Introduction to Finite Elements in Engineering. Tirupathi R. Chandrupatla Rowan University Glassboro, New Jersey
|
|
- Sara Hines
- 6 years ago
- Views:
Transcription
1 Introduction to Finite lements in ngineering Tirupthi R. Chndruptl Rown Universit Glssboro, New Jerse Ashok D. Belegundu The Pennslvni Stte Universit Universit Prk, Pennslvni Solutions Mnul Prentice Hll, Upper Sddle River, New Jerse 758 Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
2 CONTNTS Prefce Chpter Fundmentl Concepts Chpter Mtri Algebr nd Gussin limintion 8 Chpter One-Dimensionl Problems 6 Chpter Trusses 6 Chpter 5 Bems nd Frmes 86 Chpter 6 Two-Dimensionl Problems Using Constnt Strin Tringles Chpter 7 Aismmetric Solids Subjected to 5 Aismmetric Loding Chpter 8 Two-Dimensionl Isoprmetric lements 8 nd Numericl Integrtion Chpter 9 Three-Dimensionl Problems in Stress 7 Anlsis Chpter Sclr Field Problems 8 Chpter Dnmic Considertions 6 Chpter Preprocessing nd Postprocessing 8 Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
3 PRFAC This solutions mnul serves s n id to professors in teching from the book Introduction to Finite lements in ngineering, th dition. The problems in the book fll into the following ctegories:. Simple problems to understnd the concepts. Derivtions nd direct solutions. Solutions requiring computer runs. Solutions requiring progrm modifictions Our bsic philosoph in the development of this mnul is to provide complete guidnce to the techer in formulting, modeling, nd solving the problems. Complete solutions re given for problems in ll ctegories stted. For some lrger problems such s those in three dimensionl stress nlsis, complete formultion nd modeling spects re discussed. The students should be ble to proceed from the guidelines provided. For problems involving distributed nd other tpes of loding, the nodl lods re to be clculted for the input dt. The progrms do not generte the lods. This clcultion nd the boundr condition decisions enble the student to develop phsicl sense for the problems. The students m be encourged to modif the progrms to clculte the lods utomticll. The students should be introduced to the progrms in Chpter right from the point of solving problems in Chpter 6. This will enble the students to solve lrger problems with ese. The input dt file for ech progrm hs been provided. Dt for problem should follow this formt. The best strteg is to cop the emple file nd edit it for the problem under considertion. The dt from progrm MSHGN will need some editing to complete the informtion on boundr conditions, lods, nd mteril properties. We thnk ou for our enthusistic response to our first three editions of the book. We look forwrd to receive our feedbck of our eperiences, comments, nd suggestions for mking improvements to the book nd this mnul. Tirupthi R. Chndruptl P.., CMfg Deprtment of Mechnicl ngineering Rown Universit, Glssboro, NJ 88 e-mil: chndruptl@rown.edu Ashok D. Belegundu Deprtment of Mechnicl nd Nucler ngineering The Pennslvni Stte Universit Universit Prk, PA 68 e-mil: db@psu.edu Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
4 CHAPTR FUNDAMNTAL CONCPTS. We use the first three steps of q.. Adding the bove, we get ν ν ν ν ν ν Adding nd subtrcting ( ) ν ν from the first eqution, ( ) ν ν Similr epressions cn be obtined for, nd. From the reltionship for γ nd q.., τ γ etc. ( ν) Above reltions cn be written in the form D where D is the mteril propert mtri defined in q..5.. Note tht u () stisfies the ero slope boundr condition t the support. Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
5 . Plne strin condition implies tht ν ν which gives ν ( ) 6 We hve, psi psi psi ν. On substituting the vlues,. psi. Displcement field u v u v u t, ( 6) ( 6 ) ( 6) ( 6) u v v 6 9 u v ( 6 ).5 On inspection, we note tht the displcements u nd v re given b It is then es to see tht u. v Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
6 γ u v.6 The displcement field is given s u v. u 6 v 7 () The strins re then given b u 6 v u v γ (b) In order to drw the contours of the strin field using MATLAB, we need to crete script file, which m be edited s tet file nd sve with.m etension. The file for plotting is given below file probp5b.m [X,Y] meshgrid(-:.:,-:.:); Z..*X.^6.*Y.^; [C,h] contour(x,y,z); clbel(c,h); On running the progrm, the contour mp is shown s follows: Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
