Integration by Parts
|
|
- Suzan Stanley
- 6 years ago
- Views:
Transcription
1 Intgration by Parts Intgration by parts is a tchniqu primarily for valuating intgrals whos intgrand is th product of two functions whr substitution dosn t work. For ampl, sin d or d. Th rul is: u ( ) v'( ) d u( ) v( ) v( ) u '( ) d Not: With d v v'( ) d, and d u u '( ) d, th rul is also writtn mor compactly as u dv u v v du MATH 180 Lctur 8 1 of 16 Ronald Brnt 018 All rights rsrvd.
2 This coms from th product rul: ( u( ) v( ) ) u '( ) v( ) u( ) v'( ) D Intgrating both sids of Eq. with rspct to givs Or ( u( ) v( ) ) d ( u '( ) v( ) u( ) v'( ) D d ) u ( ) v( ) v( ) u '( ) d u( ) v'( ) d which can b writtn as ( ) v'( ) d u( ) v( ) u v( ) u '( ) d MATH 180 Lctur 8 of 16 Ronald Brnt 018 All rights rsrvd.
3 Evaluat d. Choos u and dv d thn u dv, thn d u d and u v v du v. d d C Evaluat cos d. Choos u and dv cos d, thn du d and v sin. thn u dv u v v du cos d sin sin (d) sin cos C MATH 180 Lctur 8 of 16 Ronald Brnt 018 All rights rsrvd.
4 Rpatd intgration-by-parts. Using this mthod on an intgral lik can gt prtty tdious. Choos u and dv d, thn du d and v. W gt d d d Th original intgral is rducd to a diffrnc of two trms. Th rsulting intgral (on th right) must also b handld by intgration by parts, but th dgr of th monomial has bn knockd down by 1. Rpating th procss with dv d givs u and d d { d } 1 MATH 180 Lctur 8 of 16 Ronald Brnt 018 All rights rsrvd. 1 Rpating this procss two mor tims rducs th intgral to d
5 d 1 d (CHECK THIS!!) Th last intgral is wll known so that d 1 C Notic what happnd. W startd out with a fourth dgr monomial tims an ponntial. Using intgration-by-parts tims taks drivativs of th original function u rducing th intgral to on asily don. Thr is a bttr way to handl ths typs of intgrals. Namly intgrals of th form n f ( ) d, whr n is any positiv intgr and f () is an asily intgrabl ponntial or trigonomtric function. MATH 180 Lctur 8 5 of 16 Ronald Brnt 018 All rights rsrvd.
6 Construct a tabl that looks lik th following: u dv MATH 180 Lctur 8 6 of 16 Ronald Brnt 018 All rights rsrvd.
7 Now in th column undr u continually tak drivativs until you gt to 0. In th column undr dv kp on intgrating th function (which always givs you hr.) W gt th following tabl: u dv 1 0 Now taking th products shown abov givs th prvious rsult d 1 C. MATH 180 Lctur 8 7 of 16 Ronald Brnt 018 All rights rsrvd.
8 As anothr ampl considr d. w construct th following tabl u dv 0 and dduc that d ( ) C C MATH 180 Lctur 8 8 of 16 Ronald Brnt 018 All rights rsrvd.
9 cos( / ) d As anothr ampl considr. w construct th following tabl u dv cos( / ) sin ( / ) 1 cos( / ) 8sin ( / ) 16cos( / ) 0 sin ( / ) and dduc that cos( / ) d sin ( / ) cos( / ) sin ( / ) 8cos( / ) 768sin ( / ) C (Chck This!) MATH 180 Lctur 8 9 of 16 Ronald Brnt 018 All rights rsrvd.
10 As anothr ampl considr d. w construct th following tabl u dv and dduc that d MATH 180 Lctur 8 10 of 16 Ronald Brnt 018 All rights rsrvd ( ) C C
11 As anothr ampl considr d. w construct th following tabl u dv and dduc that d ( ) C C MATH 180 Lctur 8 11 of 16 Ronald Brnt 018 All rights rsrvd.
12 Intgrands that nvr go away. Evaluat Lt thn cos d., u cos dv d du sin d and th intgration-by-parts formula givs v, cos d cos sin d Rpating th procss, for th intgral on th right, now with dv d, u sin du cos d v MATH 180 Lctur 8 1 of 16 Ronald Brnt 018 All rights rsrvd.
