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Exam 18: Multivariable Calculus

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If g(u,v,w)= ( If g(u,v,w)= ( + 3 (4w - 5),then (-1,1,3)= + 3 If g(u,v,w)= ( + 3 (4w - 5),then (-1,1,3)= (4w - 5),then If g(u,v,w)= ( + 3 (4w - 5),then (-1,1,3)= (-1,1,3)=

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Question 1
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E

The demand function for product A is The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither = The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither ,and the demand function for product B is The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither where The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither and The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither are the quantities demanded for A and B,respectively,and The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither and The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither are their respective prices.Determine: (a)the marginal demand for A with respect to The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither (b)the marginal demand for B with respect to The demand function for product A is = ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither (c)whether A and B are competitive,complementary,or neither

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Question 2
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(a) (a) (b)3 (c)competitive
(b)3
(c)competitive

Use the method of Lagrange multipliers to find the critical points of f(x,y,z)= 2x + 4y - 4z subject to the constraint Use the method of Lagrange multipliers to find the critical points of f(x,y,z)= 2x + 4y - 4z subject to the constraint + + = 9. + Use the method of Lagrange multipliers to find the critical points of f(x,y,z)= 2x + 4y - 4z subject to the constraint + + = 9. + Use the method of Lagrange multipliers to find the critical points of f(x,y,z)= 2x + 4y - 4z subject to the constraint + + = 9. = 9.

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Question 3
Correct Answer:
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(1,2,-2)and (-1,-2,2)

Evaluate: Evaluate:

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Question 4

Let f(x,y,z)= ln( Let f(x,y,z)= ln( + 6 )- 2 + .Find . + 6 Let f(x,y,z)= ln( + 6 )- 2 + .Find . )- 2 Let f(x,y,z)= ln( + 6 )- 2 + .Find . Let f(x,y,z)= ln( + 6 )- 2 + .Find . Let f(x,y,z)= ln( + 6 )- 2 + .Find . + Let f(x,y,z)= ln( + 6 )- 2 + .Find . Let f(x,y,z)= ln( + 6 )- 2 + .Find . .Find Let f(x,y,z)= ln( + 6 )- 2 + .Find . .

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Question 5

If f(x,y)= If f(x,y)= find find If f(x,y)= find

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Question 6

If w = f (x,y,z)= If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e) yz - If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e) + If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e) ,find: (a) If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e) (b) If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e) (c) If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e) (d) If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e) (e) If w = f (x,y,z)= yz - + ,find: (a) (b) (c) (d) (e)

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Question 7

A company's production function is given by P = 40Lk - 3 A company's production function is given by P = 40Lk - 3 - 2 + 500,where P is the total output generated by L units of labor and k units of capital.Determine: (a)the marginal production function with respect to L (b)the marginal production function with respect to k - 2 A company's production function is given by P = 40Lk - 3 - 2 + 500,where P is the total output generated by L units of labor and k units of capital.Determine: (a)the marginal production function with respect to L (b)the marginal production function with respect to k + 500,where P is the total output generated by L units of labor and k units of capital.Determine: (a)the marginal production function with respect to L (b)the marginal production function with respect to k

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Question 8

If w = f (x,y,z)= 2 If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) y + 3 If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) + 4 If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) ,find: (a) If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) (b) If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) (c) If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) (d) If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e) (e) If w = f (x,y,z)= 2 y + 3 + 4 ,find: (a) (b) (c) (d) (e)

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Question 9

A company manufactures two products,X and Y,and the joint-cost function for these products is given by A company manufactures two products,X and Y,and the joint-cost function for these products is given by where c is the total cost of producing x units of X and y units of Y.Determine the marginal cost with respect to x when and where c is the total cost of producing x units of X and y units of Y.Determine the marginal cost with respect to x when A company manufactures two products,X and Y,and the joint-cost function for these products is given by where c is the total cost of producing x units of X and y units of Y.Determine the marginal cost with respect to x when and and A company manufactures two products,X and Y,and the joint-cost function for these products is given by where c is the total cost of producing x units of X and y units of Y.Determine the marginal cost with respect to x when and

