Geometric Interpretation Of Partial Derivatives

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{5}A_{51} Base Sequence = 12735 String Sequence = 12735 - 5 - 51 **

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Geometric Interpretation Of Partial Derivatives

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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) = 60x^{3/4}y^{1/4}. 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**:
S_{7}P_{5}A_{51} (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) = 60x^{3/4}y^{1/4}

(i) *Marginal productivity of labor* = ∂f(x,y)/∂x

= 60(3/4)x^{-1/4}y^{1/4} = 45(y^{1/4})/x^{1/4}

∂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)x^{3/4}y^{-3/4} = 15(x^{3/4})/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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