7 Contours of Contours of nd γ re obtined b chnging Z in the script file. The numbers on the contours show the function vlues. (c) The mimum vlue of is t n of the corners of the squre region. The mimum vlue is..7 (, ) (u, v) ). u u. v u v u v b) γ. Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
8 .8. τ MP MP n From q..8 we get T T T n 5.67 MP τ 6. MP τ.7 MP T n n n n τ τ τ.9 MP T n n n n MP 5 MP T τ τ T n.9 From the derivtion mde in P., we hve n n n τ MP MP ( )( ) ( ν ) ν ν [ ν ν ] which cn be written in the form ( )( ) [( ν ) ] ν v ν ν nd τ γ ( ν) Lme s constnts λ nd µ re defined in the epressions τ λ λ µγ ( ν)( ν) µ ν ( ν) µ On inspection, v Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
9 µ is sme s the sher modulus G.... T GP α α T δ C du d L L 5-6 /.6 ( ) 69.6 MP du d C d L L. Following the steps of mple., we hve ( 8 5) 8 8 q 8 q 6 5 Above mtri form is sme s the set of equtions: 7 q 8 q 6 8 q 8 q 5 Solving for q nd q, we get q. mm q.87 mm Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
10 . When the wll is smooth,. T is the temperture rise. ) When the block is thin in the direction, it corresponds to plne stress condition. The rigid wlls in the direction require. The generlied Hooke s lw ields the equtions ν α T α T From the second eqution, setting, we get α T. is then clculted using the first eqution s ( ν) α T. b) When the block is ver thick in the direction, plin strin condition previls. Now we hve, in ddition to. is not ero. ν ν α T ν α T ν α T From the lst two equtions, we get α T ν α T ν ν is now obtined from the first eqution. Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
11 . For thin block, it is plne stress condition. Treting the nominl sie s, we m set the. initil strin α T in prt () of problem.. Thus...5 The potentil energ Π is given b Π du d A d ugad Consider the polnomil from mple., u du d ( ) ( ) ( ) g A On substituting the bove epressions nd integrting, the first term of becomes nd the second term Thus ugad ud ( ) Π Π u this gives ( ). 5.6 A f Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
12 We use the displcement field defined b u. u t u t We then hve u ( ), nd du/d ( ). The potentil energ is now written s Π 6 du d ( ) d ( ) 5 ( ) d ( ) 5 6 d fud d d Π This ields,. Displcemen u.( ) Stress du/d.( ).7 Let u be the displcement t mm. Piecewise liner displcement tht is continuous in the intervl 5 is represented s shown in the figure. u u u 5 Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
13 u t u u t u / u (u /) du/d u / 5 u t 5 5 u u t u u / (5/)u u (5/)u (u /) du/d u / Π l du A d d u u Π l A st A l A st A u u 5 st du A d d u u Π u A A u l st Note tht using the units MP (N/mm ) for modulus of elsticit nd mm for re nd mm for length will result in displcement in mm, nd stress in MP. Thus, l 7 MP, st, nd A 9 mm, A mm. On substituting these vlues into the bove eqution, we get u.9 mm This is precisel the solution obtined from strength of mterils pproch.8 In the Glerkin method, we strt from the equilibrium eqution d du A g d d Following the steps of mple., we get Introducing du A d dφ d d gφd Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
14 ( ) ( ) nd, φ φ u u where u nd φ re the vlues of u nd φ t respectivel, ( ) ( ) φ d d u On integrting, we get 8 φ u This is to be stisfied for ever φ, which gives the solution u.5.9 We use t t u u u This implies tht 8 nd ( ) ( ) ( ) ( ) d du u nd re considered s independent vribles in ( ) ( ) [ ] ( ) Π d on epnding nd integrting the terms, we get Π We differentite with respect to the vribles nd equte to ero. Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
15 Π Π On solving, we get.7856 nd.5. On substituting in the epression for u, t, u.79 This pproimtion is close to the vlue obtined in the emple problem.. () Π L L T Ad T ( )ud nd On substitution, Π Π 6 du A d du d d du ( 6 ) d ud ud d Tud 6 Tud (b) Since u t nd 6, nd u, we hve u du d ( 6) ( 6) On substituting nd integrting, Π Setting dπ/d gives Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
