LOCUS 1. Definite Integration CONCEPT NOTES. 01. Basic Properties. 02. More Properties. 03. Integration as Limit of a Sum
|
|
- Evelyn Nichols
- 5 years ago
- Views:
Transcription
1 LOCUS Defiie egrio CONCEPT NOTES. Bsic Properies. More Properies. egrio s Limi of Sum
2 LOCUS Defiie egrio As eplied i he chper iled egrio Bsics, he fudmel heorem of clculus ells us h o evlue he re uder curve f from o, we firs evlue he i-derivive g of f g f d d he evlue g g. Th is, re uder he curve f() from = o = is f d g g Reders who hve eve he slighes dou regrdig he discussio ove re dvised o refer o he chper o egrio Bsics efore redig o. Defiie iegrio is o ll ou jus evluig he i-derivive d susiuig he upper d lower limis. Workig hrough his chper, ou will relise h lo of echiques eis which help us i evluig he defiie iegrl wihou resorig o he (m imes edious) process of firs deermiig he i-derivive. We will develop ll hese echiques oe oe from scrch, srig wih some eremel sic properies i Secio - Secio - BASC PROPERTES () Suppose h f() < o some iervl [, ]. The, he re uder he curve = f() from = o = will e egive i sig, i.e f d This is ovious oce ou cosider how he defiie iegrl ws rrived i he firs plce; s limi of he sum of he recgles. Thus, if f() < i some iervl he he re of he recgles i h iervl will lso e egive. This proper mes h for emple, if f() hs he followig form A A A z A Fig - he will equl f d A A A A d o A A A A.
3 LOCUS f we eed o evlue A A A A (he mgiude of he ouded re), we will hve o clcule f d z f d f d z f d From his, i should lso e ovious h f d () f d The re uder he curve = f() from = o = is equl i mgiude u opposie i sig o he re uder he sme curve from = o =, i.e f d f d This proper is ovious if ou cosider he Newo-Leiiz formul. f g is he i-derivive of f d is g (f), he g while () f d is g g. The re uder he curve = f() from = o = c e wrie s he sum of he re uder he curve from = o = c d from = c o =, h is c f d f d f d c Le us cosider emple of his. Le c (, ) = f() A A c Fig - is cler h he re uder he curve from = o =, A is A + A. Noe h c eed o lie ewee d for his relio o hold rue. Suppose h c >. = f() A A Fig - c
4 LOCUS 5 Oserve h A f d A A A c c f d f d c f d f d c Alicll, his relio c e proved esil usig he Newo Leiiz s formul. () Le f g o he iervl [, ]. The, f g d. This is ecuse he curve of f() lies ove he curve of g(), or equivlel, he curve of f g lies ove he -is for [, ] = f() This is emple where f() > g() >. f ( )d = A + A = g() A A g ( )d = A while Fig - Similrl, if f g o he iervl [, ], he f d (5) g d For he iervl [, ], suppose m < f() < M. Th is, m is lower-oud for f() while M is upper oud. The, m f d M
5 LOCUS 6 This is ovious oce we cosider he figure elow: M D C = f() m A X = Fig - 5 B Y = Oserve h re rec AXYB f d re rec DXYC (6) Le us cosider he iegrl of f f from o. To evlue he re uder f f, we c seprel evlue he re uder f () d he re uder f () d dd he wo res (lgericll). Thus: f f d f d f d Now cosider he iegrl of kf() from = o =. To evlue he re uder kf(), we c firs evlue he re uder f() d he mulipl i k, h is: kf d k f d (7) Cosider odd fucio f(), i. e, f f. This mes h he grph of f() is smmeric ou he origi. - + Fig - 6 From he figure, i should e ovious h f d, ecuse he re o he lef side d h o he righ lgericll dd o.
