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We will use the concepts:

Variable/Value

English is a historical mix of two languages. So there are two words of many things. Change, for example, can also be called vary. That’s why something, that can change, is called a variable.

A variable assumes one value at a time, exclusively. It does not need to be “at a time”. It could also be “at a place”.

As a side-note: The uniqueness is expressed by the word function: We can say, the value is a function of this or that.

We can use a number to denote a value. We could also use a word, but it is easier with a number.

Variable = {value1, value2, …} Position = {10m , 20m, …, 120m, …}

Change

We assume a gradual change.

Large changes:

The change is expressed with a difference: \(-\). Difference is abbreviated with a Greek D: Δ.

\(Δy_1 = y_1 - y_0\).

The differences can be added to get the full extend of the variable. We basically undo the difference (subtraction) by addition.

\(y = Σ Δy_i = Δy_1 + Δy_2 + ...\).

Small changes:

We use d instead of Δ for very small changes.

\(dy = (y + dy) - y\).

\(d\) is called differential.

With \(dy\) we use \(∫\) for sum, and call it integral.

Velocity

How fast a value changes is again a variable. It is called speed or velocity of change of that variable.

Velocity is relative. To describe velocity of change of the value of one variable we need another variable to compare it to. Often this other variable is time, but it could be something else. If there is no other variable specified, then it is implicitly time, or better our time feeling, given by how fast our brain thinks.

Lets find the velocity by which you grow. We have two variables:

  1. Height \(y\): The distance from the floor to the top of your head.

  2. Age \(x\): The number of years that have passed since your birth.

Differences:

The average velocity over \(Δx\) is found by dividing the differences:

\(\tilde{v} = Δy/Δx\)

Why divide? Because then you can multiply to get back the difference:

\(Δy = \tilde{v} Δx\)

And you can sum the differences to get back the height:

\(y = Σ Δy = Σ \tilde{v} Δx\)

Differentials:

This velocity is at a specific \(x\) because \(dx\) is so small that we can neglect it.

\(v = dy/dx\)

and to get back y we sum the very many \(v dx\)

\(y = ∫ dy = ∫ v dx\).

Velocity is used if \(x\) is time. More generally, one calls it derivative:

  • derivative of height \(y\) with respect to age \(x\)

Summary

Change is expressed via differences,

The symbol for sum is

Velocity is a quotient between two differences