CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2414 CALCULUS II. Semester Hours Credit: 4
|
|
|
- Annabel Elinor Murphy
- 5 years ago
- Views:
Transcription
1 CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2414 CALCULUS II Smstr Hours Crdit: 4 I. INTRODUCTION A. Calculus II is a continuation of Calculus I. This cours mphasizs diffrntial quations; applications of intgration; paramtric quations and polar coordinats; tchniqus of intgration; squncs and sris; and impropr intgrals. B. This is a rquird cours for an Associat in Scinc dgr with a major in Mathmatics or Enginring. C. Prrquisit: A grad of C or highr in Math II. LEARNING OUTCOMES Upon succssful compltion of this cours, Calculus II, th studnt will: A. Us th concpts of slop filds, Eulr s Mthod, and sparation of variabls to solv diffrntial quations. (F2, F3, F4, F9) B. Us th concpts of dfinit intgrals to solv problms involving ara, volum, work and othr physical applications. (F2, F3, F4, F9) B. Us substitution, intgration by parts, trigonomtric substitution, partial fractions, and tabls of anti-drivativs to valuat dfinit and indfinit intgrals. (F2, F3, F4, F9) C. Dfin an impropr intgral. (F2) D. Apply th concpts of limits, convrgnc, and divrgnc to valuat som classs of impropr intgrals. (F2, F3, F4, F9) E. Dtrmin convrgnc or divrgnc of squncs and sris. F. Us Taylor and MacLaurin sris to rprsnt functions. (F2, F3, F4, F9) G. Us Taylor or MacLaurin sris to intgrat functions not intgrabl by convntional mthods. (F2, F3, F4, F9) H. Us th concpt of polar coordinats to find aras, lngths of curvs, and rprsntations of conic sctions. (F2, F3, F4, F9) I. Us lctronic and othr mdia, such as th computr and DVD, to rinforc and supplmnt th larning procss. (F1, F2, F3, F4, F6) III. INSTRUCTIONAL MATERIAL Th Instructional matrials idntifid for this cours ar viwabl through
2 IV. COURSE REQUIREMENTS A. Assignmnts ar givn in WbAssign and ar du as schduld by your instructor. Th instructor will monitor studnts progrss in complting th assignmnts. B. Studnts ar xpctd to attnd vry class, to arriv at ach class on tim, and rmain in class for th ntir priod. Instructors may choos to lowr a studnt's grad bcaus of tardinss. V. EXAMINATIONS A. Examinations will b givn at appropriat intrvals. Each xamination will b announcd at last on wk in advanc. Thr will b two unit xams, and a comprhnsiv final xam. B. Studnts who miss an xam should discuss with th instructor th circumstancs surrounding th absnc. Th instructor will dtrmin whthr a mak-up xam is to b givn. Mak-up xaminations ar givn by appointmnt only. VI. SEMESTER GRADE COMPUTATIONS A. Th smstr avrag is drivd from th homwork, quizzs, unit xams, and REQUIRED comprhnsiv final xam in MyMathLab. You must tak th final xam and scor at last 50% to pass th cours. Final grads will follow th grad dsignation blow: Grad Prformanc A % and 50% or bttr on th Final Exam. B 80-89% and 50% or bttr on th Final Exam. C 70-79% and 50% or bttr on th Final Exam. D 60-69% and 50% or bttr on th Final Exam. F 0-59% and 50% or bttr on th Final Exam.
