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Question : 1) f(x,y) = x^3 - 4xy + 8y : 2151823

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find values of x and y such that both f_{x}(x, y) = 0 and f_{y}(x, y) = 0.

1) f(x,y) = x^{3} - 4xy + 8y

A) x = 0, y = 0

B) x = 2, y = 3

C) x = 2, y = 2

D) x = (4/3) , y = 2

2) f(x, y) = x^{2} + xy + y^{2} - 3x + 2

A) x = 1, y = -(1/2)

B) x = -2, y = 1

C) x = 0, y = 0

D) x = 2, y = -1

3) f(x,y) = x^{3} + y^{3} - 9xy

A) x = 3, y = 3

B) x = 1, y = 1

C) x = -3, y = -3

D) x = 0, y = 0

Find the indicated derivative.

4) Find fx for f(x, y, z) = 6x^{5}y^{8} + 4x^{5}z^{3} + 7y^{9}.

A) 30x^{4}y^{8} + 20x^{4}z^{3}

B) 240x^{4}y^{7} + 60x^{4}z^{2}

C) 48x^{5}y^{7} + 12x^{5}z^{2}

D) 30x^{4} + 20x^{4}

5) Find fy for f(x, y, z) = 7x^{5}y^{5} + 6x^{3}z^{3} + 4y^{3}.

A) 35y^{4} + 12y^{2}

B) 35x^{5}y^{4} + 12y^{2}

C) 35x^{4}y^{5} + 18x^{2}z^{3}

D) 175x^{4}y^{4} + 54x^{2}z^{2} + 12y^{2}

6) Find fz for f(x, y, z) = (4x - y^{2}/9z + 10x).

A) (-36x + 9y^{2}/(9z + 10x)^{2})

B) (4x - y^{2}/(9z + 10x)^{2})

C) (36z + 10x/(9z + 10x)^{2})

D) (-36x + 9y^{2}/9z + 10x)

7) Find f_{y} for f(x, y, z) = (z/√(x + y^{2})).

A) - (yz/(x + y^{2})^{3/2})

B) - (yz/x + y^{2})

C) - (z(2y + 1)/2(x + y^{2})^{3/2})

D) - (z/2(x + y^{2})^{3/2})

8) Find fx for f(x, y, z) = ln|3x^{10} - 10xy^{4} + z^{7}|.

A) (30x^{9} - 10y^{4}/3x^{10} - 10xy^{4} + z^{7})

B) ln|30x^{9} - 10y^{4} |

C) (30x^{9} - 10xy^{4} + z^{7}/3x^{10} - 10xy^{4} + z^{7})

D) (30x^{9} - 40xy^{3}/3x^{10} - 10xy^{4} + z^{7})

9) Find fx for f(x, y, z) = 6x^{3}e(6y^{6} + 4z^{6}).

A) 18x^{2}e(6y^{6} + 4z^{6})

B) (18x^{2} + 6x^{3})e(6y^{6} + 4z^{6})

C) 18x^{2}e(36y^{5} + 24z^{5})

D) 18x^{2}

10) Find fz for f(x, y, z) = 3x^{10}e(2y^{6} + 3z^{8}).

A) 24z^{7}e^{3}z^{8}

B) 72x^{10}z^{7}e^{2}4z^{7}

C) 72x^{10}z^{7}e(2y^{6} + 3z^{8})

D) 24z^{7}e(2y^{6} + 3z^{8})

11) Find fx for f(x, y, z) = 4x^{10}y^{3}z^{3}ln|7x^{5}|.

A) 40x^{9}y^{3}z^{3}ln|35x^{4}|

B) 20x^{9}y^{3}z^{3} + 40x^{9}y^{3}z^{3}ln|7x^{5}|

C) 1400x^{9}y^{3}z^{3}ln|35x^{4}|

D) 20x^{10}y^{3}z^{3} + 40x^{9}y^{3}z^{3}ln|35x^{4}|

12) Find fzy for f(x, y, z) = ln|3xy - xz - y^{2}| .

