8.8 Taylor & Maclauren Series

8.8 Taylor & Maclauren Series - 8 .8 

T a y l...

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T a y l o r 
a n d 
M a c l a u r e n 
S e r ie s 
 Just
as
it’s
easier
to
approximate
some
numbers
 (π ≈ 3.14, 2 ≈ 1.414 ) ,
sometimes
we
want
to
 approximate
a
function
close
to
a
point
with
simpler
terms
–
 for
example
we
might
want
to
use
a
polynomial.
 Ex.
 f ( x ) = e x 
at
x
=
1
 y x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 In
general
the
formula
for
the
Taylor
series
of
a
function
f
at

 x
=
a
is
 ∞ ∑ k =0 f ( k ) ( a) f ′′ ( a) f ( n ) ( a) k 2 ( x − a) = f ( a) + f ′ (a)( x − a) + ( x − a) + ... + ( x − a) n + ....
 k! 2! n! 
 If
a
=
0,
then
the
series
is
a
Maclaurin
series.
 ∞ ∑ k =0 f ( k ) (0) k f ′′ (0) 2 f ( n ) (0) n ( x ) = f (0) + f ′ (0)( x ) + ( x ) + ... + ( x ) + ....
 k! 2! n! 
 Ex.

The
Maclaurin
series
for
 
 
 
 
 
 
 
 f ( x ) = e x 
is
 Ex.

Find
the
Maclaurin
series
for
 y = sin x .
 
 
 
 
 
 
 
 
 
 
 
 
 
 y = sin x 
at
 a = π .
 Ex.

Find
the
Taylor
series
for
 2 
 
 
 
 
 x Ex.

Find
the
Maclaurin
series
for
 f ( x ) = 2 .
 
 
 
 
 
 
 
 
 
 
 
 
 
 x Ex.

Find
the
Taylor
series
for
 f ( x ) = 2 
at
a
=
1.
 
 
 
 
 
 
 Ex.

Suppose
we
want
to
find
the
Maclaurin
series
for
 f ( x ) = 2( −7 x) .
 
 
 
 
 
 
 
 1 f ( x) = Do:
1.

Find
the
Maclaurin
series
for
 2 − x .
 
 1 f ( x) = 







2.

Find
the
Taylor
series
for
 2 − x 
at
a
=
1.
 
 
 
 
 
 ...
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