hw asst 29

course mth 272

{“\ԱyNԎ߉ʐassignment #029

029.

Applied Calculus II

08-01-2008

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09:12:01

Query problem 7.4.50 (was 7.4.46) slope in x direction and y direction for z=x^2-y^2 at (-2,1,3)

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RESPONSE -->

to find slope in x direction

hold y constant and differentiate with respect to x

fx (x,y) = 2x

at point (-2,1,3), the slope in the x-direction is

fx (-2,1,3) = -4

to find slope in y direction

hold x constant and differentiate with respect to y

fx (x,y) = 2y

at point (-2,1,3), the slope in the y-direction is

fx (-2,1,3) = 2

confidence assessment: 2

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09:12:18

What is the slope in the x direction at the given point? Describe specifically how you obtained your result.

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RESPONSE -->

previously answered.

confidence assessment:

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09:12:35

What is the slope in the y direction at the given point? Describe specifically how you obtained your result.

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RESPONSE -->

previously answered.

confidence assessment:

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09:27:31

Query problem 7.4.65 (was 7.4.61) all second partials of ln(x-y) at (2,1)

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RESPONSE -->

first find the first partial derivatives of x and y

fx(x, y) = (1/x) (ln(y)

fy(x, y) = (1/y) ln(x)

fxx(x, y) = (1) (1/y) = 1/y at (2, 1, 8) = 1

fxy(x, y) = 1 at (2, 1, 8) = 1

fyy(x, y) = (1) (1/y) = 1/x at (2, 1, 8) = 1/2

fyx(x, y) = 1 at point (2, 1, 8) = 1

confidence assessment: 1

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09:27:46

What is fxx at the given point?

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RESPONSE -->

previously shown

confidence assessment:

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09:27:55

What is fyx at the given point?

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RESPONSE -->

previously shown

confidence assessment:

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09:28:05

What is fxy at the given point?

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RESPONSE -->

previously shown

confidence assessment:

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09:28:13

What is fyy at the given point?

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RESPONSE -->

previously shown

confidence assessment:

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09:28:27

What is fx at the given point?

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RESPONSE -->

previously answered

confidence assessment:

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09:28:37

What is fy at the given point?

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RESPONSE -->

previously answered

confidence assessment:

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09:37:51

Query problem 7.4.68 R = 200 x1 + 200 x2 - 4x1^2 - 8 x1 x2 - 4 x2^2; R is revenue, x1 and x2 production of plant 1 and plant 2

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RESPONSE -->

I think you find the first partial derivative and then find second partial derivative and tehn plug in x1 and x2.

dont really know how to solve,

maybe

x1 + x2 - 8x1 - x1x2 - 8x2

that can't be right, i dont know this one

confidence assessment: 0

** The derivative of R with respect to x1 is 200 + 0 - 4 (2 x1) - 8 x2 - 0; All all derivatives treat x1 as the variable, x2 as constant. Derivatives of 200 x2 and -4 x2^2 do not involve x1 so are constant with respect to x1, hence are zero.

So the marginal revenue with respect to plant 1 is 200 - 8 x1 - 8 x2.

The derivative of R with respect to x2 is 0 + 200 - 0 - 8 x1 - 4 ( 2 x2) = 200 - 8 x1 - 8 x2; All all derivatives treat x2 as the variable, x1 as constant.

So the marginal revenue with respect to plant 2 is 200 - 8 x1 - 8 x2. **

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09:38:00

What is the marginal revenue for plant 1?

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RESPONSE -->

confidence assessment:

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09:38:08

What is the marginal revenue for plant 2?

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RESPONSE -->

confidence assessment:

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09:42:23

Why should the marginal revenue for plant 1 be the partial derivative of R with respect to x1?

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RESPONSE -->

oh, ok that makes sense

so fx1 (x1,x2) = 200 + 200x2 - 8x1 - 8x2 -4x2^2

and then

fx2 (x1,x2) = 200x1 + 200 - 4x1 - 8x1 - 8x2^2

confidence assessment: 2

** Marginal revenue is the rate at which revenue changes per unit of increased production. The increased production at plant 1 is the change in x1, so we use the derivative with respect to x1. **

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09:44:49

Why, in real-world terms, might the marginal revenue for each plant depend upon the production of the other plant?

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RESPONSE -->

because of competition in the markets. If goods are substitutes or complimentary goods.

confidence assessment: 3

The marginal revenues for each plant may depend on each other if one plant awaits shipment of a part from the other, or if one plant is somewhat slow resulting in a bottleneck.

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09:45:33

What is is about the function that ensures that the marginal revenue for each plant will depend on the production of both plants?

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RESPONSE -->

I think that when you find the partial derivative that they are both going to equal the same thing.

confidence assessment: 1

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The specific reason is that both derivatives contain x1 and x2 terms, so both marginal revenues depend on both the production of plant 1 and of plant 2.