A) (3x - 2y/(3xy - xz - y^{2})^{2})

B) (3x^{2} - 2xy/(3xy - xz - y^{2})^{2})

C) (3x^{2}/(3xy - xz - y^{2})^{2})

D) (3x^{2} - x^{2}y + xy/(3xy - xz - y^{2})^{2})

13) Find fxz for f(x, y, z) = 3x^{5}y^{5} + 3x^{7}z^{4}.

A) 4z^{3}

B) 84x^{6}z^{3}

C) 15x^{4}y^{5} + 84x^{6}z^{3}

D) 12x^{7}z^{3}

Solve the problem.

14) A company has the following production function for a certain product:

p(x, y) = 31x^{0.3}y^{0.7} .

Find the marginal productivity with fixed capital, p_{x} .

A) 9.3((x/y))^{0.7}

B) 9.3((y/x))^{1.3}

C) 9.3xy^{0.7}

D) 9.3((y/x))^{0.7}

15) A company has the following production function for a certain product:

p(x, y) = 24x^{0.3}y^{0.7} .

Find the marginal productivity with fixed labor, p_{y} .

A) 16.8((x/y))^{0.3}

B) 16.8yx^{0.3}

C) 16.8((y/x))^{0.7}

D) 16.8((y/x))^{0.3}

16) The production function z for an industrial country was estimated as z = x^{5}y^{8}, where x is the amount of labor and y the amount of capital. Find the marginal productivity of labor.

A) 8x^{5}y^{7}

B) 5x^{4}y^{8}

C) 16x^{5}y^{7}

D) 10x^{4}y^{8}

17) Suppose that the manufacturing cost of a precision instrument is approximated by M(x,y) = 30x^{2} + 50y^{2} - 6xy, where x is the cost of materials and y is the cost of labor. Find Mx(9, 4).

A) 2214

B) 584

C) 346

D) 516

18) Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its center can be approximated by the function S = (aV/0.51d^{6}), where a is an atmospheric constant, and V is the approximate volume of the tornado, in cubic feet. Assume that a is 0.78, and find S_{V}.

A) (1/d^{6})

B) (1.53/d^{6})

C) (a/1.02d^{6})

D) (0.78/d^{6})

19) Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its center can be approximated by the function S = (aV/0.51d^{6}), where a is an atmospheric constant, and V is the approximate volume of the tornado, in cubic feet. Assume that a is 0.78, and find S_{d}.

A) - (9.17V/d^{7})

B) (1.5V/d)

C) (1.5/d^{6})

D) (1.5V/d^{2})

20) Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its center can be approximated by the function S = (aV/0.51d^{4}), where a is an atmospheric constant, and V is the approximate volume of the tornado, in cubic feet. Interpret (∂S/∂V) .

A) The rate of change in speed per unit change in volume while distance is held constant.

B) The rate of change in volume per unit change in speed while distance is held constant..

C) The rate of change in speed per unit change in volume and distance.

D) The rate of change in speed per unit change in distance while volume is held constant.

21) The intelligence quotient in psychology is given by Q(m, c) = 100(m/c), where m is an individual's mental age and c is the individual's chronological, or actual, age. Find (∂Q/∂c) .

A) (100m/c)

B) 100m

C) - (100m/c^{2})

D) - (100m/c)

22) A company's monthly sales, in thousands, is given by S(x, y) = 7x^{0.4}y^{0.5}, where x is the amount spent on newspaper advertising per month in thousands of dollars and y is the amount spent on radio advertising per month in thousands of dollars. Suppose the company currently spends $4000 on newspaper advertising per month and $3000 on radio advertising per month. What would be the effect on sales if the company increases the amount spent on newspaper advertising to $5000, while the amount spent on radio advertising remains constant?

A) Sales would increase by $3518.29.

B) Sales would increase by $1846.45.

C) Sales would increase by $987.76.

D) Sales would decrease by $184.64.

23) The surface area of a certain mammal, in square meters, is approximated by A(W, H) = 0.36W^{0.42}H^{0.68}, where W is the weight of the animal in kilograms and H is the height in meters. Find (∂A/∂H).

A) 0.15W0.42H-0.32

B) 0.24W0.42H-0.32

C) 0.24W-0.58H0.68

D) 0.15W-0.58H0.68