6 LOCUS 7 Similrl, if f() ws eve, i. e, f f - + Fig - 7 f d f d ecuse he grph is smmericl ou he -is. f ou recll he discussio i he ui o fucios, fucio c lso e eve or odd ou rirr poi =. Le us suppose h f() is odd ou =, i.e f f = The pois d - lie equidis from = eiher sides of i. Fig - 8 Suppose for emple, h we eed o clcule f d. is ovious h his will e, sice we re cosiderig equl vriio o eiher side of =, i.e. he re from = o = d he re from = o = will dd lgericll o. Similrl, if f() is eve ou =, i.e. f f he we hve, for emple Fig - 9 f d f d From his discussio, ou will ge geerl ide s o how o pproch such issues regrdig eve/odd fucios.
7 LOCUS 8 (8) Le us cosider fucio f() o [, ] f() f() Fig - We w o somehow defie he verge vlue h f() kes o he iervl [, ]. Wh would e pproprie w o defie such verge? Le f v e he verge vlue h we re seekig. Le i e such h i is oied some c [, ] f() f =f(c) v f() f =f(c) v c Fig - We c mesure f v sig h he re uder f() from = o = should equl he re uder he verge vlue from = o =. This seems o e he ol logicl w o defie he verge (d his is how i is cull defied!). Thus f f d v fv f d This vlue is ied for les oe c, (uder he cosri h f is coiuous, of course).
8 LOCUS 9 Emple Fid he re uder he curve o is eire domi, i.e., evlue d Soluio: The grph for is skeched elow: Fig - Alhough he grph eeds o ifii o oh sides, he re uder he curve will sill e fiie, s we ll ow see: d Wh do we mke of? We c ke i o me Similrl, would equl. Thus, he required re is. lim k k which is. Emple Evlue he re ouded ewee d from = o =. Soluio: The give curves re skeched i he regio of ieres elow. = = = The wo curves iersec whe =, i.e = (o he posiive side). Sice < for (, ), he grph of lies elow h of Fig -
9 LOCUS The required re is A d Emple Fid he me vlue of f cos o, Soluio: Le f v e he required me vlue. fv / cos / d cos d si / f v Emple Evlue d Soluio: Therefore, he give iegrl ecomes c e rewrie s d d
10 LOCUS Someimes, o modif iegrl, pproprie susiuio hs o e used; he sme w we did i he ui o defiie egrio. For emple, iegrls coiig he epressio ( + ) c e simplified (or modified) usig he susiuio. For evluig defiie iegrl oo, we c use he pproprie susiuio, provided we chge he limis of iegrio ccordigl lso. This will ecome cler i suseque emples. Emple 5 f / d, prove h () () Soluio: () / / d d / d / / d sec d The susiuio = c ow e used o simplif his iegrl. However, we mus chge he limis of iegrio ccordig o his susiuio: sec d d f f Thus, he modified iegrl (i erms of he ew vrile ) is: d
11 LOCUS () he iegrl h we re cosiderig, he limis of iegrio re o, i. e,, his iervl, <. Thus,, From proper (), we c herefore s h: or / / / d... () Usig he resul of pr () for he firs d hird erms i (), we ge our desired resul: Emple 6 For >, le l f d. Fid he fucio f f d show h f e f. e Soluio: Oserve crefull he form of he fucio f() : is i he form of iegrl (of oher fucio), wih he lower limi eig fied d he upper limi eig he vrile. As vries, f() will correspodigl vr. Oe pproch h ou migh coemple o solve his quesio is evlue he i-derivive g() of l d he evlue g g which will ecome f(). However, his will ecome uecessril cumersome (Tr i!). We c, ised, proceed s follows: / l l f f d d = + Noice h he limis of iegrio of d re differe. f he were he sme, we could hve dded d esil. So we r o mke hem he sme: i, if we le, d vries from o, will vr from o. This susiuio will herefore mke he limis of iegrio of he sme s hose of :
