6.5 Completed Notes.pdf

6.5 Completed Notes.pdf - Approximation Consider the...

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Unformatted text preview: - Approximation Consider the function f(x) = exl. This function cannot be integrated. However, the integral can be estimated by geometric means. 1 '12 Use the midpoint rule with n=4 to estimate I9 d 0 . y f(x)=e 2’ ,g; ,. bi“: .L M’ n Li Li - x '1»; J. 3”] 1 DesiréTaylor ”7-"- EL! (4% 7 Lil 5(061ﬂ 1 A.7 The Trapezoidal Rule Area of a T rapa201d(§—-+:)h i - ' ' ' Examples: -\ L 1.) Estimate IU" by dividing the interval [0,1] into4trapezoids of equal width 0 W:ﬁ_, i1[pco')+1llﬁ)+zl(*i+ 2W5)* 31w} ‘2. _..._.._ ._... ..._.._- “w...“ __.__ __‘ _ ._..__‘...I*7- Desiré Taylor Math 1242 .9] I' ; 44444444444 +2) 2W; 121 4444mm +f(xn.2)+f(x1)+f(x )1 .;_i . : f; 44—59442 f(x,)+2 114;)4 +2f(xn2)+2f(x )4. (4)] _ B. Simpson’s Rule I - SimpSQn’s-Rule is a combination of the Midpomt rulé'aridIthe Trapezoidal rUie. V]testlmates_]f(x)dx I, ' ' bJ_—_' __-a I Where _c_I__= xi; 15:36H , Ax]: ,and n is EVEN. . . ﬂ ~17, Mk" -—[f(a)+4f(x4)+2f(x2)+4f(x3)+2f(x4)+ +2f(42)+4f(x41 b)] X“ Example: - £06) b-PA. ___ 1:25,]. x2 1.) Use Simpson 5 rule to estimate I dx with n= 6. (i 9 Find 55) [m 44(4)424;4)444(;) 24942424022] " where qFxé; 1) 4:95): ' 'anq; Ax: __ is give'njbyr -' - ' __I;--¥~-=F--Ax[r(xl)+f(x2)+ +f(x42)+f(x )+f(b)] .'1*??34);42(4);24(4);m4»2444);}4122442;426412;;2ib2'i. 'f ‘ ”WWW—”MW“ m—n —'—-{ Desiré Tayloi Math 1242 a") 0. Error Bounds Error bounds allow us to calculate the maximum amount of error between an approximation and the true integral for a set partition (n). 16(1)??? r: Ill-f}. :3 ' Midpbint Rule IE¥|< 24” where I f ( )ISk for a<x<b , w i: :0 w SlmrpsonsRule IE3 I<k (b 005 where .| fl‘U) (x) I<k for a<ic<b , 18011 ,. Example: 2 1.} For the function f(x)= e x ,find 0 £Wﬂr' b. = 1.467007 98W “’0-01‘80’17ll0l c. FinB‘t ﬂmaximum error using the error bounds and compare to the actual error. f ”JI- Ig I z. Hire? ‘ in— mm lﬁﬁx) L01] T' . _. *2 4360' a *1. K 4mm: 2e 1, W e @IETI‘A ooM‘iWSWB d. What is the smallest value of n that will yield an error within 0001? IETI 5- eﬁaﬂi i 0-00 l . #» (”411545, Y ll h p 0%[y e 02 q Ho soggy (i O)? 43.0 00) " .. a 7 Wl'llnl I: “I: r: l :6 \ CL) 4.0 ool 147" K i%‘5°ll‘-l0‘il7 “'— ﬂw w3®4 Desire Taylor - l [2,ququ 5"“ Aguasucnsza“ A5328" 0330.9a95 “‘7/7I ”FHA. 30%Qi' a. Jexzdx: qullaﬁlV‘lla f fale’ @ﬁ-lm P‘Cxi- '— 2.15311 =0 n aim,» ‘3}, 7 ﬁg [rm—Mimwzcm) i .- , .1 fr; 2.) For the function f(x) = 1n(x), find O 4—2: ln(x)dx ‘" 2 6145,7717IqLF'AX’; a? :— 7 :“ Z 8%.?— ” PM; 14,9 (l. 6)+ 2 4W) «4%? 63+2N5M Ll W5- 5) til“)? ”’1'; COM Maw @1405) +2ﬂmziit'iW/ﬂ'5) +24’hl’gltilw‘lg'ghbqul .J, 2,5‘lLlWQOQOl Tmawm MM— 39?: 23545:. . ~2 53391: c. Find the maximum error using the error bounds and compare to the actual error. C'— j' 0. 00 0 6 294 “I! 1 E..- , i 14 c 19—00 5 12-5: i4 fr”??? i) 1301a” (wafﬂlélc 1056051? ‘- '7 :: 5((5-05 “2&1 f1 W m» a; [Q 0 lo 0 O O (o “__.”.— _, «mm -—~— - /CH) -:J..J , m = 55555 cl. What is the smallest value of n that will yield an error within 0.0001? ' OJWEAEQM J . = ‘1 . s5 40 __ [Esl 4; lathe) 4 0.000: km? ng 49‘3“! L90 in. L} "v Q i. W V‘ ‘5 a , 5 i ““5 \$3 (9 (+15 .1. 2L1 .._, O 00} - L6) 7;: 7"”— 7:” I310 W’— l\ (X) x6 ; 0) . W J Li "l 2 ) :3 @5100 O) ,0! \ OM desk-555‘ \$935+ \ Desiré Taylor [(9 . 37023 _é l’l " L 051 H‘ W) = (Lat-l5 020-234 N We bwork 7. (1 pt) Extra Problem A student is speeding down Highway 16 in her fancy red Porsche when her radar system warns her of an obstacle 400 feet ahead. She immediately applies the brakes, starts to slow down, and spots a skunk in the road directly ahead of her. The "black box" in the Porsche records the car's speed every two seconds, producing the following table. The speed decreases throughout the 10 seconds it takes to stop, although not necessarily at a uniform rate. A A A A . . Time since brakes appllcd (see) En 1-.“ X C. Which one of the lollowing statements can you justify 95 45 from the information given? A. What is your best estimate of the total distance the stu- dent‘s-ear traveled before coming to rest (note that the best es- @A. The ”black hex” data is inconcluSive, The skunk limate is probably not the over 01‘ under estimate that you can may or may not have been hit. most easily ﬁnd, use the trapezoidal approximation)? 0 B. The skunk was hit by the ear. distance = W o C. The car stopped before getting to the skunk. B. Given the fact that the Porsche slows down during break~ lag. give a sharp saote ii. overestimate of the distance traveled: W a s £5 (Pram-2 1009+ 3 ex: 2 9-m.miw l Vt '2, l. tuiderem'mme of the distance traveled: Te” 1 ( (D an + zit-m «r 24243 t arcs) + 2082:.) + 1000)) ” ”a," j , [015+ 2. 35 + agar—3+ 2.254.. 2216‘ + o] :.qg6’\$e (,5: /2.\ l: «0607 t l? 1231“" PM) Witch 10(9)] 1:: more 2 l: o (a) «r @013 am» slits} +-«P 00)] 125—: ,: aaoﬁt Desiré Taylor Math 1242 ...
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