16 .5 du 6.95( 6) d Plots of displcement nd stress re given below: Displcement u Stress.. t 6 implies tht 6 ( 8 ) 6, which ields Substituting for k, h, L, nd in I, we get I I I ( ) d ( 5) [ ( 8 ) 8] ( ) d 5( 8 9) Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
17 di d di d On solving, Substituting into the epression for, we get Since u t, the displcement stisfing the boundr condition is u. Also the coordintes re, nd. The potentil energ for the problem is du π A d P u Pu d du We hve u, u,, A, nd d. Thus π ( ) d. dπ For sttionr vlue, setting, we get d, which gives.75. The pproimte solution is u.75. Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
18 . Use Glerkin pproch with pproimtion u b c to solve du u d u ( ) The week form is obtined b multipling b φ stisfing φ ( ). du φ u d d We now set u b c into the bove integrl, stisfing u ( ) nd φ. On introducing these ( )( b c b c ) d ( ) ( ) b b c c d b b c c d On integrting, we get b c c b b c c b b c b c 6 This must be stisfied for ever nd. Thus the equtions to be solved re 7 7 b c 6 b c The solution is b.957, c.8. Thus u Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
19 P The deflection nd slope t due to P re nd I nd slope t L due to lod P re ( ) P L P v I I P v I The deflection nd slope due to lod P re PL v I PL v I We then get v v v v v v P I. Using this the deflection.5 (,) (,) (,) (,) () The displcement of B is given b (.,.) nd A, C, nd D remin in their originl position. Consider displcement field of the tpe u v b b b b The four constnts cn be evluted using the known displcements Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
20 At A (, ) At B (, ) At C (, ) At D (, ) b. b b. b b b b b b The solution is,.,,. b, b., b, b u.. This gives v.. (b) The sher strin t B is u v γ... B ( ) ( ) γ.... Introduction to Finite lements in ngineering, Fourth dition, b T. R. Chndruptl nd A. D. Belegundu. ISBN Person duction, Inc., Upper Sddle River, NJ. All rights reserved. This publiction is protected b Copright nd written permission should be obtined from the publisher prior to n prohibited reproduction, storge in retrievl sstem, or trnsmission in n form or b n mens, electronic, mechnicl, photocoping, recording, or likewise. For informtion regrding permission(s), write to: Rights nd Permissions Deprtment, Person duction, Inc., Upper Sddle River, NJ 758.
Introduction to Finite Elements in Engineering. Tirupathi R. Chandrupatla Rowan University Glassboro, New Jersey
Introduction to Finite Elements in Engineering Tirupthi R. Chndruptl Rown Universit Glssboro, New Jerse Ashok D. Belegundu The Pennslvni Stte Universit Universit Prk, Pennslvni Solutions Mnul Prentice
More informationA - INTRODUCTION AND OVERVIEW
MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS A - INTRODUCTION AND OVERVIEW INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Course Content: A INTRODUCTION AND
More informationM344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
More informationStress distribution in elastic isotropic semi-space with concentrated vertical force
Bulgrin Chemicl Communictions Volume Specil Issue pp. 4 9 Stress distribution in elstic isotropic semi-spce with concentrted verticl force L. B. Petrov Deprtment of Mechnics Todor Kbleshkov Universit of
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationCalculus - Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
More informationPlate Theory. Section 11: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEMENTS Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s the thickness of the plte. A
More informationEquations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces
Equtions of Motion In Prt I, lnce of forces nd moments cting on n component ws enforced in order to ensure tht the component ws in equilirium. Here, llownce is mde for stresses which vr continuousl throughout
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More informationPlate Theory. Section 13: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s
More informationNow, given the derivative, can we find the function back? Can we antidifferenitate it?
