9.3 Taylor s Theorem: Error Analysis for Series. Tacoma Narrows Bridge: November 7, 1940

Transcription

1 9. Taylor s Theorem: Error Aalysis or Series Tacoma Narrows Bridge: November 7, 940

2 Last time i BC

3 So the Taylor Series or l x cetered at x is give by ) l x ( ) ) + ) ) + ) ) 4 Use the irst two terms o the Taylor Series or l x cetered at x to approximate: 4... l (.5 ) (.5 ) l (0.5 ) (0.5 )

4 Recall that the Taylor Series or l x cetered at x is give by (x) l x (x ) (x ) + (x ) (x ) () + (x ) Fid the maximum error boud or each approximatio. Because the series is alteratig, we ca start with l (.5 ) error actual error l(.5) l error (0.5 ) actual error l(0.5) ( 0.65) Wait! How is the actual error bigger tha the error boud or l 0.5?

5 Ad ow, the excitig coclusio o Chapter 9

6 Sice each term o a coverget alteratig series moves the partial sum a little closer to the limit: Alteratig Series Estimatio Theorem For a coverget alteratig series, the trucatio error is less tha the irst missig term, ad is the same sig as that term. This is also a good tool to remember because it is easier tha the Lagrage Error Boud which you ll id out about soo eough Muhahahahahahaaa!

7 Taylor s Theorem with Remaider I has derivatives o all orders i a ope iterval I cotaiig a, the or each positive iteger ad or each x i I: ( ( a) ) ( ) ( ) ( ) ( )( ) ( ) a ( ) x a + a x a + x a + x a + R )!! Lagrage Error Boud ( ) ( + ) ( ) ( + )! R x x a + I this case, c is the umber betwee x ad a that will give us the largest result or R (x)

8 Does ay part o this look amiliar? ( ) ( + ) ( ) ( + )! R x x a + This remaider term is just like the Alteratig Series error (ote that it uses the + term) except or the ( + ) I our Taylor Series had alteratig terms: + ( a) R ( x) a) ( + )! This is just the ext term o the series which is all we eed i it is a Alteratig Series I our Taylor Series did ot have alteratig terms: ( ) ( + ) ( ) ( + )! R x x a + ( + ) Note that workig with is the part that makes the Lagrage Error Boud more complicated.

9 Taylor s Theorem with Remaider I has derivatives o all orders i a ope iterval I cotaiig a, the or each positive iteger ad or each x i I: ( ( a) ) ( ) ( ) ( ) ( )( ) ( ) a ( ) x a + a x a + x a + x a + R )!! Lagrage Error Boud ( ) ( + ) ( ) ( + )! R x x a Why this is the case ivolves a mid-bedig proo so we just wo t do it here. Now let s go back to our last problem +

10 Recall that the Taylor Series or l x cetered at x is give by l x ) ) + ) ) 4... Fid the maximum error boud or each approximatio. 4 ( ) x + ( ) Because the series is alteratig, we ca start with l (.5 ) error actual error l(.5) l error (0.5 ) actual error l(0.5) ( 0.65) Wait! How is the actual error bigger tha the error boud or l 0.5?

11 Recall that the Taylor Series or l x cetered at x is give by l x ) ) + ) ) ( ) x + ( ) First o all, whe pluggig i ½ or x, what happes to your series? l (0.5 ) (0.5 ) l (0.5 ) ( ) (0.5) + ( 0.5) Note that whe x ½, the series is o loger alteratig. So ow what do we do? Sice the Remaider Term will work or ay Taylor Series, we ll have to use it to id our error boud i this case

12 Sice we used terms up through, we will eed to go to to id our Remaider Term(error boud): ) l x ( ) l 0 ) ( ) x ) ( ) x ( x) c x c ) l The Taylor Series or cetered at x x The third derivative gives us this coeiciet:! c! This is the part o the error boud ormula that we eed

13 l (0.5 ) error actual error l(0.5) ( 0.65) We saw that pluggig i ½ or x makes each term o the series positive ad thereore it is o loger a alteratig series. So we eed to use the Remaider Term which is also called The Lagrage Error Boud error (0.5)! (0.5 ) What value o c will give us the maximum error? )! c The third derivative o l x at x c! Normally, we would t care about the actual value o c but i this case, we eed to id out what value o c will give us the maximum value or c. ( a)

14 error (0.5 ) (0.5)! c The third derivative o l x at x c! The questio is what value o c betwee x ad a will give us the maximum error? So we are lookig or a umber or c betwee 0.5 ad. Let s rewrite it as Ad thereore c which has its largest value whe c is smallest. c 0.5 error (0.5 ) (0.5)! c (0.5)!! (0.5) Which is larger tha the actual error! actual error Ad we always wat the error boud to be larger tha the actual error l(0.5) ( 0.65)

15 Let s try usig Lagrage o a alteratig series l(+ x) x x We kow that sice this is a alteratig series, the error boud would be x But let s apply Lagrage (which works o all Taylor Series) error! The third derivative o l(+ x) is x ) ( + x) error (+ c)! x x (+ c) The value o c that will maximize the error is 0 so x (+ c) x Which is the same as the Alteratig Series error boud

16 Most text books will describe the error boud two ways: Lagrage Form o the Remaider: ( ) ad ( + ) ( ) ( + )! R x x a + ( + ) ) I M is the maximum value o o the iterval betwee a ad x, the: M Remaider Estimatio Theorem: R ( ) x x a ( + )! + Note rom the way that it is described above that M is just aother way o sayig that you have to maximize ( + ) ( c ) Remember that the oly dierece you eed to worry about betwee Alteratig Series error ad La Grage is idig ( + )

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