3 VII. NOTES AND ADDITIONAL INSTRUCTIONS A. Withdrawal from Cours: It is th studnt's rsponsibility to officially drop a class if circumstancs prvnt attndanc. Any studnt who dsirs to, or must, officially withdraw from a cours aftr th first schduld class mting must fil an Application for Withdrawal or an Application for Rfund. Th withdrawal form must b signd by th studnt. An Application for withdrawal will b accptd at any tim prior to Friday of th 12th wk of classs during th 16-wk fall and spring smstrs. Th dadlin for sssions of othr lngths is as follows. Sssion Dadlin for Withdrawal 12-wk sssion Friday of th 9 th wk 10-wk sssion Friday of th 7 th wk 8-wk sssion Friday of th 6 th wk 6-wk sssion Friday of th 4 th wk 5-wk sssion Friday of th 3 rd wk Th quivalnt dat (75% of th smstr) will b usd for sssions of othr lngths. Th spcific last day to withdraw is publishd ach smstr in th Schdul Bulltin. Studnts who officially withdraw will b awardd th grad of "W" providd th studnt's attndanc and acadmic prformanc ar satisfactory at th tim of official withdrawal. Studnts must fil a withdrawal application with th collg bfor thy may b considrd for withdrawal. A studnt may not withdraw from a class for which th instructor has prviously issud th studnt a grad of "F". B. An Incomplt Grad: Th Collg catalog stats, "An incomplt grad may b givn in thos cass whr th studnt has compltd th majority of th cours work but, bcaus of prsonal illnss, dath in th immdiat family, or military ordrs, th studnt is unabl to complt th rquirmnts for a cours..." Prior approval from th instructor is rquird bfor th grad of "I" is rcordd. A studnt who mrly fails to show for th final xamination will rciv a zro for th final and an "F" for th cours.
4 C. Cllular Phons and Bprs: Cllular phons and bprs will b turnd off whil th studnt is in th classroom or laboratory. D. Amricans With Disabilitis Act (ADA): Disability Support Srvics provid srvics to studnts who hav appropriat documntation of a disability. Studnts rquiring accommodations for class ar rsponsibl for contacting th Offic of Disability Support Srvics (DSS) locatd on th cntral campus. This srvic is availabl to all studnts, rgardlss of location. Explor th wbsit at for furthr information. Rasonabl accommodations will b givn in accordanc with th fdral and stat laws through th DSS offic. E. Civility: Individuals ar xpctd to b cognizant of what a constructiv ducational xprinc is and rspctful of thos participating in a larning nvironmnt. Failur to do so can rsult in disciplinary action up to and including xpulsion. F. Advancd Math Lab: Th Math Dpartmnt oprats an Advancd Mathmatics Lab in Building 152, Room 145. All courss offrd by th Math Dpartmnt ar supportd in th lab with appropriat tutorial softwar. Calculators ar availabl for studnt us in th lab. Studnts ar ncouragd to tak advantag of ths opportunitis. S postd hours for th Advancd Math Lab. G. Offic Hours: Full-tim instructors post offic hours outsid th door of th Mathmatics Dpartmnt (Building 152, Room 223). Part-tim instructors may b availabl by appointmnt. If you hav difficulty with th cours work, plas consult your instructor. VIII. COURSE OUTLINE A. Lsson On: Chaptr 6 Diffrntial Equations will 1. Lsson Objctivs: Upon succssful compltion of this lsson, th studnt b abl to: a. Us initial conditions to find particular solutions of diffrntial quations. b. Us slop filds to approximat solutions of diffrntial quations. c. Us Eulr s Mthod to approximat solutions of diffrntial quations. d. Us sparation of variabls to solv a simpl diffrntial quation.. Us xponntial functions to modl growth and dcay in applications.
5 f. R c o g n i z a n d s o l v d i f f r n t i a l q u a t i o n s t h a t can b solvd by sparation of variabls. g. Us diffrntial quations to modl and solv applid problms. h. Solv and analyz logistic diffrntial quations.
6 i. Solv a first-ordr linar diffrntial quation, and us linar diffrntial quations to solv applid problms. 2. Larning Activitis: a. Rad Chaptr 6, Sctions 6.1 through 6.4. b. Work problms assignd by th instructor. c. Complt assignd lab activitis. 3. Lsson Outlin: a. Sction 6.1 Slop Filds and Eulr s Mthod b. Sction 6.2 Diffrntial Equations: Growth and Dcay c. Sction 6.3 Sparation of Variabls and th Logistic Equation d. Sction 6.4 First-ordr Linar Diffrntial Equations B. Lsson Two: Chaptr 7 Applications of Intgration 1. Lsson Objctivs: Upon succssful compltion of this lsson, th studnt will b abl to: a. Find th ara of a rgion btwn two curvs using intgration. b. Find th ara of a rgion btwn two intrscting curvs using intgration. c. Dscrib intgration as an accumulation procss. d. Find th volum of a solid of rvolution using th disk mthod.. Find th volum of a solid of rvolution using th washr mthod. f. Find th volum of a solid with known cross sctions. g. Find th volum of a solid of rvolution using th shll mthod. h. Find th arc lngth of a smooth curv. i. Find th ara of a surfac of rvolution. j. Find th work don by a constant forc. k. Find th work don by a variabl forc. l. Undrstand th dimnsion of mass. m. Find th cntr of mass in a on-dimnsional systm. n. Find th cntr of mass in a two-dimnsional systm. o. Find th cntr of mass of a planar lamina. p. Us th Fundamntal Thorm of Pappus to find th volum of a solid of rvolution. q. Find fluid prssur and fluid forc. 2. Larning Activitis: a. b. c. 3. Lsson Outlin: Rad Chaptr 7, Sctions 7.1 through 7.7. Work problms assignd by th instructor. Complt assignd lab activitis.
