ENGR 1990 Engineering Mathematics The Integral of a Function as a Function


 Candace Ramsey
 5 years ago
 Views:
Transcription
1 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 f( ) from o. We now consider he concep of niderivives nd how hey re used o clcule he inegrl of funcion. This ide lso eposes he direc link eween he conceps of differeniion nd inegrion. Aniderivives A funcion G ( ) is clled n niderivive of funcion f( ) over some inervl if nd only if G ( ) is coninuous nd G( ) f ( ) on he inervl. The following le gives niderivives of some common funcions used in engineering. Nme Funcion, ( ) f Aniderivive, G( ) G( ) f ( ) Consn Polynomil erms n n1 n 1 Eponenil e e Sine sin( ) cos( ) Cosine cos( ) sin( ) Noe h he niderivive is no unique, ecuse we cn dd consn o ny known niderivive o produce noher niderivive. Recll h he derivive of consn is zero. Fundmenl Theorem of Inegrl Clculus If f( ) is coninuous funcion over some inervl, nd G ( ) is n niderivive of f( ) on h sme inervl, hen f ( ) d G ( ) d G( ) G( ) G( ) (1) So, if we know he funcion nd n niderivive, hen we cn clcule he inegrl direcly. We do no hve o pproime i y summing res. Kmmn ENGR 1990 Engineering Mhemics pge: 1/6
2 Inegrls of Funcions s Funcions The resul of Eq. (1) is numer h represens he re under funcion f( ) from o. These resuls cn e mde more generl o pply n rirry ending poin of y simply replcing he upper limi of he inegrl wih he vrile. f ( ) d G ( ) d G( ) G( ) G( ) () This resul my e used o clcule he inegrl for sring poin of o ny ending poin. Indefinie Inegrls When we do no specify he inervl over which he inegrl is o e evlued (h is, we do no specify he limis of he inegrl), we cll he inegrl indefinie. Effecively, we re using rirry upper nd rirry lower limis. We wrie, f ( ) d G( ) D (3) Here, D is n rirry consn. If G ( ) is n niderivive of f( ) on he inervl, we cn evlue he resul of Eq. (3) over h inervl. As epeced, we ge he sme resul s in Eq. (1). f ( ) d G( ) D G( ) D G( ) D G( ) G( ) (4) If we use Eq. (3) nd evlue over n rirry upper limi, we ge he sme resul s in Eq. (). f ( ) d G( ) D G( ) D G( ) D G( ) G( ) Kmmn ENGR 1990 Engineering Mhemics pge: /6
3 Emple 1: Given: The displcemen of cr s i moves wih velociy from ime 1 o is he inegrl of over h period of ime. s v() d 1 The displcemen cn e posiive or negive depending on wheher is posiive or negive. Find: Assuming he cr hs velociy v ( ) 7. (f/s ), () find he displcemen of he cr from o seconds; () find he ol disnce rveled from o seconds. Soluion: () An niderivive of v( ) 7. is G( ) , so s v( ) d 7. d (f) () As efore, ecuse is posiive in he rnge, he ol disnce rveled is d s 78.7 (f). Emple : Given: The velociy of ll for cerin period of ime fer i is hrown upwrd is v( ) (f/s) Find: () he vericl displcemen of he ll from 0 o seconds; nd () he ol disnce rveled y he ll from 0 o seconds. 3 Kmmn ENGR 1990 Engineering Mhemics pge: 3/6
4 Soluion: () An niderivive of v( ) is G( ) , so s d (f) () To find he ol disnce rveled, we mus o rek he inervl ino wo segmens: 0 3, nd s d d (f) Emple 3: Given: The velociy of cr over he ime inervl from 0 o seconds is v ( ) (f/s). Find: The disnce rveled y he cr from o seconds. Soluion: Since he funcion is posiive hroughou he enire rnge of, he ol disnce rveled is equl o he displcemen. We cn clcule he displcemen y noing h n niderivive of v( ) G(). So, 3 3 is d s d (f) Kmmn ENGR 1990 Engineering Mhemics pge: 4/6
5 Emple 4: Given: The velociy of ll for cerin period of ime fer i is hrown upwrd is v( ) (f/s) Find: () s (), he vericl displcemen of he ll s funcion of ime; () he ime required for he ll o is sring poin; nd (c) he mimum heigh of he ll. 3 Soluion: () The displcemen funcion relive o he sring poin my e found y inegring o n rirry upper limi. () s( ) v( ) d d In his resul, he displcemen funcion is zero 0. If we wn o specify nonzero displcemen 0, sy s(0) 0, hen we ke slighly differen pproch. Firs, we se s( ) v( ) d D Then, we solve for D o sisfy he iniil condiion. In his cse, D 0. () To find he ime required for he ll o reurn o is sring poin, se s () from Eq. () o zero, nd solve for. s( ) or 0 0 (sec) 6 (sec) (c) Recll h mim or minim of funcions occur when heir derivives re zero. d s d ( ) (16.1 ) or 3 (sec) m s (96.6 3) (f) 3 This corresponds physiclly o he condiion h he ll s velociy e zero. Kmmn ENGR 1990 Engineering Mhemics pge: /6
