Calculus 2 – Fall 2008
Dr. Hamblin
Section 9.2: Taylor’s Theorem
Recall Taylor’s Theorem, which states that if
centered at
for all
, and for all
is the
th
-order Taylor polynomial for a function
in an interval ,
, then
in .
1. Let
and let
be the third-order Taylor polynomial for
centered at
What does Taylor’s Theorem imply about the maximum approximation error committed by
on the interval
?
2. Let
and let
be the sixth-order Maclaurin polynomial for
Taylor’s Theorem imply about the maximum approximation error committed by
interval
?
3. Let
as above. For what value of
accurately to 3 decimal places on the interval
is
. What does
on the
guaranteed to approximate
?
.
Calculus 2 – Fall 2008
Dr. Hamblin
Solutions
1. Using
2. Using
, we get
, we get
3. We can use
for all values of , and
error bound is less than
.
.
.
is the smallest value of
for which the