7 a. Sction 7.1 Ara of a Rgion Btwn Two Curvs b. Sction 7.2 Volum: Th Disk Mthod c. Sction 7.3 Volum: Th Shll Mthod d. Sction 7.4 Arc Lngth and Surfacs of Rvolution. Sction 7.5 Work f. Sction 7.6 Momnts, Cntrs of Mass, Cntroids g. Sction 7.7 Fluid Prssur and Fluid Forc C. Lsson Thr: Chaptr 8 Intgration Tchniqus, L Hopital s Rul, and Impropr Intgrals a. Lsson Objctivs: Upon succssful compltion of this lsson, th studnt will b abl to: a. Rviw procdurs for fitting an intgrand to on of th basic intgration ruls. b. Find an antidrivativ using intgration by parts. c. Solv trigonomtric intgrals involving powrs of sin and cosin. d. Solv trigonomtric intgrals involving powrs of scant and tangnt.. Solv trigonomtric intgrals involving sin-cosin products with diffrnt angls. f. Us trigonomtric substitution to solv an intgral. g. Us intgrals to modl and solv ral-lif applications. h. Undrstand th concpt of partial fraction dcomposition. i. Us partial fraction dcomposition with linar factors to intgrat rational functions. j. Us partial fraction dcomposition with quadratic factors to intgrat rational functions. k. Evaluat an indfinit intgral using a tabl of intgrals. l. Evaluat an indfinit intgral using rduction formulas. m. Evaluat an indfinit intgral involving rational functions of sin and cosin. n. Rcogniz limits that produc indtrminat forms. o. Apply L Hopital s Rul to valuat a limit. p. Evaluat an impropr intgral that has an infinit limit of intgration. q. Evaluat an impropr intgral that has an infinit discontinuity. 2. Larning Activitis: a. Rad Chaptr 8, Sctions 8.1 through 8.8. b. Work problms assignd by th instructor. c. Complt assignd lab activitis. 3. Lsson Outlin:
8 a. Sction 8.1 Basic Intgration Ruls b. Sction 8.2 Intgration by Parts c. Sction 8.3 Trigonomtric Intgrals d. Sction 8.4 Trigonomtric Substitution. Sction 8.5 Partial Fractions f. Sction 8.6 Intgration by Tabls and Othr Intgration Tchniqus g. Sction 8.7 Indtrminat Forms and L Hopital s Rul h. Sction 8.8 Impropr Intgrals D. Lsson Four: Chaptr 9 Infinit Sris 1. Lsson Objctivs: Upon succssful compltion of this lsson, th studnt will b abl to: a. List th trms of a squnc. b. Dtrmin whthr a squnc convrgs or divrgs. c. Writ a formula for th nth trm of a squnc. d. Us proprtis of monotonic squncs and boundd squncs.. Undrstand th dfinition of a convrgnt infinit sris. f. Us proprtis of infinit gomtric sris. g. Us th nth-trm Tst for Divrgnc of an infinit sris. h. Us th Intgral Tst to dtrmin whthr a sris convrgs or divrgs. i. Us proprtis of p-sris and harmonic sris. j. Us th Dirct Comparison Tst to dtrmin whthr a sris convrgs or divrgs. k. Us th Limit Comparison Tst to dtrmin whthr a sris convrgs or divrgs. l. Us th Altrnating Sris Tst to dtrmin whthr an infinit sris convrgs. m. Us th Altrnating Sris Rmaindr to approximat th sum of an altrnating sris. n. Classify a convrgnt sris as absolutly or conditionally convrgnt. o. Rarrang an infinit sris to obtain a sum. p. Us th Ratio Tst to dtrmin whthr a sris convrgs or divrgs. q. Us th Root Tst to dtrmin whthr a sris convrgs or divrgs. r. Rviw th tsts for convrgnc and divrgnc of an infinit sris. s. Find Taylor approximations of lmntary functions and compar thm with lmntary functions. t. Find Taylor and Maclaurin polynomial approximations of lmntary functions. u. Us th rmaindr of a Taylor polynomial.