12 LOCUS d d / l d l / d / d c ow e esil dded: l d l l d l l d l d We used ised of i. This does' mke differece; is he vrile of iegrio; i c e replced wih oher vrile s log s he limis of iegrio re he sme. The fil epressio shows how simplified hs ecome. We le l z d dz d he limis of iegrio ecome o l. l z dz l Thus, f e f l e e
13 LOCUS Emple 7 Evlue / si cos d Soluio: The give iegrl c e modified io (esil) iegrle form epressig i i form ivolvig d sec. / si cos d / cos d / sec d The susiuio = c ow e used. sec d d d d d
14 LOCUS 5 Emple 8 Deermie posiive ieger 5 such h e d 6 6e Soluio: Sice will o ur o e ver lrge ieger, oe migh e emped o r ou vrious vlues of i he give relio, srig from owrds, d see which oe fis. This ril-d-error pproch migh quickl give resul i his priculr emple, u wh would we do i ws possil lrger? The geerll followed pproch i such emples, where he iegrl c e chrcerised posiive ieger (clled he order of he iegrl), is o epress i i erms of lower order iegrl. f we deoe he h order iegrl, we should r o epress i erms of k where k <. Such relio is clled recursive relio. We c he simpl use his relio repeedl o order d oi he iegrl of he e order (ised of ever ime repeig he clculio of iegrio gi) We will ow use his pproch o he curre emple: Le e d To simplif, firs of ll le d d d he limis ecome o. Thus, e d e e d We ow use iegrio prs o solve his iegrl, kig s he firs fucio: e e e d d e e d e We hve hus eslished he relio ewee d i () Now oserve h c esil e evlued:...() e e d e
15 LOCUS 6 Usig () repeedl, we c ow oi ll he higher order iegrls:.. 5 e. 6 6e = is herefore he posiive ieger we hd se ou o deermie. Noice he power of he recursive relio h we oied i (). Usig h relio, i ws jus mer of mior clculios o successivel deermie, d from. Wihou (), we would hve o ppl iegrio prs everime, hd we used he ril-d-error pproch. Les look oher emple of his sor. e Emple 9 Evlue e d Soluio: Noice h o mer wh e, lim e so h we will oi fiie re uder he curve. Le We ppl iegrio prs o : Thus, our recursive relio is We use his repeedl ow: e d e e d is simple o deermie:! e d e = Thus,!
16 LOCUS 7 Emple Prove h si d for ll si Soluio: Le Wh is? si si d. si d si si d cos d si Thus, = sisfies he sed proper. How do we pproch he geerl cse? Some reflecio o he ure of he iegrl will show ou h evluig iself would e edius. Wh we could ised do is his: We hve lred show h. f we show h, our sk would e ccomplished, sice he, d ll he higher order iegrls ecome equl o, which is ; his is wh we w o prove. si si d si si si cos d si si = Therefore, N
17 LOCUS 8 Emple / l d Evlue, ecuse, Soluio: Creful oservio will show h he fucio l is odd ou l l co l l Thus, s discussed i proper 7, he give iegrl will ecome. Emple d. Evlue Soluio: The fucio o e iegred s ee skeched elow i he regio of ieres: f() - Fig - This fucio is discoiuous; s discussed i proper-, we c spli he required iervl of iegrio. We will do i i such w so h i ech of he su-iervls h we oi, he fucio is coiuous d c e iegred. f, he Thus, if f d f d f d d d d f d d f d geerl, for discoiuous fucio f() whose iegrl we eed o evlue, he pproch descried ove is followed. f() is seprel iegred i su iervls where i is coiuous d he resuls so oied re dded.
18 LOCUS 9 TRY YOURSELF - Q. Prove h si d si Q. f is odd posiive ieger, prove h / si d si Q. Show h / cos / cos d 6 Q. Show h / d l sec 8 l z Q. 5 For >, if f dz, show h z z f f Q. 6 f cos cos d where is o-egive ieger, show h, d re i A.P. Q. 7 Show h d, is equl o. Q. 8 f = / cos d, N, show h Q. 9 Prove h Q. Prove h d si cos 5 l 6 * You mus hve used susiuios i some of he quesios ove. Thik ou he vlidi of hese susiuios. s susiuio lws vlid? Or do we eed o fulfill ceri requiremes if susiuio is o e vlid?
