Pj Problems - Overview
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States Of Matter
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The Human-Body Systems
Faith, Love, Charity
z = f(x,y) is the three dimensional surface illustrated in figure 114.7.
(a) What is the meaning of ∂f(x,y)/∂x?
(b) Supppose the variables x, y and z represent the length, width and height of a building respectively and the heat loss function for the building is:
f(x,y,z) = 11xy + 14yz + 15xz
(c) Suppose the production function of a manufacturer is:
f(x,y) = 60x3/4y1/4. Where x and y are units of labor and capital respectively
(i) What is the marginal productivity of labor for f(81,16)?
(ii) What is the marginal productivity of capital for f(81,16)?
S7P5A51 (Change - Physical Change).
Pj Problem of Interest is of type change (physical change). The derivatives of functions usually measure the rate of change of the dependent variable with respect to the independent variable.
(a) ∂f(x,y)/∂x means the rate at which f(x,y) will change with respect to x if y is kept constant. Similarly, ∂f(x,y)/∂y means the rate at which f(x,y) will change with respect to y if x is kept constant. For example, in figure 114.7, y is held constant at y = b and the red curve describes the curve x = f(x,b) on the surface. The value of ∂ f(a,b)/∂x is the slope of the tangent line to the curve at the point x = a and y = b.
(b) f(x,y,z) = 11xy + 14yz + 15xz
So, ∂f(x,y,z)/∂x = 11y + 15z
So, ∂f(x,y,z)/∂x = 11(7) + 15(7) = 182.
∂f(x,y,z)/∂x is referred to as the marginal heat loss with respect to x. In other words, if x is changed by a small unit h, then the amount of heat loss will change by approximately 182h units.
(c) f(x,y) = 60x3/4y1/4
(i) Marginal productivity of labor = ∂f(x,y)/∂x
= 60(3/4)x-1/4y1/4 = 45(y1/4)/x1/4
∂f(81,16)/∂x = 30. This means if capital is fixed at 16, and labor is increased by 1 unit, the quantity of goods produced will increase by approximately 30 units.
(ii) Marginal productivity of capital = ∂f(x,y)/∂y
= 60(1/4)x3/4y-3/4 = 15(x3/4)/y-3/4
∂f(81,16)/∂x = 405/8 = 50(5/8)
This means if labor is fixed at 81, and capital is increased by 1 unit, the quantity of goods produced will increase by approximately 50(5/8) units.
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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