6 Recll lso h for he funcion o hve mimum, he second derivive should e negive. d s( ) (checks) d Emple : Given: A ll is hrown upwrd wih n iniil velociy of 0 (f/s) from n iniil heigh of 6 (f). The ll hs downwrd ccelerion of 3. (f/s ). v( ) ( ) d s( ) v( ) d Find: () he velociy funcion ; nd () he displcemen funcion s (). Soluion: () Using he indefinie form of inegrion nd he iniil condiion v(0) 0 (f/s) v( ) ( ) d 3.d 3. D v( ) 0 3. (f/s) () Using he indefinie form of inegrion nd he iniil condiion s(0) 6 (f) s( ) v( ) d 0 3. d D s( ) (f) Kmmn ENGR 1990 Engineering Mhemics pge: 6/6
Chapter Direct Method of Interpolation
Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o
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 33 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationMotion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v  vo = Δv Δ ccelerion = = v  vo chnge of velociy elpsed ime Accelerion is vecor, lhough in onedimensionl
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 informationf t f a f x dx By Lin McMullin f x dx= f b f a. 2
Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes
More informationChapter 2. Motion along a straight line. 9/9/2015 Physics 218
Chper Moion long srigh line 9/9/05 Physics 8 Gols for Chper How o describe srigh line moion in erms of displcemen nd erge elociy. The mening of insnneous elociy nd speed. Aerge elociy/insnneous elociy
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 informationAverage & instantaneous velocity and acceleration Motion with constant acceleration
Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission
More informationFM Applications of Integration 1.Centroid of Area
FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is
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 information(b) 10 yr. (b) 13 m. 1.6 m s, m s m s (c) 13.1 s. 32. (a) 20.0 s (b) No, the minimum distance to stop = 1.00 km. 1.
Answers o Een Numbered Problems Chper. () 7 m s, 6 m s (b) 8 5 yr 4.. m ih 6. () 5. m s (b).5 m s (c).5 m s (d) 3.33 m s (e) 8. ().3 min (b) 64 mi..3 h. ().3 s (b) 3 m 4..8 mi wes of he flgpole 6. (b)
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 informationINTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).
INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely
More informationPHYSICS 1210 Exam 1 University of Wyoming 14 February points
PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is opennoe nd closedbook. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher
More informationContraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 4953 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 information3 Motion with constant acceleration: Linear and projectile motion
3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr
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 information1. Consider a PSA initially at rest in the beginning of the lefthand end of a long ISS corridor. Assume xo = 0 on the left end of the ISS corridor.
In Eercise 1, use sndrd recngulr Cresin coordine sysem. Le ime be represened long he horizonl is. Assume ll ccelerions nd decelerions re consn. 1. Consider PSA iniilly res in he beginning of he lefhnd
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss Mes. 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 informationHonours Introductory Maths Course 2011 Integration, Differential and Difference Equations
Honours Inroducory Mhs Course 0 Inegrion, Differenil nd Difference Equions Reding: Ching Chper 4 Noe: These noes do no fully cover he meril in Ching, u re men o supplemen your reding in Ching. Thus fr
More informationMAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017
MAT 66 Clculus for Engineers II Noes on Chper 6 Professor: John Quigg Semeser: spring 7 Secion 6.: Inegrion by prs The Produc Rule is d d f()g() = f()g () + f ()g() Tking indefinie inegrls gives [f()g
More informationMATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)
MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide
More information2D Motion WS. A horizontally launched projectile s initial vertical velocity is zero. Solve the following problems with this information.
