The normal way of discussing these changes is to consider

*initial*and

*final*instances. Consider this example:

A runner begins moving at a distance of 20 meters from the picnic table. A moment later the runner is 50 meters from the picnic table. This means there was a change in distance from the picnic table of 30 meters.

In this case:

The initial distance (from the picnic table) is 20 meters

The final distance from the picnic table is 50 meters

Whereas in science and engineering, using a few numbers and letters as possible is the norm, and whereas different measures have "normal" symbols, the above way of describing the situation is never used in the actual solution of a problem.The change in distance is 30 meters.

The normal symbol for distance is

*d*(but sometimes

*s*for reasons I cannot remember), so it would be expected to use the

*d*for the various distances. But, how can you tell WHICH of the distances being referenced? There must be some other means.

There is.

There are two common ways to distinguish between various distances (or anything else) in the same discussion (a lab writeup, a word problem, etc.). Both ways rely on putting a subscript after the variable symbol.

That will look like this:

So if there are more than once distance, one way is to use numbers for the different distances. Distance 1… Distance 2… Distance 3… That makes sense.VariableSymbolSubscript

**Example time**:

A park has three water fountains. One is 20 meters from the pavilion. Another is 150 meters from the pavilion, The last one is 285 meters from the pavilion. Therefore…

d1= 20 m

d2= 150 m

Another way to distinguish between different measures of the same quantity is to look at the "before and after" information. In such a case, the idea if "initial" and "final" are represented by the subscripts ofd3= 285 m

*i*and

*f*.

The usual symbol for temperature is

*T…*

**Example time**:

A ball is kept in a refrigerator where the temperature is 10 C. It is moved out into the room where the temperature is 23 C.

Therefore…

Ti= 10 C

It is not unheard of to use 1 and 2 as the subscripts, even when it is clearly a before and after situation. Thus, the above information could be written as:Tf= 23 C

T1= 10 C

T2= 23 C

**INTRODUCING THE ∆**

In many, many cases, the most interesting part of a situation is how the variable changes.

**In science and engineering, the change in something is represented by the**

*∆***symbol before the normal quantity symbol.**Therefore, given the typical symbols for distance and temperature:

The amount that distance changed would be written as∆d

and

the amount that temperature changed would be written as∆T.

*The concept of change is not limited to distance or temperature.*Any measured quantity works.

**If something changes value, the ∆ can be used to represent the change.**

**Some "Change" Words**

In stories or word problems, s

**ome words strongly represent a change in the measure**. When encountered, these words will correspond to the ∆. This is NOT EVERY way to indicate use of the ∆, but usually, when you see these words, it will be not the initial or final value, but the ∆ value:

increased by

decreased by

changed by

moved

farther

closer

more

an additional

added (e.g. he added 5 more…)

**Calculating Change**

Finding the change in something is really easy. How much did you start with? How much did you end up with? What's the difference.

If you had 8 apples to start with and ended up with 12, the number of apples you had changed by 4.

It works the same way with distance, pressure, velocity, time, temperature, volume… it works the same with any measured quantity.

As a formula, if you are looking at distance, it looks like this.

∆d = df - di

NOTE: if you are using 1s and 2s as the subscripts, you need to make sure you know which was the final and which were the initial values.

SUMMARY

1. When more than one measure of the same quantity is required, the different measures are indicated by subscripts.

2. Subscripts can be i and f for initial and final, or can be numbers representing different measures.

3. The change in the measured thing is represented by the ∆ followed by the normal symbol.