Grandpa Ken’s

Glossary

Series

Glossary of Words

for

Differential Calculus

A Kenamar Videobook

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Written , Narrated and Produced by

Kenneth Kunz

© Kenamar Corp.

Glossary of Wordsfor

Differential Calculus

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Forward:

There are several steps to learning a new subject:

1. Learning the definitions of the special words (jargon.)

2. The concepts and relationshipsof the elements.

3. Proficiency in applying the concepts to real problems.(Engineering)

This monograph is only concerned with Item 1.

It is absolutely necessary to know this material first,

or you will forever ]be confused and flustered in your study of this subject.

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A glossary or vocabulary is an alphabetical list of words or terms in a particular area of knowledge with the definitions relevant to that area.

The words may mean something entirely different when used in other contexts.

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“Differential Calculus”

Differ: From Latin differre "to set apart.“A moving object keeps changing its position to a different position.

Calculus: From Latin calx “chalk” or “limestone.”

A “Calculus” was a pebbleused to count or figure outor calculate.

Differential Calculus is the figuring out of how things change.

For example, things move (change in position) as time moves on (change in time.)

The calculus is the study of the relationships of these changes.

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Variable

A Variable is something that varies.

Vary is from Latin verruca "wart.”Evidently warts changed over time and thus were a part of the skin that varied.

Variables can change inherently, like time, which waits for no one.

Such variables are called Independent Variables.

Other variables are called Dependent Variables because they depend on something else.

Like my bank account, which depends on the interest rate and how much I spend.

Variables which don’t vary are called Constants.

Latin constare, from com- "together“ + stare "to stand.“

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Naming of variables

It is conventional(from Latin com- "together“ + venire "to come”) or general agreement that variables are usually given symbols or letters to denote their values.

The letters can refer to their names or not.

They should be obvious from the context, or be explicitly definedso that everybody knows who is who.

x, y, and zThese are usually independent variables and can refer to :

Graph on paperx-across (horizontal) axisy-up (vertical) axisz-in/out axis

Mapx-east/westy-north/southz-up/down

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Naming of variables (continued)

Timet -on the horizontal (ordinate) axis;the dependent variable -on the vertical axis(abscissa)

Ordinate: Latin ordinare “order”Abscissa: Latin ab+ scindere “cut off”

Time is often called a Parameter.

A Parameter is a "measurable factor which helps to define a particular system.”

a, b, c, d, k, A, B, C, DThese are usually constants.

m, n, p, q, r, sThese are usually dependent variables.

fmay be a function of many variablesf(x) is a function of the variable x;f(x,y) is a function of both x and y;x=f(t) is how x changes with time.

mis often used for the slope;m(x,y)

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Function

A function is a relation between a set of inputs and a set of outputs

with the property that each input is related to exactly one output.

The generic function is called “f.”

Other letters used are g,u,v,w.

Example:

The rule f(x) = x2, which relates an input x to its square, (read "f of x equal x squared")

The output of the function f corresponding to an input xis denoted by f(x) .

If the input is 3, then the output is 9, and we may write f(3) = 9.

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Analytic Function

Analytic functions are those that can be expressed by terms using powers of the variables.

Such functions can be plotted as smooth curves without gaps or sharp corners.

Analytic: From Greek: analytos "dissolved" analysis: resolution of anything complex into simple elements.

A square is not represented by an analytic function, because we don’t know how to represent the corners.

A dashed line is non-analytic.

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Example of the graph of a function

Here is a graph of the function:

y = f(x) = x³ - 9x

x being the dependent variable is on the horizontal axis (ordinate.)

y being the dependent variableis on the vertical axis (abscissa.)

Here the numerical values of x and y are given in dimensionless units.

http://upload.wikimedia.org/wikipedia/commons/d/d1/Cubicpoly.png

We don’t know if the units should be feet, seconds or what.

Probably not pounds, since there are negative numbers.

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Graph of a Function

Graph: from Latin graphicus “drawing, picturesque.“

It could be the path of a finger in the sand.