Nme D Moion WS The equions of moion h rele o projeciles were discussed in he Projecile Moion Anlsis Acii. ou found h projecile moes wih consn eloci in he horizonl direcion nd consn ccelerion in he ericl
More informationPhysic 231 Lecture 4. Mi it ftd l t. Main points of today s lecture: Example: addition of velocities Trajectories of objects in 2 = =
Mi i fd l Phsic 3 Lecure 4 Min poins of od s lecure: Emple: ddiion of elociies Trjecories of objecs in dimensions: dimensions: g 9.8m/s downwrds ( ) g o g g Emple: A foobll pler runs he pern gien in he
More information5.1The InitialValue Problems For Ordinary Differential Equations
5.The IniilVlue Problems For Ordinry Differenil Equions Consider solving iniilvlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
More informationA 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m
PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl
More informationSolutions to Problems from Chapter 2
Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5
More information1.0 Electrical Systems
. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,
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 informationAn integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples.
Improper Inegrls To his poin we hve only considered inegrls f(x) wih he is of inegrion nd b finie nd he inegrnd f(x) bounded (nd in fc coninuous excep possibly for finiely mny jump disconinuiies) An inegrl
More informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More informationQuestion Details Int Vocab 1 [ ] Question Details Int Vocab 2 [ ]
/3/5 Assignmen Previewer 3 Bsic: Definie Inegrls (67795) Due: Wed Apr 5 5 9: AM MDT Quesion 3 5 6 7 8 9 3 5 6 7 8 9 3 5 6 Insrucions Red ody's Noes nd Lerning Gols. Quesion Deils In Vocb [37897] The chnge
More informationHermiteHadamardFejér type inequalities for convex functions via fractional integrals
Sud. Univ. BeşBolyi Mh. 6(5, No. 3, 355 366 HermieHdmrdFejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie HdmrdFejér inequliy for
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:96377679,735 Emil:hf@scsne.org Commens: 3 ges SubjClss: Funcionl nlsis, comle
More informationMinimum Squared Error
Minimum Squred Error LDF: Minimum SquredError Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > for ll smples y i solve sysem of liner inequliies MSE procedure y i = i for ll smples
More informationMinimum Squared Error
Minimum Squred Error LDF: Minimum SquredError Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > 0 for ll smples y i solve sysem of liner inequliies MSE procedure y i i for ll smples
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 informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 6, 8 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More informationIX.2 THE FOURIER TRANSFORM
Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 7 IX. THE FOURIER TRANSFORM IX.. The Fourier Trnsform Definiion 7 IX.. Properies 73 IX..3 Emples 74 IX..4 Soluion of ODE 76 IX..5
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 information14. The fundamental theorem of the calculus
4. The funmenl heorem of he clculus V 20 00 80 60 40 20 0 0 0.2 0.4 0.6 0.8 v 400 200 0 0 0.2 0.5 0.8 200 400 Figure : () Venriculr volume for subjecs wih cpciies C = 24 ml, C = 20 ml, C = 2 ml n (b) he
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECONDORDER ITERATIVE BOUNDARYVALUE PROBLEM
Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: 072669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECONDORDER ITERATIVE
More informationA Kalman filtering simulation
A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer
More informationForms of Energy. Mass = Energy. Page 1. SPH4U: Introduction to Work. Work & Energy. Particle Physics:
SPH4U: Inroducion o ork ork & Energy ork & Energy Discussion Definiion Do Produc ork of consn force ork/kineic energy heore ork of uliple consn forces Coens One of he os iporn conceps in physics Alernive
More informationCHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES
CHAPTER PARAMETRIC EQUATIONS AND POLAR COORDINATES. PARAMETRIZATIONS OF PLANE CURVES., 9, _ _ Ê.,, Ê or, Ÿ. 5, 7, _ _.,, Ÿ Ÿ Ê Ê 5 Ê ( 5) Ê ˆ Ê 6 Ê ( 5) 7 Ê Ê, Ÿ Ÿ $ 5. cos, sin, Ÿ Ÿ 6. cos ( ), sin (
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 GussJordn 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 informationThe order of reaction is defined as the number of atoms or molecules whose concentration change during the chemical reaction.
