Pj Problems - Overview
The Number Line
7 Spaces Of Interest - Overview
Triadic Unit Mesh
States Of Matter
COHN - Natures Engineering Of The Human Body
The Human-Body Systems
Faith, Love, Charity
The derivatives and anti-derivatives of functions are very important analytical tools. They form the branch of mathematics generally known as calculus. Calculus has been and continue to be an indispensable tool in the analysis of existential problems. The great Isaac Newton (1642-1727 A.D.) formulated the calculus. He called his formulation the method of Fluxions.
(a) What work does the derivative do?
(b) Use PjProblemStrings Sequencing to broadly describe this work of the derivative.
(c) What is the geometric interpretation of the derivative of a single variable function and a multi-variable function.
(d) Is the geometric interpretation consistent with PjProblemStrings Sequencing?
(e) What are some of the applications of the derivative?
S7P3A3k (force. k =1, 2), S7P4A4k (motion. k =1, 2, 3,4), S7P5A5k (change. k =1, 2). Where k is not a multiplicand.
All Pj Problems are at play. However, Pj Problems of Interest are of types force, motion and change.
(a)The derivative as an analytic tool, measures relational dynamism. In other words, a variable change (change in a dependent variable) that is caused by another variable change (change in an independent variable). The process of deriving a derivative is called differentiation.
(b)Space dynamism is a fundamental existential phenomenon. It is caused by forces. A force is either a pull or a push. The PjProblemStrings representing these forces are:
- S7P3A31 (a multi-matter dynamic space in which the dynamism of interest is caused by pulls)
- S7P3A32 (a multi-matter dynamic space in which the dynamism of interest is caused by pushes).
The dynamism of a space may be caused by pushes and pulls acting concurrently.
PjProblemStrings sequencing describes dynamism as motion sandwiched between two forces. The PjProblemStrings representation is:
- SiP3A3kSiP4A4kSiP3A3k. Where SiP4A4k represent motion and k is not a multiplicand.
The first force is the active force and the second is resistive force. The difference between the active force and the resistive force is the resultant force (driving force) causing the motion. In some scenarios, the active force is deliberately reduced in order to reduce or stop the motion (e.g. a runner reduces speed after crossing the finishing line). When the resultant force is zero, there is static equilibrium. When the resultant force is constant, there is dynamic equilibrium. The PjproblemStrings of change are implicit in the dynamism.
(c)The derivative is defined as the slope of a function at a point.
In figure 1a, there is no change in the dependent variable relative to the change in the independent variable. So, the slope is zero and by definition, the derivative is zero.
In figure 1b, the change in the dependent variable relative to the change in the independent variable is constant. So, the slope is a constant and so is the derivative.
In figure 1c, the dependent variable changes at every point relative to the change in the independent variable. So the slope varies at every point of the curve. Therefore, the slope at a point is the slope of a line tangent to the point. The slope of the tangent line is the slope of the secant line as the change in the independent variable tends to zero.
Figure 1d is a geometric interpretation of the partial derivative at a point of a multi-variable function. The derivative is a partial derivative because there are other variables at play. Only one independent variable is allowed to change at a time, relative to the change in the dependent variable. See geometric interpretation of the partial derivative.
(d)Yes. In all of the cases in figure 1, one can imagine the PjProblemStrings Sequencing necessary to realize the desired slope. For example, the derivative (slope) at point P is realized by the movement of point Q towards point P as the secant PQ tends towards the tangent line at P.
(e)The derivative is used to measure rate of change. For example, instantaneous velocity and acceleration. In economics it is used to measure marginal cost, marginal profit and many other optimization problems in various disciplines.
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 = x2 About The X Axis
Single Variable Functions
Absolute Value Functions
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
Derivation Of Heat Equation For A One-Dimensional Heat Flow
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.
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