A farmer can produce f(x,y)=250sqrt(4x^2+y^2) units of produce by utilizing x units of labor and y units of capital.

?(a)

Calculate the marginal productivities of labor and capital when x=12 and y=7.

?(b)

Let h be a small number. Use the result of part? (a) to determine the approximate effect on production of changing labor from 12 to 12+h units while keeping capital fixed at 7 units.

?(c)

Use part? (b) to estimate the change in production when labor decreases from 12 to 10.5 units and capital stays fixed at 7 units.

?(a) The marginal productivity of labor when x=12 and y=7 is _________.

?(Simplify your? answer.)

The marginal productivity of capital when x=12 and y=7 is ___________.

?(Simplify your? answer.)

?(b) Let h be a small number. Use the result of part? (a) to determine the approximate effect on production of changing labor from 12 to 12+h units while keeping capital fixed at 7 units.

Production will change approximately ________ units.

?(c) Use part? (b) to estimate the change in production when labor decreases from 12 to 10.5 units and capital stays fixed at 7 units.

The production will decrease approximately ________ units.

Summary:

Using partial derivatives as well as differences of functions at two values of its parameters we find the marginal values of a two variable function as well as the change in the function value when its parameters change value.

Step-by-step explanation

Answer:

(a) Since the variable x represents labor and y represents capital, the marginal productivity of labor at (x, y) = (12, 7) is given by the partial derivative of f with respect to x evaluated at (x, y) = (12, 7). So first we calculate the partial derivative of f with respect to x as follows:

?fx?=24x2+y2?250(8x)?=4x2+y2?1000x?(1)?

So, using (1), we get

?fx?(12,7)=4(122)+72?1000(12)?=480.?

So the marginal productivity of labor when x = 12 and y = 7 is 480.

Similarly, the marginal productivity of capital (y) at (x, y) = (12, 7) will be given by the partial derivative of f with respect to y at the point (x, y) = (12, 7). First we get the partial derivative of f with respect to y as follows:

?fy?=4x2+y2?250y?(2)?

From (2) above,

?fy?(12,7)=4(122)+72?250(7)?=70.?

The marginal productivity of capital when x = 12 and y = 7 is 70.

(b) The change in productivity when labor changes from x = 12 to x = (12+h), keeping capital fixed at y = 7 is given by

?f(12+h,7)−f(12,7)=2504(12+h)2+72?−2504(122)+72?=250(4(122+h2+24h)+49?−25)? .

Doing some algebra with the right hand side of the above expression gives us

??f(12+h,7)−f(12,7)=250(625(1+6254h2+96h?)?−25)??

Pulling out 625 from under the square root as 25 gives us

?f(12+h,7)−f(12,7)=250(25)(1+6254h2+96h??−1)?

In the above expression we use the approximation ?1+a?≈1+2a?? to get that

?f(12+h,7)−f(12,7)≈250(25)(1+2(625)4h2+96h?−1)? = 250(25)?2(625)4h2+96h?=5(4h2+96h).?

So the approximate difference is ?5(4h2+96h).?

(c) When labor decreases from 12 to 10.5 units, with capital being fixed at y = 7, then the change in productivity is given by

f(12, 7) - f(10.5, 7) = f(12, 7) - f(12-1.5, 7) = -[ f(12-1.5, 7) - f(12, 7) ].

The result in part (b) can be applied with h = -1.5 to the quantity above to get that

-[ f(12-1.5, 7) - f(12, 7) ] ?≈−5[4(−1.5)2+96(−1.5)]=675.?

Hence the change is about 675.