Mth4 Project I (TI 89) Nme: Riem Sums d Defiite Itegrls The re uder the grph of positive fuctio is give y the defiite itegrl of the fuctio. The defiite itegrl c e pproimted y the followig sums: Left Riem Sum: f ) d L [ f ( ) f ( )... f ( )] ( Right Riem Sum: f ) d R [ f ( ) f ( )... f ( )] ( Midpoit Rule: f ) d M [ f ( ) f ( )... f ( )] ( Trpezoidl Rule: f ( ) d T [ f ( ) f ( )... f ( ) f ( )] Simpso s Rule: f ( ) d S [ f ( ) 4 f ( ) f ( )... 4 f ( ) f ( )] Where, i i, i the midpoit of the ith suitervl, d is eve for Simpso s Rule. L R It turs out tht the Trpezoidl pproimtio T d Simpso s pproimtio T / M / S I ll of these methods we get more ccurte pproimtios whe we icrese the vlue of. The Error i usig pproimtio is: Error = Actul vlue of the itegrl - Approimtio = f ( ) d - Approimtio Error Bouds for Midpoit d Trpezoidl Rules: K( ) K( ) Suppose tht f ( ) K for. The E M d E T 4 Error Bouds for Simpso s Rules: K ( ) Suppose tht f ( ) K for. The E S 4 8 These Error Bouds re very useful to estimte the errors d the ccurcy of the pproimtios without hvig to fid the vlue of these pproimtios, especilly for lrge s. These Error Bouds re lso helpful i estimtig the umer of prtitios required to gurtee specific ccurcy whe pproimtig itegrl. 5
The ove Approimtig Sums c e foud usig progrm o the clcultor clled riem. riem( ) Prgm Locl md,,,,h,s,c,l,r,t,m,i getmode( Ect/Appro ) md setmode( Ect/Appro, APPROXIMATE ) setmode( Disply Digits, FLOAT ) ClrIO Prompt,, ClrIO Output,,,,N Output,4, Output, 8, Output,, (-)/ h s For i,,- +h*i c s+y(c) s EdFor (y()+s)*h L (s+y())*h r (L+r)/ t Output,, L Output, 4, L Output,, R Output, 4, r Output 4,, T Output 4, 4, t m +h/c For i,, m+y(c) m c+h c EdFor m*h m Output 5,, M Output 5, 4, m Output 6,, S(N) Output 6, 4, (t+*m)/ setmode( Ect/Appro, md) EdPrgm To crete this progrm o your clcultor: Go to the Progrm Editor (fter pressig APPS). Press to crete ew progrm. Go dow to Vrile d type riem. Whe doe press ENTER twice. The you type the followig progrm etwee Prgm d EdPrgm. The ALPHA key lets you eter the lphetic chrcters. To eter r, for emple, press ALPHA d the press. If you hve to eter severl lphetic chrcters, press d ALPHA to get the ALOCK so tht you void pressig the ALPHA key my times. To get the cpitl letters press ALPHA To hve setmode( Ect/Appro, APPROXIMATE ), press d F to get F6, press D to get Ect/Appro, press ENTER to get APPROXIMATE To hve getmode( Ect/Appro ), modify the previous istructios y chgig s to g d delete APPROXIMATE To hve setmode( Disply Digits, FLOAT ), press d F scroll dow to E, press ENTER. The commds Prompt, Output re i F the progrm iput/output meu. The commds For d EdFor re i F, the commd Locl is i F4 The commds getmode, setmode, ClrIO re i CATALOG is d d is the STO key. To isert ew lie press ENTER or d ENTER Whe doe typig the progrm, Press HOME to leve the progrm editor. We c use the riem progrm to pproimte the followig itegrl: Press Y= d set y = ^. Press HOME to go to the home scree. Type riem( ) ENTER to eecute the progrm. The progrm will prompt you for, the left edpoit of the itervl,, the right edpoit of the itervl, d, the umer of prtitio. It gives you the Left Riem Sum L, the right Riem Sum R, the midpoit pproimtio M, the Trpezoidl pproimtio T, d Simpso s pproimtio S. d
Prolem Use the RIEMANN progrm to pproimte d. Set Y = X, A =, B =, d N =. Get the followig: L 6.84, R 9.4, T 8.4, M 7.98, S 8. Note tht to fid S, you tke N = 5. To get etter pproimtio of the itegrl, you icrese the umer of prtitios N. Fill i the followig tle to pproimte the itegrl d : (swers correct to 6 deciml plces) N L N R N T N M N S N 6.84 9.4 8.4 7.98 8 4 8 Give tht d 8 d usig the tle ove, Which method gve the est pproimtios? Which vlue of N gve the est pproimtios? Which method(s) gve uderestimte of the itegrl? Which method(s) gve overestimte of the itegrl? The error i the ove pproimtios is ERROR = Actul vlue of itegrl Approimtio With N =, the error i the left Riem sum is E L = 8-6.84 =.6 Fill i the tle elow to fid the errors i the ove pproimtios (use 6 deciml plces) N E L E R E T E M E S.6 -.4 -.4. 4 8 Which method gve the lest errors? Which vlue of N gve the lest errors? Note tht the Trpezoidl d Midpoit Rules re much more ccurte th the edpoit pproimtios. The size of the error i the Midpoit Rule is out hlf the size of the error i the Trpezoidl Rule. Simpso s Approimtios re the most ccurte.
