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(a) Suppose a company has a production function $f(x, y)=100 x^{\frac{1}{4}} y^{\frac{3}{4}}$. Show that the marginal product of labor is positive and that the marginal product is a decreasing function of labor for constant capital. (This may be interpreted that as labor increases, production increases, but at a slower rate.)

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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function. If is given us the function of the amount of labor, eggs and the amount of capital. Why, as one forthe next to the negative from fourth plus 3/4 y to the negative 1/4 and all that negative four. So you had to find a bar a the number of units produced when 60 units of laborers and 81 units of capital re areas. So but we get to finding these for a is the function. If at the 0.16 81 and this is equal to 1/4 X, that is 16 to the negative 1/4 plus through fourth What's the negative one force that is 81 to the native, 1/4 to the negative four. So this is equal to 1/4 16 is two to fourth and that the negative 1/4 plus 3/4 and 81 is three to the fourth and that to the negative 1/4 in all that to negative four. And that's equal to 1/4 times two to the fourth to the negative. 1/4 is too negative. One plus three forth times three to the fourth to the negative. 1/4 is three to the negative one and hold that to the negative four. And that's equal to 1/4. Time's 1/2 is 1/8 Bless 3/4 Time's 1/3 is 1/4 to the negative four before in that single to three aides to negative fourth, which is equal to aid to the fourth, divided by three to the fourth and that is equal to 4000 95 96. You added by 81 that's every similarly equal to 50 point 57 and that that the production in hundreds of units. So he's 5057 units. So the production is 5057 units when the amount of labor is 16 units and the amount of capital is 81 units. Now, Burt be you got to find the partial derivatives of If with Respect to X, and we respect why at the 0.16 81 and we inter bread that that written those results. So first we get to find partial derivative off F respect to eggs at any point x y, and that's equal to We have a power here, so it's a exponents. Negative four times based 1/4 X to the negative 1/4 Love three forth Why to the negative 1/4 and that to the 94 minutes. One is near the five times the partial derivative of the base respective IX. So the second term 3/4 Why to them negative 1/4. It's a relative zero Respect to x o. You find the derivative respect to exit. The first term is for 1/4 time's The exponents Negative went forth times same base egg eggs to the negative or fourth Um, mine's one is negative. 5/4 that's equal to we have here. Negative 1/4 times before is one. So you see with the 1/4 which is this one here two times x to the name Negative 5/4 kisses. One here times All this expression 1/4 ex to the negative 1/4 plus 3/4. Why to the narrative 1/4 and all that to the negative five. And that's equal to Well, this expression can be left a sees right now this way, and we get to you. Read that. But the 0.16 81 So he's partial derivative of if respect to extract at the 0.16 81 and that's equal to 1/4 16 to the narrative. Five. Fourth 16 is to to the fourth and that to the negative forth times. And no, when we evaluate his space, 1/4 ex to the negative 1/4 plus 3/4 wine it to the native 1/4 at the 0.16. Anyone we know, How much is this oppression? How is this expression? Because we have calculated here the discretion here we have calculated before exactly the same thing. So we had at the time, 3/8. So the result here it's going to be three dates in that to the my negative Fifth 85. So he's, uh, here we have 12 went over four, east to square, and this one here to to the fourth to the negative five. Fourth is, too, to the negative five, and this one is eight to the fifth, divided by three to the fifth, and that's equal to one divided by to square times to the fifties to to the seventh. And in the numerator we have three to the third to the fifth. So is two to the 15th. Fight it by three to the fifth. And finally, this is equal to, um it's going to be able to to the eighth divided by three to the fifth. And that's equal to 256 divided by 243. And that's approximately equal to 0.1 point 053 And that is its in hundreds of units. Always 1000 a 100 Sorry, 105 units, more or less so the change in production. The change of the function if that is a change in production when the amount of labor eggs changes from 16 to 17 is 105 units in that case labor. But, sir, the capital amount of capital is hell constant and 81. So the change of the rate of change in production is 105 units. When the amount of labor changing one unit from 16 to 17 Andi capital is held constant at 81 unit to know we're going to calculate the partial derivative off F respect. Why at the 0.16 81. So first we get to calculate partial derivative off f respect toe. Why at any point, ex what and that's equal to We have a power. So is this opponent negative? Four times the same base, 1/4 extra. The negative 1/4 Placide 3/4. Why to the negative 1/4 Raised to negative four murders. One is 95 times a partial derivative of the base we respect. Why so The Pfister First term. It's a riveting zero. Respect to actor. Why? And the second term is 3/4 times. Explore int negative 1/4 times. Why to the negative one for when fourth minus one is negative. Ah, five. Fourth. And that's equal to So we have before the negative one. Forced time. They got two fours. One. So this is 3/4. Why? To the negative. 5/4 times, 1/4 x to the negative 1/4 plus 3/4. Why to the negative 1/4? Well, they're to the negative five. And that's equal to well, that can be held that way. Asses. So now we have to evaluate his partial derivative off if we respect to Why at the points. 16 81 that's equal to 3/4 times. Why to the negative 5/4. That is, um, 81 to the negative 5/4 but 81 23 to the fourth and that to the negative for fourth times on. We know that again, this expression beg Inside. The branch offices are equal to three decades whenever elated at the 30.16 80 81. So is three gates to the negative five. That's equal to 3/4 times, three to the fourth negative five. Fourth is three 2 95 times eight to the fifth. You write it by three to the faith, and that's equal to so we have three divided by two square times one word by 3/5 times aid to the fifth. You added by three to the fifth in the tickle, too. Okay, here we have three to the third to the fifth to do the 15 time three you guided by to square times three to the dance, and that's final. Legal, too, Um, to the 13 divided by 327 That's Ari threw into the night to the nine ninth Power so and that if we made all the calculations here we get a sequel to 1000 101 192 divided by 1919 7 683 and that silver timidly equal to zero point 4162 and that's in hundreds of units again. So we have 41 0.62 units. So the production in changing in 41 0.6 units When, uh, the amount of capital is changing from 81 units to fade into units while the amount of laborers 16 is hell constant. That's the interpretation of the part. Um and no, But see, we get to fine what will be the approximate effect on production of increasing labor one by one unit from 16 units of Labor E and with 81 years of capital. So here what is changing is the labor, the units of labor. So he's the expert in changing. So we want to use We must seize this partial derivative off everything to eggs. And what is the point where you this Abe awaited world the, uh, the laborer changing for all 16 to 17. So they were resisting and the capital is fixed at 81. So we get to find the partial derivative of faith, respect to eggs at the 0.16 81 and that has been regulated Barbie So that waas 105 so disease 100 five units. So the effect off increasing labor by one unit from 60 units of labor with 81 years of capital, he's 100 five units into production. That is changing production. When we increased, Labor will buy one unit from 60 units of labor to 17 units of labor and hell constantly capital at 81 units.

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