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(Section 1.10: Difference Quotients)
1.10.11
FOOTNOTES
1.
Assumptions made about a function.
When defining the average rate of change of a
function
f
on an interval
a
,
b
±
²
³
´
, where
a
<
b
, sources typically do not state the assumptions
made about
f
. The formula
fb
()
±
fa
b
±
a
seems only to require the existence of
and
, but we typically assume more than just that.
• Although the slope of the secant line on
a
,
b
±
²
³
´
can still be defined, we need more for
the existence of derivatives
(i.e., the differentiability
of
f
) and the existence of
nonvertical tangent lines, as we will see in Section 1.11.
• We ordinarily assume that
f
is continuous
on
a
,
b
±
²
³
´
. Then, there are no holes, jumps,
or vertical asymptotes on
a
,
b
±
²
³
´
when
f
is graphed. (See Section 1.5.)
• We may also assume that
f
is differentiable
on
a
,
b
±
²
³
´
. Then, the graph of
f
makes no
sharp turns and does not exhibit “infinite steepness” (corresponding to vertical tangent
lines). However, this assumption may lead to circular reasoning, because the ideas of
secant lines and average rate of change are used to develop the ideas of derivatives,
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This document was uploaded on 12/29/2011.
 Spring '09
 Rate Of Change

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