Geometric Interpretation Of Partial Derivatives
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Geometric Interpretation Of Partial Derivatives

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
Interprete ∂f(10,7,7)/∂x
(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)?

The strings: S7P5A51 (Change - Physical Change).

The math:
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.

Blessed are they that have not seen, and yet have believed. John 20:29