warm up construct the taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. graph f...
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Warm up Construct the Taylor polynomial of degree
5 about x = 0 for the function f(x)=ex. Graph f and your approximation function
for a graphical comparison. To check for accuracy, find f(1) and P5(1).
!5!4!3!21)(
5432
5xxxx
xxP
Taylor Polynomials The polynomial Pn(x) which agrees at
x = 0 with f and its n derivatives is called a Taylor Polynomial at x = 0.
Taylor polynomials at x = 0 are called Maclaurin polynomials.
nn
xn
fx
fx
fxff
!)0(
...!3)0(
!2)0(
)0()0()(
32
)()( xPxf n
Polynomials not centered at x = 0 Suppose we want to approximate
f(x) = ln x by a Taylor polynomial. The function is not defined for x < 0.
How can we write a polynomial to approximate a function about a point other than x = 0?
Polynomials not centered at x = 0 We modify the definition of a Taylor
approximation of f in two ways. The graph of P must be shifted horizontally.
This is accomplished by replacing x with x – a. The function value and the derivative values
must be evaluated at x = a rather than at x = 0.
nn
axnaf
axaf
axafaf )(!)(
...)(!2)(
))(()()(
2
)()( xPxf n
Taylor Polynomial of degree n approximating f(x) near x = a
Construct the Taylor polynomial of degree 4 approximating the function f(x) = ln x for x near 1.
nn
axnaf
axaf
axafaf )(!)(
...)(!2)(
))(()()(
2
)()( xPxf n
4)1(
3)1(
2)1(
)1(ln432 xxx
xx
How does the graph look? Graph y1 = ln x Graph Taylor polynomial of degree 4
approximating ln x for x near 1:
Graph each of the following one at a time to see what is happening around x = 1. y5 = y4(x) + ?? y6 = y5(x) + ?? Y7 = y6(x) + ??
4)1(
3)1(
2)1(
)1(4432 xxx
xy
Replace ?? with last term in
the Taylor polynomialof next degree
Conclusions Taylor polynomials centered at x = a give
good approximations to f(x) for x near a. Farther away, they may or may not be good.
The higher the degree of the Taylor polynomial, the larger the interval over which it fits the function closely.
Taylor Polynomials to Taylor Series Recall the Taylor polynomials centered at
x = 0 for cos x:
The more terms we added the better the approximation.
!4!30
!2!101)(cos
2101)(cos
1)(cos
432
4
2
2
1
xxxxxPx
xxxPx
xPx
Taylor Series or Taylor expansion For an infinite number of terms we can
represent the whole sequence by writing a Taylor series for cos x:
How would represent the series for ex?
...!8!6!4!2
18642
xxxx
...!4!3!2
1432
xxxx
Taylor Series for sin x To get the Taylor series for sin x take the
derivative of both sides.
...!8!6!4!2
1cos8642
xxxxx
...!8
8!6
6!4
4!22
0sin7531
xxxxx
...!7!5!3
sin753
xxxxx
Taylor expansions About x = 0
About x = 1
...!7!5!3
sin753 xxx
xx
...!8!6!4!2
1cos8642 xxxx
x ...!4!3!2
1432 xxx
xex
...111 32
xxxx
■
...4)1(
3)1(
2)1(
)1(ln432 xxx
xx