Tangent line

http://www.leosoderman.com/wp-content/uploads/2011/08/Drawing-A-Line-In-The-SandWM900-800x532.jpg

Graph of a Function

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Graph of a Function

Graph: from Latin graphicus “drawing, picturesque.“

The graph of this functionis a picture representation of how ‘y’ varies with the changes of ‘x.’ Ordinate (x)

Tangent line

(0,0)

By convention,the independent variable is x,and the dependent variable is y.

f(x,y)

Ordinate: Latin ordinare “order”Abscissa: Latin ab+ scindere “cut off”

t -on the horizontal (ordinate) axisthe dependent variable -on the vertical axis(abscissa)

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Tangent

The tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point.

It is a continuation (in both directions) of the curve at that point.It just doesn’t curve.

Ordinate (x)

Tangent line

(0,0)

f(x,y)

Ordinate: Latin ordinare “order”Abscissa: Latin ab+ scindere “cut off”

Time graphs:t is on the horizontal (ordinate) axis,and the dependent variable is on the vertical axis (abscissa.)

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Difference

A variable changes from one to another condition. The change in condition from the original to the new is called the Difference.

The large letter is used for a change which is “large.”It is represented by the Greek capital letter Delta ∆, for ‘D’ (difference.)

For changes which are “infinitely small,” the small letter “d” is used. It is just called “d.”

This is read as:

m equals the change in ydivided by the change in xequals delta y over delta x.

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Derivative

The change in y is just the difference between its state at instance 2 and its state at instance 1:

y(2) – y(1) or ∆y

the change in x at for the same instances is:

x(2) – x(1) or ∆x

Now if the change keeps getting smaller and smaller until the ratio ∆y/∆x seems to stay the same, we call this the infinitesimal limit and change the nomenclature to dy/dx and give the ratio a name: “the derivative .”

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Partial Derivative

If y depends on more than just x, then we call it:

the partial derivative with respect to x:

∂y/∂x

using the small Greek letter for ∆.

We calculate this value by treating all the other variables as invariables or constantswith their current values.

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Square and Quadratic

Square: Latin ex- "out“ + quattuor "four.”

The idea is that a square has four sides.

Quadratic equations got that name in 1660, because they involve the square of x.

X times X

or X (read x-squared)

The famous quadratic equation:

f(x) = a x + bx + c

a, b, c are constants.

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Quadratic function

We can plot the function and make a graph as an example.

Plot: Originally it meant small piece of ground.

In mathematics,it means to make a map or diagram.

So here is the “burial” plot of our famous function!

Plotting the quadratic equation:

y = f(x) = a x + bx + c

a = 1, b = -1, c = -2

x

y

RIPBeloved Quaddie

f(x)

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The cubic equation (degree 3)

has the discriminant:

It really gets complicated after that!

A quartic (degree 4) has 16 terms,and the 6th degree has 246 terms.

Discriminant

Discriminant is a French word meaning discriminating.

From Latin: dis- "off, away" + cernere "distinguish, separate“

The discriminant is unique for every quadratic equation.

The quadratic equation:f(x) = a x + bx +

c

has the discriminant:∆ = b - 4ac

                       is

                                               

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Polynomial

Poly-many + nomial-number

A mon-nomial has one term such as:

x4y +3

A binomial has two terms such as:

x + y3az + 2w + 8

Constants don’t count!

Degree of a Polynomial

The degree of a term is determined by the number of times the variables are multiplied together.

7xyz, x³, 32Axyare all of degree 3.

The degree of a polynomial is the highest degree of any of its terms.

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Maximum, Minimum and Inflection Point

A maximum is like standing on the top of a hill, where every direction you look is down.

It is a local maximum if there are other higher hills.

If you are on top of the highest hill, then we call it a global maximum.

Same thing with minima, only you are in a valley.

An inflection point,is like you are standing on a flat ledge, where it is down one way and up the other way.

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Gradient

From Latin gradientem, prp. of gradi "to walk.”

Refers to the slope of a hill where you either step up or down.

Often expressed as:

m = dy/dx.

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Related to Degree From Latin de- "down“+ gradus "step"

Take little steps such as thosemarked out on a thermometer.

It’s really not that complicated.

You've been walking up and down hills- that’s differentials in height.

In fact if you walk north you get one slope and if you walk east, you get another slope. Well, that’s partial differentials.

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You’ve dealt with speed, s = dx/dt,

and acceleration, a = ds/dt.

All differential calculus is , is to organize the thoughts so you can put numbers to it and make things do what you want.

It’s just an analytic ski park!

Have fun!

A Kenamar Videobook

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