www.hechemisryguru.com Re Lw Expression Order of Recion The order of recion is defined s he number of oms or molecules whose concenrion chnge during he chemicl recion. Or The ol number of molecules or
More informationCollision Detection and Bouncing
Collision Deecion nd Bouncing Collisions re Hndled in Two Prs. Deecing he collision Mike Biley mj@cs.oregonse.edu. Hndling he physics of he collision collisionouncing.ppx If You re Lucky, You Cn Deec
More informationPhysics 101 Lecture 4 Motion in 2D and 3D
Phsics 11 Lecure 4 Moion in D nd 3D Dr. Ali ÖVGÜN EMU Phsics Deprmen www.ogun.com Vecor nd is componens The componens re he legs of he righ ringle whose hpoenuse is A A A A A n ( θ ) A Acos( θ) A A A nd
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:56 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More informationPhysics Worksheet Lesson 4: Linear Motion Section: Name:
Physics Workshee Lesson 4: Liner Moion Secion: Nme: 1. Relie Moion:. All moion is. b. is n rbirry coorine sysem wih reference o which he posiion or moion of somehing is escribe or physicl lws re formule.
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 informationIntegral Transform. Definitions. Function Space. Linear Mapping. Integral Transform
Inegrl Trnsform Definiions Funcion Spce funcion spce A funcion spce is liner spce of funcions defined on he sme domins & rnges. Liner Mpping liner mpping Le VF, WF e liner spces over he field F. A mpping
More information3. Renewal Limit Theorems
Virul Lborories > 14. Renewl Processes > 1 2 3 3. Renewl Limi Theorems In he inroducion o renewl processes, we noed h he rrivl ime process nd he couning process re inverses, in sens The rrivl ime process
More informationGreen s Functions and Comparison Theorems for Differential Equations on Measure Chains
Green s Funcions nd Comprison Theorems for Differenil Equions on Mesure Chins Lynn Erbe nd Alln Peerson Deprmen of Mhemics nd Sisics, Universiy of NebrskLincoln Lincoln,NE 685880323 lerbe@@mh.unl.edu
More informationConvergence of Singular Integral Operators in Weighted Lebesgue Spaces
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335347 ISSN 13075543 www.ejpm.com Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue
More informationTEACHING STUDENTS TO PROVE BY USING ONLINE HOMEWORK Buma Abramovitz 1, Miryam Berezina 1, Abraham Berman 2
TEACHING STUDENTS TO PROVE BY USING ONLINE HOMEWORK Bum Armoviz 1, Mirym Berezin 1, Arhm Bermn 1 Deprmen of Mhemics, ORT Brude College, Krmiel, Isrel Deprmen of Mhemics, Technion IIT, Hf, Isrel Asrc We
More informationSome Inequalities variations on a common theme Lecture I, UL 2007
Some Inequliies vriions on common heme Lecure I, UL 2007 Finbrr Hollnd, Deprmen of Mhemics, Universiy College Cork, fhollnd@uccie; July 2, 2007 Three Problems Problem Assume i, b i, c i, i =, 2, 3 re rel
More informationgraph of unit step function t
.5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion"
More informationMotion in a Straight Line
Moion in Srigh Line. Preei reched he mero sion nd found h he esclor ws no working. She wlked up he sionry esclor in ime. On oher dys, if she remins sionry on he moing esclor, hen he esclor kes her up in
More informationOn the PseudoSpectral Method of Solving Linear Ordinary Differential Equations
Journl of Mhemics nd Sisics 5 ():13614, 9 ISS 15493644 9 Science Publicions On he PseudoSpecrl Mehod of Solving Liner Ordinry Differenil Equions B.S. Ogundre Deprmen of Pure nd Applied Mhemics, Universiy
More informationPhys 110. Answers to even numbered problems on Midterm Map
Phys Answers o een numbered problems on Miderm Mp. REASONING The word per indices rio, so.35 mm per dy mens.35 mm/d, which is o be epressed s re in f/cenury. These unis differ from he gien unis in boh
More informationChapter 2 PROBLEM SOLUTIONS
Chper PROBLEM SOLUTIONS. We ssume h you re pproximely m ll nd h he nere impulse rels uniform speed. The elpsed ime is hen Δ x m Δ = m s s. s.3 Disnces reled beween pirs of ciies re ( ) Δx = Δ = 8. km h.5