Prolem 4 I this prolem we will use the RIEMANN progrm to pproimte the vlue of = d Use your clcultor with Y = 4 / (+ X ) to fill i the followig tle. (Aswers to 6 deciml plces) N L N R N T N M N S N 4 8 6 Usig the tle, pproimtely, how my prtitios re eeded to pproimte to withi.5: whe usig the Midpoit Rule? whe usig the Trpezoidl Rule? whe usig the Simpso s Rule? Note tht your swers might ot e the smllest umer of prtitios tht will give you such precisio. I prolem, we ler how to fid etter estimtes of the umer of prtitios y usig the Error Boud Formuls. Prolem I this prolem we del with the Actul Errors = Actul vlue of itegrl Approimtios, d the Estimtes of Errors usig the Error Bouds give o the first pge of this project. Cosider the fuctio f ( ) d the itegrl d 4. (Give swers with 6 deciml plces) A) I this prt we fid the ctul vlue of the errors whe pproimtig 4 d. (i) Fid M =, T =, d S = (ii) You c evlute the itegrl 4 d usig F i your clcultor 4 ( /,,,4 ) = or y hd d l 4 (iii) For =, fid the ctul error E M = 4 d M = the ctul error E T =, d the ctul error E S =
B) It is possile to estimte these Errors without fidig the pproimtios M, T, d S. I this prt we fid estimte of the errors usig the Error Bouds formuls. Error Bouds for Midpoit d Trpezoidl Rules: K( ) K( ) Suppose tht f ( ) K for. The E M d E T 4 Error Bouds for Simpso s Rules: K ( ) Suppose tht f ( ) K for. The E S 4 8 (i) Fid the followig derivtives of f ( ) : () f (), f ( ), f ( ), f ( ) (ii) To fid K, sketch the grph of y = f () o the itervl [, 4] y y= s( / ) 5 The mimum vlue of f () is K =. Or use the followig iequlities: So f ( ) K 4 64. 64 (iii) With = prtitios d usig the ove formuls for Error Bouds, fid ( Show your work) K( ) (4 ) E M 4 4(), d E T (iv) Sketch the grph of y = f ( ) o the itervl [, 4] to fid K Upper Boud (or Mimum) of f ( ), K d E S (v) Are the Actul Errors foud i prt A) comptile with the Error Bouds i prt B)?
C) (i) Use the Error Boud formuls to fid the mimum possile error (i.e. upper oud for the error) i pproimtig 4 d with = 5 d usig the Trpezoidl rule. E T (ii) Use the Error Boud formuls to fid the mimum possile error i pproimtig with = usig the Simpso s rule. E S 4 d (iii) Usig your swers to prt (i) d (ii), the umer of prtitios eeded to pproimte 4 d correct to deciml plces is pproimtely: = with the Trpezoidl rule, d = with the Simpso s rule. D) Use the Error Boud formuls to fid how lrge do we hve to choose so tht the pproimtios T, M, d S to the itegrl 4 d re ccurte to withi.: K( ) (4 ) Trpezoidl rule: E T.. () > = (.) Midpoit rule: = (show work) Simpso s rule: = (show work)