13 cos d cos sin cos d Add th last intgral to th LHS. cos d cos d cos sin or 5 cos d cos sin Divid by 5 cos cos d 5 sin C MATH 180 Lctur 8 1 of 16 Ronald Brnt 018 All rights rsrvd.
14 Intgrands that hav only on function. Whn th intgrand has only on function, just rmmbr that you can always choos dv d. Hr choos in which cas Equation givs ln d u ln dv d, du ln 1 d v. d ln d ln C MATH 180 Lctur 8 1 of 16 Ronald Brnt 018 All rights rsrvd.
15 Hr choos in which cas Equation givs u ln ln d 1 dv du ln 1 d, v. d ln 1 d ln 9 C MATH 180 Lctur 8 15 of 16 Ronald Brnt 018 All rights rsrvd.
16 Choic of u and dv. Us th following rul-of-thumb: (IT DOESN T ALWAYS WORK!) u dv L I A T E Whr L Logarithmic Functions I Invrs Functions (invrs trig, or hyprbolic) A Algbraic Functions (Polynomial, rational, root functions tc.) T Trigonomtric Functions (Primarily sin and cosin.) E Eponntial Functions MATH 180 Lctur 8 16 of 16 Ronald Brnt 018 All rights rsrvd.
INTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More information6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.
6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b
More information3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d www.clas.ucsb.du/staff/vinc
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationPROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS
Intrnational Journal Of Advanc Rsarch In Scinc And Enginring http://www.ijars.com IJARSE, Vol. No., Issu No., Fbruary, 013 ISSN-319-8354(E) PROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS 1 Dharmndra
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationdx equation it is called a second order differential equation.
TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More informationChapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.
Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation
More informationReview of Exponentials and Logarithms - Classwork
Rviw of Eponntials and Logarithms - Classwork In our stud of calculus, w hav amind drivativs and intgrals of polnomial prssions, rational prssions, and trignomtric prssions. What w hav not amind ar ponntial
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationWhere k is either given or determined from the data and c is an arbitrary constant.
Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is
More information4 x 4, and. where x is Town Square
Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and
More informationMath 120 Answers for Homework 14
Math 0 Answrs for Homwork. Substitutions u = du = d d = du a d = du = du = u du = u + C = u = arctany du = +y dy dy = + y du b arctany arctany dy = + y du = + y + y arctany du = u du = u + C = arctan y
More informationExponential Functions
Eponntial Functions Dinition: An Eponntial Function is an unction tat as t orm a, wr a > 0. T numbr a is calld t bas. Eampl: Lt i.. at intgrs. It is clar wat t unction mans or som valus o. 0 0,,, 8,,.,.
More information1 General boundary conditions in diffusion
Gnral boundary conditions in diffusion Πρόβλημα 4.8 : Δίνεται μονοδιάτατη πλάκα πάχους, που το ένα άκρο της κρατιέται ε θερμοκραία T t και το άλλο ε θερμοκραία T 2 t. Αν η αρχική θερμοκραία της πλάκας
More informationAP Calculus BC Problem Drill 16: Indeterminate Forms, L Hopital s Rule, & Improper Intergals
AP Calulus BC Problm Drill 6: Indtrminat Forms, L Hopital s Rul, & Impropr Intrgals Qustion No. of Instrutions: () Rad th problm and answr hois arfully () Work th problms on papr as ndd () Pik th answr
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationEngineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12
Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th
More informationThings I Should Know Before I Get to Calculus Class
Things I Should Know Bfor I Gt to Calculus Class Quadratic Formula = b± b 4ac a sin + cos = + tan = sc + cot = csc sin( ± y ) = sin cos y ± cos sin y cos( + y ) = cos cos y sin sin y cos( y ) = cos cos
More informationPHA 5127 Answers Homework 2 Fall 2001
PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring
More informationCalculus Revision A2 Level
alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationFor more important questions visit :
For mor important qustions visit : www4onocom CHAPTER 5 CONTINUITY AND DIFFERENTIATION POINTS TO REMEMBER A function f() is said to b continuous at = c iff lim f f c c i, lim f lim f f c c c f() is continuous
More informationAs the matrix of operator B is Hermitian so its eigenvalues must be real. It only remains to diagonalize the minor M 11 of matrix B.
7636S ADVANCED QUANTUM MECHANICS Solutions Spring. Considr a thr dimnsional kt spac. If a crtain st of orthonormal kts, say, and 3 ar usd as th bas kts, thn th oprators A and B ar rprsntd by a b A a and
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationnd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1).
