Transcript 9-10 Taylor and Maclaurin series

DEFINITIONS OF TAYLOR AND MACLAURIN SERIES
9-2
FORMING A POWER SERIES
Use the function f(x)=sinx to form a MacLaurin series and
determine the interval of convergence
THEOREM 9.23 CONVERGENCE OF TAYLOR SERIES
9-4
A CONVERGENT MACLAURIN SERIES
Show that the MacLaurin series for f(x) = sinx converges to
sinx for all x.
GUIDELINES FOR FINDING A TAYLOR SERIES
9-6
MACLAURIN SERIES FOR A COMPOSITE FUNCTION
Find the Maclaurin series for f(x) = sinx2
POWER SERIES YOU NEED TO MEMORIZE
x
x
sin x = x - + - ...
3! 5!
2
4
x
x
cos x = 1- + - ...
2! 4!
2
3
x
x
x
e = 1+ x + + + ...
2! 3!
1
2
3
= 1+ x + x + x + ...
1- x
3
5
FIND THE MACLAURIN SERIES
f (x) = cos x
POWER SERIES FOR ELEMENTARY FUNCTIONS
9-10
MULTIPLICATION/ DIVISION OF POWER SERIES
Find the first 3 nonzero terms in the
Maclaurin series tanx.
A POWER SERIES FOR SIN2X
Find the power series for f(x) = sin2x
POWER SERIES APPROXIMATION OF A DEFINITE INTEGRAL
Use a power series to approximate
ò
1 - x2
0
e
dx
TRY ON YOUR OWN
Find the Taylor series.
1) f(x)=x -1 c=1
Find the Maclaurin series using the power
series you need to memorize.
x 2 /2
2) f (x) = e
3) g(x) = sin 3x