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 informationMagnetostatics Bar Magnet. Magnetostatics Oersted s Experiment
Mgneosics Br Mgne As fr bck s 4500 yers go, he Chinese discovered h cerin ypes of iron ore could rc ech oher nd cerin mels. Iron filings "mp" of br mgne s field Crefully suspended slivers of his mel were
More information23.5. HalfRange Series. Introduction. Prerequisites. Learning Outcomes
HalfRange Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationPhysics 201, Lecture 5
Phsics 1 Lecue 5 Tod s Topics n Moion in D (Chp 4.14.3): n D Kinemicl Quniies (sec. 4.1) n D Kinemics wih Consn Acceleion (sec. 4.) n D Pojecile (Sec 4.3) n Epeced fom Peiew: n Displcemen eloci cceleion
More informationECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 10. Waveguides Part 7: Transverse Equivalent Network (TEN)
EE 537635 Microwve Engineering Fll 7 Prof. Dvid R. Jcson Dep. of EE Noes Wveguides Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for
More informationM r. d 2. R t a M. Structural Mechanics Section. Exam CT5141 Theory of Elasticity Friday 31 October 2003, 9:00 12:00 hours. Problem 1 (3 points)
Delf Universiy of Technology Fculy of Civil Engineering nd Geosciences Srucurl echnics Secion Wrie your nme nd sudy numer he op righhnd of your work. Exm CT5 Theory of Elsiciy Fridy Ocoer 00, 9:00 :00
More informationAP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationLecture 3: 1D Kinematics. This Week s Announcements: Class Webpage: visit regularly
Lecure 3: 1D Kinemics This Week s Announcemens: Clss Webpge: hp://kesrel.nm.edu/~dmeier/phys121/phys121.hml isi regulrly Our TA is Lorrine Bowmn Week 2 Reding: Chper 2  Gincoli Week 2 Assignmens: Due:
More informationWhat distance must an airliner travel down a runway before reaching
2 LEARNING GALS By sudying his chper, you will lern: How o describe srighline moion in erms of erge elociy, insnneous elociy, erge ccelerion, nd insnneous ccelerion. How o inerpre grphs of posiion ersus
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddhrh Ngr, Nrynvnm Rod 5758 QUESTION BANK (DESCRIPTIVE) Subjec wih Code :Engineering MhemicI (6HS6) Coure & Brnch: B.Tech Com o ll Yer & Sem:
More informationf(x) dx with An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples dx x x 2
Impope Inegls To his poin we hve only consideed inegls f() wih he is of inegion nd b finie nd he inegnd f() bounded (nd in fc coninuous ecep possibly fo finiely mny jump disconinuiies) An inegl hving eihe
More informationName: Per: L o s A l t o s H i g h S c h o o l. Physics Unit 1 Workbook. 1D Kinematics. Mr. Randall Room 705
Nme: Per: L o s A l o s H i g h S c h o o l Physics Uni 1 Workbook 1D Kinemics Mr. Rndll Room 705 Adm.Rndll@ml.ne www.laphysics.com Uni 1  Objecies Te: Physics 6 h Ediion Cunel & Johnson The objecies
More informationCHAPTER 2 KINEMATICS IN ONE DIMENSION ANSWERS TO FOCUS ON CONCEPTS QUESTIONS
Physics h Ediion Cunell Johnson Young Sdler Soluions Mnul Soluions Mnul, Answer keys, Insrucor's Resource Mnul for ll chpers re included. Compleed downlod links: hps://esbnkre.com/downlod/physicshediionsoluionsmnulcunelljohnsonyoungsdler/
More information3D Transformations. Computer Graphics COMP 770 (236) Spring Instructor: Brandon Lloyd 1/26/07 1
D Trnsformions Compuer Grphics COMP 770 (6) Spring 007 Insrucor: Brndon Lloyd /6/07 Geomery Geomeric eniies, such s poins in spce, exis wihou numers. Coordines re nming scheme. The sme poin cn e descried
More information() t. () t r () t or v. ( t) () () ( ) = ( ) or ( ) () () () t or dv () () Section 10.4 Motion in Space: Velocity and Acceleration
Secion 1.4 Moion in Spce: Velociy nd Acceleion We e going o dive lile deepe ino somehing we ve ledy inoduced, nmely () nd (). Discuss wih you neighbo he elionships beween posiion, velociy nd cceleion you
More informationLECTURE 5. is defined by the position vectors r, 1. and. The displacement vector (from P 1 to P 2 ) is defined through r and 1.
