Taylor Polyomials ad Approimatios - Classwork Suppose you were asked to id si 37 o. You have o calculator other tha oe that ca do simple additio, subtractio, multiplicatio, or divisio. Fareched\ Not really. The advaced calculators you ow ca ot do ay more tha these simple operatios. I order to uderstad how calculus allows us to calculate si 37 o, we must start with a ew basics. First, you must uderstad the cocept o actorial, which is primarily used i the theory o probability.!! =! 2... 3 2 So 4! = 4 R 3R 2 R = 24, 5! = 5 R 4! = 20, etc. We deie! = ad 0! = as well. Our goal is to id a polyomial uctio P that approimates aother uctio. We begi by choosig some umber c such that the value o ad p at c are the same - P! c) =! c). Not oly that must be true, but at poit c P c c. I this is true, we say that the approimatig polyomial P is epaded about or cetered at c. c, ' = ' =! c) ' = ' P c P c c! c,! c) ) P With these two requiremets, we ca obtai a simple liear approimatio o usig a irst degree polyomial uctio. =, id a irst-degree polyomial uctio P = a $ a Eample ) For the uctio e 0 (a lie) whose value ad slope agree with the slope o at = 0. (Note: I kow you are ot wild about the coeiciets a ad a but, as usual, we have a good reaso or usig them) 0 I the equatio o the polyomial we search or is P = a $ a0 P! 0) =! 0). So what does that give you or a 0 \ kkkkkkkkkkkkkkkkkk I the slope o the polyomial P'! 0 ) must equal the slope o the uctio '! 0 ), what does that give you or a \ kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk So your st degree polyomial approimatio is P =kkkkkkkkkkkk. Traph it agaist = e., we ca impose the coditios that You ca tell that at poits ear 0, the graph o the uctio approimatio P ad its are very close. However, as we move away rom 0,, the approimatio is ot good ad the two graphs move urther apart. To improve the approimatio, we ca urther require that the values o the secod derivatives o P ad agree at = 0. Assume the uctio is a quadratic. MasterMathMetor.com - 82 - Stu Schwartz
= $ $ 2 So letgs assume that P a a a 2 0 with = e I P! 0) =! 0), the solve or a 0 I P'! 0) = '! 0 ), the solve or a I P''! 0) = ''! 0 ), the solve or a 2 So your polyomial is P =. = kkkkkkkkkkkkkkkkkkkkkkkkk Now graph it agaist e Not bad. But, it still is't great. Let's get a eve better approimatio by (you guessed it), makig the uctio a cubic ad makig the third derivatives equal. = (see the reasos or those coeiciets ow\) 3 2 So letgs assume that a3 $ a2 $ a $ a0 with e I P! 0) =! 0), the solve or a 0 I P'! 0) = '! 0 ), the solve or a I P''! 0) = ''! 0 ), the solve or a 2 P 0 0, the solve or a 3 I ''' = ''' So your polyomial is P = kkkkkkkkkkkkkkkkkkkkkkkkk Now graph it agaist = e. I you cotiue this patter by takig more ad more derivatives, you get better approimatios all the time. 2 3 4 I we take derivatives, we get P 2 3! 4!! = $ $ $ $ $)$ 6 e MasterMathMetor.com - 83 - Stu Schwartz
Set your calculator to ull decimal place accuracy ad compare the value o e to P3! " -.0-0.2 -. 0. 0.2 e P 3 This does ot have to oly work at cetered at = 0. It ca also be cetered at some umber c. This work was doe by Eglish mathematicias Brook Taylor (685-73) ad Coli Maclauri (698-746). Deiitio o th Taylor ad Maclauri Polyomials I has derivatives at c, the the polyomial c c c P! " =! c" $!! c"! " c" $!!! " c" $!!! 2 3! " c" $%$ 2! 3!! is called the th Taylor polyomial or at c. I c = 0 (cetered at 0), the P! " = $! $!! $!!! 0 2! 0" 3 0 0 0 $%$ 2! 3!! is called the th Maclauri polyomial or at c. Eample 2) Fid the Taylor polyomials P, P2, P3, P4 or l cetered at c = 2 = Now id!! " = ad! = " To start, id 2 2 =!!! " = ad!!! 2 " =!!!! " = ad!!!! 2 " =! 4 "! " = ad 4 2 So ow we ca write =! " c" P! " = P2! " = P3! " = P4! " = Graph P! " ad P2! " o this graph ad graph P3! " ad P4! " o this ais www.mastermathmetor.com BC Solutios - 84 - Illegal to post o Iteret
= cos, dse the result to approimate cos 0.2. Eample 3) Fid the Maclauri polyomials P0, P2, P4, P6 or Sice it is a Maclauri polyomial, you will epad about = c = 0 = cos ad! 0) = kkkkkkkkkkkkkkkkkkkk ''! ) = kkkkkkkkkkkkkkkkkk ad ''! ) = kkkkkkkkkkkkkkkkkkkk '''! ) = kkkkkkkkkkkkkkkkkkk ad '''! ) = kkkkkkkkkkkkkkkkkkkk! 4 ) = kkkkkkkkkkkkkkkkkk ad 4 = kkkkkkkkkkkkkkkkkkk Repeated dieretiatio gives you the patter: kkkkkkkkkkkkkkkkkkkkkkkkk So ow we ca write P 0 = k 2 = k 4 = k P P P6= k So calculate with ull accuracy cos.2. Fid the dierece betwee that ad usig the cos key. kkkk = si, dse the result to approimate si.2. Eample 4) Fid the Maclauri polyomials P, P3, P5, P7 or Qlso graph the uctio to covice yoursel that this polyomial eacts like si close to Aero. Eample 5) Fid the 4th Maclauri polyomial to approimate the value o l!. ). = l ad ceter it aroud =. Ceterig it aroud = 0 is too ar rom.. = l cetered at You could use @ut you would preer to use Maclauri polyomials. So istead o usig =, you ca use kkkkkkkkkkkkkkk cetered at = 0. Do it ad the calculate l!. ). Eample 6) Iet t be a uctio that has derivatives or all orders or all real umbers. I = ' =! '' = ''' = 0, 0 4, write the 3rd degree Maclauri polyomial or ad t dt.. 0 0 5, 0 3, MasterMathMetor.com - 85 - Stu Schwartz
