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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Transcript of Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f...
![Page 1: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/1.jpg)
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
![Page 2: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/2.jpg)
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
![Page 3: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/3.jpg)
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?
![Page 4: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/4.jpg)
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
![Page 5: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/5.jpg)
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
![Page 6: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/6.jpg)
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
![Page 7: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/7.jpg)
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.
![Page 8: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/8.jpg)
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
![Page 9: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/9.jpg)
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
![Page 10: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/10.jpg)
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
![Page 11: Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649efb5503460f94c0d6ec/html5/thumbnails/11.jpg)
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