LECTURE 5 ] DESCRIPTION OF PARTICLE MOTION IN SPACE The displcemen, veloci nd cceleion in D moion evel hei veco nue (diecion) houh he cuion h one mus p o hei sin. Thei full veco menin ppes when he picle
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08302010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
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 informationSolutions for Nonlinear Partial Differential Equations By TanCot Method
IOSR Journl of Mhemics (IOSRJM) eissn: 78578. Volume 5, Issue 3 (Jn.  Feb. 13), PP 611 Soluions for Nonliner Pril Differenil Equions By TnCo Mehod Mhmood Jwd Abdul Rsool Abu AlSheer Al Rfidin Universiy
More informationBoyce/DiPrima 9 th ed, Ch 6.1: Definition of. Laplace Transform. In this chapter we use the Laplace transform to convert a
Boye/DiPrima 9 h ed, Ch 6.: Definiion of Laplae Transform Elemenary Differenial Equaions and Boundary Value Problems, 9 h ediion, by William E. Boye and Rihard C. DiPrima, 2009 by John Wiley & Sons, In.
More informationPositive and negative solutions of a boundary value problem for a
Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM) ISSN: 24553689 www.ijrem.com Volume 2 Iue 9 ǁ Sepemer 28 ǁ PP 7383 Poiive nd negive oluion of oundry vlue prolem for frcionl, difference
More informationIB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF sshifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationA LIMITPOINT CRITERION FOR A SECONDORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES
A LIMITPOINT CRITERION FOR A SECONDORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is relvlued nd loclly
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 9: The High Beta Tokamak
.65, MHD Theory of Fusion Sysems Prof. Freidberg Lecure 9: The High e Tokmk Summry of he Properies of n Ohmic Tokmk. Advnges:. good euilibrium (smll shif) b. good sbiliy ( ) c. good confinemen ( τ nr )
More informationS Radio transmission and network access Exercise 12
S7.330 Rdio rnsmission nd nework ccess Exercise 1  P1 In foursymbol digil sysem wih eqully probble symbols he pulses in he figure re used in rnsmission over AWGNchnnel. s () s () s () s () 1 3 4 )
More informationANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2
ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise : Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen
More informationAnswers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
More informationFractional Calculus. Connor Wiegand. 6 th June 2017
Frcionl Clculus Connor Wiegnd 6 h June 217 Absrc This pper ims o give he reder comforble inroducion o Frcionl Clculus. Frcionl Derivives nd Inegrls re defined in muliple wys nd hen conneced o ech oher
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationLAPLACE TRANSFORMS. 1. Basic transforms
LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming
More informationON THE OSTROWSKIGRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS
Hceepe Journl of Mhemics nd Sisics Volume 45) 0), 65 655 ON THE OSTROWSKIGRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS M Emin Özdemir, Ahme Ock Akdemir nd Erhn Se Received 6:06:0 : Acceped
More informationSph3u Practice Unit Test: Kinematics (Solutions) LoRusso
Sph3u Prcice Uni Te: Kinemic (Soluion) LoRuo Nme: Tuey, Ocober 3, 07 Ku: /45 pp: /0 T&I: / Com: Thi i copy of uni e from 008. Thi will be imilr o he uni e you will be wriing nex Mony. you cn ee here re
More informationChapter 2 : Fourier Series. Chapter 3 : Fourier Series
Chaper 2 : Fourier Series.0 Inroducion Fourier Series : represenaion of periodic signals as weighed sums of harmonically relaed frequencies. If a signal x() is periodic signal, hen x() can be represened
More informationECE Microwave Engineering
EE 537635 Microwve Engineering Adped from noes y Prof. Jeffery T. Willims Fll 8 Prof. Dvid R. Jcson Dep. of EE Noes Wveguiding Srucures Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II  0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMAprocesses. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More information