Fundmentl Theorem of Clculus. Prt I Connection between integrtion nd differentition. Tody we will discuss reltionship between two mjor concepts of Clculus: integrtion nd differentition. We will show tht
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationMath 113 Exam 1-Review
Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
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 information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More informationChapter Direct Method of Interpolation More Examples Electrical Engineering
Chpter. Direct Method of Interpoltion More Emples Electricl Engineering Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More information4.6 Numerical Integration
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
More informationu t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationThe Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11
The Islmic University of Gz Fculty of Engineering Civil Engineering Deprtment Numericl Anlysis ECIV 6 Chpter Specil Mtrices nd Guss-Siedel Associte Prof Mzen Abultyef Civil Engineering Deprtment, The Islmic
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
More informationHOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016
HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M
More informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationMATH , Calculus 2, Fall 2018
MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationSome Methods in the Calculus of Variations
CHAPTER 6 Some Methods in the Clculus of Vritions 6-. If we use the vried function ( α, ) α sin( ) + () Then d α cos ( ) () d Thus, the totl length of the pth is d S + d d α cos ( ) + α cos ( ) d Setting
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationModule 1. Energy Methods in Structural Analysis
Module 1 Energy Methods in Structurl Anlysis Lesson 4 Theorem of Lest Work Instructionl Objectives After reding this lesson, the reder will be ble to: 1. Stte nd prove theorem of Lest Work.. Anlyse stticlly
More informationPhysics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15
Physics H - Introductory Quntum Physics I Homework #8 - Solutions Fll 4 Due 5:1 PM, Mondy 4/11/15 [55 points totl] Journl questions. Briefly shre your thoughts on the following questions: Of the mteril
More informationFinal Exam - Review MATH Spring 2017
Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More information1 Bending of a beam with a rectangular section
1 Bending of bem with rectngulr section x3 Episseur b M x 2 x x 1 2h M Figure 1 : Geometry of the bem nd pplied lod The bem in figure 1 hs rectngur section (thickness 2h, width b. The pplied lod is pure
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More information6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection
More informationTable of Contents. 1. Limits The Formal Definition of a Limit The Squeeze Theorem Area of a Circle
Tble of Contents INTRODUCTION 5 CROSS REFERENCE TABLE 13 1. Limits 1 1.1 The Forml Definition of Limit 1. The Squeeze Theorem 34 1.3 Are of Circle 43. Derivtives 53.1 Eploring Tngent Lines 54. Men Vlue
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More informationElectromagnetics P5-1. 1) Physical quantities in EM could be scalar (charge, current, energy) or vector (EM fields).
Electromgnetics 5- Lesson 5 Vector nlsis Introduction ) hsicl quntities in EM could be sclr (chrge current energ) or ector (EM fields) ) Specifing ector in -D spce requires three numbers depending on the
More informationPlates on elastic foundation
Pltes on elstic foundtion Circulr elstic plte, xil-symmetric lod, Winkler soil (fter Timoshenko & Woinowsky-Krieger (1959) - Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016) Introduction Winkler
More informationSECTION 9-4 Translation of Axes
9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationThe Trapezoidal Rule
SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion Approimte
More informationKirchhoff and Mindlin Plates
Kirchhoff nd Mindlin Pltes A plte significntly longer in two directions compred with the third, nd it crries lod perpendiculr to tht plne. The theory for pltes cn be regrded s n extension of bem theory,
More informationQuotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m
Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble
More informationINTERFACE DESIGN OF CORD-RUBBER COMPOSITES
8 TH INTENATIONAL CONFEENCE ON COMPOSITE MATEIALS INTEFACE DESIGN OF COD-UBBE COMPOSITES Z. Xie*, H. Du, Y. Weng, X. Li Ntionl Ke Lbortor of Science nd Technolog on Advnced Composites in Specil Environment,
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationES.182A Topic 30 Notes Jeremy Orloff
ES82A opic 3 Notes Jerem Orloff 3 Non-independent vribles: chin rule Recll the chin rule: If w = f, ; nd = r, t, = r, t then = + r t r t r t = + t t t r nfortuntel, sometimes there re more complicted reltions
More informationPrecalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.
Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationReference. Vector Analysis Chapter 2
Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter
More informationBME 207 Introduction to Biomechanics Spring 2018
April 6, 28 UNIVERSITY O RHODE ISAND Deprtment of Electricl, Computer nd Biomedicl Engineering BME 27 Introduction to Biomechnics Spring 28 Homework 8 Prolem 14.6 in the textook. In ddition to prts -e,
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationStudent Handbook for MATH 3300
Student Hndbook for MATH 3300 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 0.5 0 0.5 0.5 0 0.5 If people do not believe tht mthemtics is simple, it is only becuse they do not relize how complicted life is. John Louis
More informationA-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)
A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationx ) dx dx x sec x over the interval (, ).
Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationCHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS
CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6. VOLUMES USING CROSS-SECTIONS. A() ;, ; (digonl) ˆ Ȉ È V A() d d c d 6 (dimeter) c d c d c ˆ 6. A() ;, ; V A() d d. A() (edge) È Š È Š È ;, ; V A() d d 8
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationAPPM 1360 Exam 2 Spring 2016
APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationObjectives. Materials
Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 8 The Force Method of Anlysis: Bems Version CE IIT, Khrgpur Instructionl Objectives After reding
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More information