Week 8 Lecture 3: Problems 49, 50 Fourier analysis Courseware pp (don t look at French very confusing look in the Courseware instead)
Week 8 Lecure 3: Problems 49, 5 Fourier lysis Coursewre pp 6-7 (do look Frech very cofusig look i he Coursewre ised) Fourier lysis ivolves ddig wves d heir hrmoics, so i would hve urlly followed fer he
More informationF.Y. Diploma : Sem. II [CE/CR/CS] Applied Mathematics
F.Y. Diplom : Sem. II [CE/CR/CS] Applied Mhemics Prelim Quesio Pper Soluio Q. Aemp y FIVE of he followig : [0] Q. () Defie Eve d odd fucios. [] As.: A fucio f() is sid o e eve fucio if f() f() A fucio
More informationHOMEWORK 6 - INTEGRATION. READING: Read the following parts from the Calculus Biographies that I have given (online supplement of our textbook):
MAT 3 CALCULUS I 5.. Dokuz Eylül Uiversiy Fculy of Sciece Deprme of Mhemics Isrucors: Egi Mermu d Cell Cem Srıoğlu HOMEWORK 6 - INTEGRATION web: hp://kisi.deu.edu.r/egi.mermu/ Tebook: Uiversiy Clculus,
More informationReinforcement Learning
Reiforceme Corol lerig Corol polices h choose opiml cios Q lerig Covergece Chper 13 Reiforceme 1 Corol Cosider lerig o choose cios, e.g., Robo lerig o dock o bery chrger o choose cios o opimize fcory oupu
More informationSupplement: Gauss-Jordan Reduction
Suppleme: Guss-Jord Reducio. Coefficie mri d ugmeed mri: The coefficie mri derived from sysem of lier equios m m m m is m m m A O d he ugmeed mri derived from he ove sysem of lier equios is [ ] m m m m
More informationIntegration and Differentiation
ome Clculus bckgroud ou should be fmilir wih, or review, for Mh 404 I will be, for he mos pr, ssumed ou hve our figerips he bsics of (mulivrible) fucios, clculus, d elemer differeil equios If here hs bee
More informationSpecial Functions. Leon M. Hall. Professor of Mathematics University of Missouri-Rolla. Copyright c 1995 by Leon M. Hall. All rights reserved.
Specil Fucios Leo M. Hll Professor of Mhemics Uiversiy of Missouri-Roll Copyrigh c 995 y Leo M. Hll. All righs reserved. Chper 5. Orhogol Fucios 5.. Geerig Fucios Cosider fucio f of wo vriles, ( x,), d
More informationSuggested Solution for Pure Mathematics 2011 By Y.K. Ng (last update: 8/4/2011) Paper I. (b) (c)
per I. Le α 7 d β 7. The α d β re he roos o he equio, such h α α, β β, --- α β d αβ. For, α β For, α β α β αβ 66 The seme is rue or,. ssume Cosider, α β d α β y deiiio α α α α β or some posiive ieer.
More informationExistence Of Solutions For Nonlinear Fractional Differential Equation With Integral Boundary Conditions
Reserch Ivey: Ieriol Jourl Of Egieerig Ad Sciece Vol., Issue (April 3), Pp 8- Iss(e): 78-47, Iss(p):39-6483, Www.Reserchivey.Com Exisece Of Soluios For Nolier Frciol Differeil Equio Wih Iegrl Boudry Codiios,
More informationERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION
ERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA Absrc. I his pper we poi ou pproximio for he Fourier rsform for fucios
More informationFunctions, Limit, And Continuity
Fucios, Limi d coiuiy of fucio Fucios, Limi, Ad Coiuiy. Defiiio of fucio A fucio is rule of correspodece h ssocies wih ech ojec i oe se clled f from secod se. The se of ll vlues so oied is he domi, sigle
More informationSection - 2 MORE PROPERTIES
LOCUS Section - MORE PROPERTES n section -, we delt with some sic properties tht definite integrls stisf. This section continues with the development of some more properties tht re not so trivil, nd, when
More information2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)
Mah PracTes Be sure o review Lab (ad all labs) There are los of good quesios o i a) Sae he Mea Value Theorem ad draw a graph ha illusraes b) Name a impora heorem where he Mea Value Theorem was used i he
More informationReview for the Midterm Exam.
