Constructing Taylor Series
Let
Find
Find
be an infinite series.
.
. Study it to make an observation. What can you conclude about f(x)?
What function f(x) would have its own derivative f’(x)? What function must f(x) be?
Construct a new fourth degree polynomial P(x) such where
______________________________________________
Math Through Discovery LLC © 2017
Use the same technique used in the previous exercise to construct a polynomial P(x) of degree 4
that matches the behavior of ln(1+x) at x = 0 through the first four derivatives.
______________________________________________
Use the same technique to construct a polynomial P(x) of degree 4 that matches the behavior
of
at x = 0 through the first four derivatives.
______________________________________________
Math Through Discovery LLC © 2017
Use this same technique to construct a polynomial of degree 7 that matches the behavior of
sin(x) at x = 0 through the first four derivatives.
______________________________________________
Math Through Discovery LLC © 2017
Take your polynomial for sin x. Graph
against (y2) a one term polynomial,
(y3) a two term polynomial, (y4) a three term polynomial, and (y5) a five term polynomial to see
how the polynomials compare to f(x) as you add more terms.
One-term Polynomial
Three-term Polynomial
Math Through Discovery LLC © 2017
Two-term Polynomial
Four-term Polynomial
Create four sets of table to compare the y values for
against (y2) a one
term polynomial, (y3) a two term polynomial, (y4) a three term polynomial, and (y5) a five term
polynomial to see how the polynomials compare to f(x) as you add more terms. Set the minimum
x value as -0.04 and the steps to 0.01. What do you notice from the table values?
x
y1=sin x
y2
x
-0.04
-0.04
-0.03
-0.03
-0.02
-0.02
-0.01
-0.01
0
0
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
x
y1=sin x
y4
x
-0.04
-0.04
-0.03
-0.03
-0.02
-0.02
-0.01
-0.01
0
0
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
Math Through Discovery LLC © 2017
y1=sin x
y3
y1=sin x
y5
Use the same technique to construct a polynomial P(x) of degree 4 for f(x) = cos x.
______________________________________________
Math Through Discovery LLC © 2017
Look back at the previous work you completed on writing polynomials P(x) for the various functions. Write out a
summary of the polynomials you have worked with so far and write an infinite series for each function by
generalizing the finite series to an infinite series as illustrated in the first example. If you haven’t already
written a series for the function, develop the series. Include what the general term looks like. The polynomial
for
has been completed.
•
•
•
•
•
•
•
•
Math Through Discovery LLC © 2017
Try to find a Taylor Polynomial P(x) with n terms that can be used to approximate sin (x) so that
for
Recall that
series for
.
. Use this Taylor series to write a Taylor
.
Math Through Discovery LLC © 2017
Approximate
over the interval (-1,1) with a 4 term Taylor polynomial.
Determine the error created by approximating
Math Through Discovery LLC © 2017
with a 4 term Taylor polynomial.
Find a formula for the truncated error if we use P6(x) to approximate
.
Math Through Discovery LLC © 2017
in the interval
Constructing Taylor Series - ANSWERS
Let
Find
Find
be an infinite series.
.
. Study it to make an observation. What can you conclude about f(x)?
What function f(x) would have its own derivative f’(x)? What function must f(x) be?
Only the function
therefore,
Construct a new fourth degree polynomial P(x) such where
Math Through Discovery LLC © 2017
Use the same technique used in the previous exercise to construct a polynomial P(x) of degree 4
that matches the behavior of ln(1+x) at x = 0 through the first four derivatives.
Math Through Discovery LLC © 2017
Use the same technique to construct a polynomial P(x) of degree 4 that matches the behavior
of
Math Through Discovery LLC © 2017
at x = 0 through the first four derivatives.
Use this same technique to construct a polynomial of degree 7 that matches the behavior of
sin(x) at x = 0 through the first four derivatives.
Math Through Discovery LLC © 2017
Take your polynomial for sin x. Graph
against (y2) a one term polynomial,
(y3) a two term polynomial, (y4) a three term polynomial, and (y5) a five term polynomial to see
how the polynomials compare to f(x) as you add more terms.
One-term Polynomial
Three-term Polynomial
Math Through Discovery LLC © 2017
Two-term Polynomial
Four-term Polynomial
Create four sets of table to compare the y values for
against (y2) a one
term polynomial, (y3) a two term polynomial, (y4) a three term polynomial, and (y5) a five term
polynomial to see how the polynomials compare to f(x) as you add more terms. Set the minimum
x value as -0.04 and the steps to 0.01. What do you notice from the table values?
As x stay close to zero the y-values are very close to the sin x.
x
y1=sin x
y2
x
y1=sin x
y3
-0.04
-0.03998
-0.04
-0.04
-0.03998
-0.03997
-0.03
-0.02999
-0.03
-0.03
-0.02999
-0.02999
-0.02
-0.01999
-0.02
-0.02
-0.01999
-0.01999
-0.01
-0.00999
-0.01
-0.01
-0.00999
-0.00999
0
0
0
0
0
0
0.01
0.00999
0.01
0.01
0.00999
0.00999
0.02
0.01999
0.02
0.02
0.01999
0.01999
0.03
0.02999
0.03
0.03
0.02999
0.02999
0.04
0.03998
0.04
0.04
0.03998
0.03997
x
y1=sin x
y4
x
y1=sin x
y5
-0.04
-0.03998
-0.03997
-0.04
-0.03998
-0.03997
-0.03
-0.02999
-0.02999
-0.03
-0.02999
-0.02999
-0.02
-0.01999
-0.01999
-0.02
-0.01999
-0.01999
-0.01
-0.00999
-0.00999
-0.01
-0.00999
-0.00999
0
0
0
0
0
0
0.01
0.00999
0.00999
0.01
0.00999
0.00999
0.02
0.01999
0.01999
0.02
0.01999
0.01999
0.03
0.02999
0.02999
0.03
0.02999
0.02999
0.04
0.03998
0.03997
0.04
0.03998
0.03997
Math Through Discovery LLC © 2017
Use the same technique to construct a polynomial P(x) of degree 4 for f(x) = cos x.
Math Through Discovery LLC © 2017
Look back at the previous work you completed on writing polynomials P(x) for the various functions. Write out a
summary of the polynomials you have worked with so far and write an infinite series for each function by
generalizing the finite series to an infinite series as illustrated in the first example. If you haven’t already
written a series for the function, develop the series. Include what the general term looks like. Write the series
using sigma notation. The polynomial for
•
•
•
•
•
•
•
•
Math Through Discovery LLC © 2017
has been completed.
Try to find a Taylor Polynomial P(x) with n terms that can be used to approximate sin (x) so that
for
.
We want
since
We can use the graphing calculator to determine how many terms we need to use so
A seven termed polynomial will do it:
Recall that
series for
Approximate
. Use this Taylor series to write a Taylor
.
over the interval (-1,1) with a 4 term Taylor polynomial.
Determine the error created by approximating
A 4 term Taylor polynomials for
By division
Math Through Discovery LLC © 2017
with a 4 term Taylor polynomial.
The error
is a geometric series whose first term is
would be
and
. The absolute value of the sum
.
Find a formula for the truncated error if we use P6(x) to approximate
.
This difference is a geometric series with the first term
Math Through Discovery LLC © 2017
and
. Therefore,
in the interval