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Question 10

If f(x,y)= 4 If f(x,y)= 4 + 3 - 7 + 4x - 3y + 2,find (a) (x,y)and (b) (x,y). If f(x,y)= 4 + 3 - 7 + 4x - 3y + 2,find (a) (x,y)and (b) (x,y). + 3 If f(x,y)= 4 + 3 - 7 + 4x - 3y + 2,find (a) (x,y)and (b) (x,y). If f(x,y)= 4 + 3 - 7 + 4x - 3y + 2,find (a) (x,y)and (b) (x,y). - 7 If f(x,y)= 4 + 3 - 7 + 4x - 3y + 2,find (a) (x,y)and (b) (x,y). + 4x - 3y + 2,find (a) If f(x,y)= 4 + 3 - 7 + 4x - 3y + 2,find (a) (x,y)and (b) (x,y). (x,y)and (b) If f(x,y)= 4 + 3 - 7 + 4x - 3y + 2,find (a) (x,y)and (b) (x,y). (x,y).

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Question 11

If f(x,y)= If f(x,y)= y,then (1,0)= y,then If f(x,y)= y,then (1,0)= (1,0)=

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Question 12

If z = If z = ,then = ,then If z = ,then = =

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Question 13

Let f(x,y)= 3 Let f(x,y)= 3 + 5 .Find: , , + 5 Let f(x,y)= 3 + 5 .Find: , , .Find: Let f(x,y)= 3 + 5 .Find: , , , Let f(x,y)= 3 + 5 .Find: , , , Let f(x,y)= 3 + 5 .Find: , ,

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Question 14

The demand function for product A is The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither ,and the demand function for product B is The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither where The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither and The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither are the quantities demanded for A and B,respectively,and The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither and The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither are their respective prices.Determine: (a)the marginal demand for A with respect to The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither (b)the marginal demand for B with respect to The demand function for product A is ,and the demand function for product B is where and are the quantities demanded for A and B,respectively,and and are their respective prices.Determine: (a)the marginal demand for A with respect to (b)the marginal demand for B with respect to (c)whether A and B are competitive,complementary,or neither (c)whether A and B are competitive,complementary,or neither

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Question 15

Evaluate Evaluate

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Question 16

Let Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. = 50 - 5 Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. + 6 Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. and Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. = 20 Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. × Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. be demand functions,where Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. and Let = 50 - 5 + 6 and = 20 × be demand functions,where and are prices for products A and B,respectively.Find all four marginal demand functions. are prices for products A and B,respectively.Find all four marginal demand functions.

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Question 17

If f(x,y)= If f(x,y)= find find If f(x,y)= find

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Question 18

Evaluate: Evaluate:

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Question 19

Determine all of the critical points of f(x,y)= Determine all of the critical points of f(x,y)= + - 3x + - 4y.Also use the second derivative test to determine,if possible,whether a maximum,minimum or saddle point occurs at each of these critical points. Determine all of the critical points of f(x,y)= + - 3x + - 4y.Also use the second derivative test to determine,if possible,whether a maximum,minimum or saddle point occurs at each of these critical points. + Determine all of the critical points of f(x,y)= + - 3x + - 4y.Also use the second derivative test to determine,if possible,whether a maximum,minimum or saddle point occurs at each of these critical points. - 3x + Determine all of the critical points of f(x,y)= + - 3x + - 4y.Also use the second derivative test to determine,if possible,whether a maximum,minimum or saddle point occurs at each of these critical points. Determine all of the critical points of f(x,y)= + - 3x + - 4y.Also use the second derivative test to determine,if possible,whether a maximum,minimum or saddle point occurs at each of these critical points. - 4y.Also use the second derivative test to determine,if possible,whether a maximum,minimum or saddle point occurs at each of these critical points.

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Question 20
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