9 v. Undrstand th dfinition of a powr sris. w. Find th radius and intrval of convrgnc of a powr sris. x. Dtrmin th ndpoint of convrgnc of a powr sris. y. Diffrntiat and intgrat a powr sris. z. Find a gomtric powr sris that rprsnts a function. aa. Construct a powr sris using sris approximations. bb. Find a Taylor or Maclaurin sris for a function. cc. Find a binomial sris. dd. Us a basic list of Taylor sris to find othr Taylor sris. 2. Larning Activitis: a. Rad Chaptr 9, Sctions 9.1 through b. Work problms assignd by th instructor. c. Complt assignd lab activitis. 3. Lsson Outlin: a. Sction 9.1 Squncs b. Sction 9.2 Sris and Convrgnc b. Sction 9.3 Th Intgral Tst and p-sris c. Sction 9.4 Comparisons of Sris d. Sction 9.5 Altrnating Sris. Sction 9.6 Th Ratio and Root Tsts f. Sction 9.7 Taylor Polynomials and Approximations g. Sction 9.8 Powr Sris h. Sction 9.9 Rprsntation of Functions by Powr Sris i. Sction 9.10 Taylor and Maclaurin Sris E. Lsson Fiv: Chaptr 10 Conics, Paramtric Equations, and Polar Coordinats 1. Lsson Objctivs: Upon succssful compltion of this lsson, th studnt will b abl to: a. Undrstand th dfinition of a conic sction. b. Analyz and writ quations of parabolas using proprtis of parabolas. c. Analyz and writ quations of llipss using proprtis of llipss. d. Analyz and writ quations of hyprbolas using proprtis of hyprbolas.. Sktch th graph of a curv givn by a st of paramtric quations. f. Eliminat th paramtr in a st of paramtric quations. g. Find a st of paramtric quations to rprsnt a curv. h. Undrstand two classic calculus problms, th tautochron and brachistochron problms. i. Find th slop of a tangnt lin to a curv givn by a st of
10 paramtric quations. j. Find th arc lngth of a curv givn by a st of paramtric quations. k. Find th ara of a surfac of rvolution (paramtric form). l. Undrstand th polar coordinat systm. m. Rwrit rctangular coordinats and quations in polar form and vic vrsa. n. Sktch th graph of an quation givn in polar form. o. Find th slop of a tangnt lin to a polar graph. p. Idntify svral typs of spcial polar graphs. q. Find th ara of a rgion boundd by a polar graph. r. Find th points of intrsction of two polar graphs. s. Find th arc lngth of a polar graph. t. Find th ara of a surfac of rvolution (polar form). u. Analyz and writ polar quations of conics. v. Undrstand and us Kplr s Laws of plantary motion. 2. Larning Activitis: a. Rad Chaptr 10, Sctions 10.1 through b. Work problms assignd by th instructor. c. Complt assignd lab activitis. 3. Lsson Outlin: a. Sction 10.1 Conics and Calculus b. Sction 10.2 Plan Curvs and Paramtric Equations c. Sction 10.3 Paramtric Equations and Calculus d. Sction 10.4 Polar Coordinats and Polar Graphs. Sction 10.5 Ara and Arc Lngth in Polar Coordinats f. Sction 10.6 Polar Equations of Conics and Kplr s Law
MATH 1080 Test 2-SOLUTIONS Spring
MATH Tst -SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. =
San José State University Aerospace Engineering AE 138 Vector-Based Dynamics for Aerospace Applications, Fall 2016
San José Stat Univrsity Arospac Enginring AE 138 Vctor-Basd Dynamics for Arospac Applications, Fall 2016 Instructor: Offic Location: Email: Offic Hours: Class Days/Tim: Classroom: Prof. J.M. Huntr E272F
Solution: 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
Content Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles.