A approimatio techique is o little value uless we have some idea o how accurate it is. To measure the accuracy o a Taylor or Maclauri polyomial or a uctio! ", we use the cocept o a remaider R! " deied as ollows: = $ or the eact value o the uctio = the approimate value plus a remaider. P R Thus R P =! ad we call the absolute value o R! " the error associated with the approimatio. =!. This is called the Lagrage error boud. So the error = R P This ca be stated i a theorem called Taylor s Theorem. It says the ollowig: I a uctio is dieretiable through order $ i a iterval I cotaiig c, the or each i I, there eists a z betwee ad c such that c c c! " =! c" $ "! c"!! c" $ ""!! c" $ """ 2 3!! c" $%$ 2! 3!!! $ "! z" $ where R! " =!! c" $!!! c" $ R! ". R! " is called the Lagrage orm o the remaider. While the Lagrage remaider ca be positive or egative, the Lagrage error is R! ". The z is some value betwee ad c.you will ot id z. For ay th degree Taylor polyomial, you will eed to id the maimum value o the! $ " st derivative at ay z to calculate the Lagrage remaider. The Lagrage error boud thus represets the maimum error betwee the eact value o the uctio ad the approimatio. There are geerally two types o problems that ask or the Lagrage remaider or error. the maimum value o the! $ " st derivative at z is easily determied (usually trig uctios). the maimum value o the $ st derivative at z is give to you. Eample 7) Fid the accuracy o the third Maclauri polyomial or si! 0. " First, id the third Maclauri polyomial or si amely P3! " Usig Taylor s Theorem, we ca write si = Now, usig P3! ", we ca estimate si 0. as (ull accuracy) So we ow eed R3! ", the Lagrage orm o the remaider =? I the iterval [0, 0.], we kow that 0 4 What is z # siz #. So the largest possible value o the 4th derivative o is. It may ever actually be, but is the coveiet to use outer boud o this 4th derivative. So, we ca say that 0 R3 which says that 0 < $ # So ially, we ca coclude that < si! 0. " < www.mastermathmetor.com BC Solutios - 86 - Illegal to post o Iteret
Eample 8) Fid the accuracy o the ourth Maclauri polyomial or e!0. Eample 9) Let be a uctio give by! " = & $ % ) cos ' ( 2 + 6* & about at = 0. Use the Lagrage error boud to show that ( ) P ' 3* +! & ' ( ) 3* + &. 00 ad let P! " be the third-degree Taylor polyomial Eample 0) Let be a uctio havig derivatives or all orders at all real umbers. The third-degree Taylor 5 2 20 3 polyomial or about = -2 is give by: P! " = 9! 4! $ 2" $! $ 2"!! $ 2". The 3 7! 4 ourth derivative or satisies the iequality that "! " # 25 or all i the iterval! 2, 0. Fid the Lagrage error boud or the approimatio to!. 6. ' ( www.mastermathmetor.com BC Solutios - 87 - Illegal to post o Iteret
Taylor Polyomials ad Approimatios - Homework For the ollowig, id the Maclauri polyomial o degree or the give uctio ad evaluate at = 0...! " = e!, = 5 2.! " =, = 5! 3.! " = % = = = si, 3 4. e, 3 5.! " = = = =! ta, 3 6. ta, 3 For the ollowig, id the Taylor polyomial degree or the give uctio cetered at = c 7.! " =, = 4, c = 8.! " = cos, = 2, c = % 9. Estimate the error i approimatig si 0. 5 Lagrage remaider. by usig the ith-degree Maclauri polyomial by idig the www.mastermathmetor.com BC Solutios - 88 - Illegal to post o Iteret
0. A studet eeds to calculate o a our-uctio calculator. He uses a third-degree Taylor polyomial e or y = e ad adds o the Lagrage remaider. What is the result o his calculatio?. Let be a uctio give by! " = &! % ) cos ' ( 3 + 0* & ) at = 0. Use the Lagrage error boud to show that ( +! P & ' 4* ' ( ) + &. 4* 500 ad let P! " be the ourth-degree Taylor polyomial about 2. Let be a uctio havig derivatives or all orders o real umbers. The irst our derivatives o at = 2 are give i the table below. The ourth derivative o satisies the iequality # ' ( 4 50 or all i the iterval 2,2.5. Use the Lagrage error boud o this approimatio to to eplai why! 2 " & 8 5 2....! " "! " ""! " """! " 2 8 4 6 8 www.mastermathmetor.com BC Solutios - 89 - Illegal to post o Iteret