Eamn EDO. Givn th family of curvs y + C nd th particular orthogonal trajctory from th family of orthogonal trajctoris passing through point (0; ). Solution: In th rst plac, lt us calculat th di rntial
More informationIntegral Calculus What is integral calculus?
Intgral Calulus What is intgral alulus? In diffrntial alulus w diffrntiat a funtion to obtain anothr funtion alld drivativ. Intgral alulus is onrnd with th opposit pross. Rvrsing th pross of diffrntiation
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationDSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer
DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016 LECTURE
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More information( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationVTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationHow would you do the following integral? dx.
How would you do th following intgral? d 7 Wll of cours today you would us on of th modrn computr packags, such as Mathmatica or Mapl or Wolfram, that do calculus and algbraic manipulations for you Som
More informationSouthern Taiwan University
Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution:
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationTOPIC 5: INTEGRATION
TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationHyperbolic Functions Mixed Exercise 6
Hyprbolic Functions Mid Ercis 6 a b c ln ln sinh(ln ) ln ln + cosh(ln ) + ln tanh ln ln + ( 6 ) ( + ) 6 7 ln ln ln,and ln ln ln,and ln ln 6 artanh artanhy + + y ln ln y + y ln + y + y y ln + y y + y y
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationDISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P
DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,
More informationCalculus II Solutions review final problems
Calculus II Solutions rviw final problms MTH 5 Dcmbr 9, 007. B abl to utiliz all tchniqus of intgration to solv both dfinit and indfinit intgrals. Hr ar som intgrals for practic. Good luck stuing!!! (a)
More informationText: WMM, Chapter 5. Sections , ,
Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationMath-3. Lesson 5-6 Euler s Number e Logarithmic and Exponential Modeling (Newton s Law of Cooling)
Math-3 Lsson 5-6 Eulr s Numbr Logarithmic and Eponntial Modling (Nwton s Law of Cooling) f ( ) What is th numbr? is th horizontal asymptot of th function: 1 1 ~ 2.718... On my 3rd submarin (USS Springfild,
More informationLie Groups HW7. Wang Shuai. November 2015
Li roups HW7 Wang Shuai Novmbr 015 1 Lt (π, V b a complx rprsntation of a compact group, show that V has an invariant non-dgnratd Hrmitian form. For any givn Hrmitian form on V, (for xampl (u, v = i u
More informationINTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS
adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationLogarithms. Secondary Mathematics 3 Page 164 Jordan School District
Logarithms Sondary Mathmatis Pag 6 Jordan Shool Distrit Unit Clustr 6 (F.LE. and F.BF.): Logarithms Clustr 6: Logarithms.6 For ponntial modls, prss as a arithm th solution to a and d ar numrs and th as
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by:
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationComplex Powers and Logs (5A) Young Won Lim 10/17/13
Complx Powrs and Logs (5A) Copyright (c) 202, 203 Young W. Lim. Prmission is grantd to copy, distribut and/or modify this documnt undr th trms of th GNU Fr Documntation Licns, Vrsion.2 or any latr vrsion
More informationChapter two Functions
Chaptr two Functions -- Eponntial Logarithm functions Eponntial functions If a is a positiv numbr is an numbr, w dfin th ponntial function as = a with domain - < < ang > Th proprtis of th ponntial functions
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More informationLegendre Wavelets for Systems of Fredholm Integral Equations of the Second Kind
World Applid Scincs Journal 9 (9): 8-, ISSN 88-495 IDOSI Publications, Lgndr Wavlts for Systs of Frdhol Intgral Equations of th Scond Kind a,b tb (t)= a, a,b a R, a. J. Biazar and H. Ebrahii Dpartnt of
More informationAS 5850 Finite Element Analysis
AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationElectromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology
Elctromagntic scattring Graduat Cours Elctrical Enginring (Communications) 1 st Smstr, 1388-1389 Sharif Univrsity of Tchnology Contnts of lctur 8 Contnts of lctur 8: Scattring from small dilctric objcts
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationInstructions for Section 1
Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationINTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)
Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationGrade 12 (MCV4UE) AP Calculus Page 1 of 5 Derivative of a Function & Differentiability
Gra (MCV4UE) AP Calculus Pag of 5 Drivativ of a Function & Diffrntiabilit Th Drivativ at a Point f ( a h) f ( a) Rcall, lim provis th slop of h0 h th tangnt to th graph f ( at th point a, f ( a), an th
More information