Review for he iderm Exm Rememer! Gross re e re Vriles suh s,, /, p / p, r, d R re gross res 2 You should kow he disiio ewee he fesile se d he udge se, d kow how o derive hem The Fesile Se Wihou goverme
More informationDIFFERENCE EQUATIONS
DIFFERECE EQUATIOS Lier Cos-Coeffiie Differee Eqios Differee Eqios I disree-ime ssems, esseil feres of ip d op sigls pper ol speifi iss of ime, d he m o e defied ewee disree ime seps or he m e os. These
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationONE RANDOM VARIABLE F ( ) [ ] x P X x x x 3
The Cumulive Disribuio Fucio (cd) ONE RANDOM VARIABLE cd is deied s he probbiliy o he eve { x}: F ( ) [ ] x P x x - Applies o discree s well s coiuous RV. Exmple: hree osses o coi x 8 3 x 8 8 F 3 3 7 x
More informationENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
More informationCalculus Limits. Limit of a function.. 1. One-Sided Limits...1. Infinite limits 2. Vertical Asymptotes...3. Calculating Limits Using the Limit Laws.
Limi of a fucio.. Oe-Sided..... Ifiie limis Verical Asympoes... Calculaig Usig he Limi Laws.5 The Squeeze Theorem.6 The Precise Defiiio of a Limi......7 Coiuiy.8 Iermediae Value Theorem..9 Refereces..
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationUsing Linnik's Identity to Approximate the Prime Counting Function with the Logarithmic Integral
Usig Lii's Ideiy o Approimae he Prime Couig Fucio wih he Logarihmic Iegral Naha McKezie /26/2 aha@icecreambreafas.com Summary:This paper will show ha summig Lii's ideiy from 2 o ad arragig erms i a cerai
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informationNOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA. B r = [m = 0] r
NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA MARK WILDON. Beroulli umbers.. Defiiio. We defie he Beroulli umbers B m for m by m ( m + ( B r [m ] r r Beroulli umbers re med fer Joh Beroulli
More informationMath 2414 Homework Set 7 Solutions 10 Points
Mah Homework Se 7 Soluios 0 Pois #. ( ps) Firs verify ha we ca use he iegral es. The erms are clearly posiive (he epoeial is always posiive ad + is posiive if >, which i is i his case). For decreasig we
More informationPre-Calculus - Chapter 3 Sections Notes
Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationDavid Randall. ( )e ikx. k = u x,t. u( x,t)e ikx dx L. x L /2. Recall that the proof of (1) and (2) involves use of the orthogonality condition.
! Revised April 21, 2010 1:27 P! 1 Fourier Series David Radall Assume ha u( x,) is real ad iegrable If he domai is periodic, wih period L, we ca express u( x,) exacly by a Fourier series expasio: ( ) =
More informationECE-314 Fall 2012 Review Questions
ECE-34 Fall 0 Review Quesios. A liear ime-ivaria sysem has he ipu-oupu characerisics show i he firs row of he diagram below. Deermie he oupu for he ipu show o he secod row of he diagram. Jusify your aswer.
More informationSolutions to selected problems from the midterm exam Math 222 Winter 2015
Soluios o seleced problems from he miderm eam Mah Wier 5. Derive he Maclauri series for he followig fucios. (cf. Pracice Problem 4 log( + (a L( d. Soluio: We have he Maclauri series log( + + 3 3 4 4 +...,
More informationIdeal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory
Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable
More informationSLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO SOME PROBLEMS IN NUMBER THEORY
VOL. 8, NO. 7, JULY 03 ISSN 89-6608 ARPN Jourl of Egieerig d Applied Sciece 006-03 Ai Reerch Publihig Nework (ARPN). All righ reerved. www.rpjourl.com SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO
More informationProperties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)
Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 =
More informationREAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
More information[Nachlass] of the Theory of the Arithmetic-Geometric Mean and the Modulus Function.
