B)

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Provide an appropriate response. -If we wish to locate the candidates for local extrema and saddle points in the interior of the domain of a function f(x,y) f(x, y) , then how should we proceed? (i) Set the partial derivatives equal to each other. (ii) Set the partial derivatives equal to zero. (iii) Set the partial derivatives equal to the critical values.


A) Only (iii) is correct.
B) Only (ii) is correct.
C) Only (i) is correct.
D) None of the above is correct.

E) All of the above
F) None of the above

Correct Answer

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Solve the problem. -Assuming that a cylindrical container can be mailed only if the sum of its height and circumference do not exceed 270 centimeters, what are the dimensions of the cylinder with the largest volume that can be mailed?


A) Height 180 centimeters and radius 90 centimeters
B) Height 90 centimeters and radius 90/π90 / \pi centimeters
C) Height 270 centimeters and radius 90/π90 / \pi centimeters
D) Height 90 centimeters and radius 270/π270 / \pi centimeters

E) A) and B)
F) A) and C)

Correct Answer

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Find the indicated maxima or minima. -Minimum of f(x,y) =x2+y2xyf(x, y) =x^{2}+y^{2}-x y , subject to xy=10x-y=10


A) 75 when x=5,y=5x=5, y=-5
B) 7 when x=2,y=1x=2, y=-1
C) 3 when x=1,y=2x=1, y=2
D) 25 when x=5,y=5x=5, y=5

E) B) and D)
F) None of the above

Correct Answer

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Evaluate the required second order partial derivative. -Find fxy(2,5) f_{x y}(-2,-5) , if f(x,y) =(x4) ln(xy5) f(x, y) =(x-4) \ln \left(x y^{5}\right) .


A) 7.5
B) -2.5
C) -1.25
D) -1

E) A) and B)
F) A) and C)

Correct Answer

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Solve the problem. -The material for the bottom of a rectangular box costs $3\$ 3 per square foot while the material for the sides and top costs $1\$ 1 per square foot. Find the greatest capacity such a box can have if the total amount available for material is $12\$ 12 .


A) 2ft32 \mathrm{ft}^{3}
B) 3ft33 \mathrm{ft}^{3}
C) 4ft34 \mathrm{ft}^{3}
D) 1ft31 \mathrm{ft}^{3}

E) A) and D)
F) A) and C)

Correct Answer

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Sketch the indicated plane. - f(x,y) =x+y21f(x, y) =x+\frac{y}{2}-1  Sketch the indicated plane. - f(x, y) =x+\frac{y}{2}-1     A)     B)    C)     D)


A)  Sketch the indicated plane. - f(x, y) =x+\frac{y}{2}-1     A)     B)    C)     D)
B)  Sketch the indicated plane. - f(x, y) =x+\frac{y}{2}-1     A)     B)    C)     D)
C)  Sketch the indicated plane. - f(x, y) =x+\frac{y}{2}-1     A)     B)    C)     D)
D)  Sketch the indicated plane. - f(x, y) =x+\frac{y}{2}-1     A)     B)    C)     D)

E) B) and C)
F) C) and D)

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C

Find the partial derivative. -Let z=g(x,y) =2x+3x2y22y2z=g(x, y) =2 x+3 x^{2} y^{2}-2 y^{2} . Find zy\frac{\partial z}{\partial y} .


A) 2+6x2y2+6 x^{2} y
B) 6xy24y6 x y^{2}-4 y
C) 6x2y4y6 x^{2} y-4 y
D) 2+6xy22+6 x y^{2}

E) A) and D)
F) None of the above

Correct Answer

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Find the partial derivative. -Let z=g(x,y) =4x+2x2y24y2z=g(x, y) =4 x+2 x^{2} y^{2}-4 y^{2} . Find zx\frac{\partial z}{\partial x} .


A) 4+4x2y4+4 x^{2} y
B) 4+4xy24+4 x y^{2}
C) 4xy28y4 x y^{2}-8 y
D) 4x2y8y4 x^{2} y-8 y

E) None of the above
F) All of the above

Correct Answer

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Locate and classify all critical points for the given function. - f(x,y) =x2+x+y2y5f(x, y) =x^{2}+x+y^{2}-y-5


A) Minimum of 72-\frac{7}{2} at (12,12) \left(-\frac{1}{2},-\frac{1}{2}\right)
B) Maximum of 72-\frac{7}{2} at (12,12) \left(-\frac{1}{2},-\frac{1}{2}\right)
C) Minimum of 72-\frac{7}{2} at (12,12) \left(-\frac{1}{2}, \frac{1}{2}\right)
D) Maximum of 72-\frac{7}{2} at (12,12) \left(-\frac{1}{2}, \frac{1}{2}\right)

E) B) and D)
F) All of the above

Correct Answer

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Locate and classify all critical points for the given function. - f(x,y) =1(x2y2) f(x, y) =1-\left(x^{2}-y^{2}\right)


A) Minimum of 1 at (0,0) (0,0)
B) Saddle point at (0,0) (0,0)
C) Maximum of 1 at (0,0) (0,0)
D) None of the above is correct.

E) C) and D)
F) B) and D)

Correct Answer

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Choose the graph of the equation. - z=y2x2z=y^{2}-x^{2}


A)  Choose the graph of the equation. - z=y^{2}-x^{2}  A)     B)    C)     D)
B)  Choose the graph of the equation. - z=y^{2}-x^{2}  A)     B)    C)     D)
C)  Choose the graph of the equation. - z=y^{2}-x^{2}  A)     B)    C)     D)
D)  Choose the graph of the equation. - z=y^{2}-x^{2}  A)     B)    C)     D)

E) A) and B)
F) A) and C)

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Choose the graph of the equation. - z2=y2+x2z^{2}=y^{2}+x^{2}


A)  Choose the graph of the equation. - z^{2}=y^{2}+x^{2}  A)     B)    C)     D)
B)  Choose the graph of the equation. - z^{2}=y^{2}+x^{2}  A)     B)    C)     D)
C)  Choose the graph of the equation. - z^{2}=y^{2}+x^{2}  A)     B)    C)     D)
D)  Choose the graph of the equation. - z^{2}=y^{2}+x^{2}  A)     B)    C)     D)

E) All of the above
F) B) and C)

Correct Answer

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B

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