Finding K
Derrick Smith
Introduction
The error bounds for the integration approximation techniques you learned in class need
you to find a value of
K
such that
( )
f
x
K
′
≤
,
( )
f
x
K
′′
≤
or
( )
4
( )
f
x
K
≤
for all
x
in
[ , ]
a b
.
After finishing this supplement, you will know the two ways to find
K
: the High
Road and the Quick and Dirty Method.
The High Road
All of the approximation methods require finding an upper bound on some function.
For
now, forget that the function is some derivative and just concentrate on how to find the
upper bond of a function.
The High Road is based on the following theorem from Calculus 1:
THE MAXIMUM VALUE THEOREM
If
f
is continuous on the closed interval[ , ]
a b
, then
f
attains is absolute maximum
value
( )
f c
at some number
c
in the interval.
Moreover, either
c
is one of the
endpoints of the interval, or
c
is a critical number of
f
.
(Recall that
c
is
a critical number of
f
iff
( )
0
f c
′
=
or
( )
f c
′
is undefined.)
So, to find
K
for an arbitrary function
f
on a closed interval, you need to do two things:
1)
Find all places in the interval where the derivative of
f
is 0.
2)
Look at values of
f
for all the points you found in 1) and at the endpoints.
3)
Let
K
be the absolute value of
f
that has the largest magnitude.
Let’s do a few examples.
Example 1:
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 Spring '10
 Williamson
 Calculus, Approximation, Critical Point, Continuous function, Order theory

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