[Nchlss] of he Theory of he Arihmeic-Geomeric Me d he Modulus Fucio Defiiio d Covergece of he Algorihm [III 6] Le d e wo posiive rel mgiudes d le From hem we form he wo sequeces: i such wy h y wo correspodig
More informationN! AND THE GAMMA FUNCTION
N! AND THE GAMMA FUNCTION Cosider he produc of he firs posiive iegers- 3 4 5 6 (-) =! Oe calls his produc he facorial ad has ha produc of he firs five iegers equals 5!=0. Direcly relaed o he discree! fucio
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 4 (R) Wier 008 Iermediae Calculus I Soluios o Problem Se # Due: Friday Jauary 8, 008 Deparme of Mahemaical ad Saisical Scieces Uiversiy of Albera Quesio. [Sec.., #] Fid a formula for he geeral erm
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationExtension of Hardy Inequality on Weighted Sequence Spaces
Jourl of Scieces Islic Reublic of Ir 20(2): 59-66 (2009) Uiversiy of ehr ISS 06-04 h://sciecesucir Eesio of Hrdy Iequliy o Weighed Sequece Sces R Lshriour d D Foroui 2 Dere of Mheics Fculy of Mheics Uiversiy
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationK3 p K2 p Kp 0 p 2 p 3 p
Mah 80-00 Mo Ar 0 Chaer 9 Fourier Series ad alicaios o differeial equaios (ad arial differeial equaios) 9.-9. Fourier series defiiio ad covergece. The idea of Fourier series is relaed o he liear algebra
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationManipulations involving the signal amplitude (dependent variable).
Oulie Maipulaio of discree ime sigals: Maipulaios ivolvig he idepede variable : Shifed i ime Operaios. Foldig, reflecio or ime reversal. Time Scalig. Maipulaios ivolvig he sigal ampliude (depede variable).
More informationCalculus BC 2015 Scoring Guidelines
AP Calculus BC 5 Scorig Guidelies 5 The College Board. College Board, Advaced Placeme Program, AP, AP Ceral, ad he acor logo are regisered rademarks of he College Board. AP Ceral is he official olie home
More informationAppendix A Examples for Labs 1, 2, 3 1. FACTORING POLYNOMIALS
Appedi A Emples for Ls,,. FACTORING POLYNOMIALS Tere re m stdrd metods of fctorig tt ou ve lered i previous courses. You will uild o tese fctorig metods i our preclculus course to ele ou to fctor epressios
More informationName: Period: Date: 2.1 Rules of Exponents
SM NOTES Ne: Period: Dte:.1 Rules of Epoets The followig properties re true for ll rel ubers d b d ll itegers d, provided tht o deoitors re 0 d tht 0 0 is ot cosidered. 1 s epoet: 1 1 1 = e.g.) 7 = 7,
More informationLIMITS OF FUNCTIONS (I)
LIMITS OF FUNCTIO (I ELEMENTARY FUNCTIO: (Elemeary fucios are NOT piecewise fucios Cosa Fucios: f(x k, where k R Polyomials: f(x a + a x + a x + a x + + a x, where a, a,..., a R Raioal Fucios: f(x P (x,
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationMath 153: Lecture Notes For Chapter 1
Mth : Lecture Notes For Chpter Sectio.: Rel Nubers Additio d subtrctios : Se Sigs: Add Eples: = - - = - Diff. Sigs: Subtrct d put the sig of the uber with lrger bsolute vlue Eples: - = - = - Multiplictio
More informationNotes 03 largely plagiarized by %khc
1 1 Discree-Time Covoluio Noes 03 largely plagiarized by %khc Le s begi our discussio of covoluio i discree-ime, sice life is somewha easier i ha domai. We sar wih a sigal x[] ha will be he ipu io our
More informationSection P.1 Notes Page 1 Section P.1 Precalculus and Trigonometry Review
Secion P Noe Pge Secion P Preclculu nd Trigonomer Review ALGEBRA AND PRECALCULUS Eponen Lw: Emple: 8 Emple: Emple: Emple: b b Emple: 9 EXAMPLE: Simplif: nd wrie wi poiive eponen Fir I will flip e frcion
More informationSection 8 Convolution and Deconvolution
APPLICATIONS IN SIGNAL PROCESSING Secio 8 Covoluio ad Decovoluio This docume illusraes several echiques for carryig ou covoluio ad decovoluio i Mahcad. There are several operaors available for hese fucios:
More informationExtremal graph theory II: K t and K t,t
Exremal graph heory II: K ad K, Lecure Graph Theory 06 EPFL Frak de Zeeuw I his lecure, we geeralize he wo mai heorems from he las lecure, from riagles K 3 o complee graphs K, ad from squares K, o complee
More informationON BILATERAL GENERATING FUNCTIONS INVOLVING MODIFIED JACOBI POLYNOMIALS
Jourl of Sciece d Ars Yer 4 No 227-6 24 ORIINAL AER ON BILATERAL ENERATIN FUNCTIONS INVOLVIN MODIFIED JACOBI OLYNOMIALS CHANDRA SEKHAR BERA Muscri received: 424; Acceed er: 3524; ublished olie: 3624 Absrc
More informationMoment Generating Function
1 Mome Geeraig Fucio m h mome m m m E[ ] x f ( x) dx m h ceral mome m m m E[( ) ] ( ) ( x ) f ( x) dx Mome Geeraig Fucio For a real, M () E[ e ] e k x k e p ( x ) discree x k e f ( x) dx coiuous Example
More informationReview Exercises for Chapter 9
0_090R.qd //0 : PM Page 88 88 CHAPTER 9 Ifiie Series I Eercises ad, wrie a epressio for he h erm of he sequece..,., 5, 0,,,, 0,... 7,... I Eercises, mach he sequece wih is graph. [The graphs are labeled
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationExperiment 6: Fourier Series
Fourier Series Experime 6: Fourier Series Theory A Fourier series is ifiie sum of hrmoic fucios (sies d cosies) wih every erm i he series hvig frequecy which is iegrl muliple of some pricipl frequecy d
More informationComparison between Fourier and Corrected Fourier Series Methods
Malaysia Joural of Mahemaical Scieces 7(): 73-8 (13) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/oural Compariso bewee Fourier ad Correced Fourier Series Mehods 1
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationBINOMIAL THEOREM OBJECTIVE PROBLEMS in the expansion of ( 3 +kx ) are equal. Then k =
wwwskshieduciocom BINOMIAL HEOREM OBJEIVE PROBLEMS he coefficies of, i e esio of k e equl he k /7 If e coefficie of, d ems i e i AP, e e vlue of is he coefficies i e,, 7 ems i e esio of e i AP he 7 7 em
More information( ) ( ) ( ) ( ) ( ) ( y )
8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll
More informationLinford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)
Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem
More information12 Getting Started With Fourier Analysis
Commuicaios Egieerig MSc - Prelimiary Readig Geig Sared Wih Fourier Aalysis Fourier aalysis is cocered wih he represeaio of sigals i erms of he sums of sie, cosie or complex oscillaio waveforms. We ll
More informationPhysics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationHow to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
More informationThe solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.
[~ o o :- o o ill] i 1. Mrices, Vecors, nd Guss-Jordn Eliminion 1 x y = = - z= The soluion is ofen represened s vecor: n his exmple, he process of eliminion works very smoohly. We cn elimine ll enries
More informationSTK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationBEST LINEAR FORECASTS VS. BEST POSSIBLE FORECASTS
BEST LINEAR FORECASTS VS. BEST POSSIBLE FORECASTS Opimal ear Forecasig Alhough we have o meioed hem explicily so far i he course, here are geeral saisical priciples for derivig he bes liear forecas, ad
More informationth m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)
1 Trasform Techiques h m m m m mome : E[ ] x f ( x) dx h m m m m ceral mome : E[( ) ] ( ) ( x) f ( x) dx A coveie wa of fidig he momes of a radom variable is he mome geeraig fucio (MGF). Oher rasform echiques
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationOn Absolute Indexed Riesz Summability of Orthogonal Series
Ieriol Jourl of Couiol d Alied Mheics. ISSN 89-4966 Volue 3 Nuer (8). 55-6 eserch Idi Pulicios h:www.riulicio.co O Asolue Ideed iesz Suiliy of Orhogol Series L. D. Je S. K. Piry *. K. Ji 3 d. Sl 4 eserch
More informationSUMMATION OF INFINITE SERIES REVISITED
SUMMATION OF INFINITE SERIES REVISITED I several aricles over he las decade o his web page we have show how o sum cerai iiie series icludig he geomeric series. We wa here o eed his discussio o he geeral
More informationSeptember 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
More informationSampling Example. ( ) δ ( f 1) (1/2)cos(12πt), T 0 = 1
Samplig Example Le x = cos( 4π)cos( π). The fudameal frequecy of cos 4π fudameal frequecy of cos π is Hz. The ( f ) = ( / ) δ ( f 7) + δ ( f + 7) / δ ( f ) + δ ( f + ). ( f ) = ( / 4) δ ( f 8) + δ ( f
More informationTransient Solution of the M/M/C 1 Queue with Additional C 2 Servers for Longer Queues and Balking
Jourl of Mhemics d Sisics 4 (): 2-25, 28 ISSN 549-3644 28 Sciece ublicios Trsie Soluio of he M/M/C Queue wih Addiiol C 2 Servers for Loger Queues d Blkig R. O. Al-Seedy, A. A. El-Sherbiy,,2 S. A. EL-Shehwy
More informationContraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More informationRiemann Integral and Bounded function. Ng Tze Beng
Riem Itegrl d Bouded fuctio. Ng Tze Beg I geerlistio of re uder grph of fuctio, it is ormlly ssumed tht the fuctio uder cosidertio e ouded. For ouded fuctio, the rge of the fuctio is ouded d hece y suset
More informationNext we encountered the exponent equaled 1, so we take a leap of faith and generalize that for any x (that s not zero),
79 CH 0 MORE EXPONENTS Itroductio T his chpter is cotiutio of the epoet ides we ve used m times efore. Our gol is to comie epressios with epoets i them. First, quick review of epoets: 0 0 () () 0 ( ) 0
More informationA LOG IS AN EXPONENT.
Ojeives: n nlze nd inerpre he ehvior of rihmi funions, inluding end ehvior nd smpoes. n solve rihmi equions nlill nd grphill. n grph rihmi funions. n deermine he domin nd rnge of rihmi funions. n deermine
More information( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
More informationAn interesting result about subset sums. Nitu Kitchloo. Lior Pachter. November 27, Abstract
A ieresig resul abou subse sums Niu Kichloo Lior Pacher November 27, 1993 Absrac We cosider he problem of deermiig he umber of subses B f1; 2; : : :; g such ha P b2b b k mod, where k is a residue class
More informationλiv Av = 0 or ( λi Av ) = 0. In order for a vector v to be an eigenvector, it must be in the kernel of λi
Liear lgebra Lecure #9 Noes This week s lecure focuses o wha migh be called he srucural aalysis of liear rasformaios Wha are he irisic properies of a liear rasformaio? re here ay fixed direcios? The discussio
More information1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4)
7 Differeial equaios Review Solve by he mehod of udeermied coefficies ad by he mehod of variaio of parameers (4) y y = si Soluio; we firs solve he homogeeous equaio (4) y y = 4 The correspodig characerisic
More information1. Six acceleration vectors are shown for the car whose velocity vector is directed forward. For each acceleration vector describe in words the
Si ccelerio ecors re show for he cr whose eloci ecor is direced forwrd For ech ccelerio ecor describe i words he iseous moio of he cr A ri eers cured horizol secio of rck speed of 00 km/h d slows dow wih
More informationChapter 2: Evaluative Feedback
Chper 2: Evluive Feedbck Evluing cions vs. insrucing by giving correc cions Pure evluive feedbck depends olly on he cion ken. Pure insrucive feedbck depends no ll on he cion ken. Supervised lerning is
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More informationAlgebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents
Algebr II, Chpter 7 Hrdig Chrter Prep 06-07 Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit
More informationIntroduction to Matrix Algebra
Itrodutio to Mtri Alger George H Olso, Ph D Dotorl Progrm i Edutiol Ledership Applhi Stte Uiversit Septemer Wht is mtri? Dimesios d order of mtri A p q dimesioed mtri is p (rows) q (olums) rr of umers,
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More information5.1-The Initial-Value Problems For Ordinary Differential Equations
5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
More information