**Making and Discuss Predictions with**

*Trajectories*Let's apply the following to:

. . .

__Topics of your choice__

or, as other examples -

. . . Cars of the future

. . . Going to college in the future

. . . The lowly(?) pencil

**Method: Trajectories of change**

. . . in the short term, change appears

**linear**

Example:

Last year you had 1 or 2 compact fluorescent bulbs

This year you will "probably have 1-2 more"

In the longer term, change looks

**exponential**

Lightbulb example:

. . . you start with 1-2, but after a couple of years you've got a bunch

. . . change accelerates, in this case

. . . if you look at an exponential curve with a microscope, what does it look like?

. . . "Exponential": complicated word, tricky math, simple concept

. . . . . . goes up faster and faster

. . . . . . has a doubling time

**Exponential curves explained**

. . . Suppose something doubles every 3 years

. . . Popular example: computer CPU complexity doubles every 2 years

. . . new value after

*t*years is original value

*v*times 2^(t/3)

. . .

*f*(

*t*)=

*t*o * 2^(

*t*/3)

. . . . . . where does the "doubles" appear?

. . . . . . where does the "every 3 years appear?

. . . . . . so it works for any factor of increase and any time constant

**Longer term, things "Level Off": the S-curve**

Also called "logistic curve"

Sort of "linear" early on

Then looks "exponential"

Then levels off

**Do you think an even longer-term view will look like a plateau curve?**

Think about pencils, compact fluorescents, and college, etc., etc.

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