Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

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Energy

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Sound

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Molecular Shapes

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Human-cells

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COHN - Natures Engineering Of The Human Body

The Human-Body Systems

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Photosynthesis

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Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

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) = 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.

Math

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.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings