Here is an index card
that goes with this lesson (side 1a only); the ideas are discussed more fully below.
You may want to copy (handwrite) this information onto your own index card, to have an inanutshell
summary of important calculus ideas.
Dr. Burns talks about card 1a (video, 4 mins, 51 seconds)
Recall that a function is a rule that takes an input, does something to it,
and gives a corresponding output.
Function notation is extremely useful for describing this input/output relationship:
if the function name (the rule) is
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$\,f\,$, and the input is
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$\,x\,$, then the corresponding output is
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$\,f(x)\,$.
That is, the notation
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$\,f(x)\,$ (read as ‘$f$ of $x$’) represents the output from the function
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$\,f\,$ when the input is $\,x\,$.
As inputs change, you often want to know how the corresponding outputs (function values) change.
That is, as inputs change from $\,a\,$ to $\,b\,$, how is
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$\,f(a)\,$ changing to
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$\,f(b)\,$? Are these outputs increasing? Decreasing? Quickly? Slowly?
Here's an example.
Suppose you have a shower dial that controls the temperature of the water.
How do adjustments of the dial (changes in the input) affect the temperature of the water (changes in the output)?
When you move the dial from (say) position $\,a\,$ to position $\,b\,$, does the water get hotter or colder?
(Perhaps the dial is broken, and the temperature doesn't change at all!)
Does moving the dial from $a$ to $b$ change the temperature a lot, or only a little?
These are the sorts of questions we want answered, and the concept of average rate of change comes to the rescue.
DEFINITION
average rate of change of a function $\,f\,$ from $\,a\,$ to $\,b\,$
The average rate of change of $\,f\,$ from $\,a\,$ to $\,b\,$ is:
$$\frac{f(b)  f(a)}{b  a}
$$
Observations on the definition:

$t$  0  1  2  3  4  5 
$s(t)$  10  9  7  4  0  5 