Tachr: CORE TRIGONOMETRY Yar: 2012-13 Cours: TRIGONOMETRY Month: All Months S p t m b r Angls Essntial Qustions Can I idntify draw ngativ positiv angls in stard position? Do I hav a working knowldg of
Calculus 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.
Differentiation 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
Exercise 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
CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2415 CALCULUS III. Semester Hours Credit: 4
CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2415 CALCULUS III Semester Hours Credit: 4 I. INTRODUCTION A. Calculus III is a continuation course from Calculus II, which includes advanced topics in calculus,
6.1 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
JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH 241) Final Review Fall 2016
JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH ) Final Rviw Fall 06 Th Final Rviw is a starting point as you study for th final am. You should also study your ams and homwork. All topics listd in th
4 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
Mathematics. 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
2008 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
ECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
Higher 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
General Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
1. I = 2 3. I = 4 5. I = I = 5 2
Vrsion Homwork 2 grandi This print-out should hav 24 qustions Multipl-choic qustions may continu on th nxt column or pag find all choics bfor answring If CalC5c9c points Fx = x 5 8sin2 d, find th valu
SECTION 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
Math 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
MA 262, Spring 2018, Final exam Version 01 (Green)
MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in
AP 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
CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2318 Linear Algebra. Semester Hours Credit: 3
CENTRAL TEXAS COLLEGE SYLLABUS FOR MATH 2318 Linear Algebra Semester Hours Credit: 3 I. INTRODUCTION A. Linear Algebra is a three semester-hour course. This course introduces and provides models for application
Mor 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
u 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
5. 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
10. 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
Calculus 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)
u 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
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
The Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
3 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
Sundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A, B and C.
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Tim: 3hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A, B and C. SECTION -A Vry Short Answr Typ Qustions. 0 X = 0. Find th condition
1973 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=
The Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
Prelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours
Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd - hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn
MATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
Application of Vague Soft Sets in students evaluation
Availabl onlin at www.plagiarsarchlibrary.com Advancs in Applid Scinc Rsarch, 0, (6):48-43 ISSN: 0976-860 CODEN (USA): AASRFC Application of Vagu Soft Sts in studnts valuation B. Chtia*and P. K. Das Dpartmnt
3 Finite Element Parametric Geometry
3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,
Calculus 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
Announce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review
ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG
CENTRAL TEXAS COLLEGE SYLLABUS FOR ENGR 2301 Engineering Mechanics - Statics. Semester Hours Credit: 3
CENTRAL TEXAS COLLEGE SYLLABUS FOR ENGR 2301 Engineering Mechanics - Statics Semester Hours Credit: 3 I. INTRODUCTION A. ENGR 2301, Engineering Mechanics Statics, is a three-semester hour course that is
Middle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME 43 Introduction to Finit Elmnt Analysis Chaptr 3 Computr Implmntation of D FEM Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
A Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
Fourier 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
(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
a 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
BSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
First 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
10. 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
ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
NEW 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
PROOF 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
A=P=E M-A=N Alpha particle Beta Particle. Periodic table
Nam Pr. Atomic Structur/Nuclar Chmistry (Ch. 3 & 21) OTHS Acadmic Chmistry Objctivs: Undrstand th xprimntal dsign and conclusions usd in th dvlopmnt of modrn atomic thory, including Dalton's Postulats,
Basic 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
Applied Statistics II - Categorical Data Analysis Data analysis using Genstat - Exercise 2 Logistic regression
Applid Statistics II - Catgorical Data Analysis Data analysis using Gnstat - Exrcis 2 Logistic rgrssion Analysis 2. Logistic rgrssion for a 2 x k tabl. Th tabl blow shows th numbr of aphids aliv and dad
DIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
Calculus 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
Things 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
Deift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
MSLC 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
Review 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
First 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
COMPUTER 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
2013 Specialist Mathematics GA 3: Written examination 2
0 0 Spcialist Mathmatics GA : Writtn xamination GENERAL COMMENTS Th 0 Spcialist Mathmatics xamination comprisd multipl-choic qustions (worth marks) and fiv xtndd qustions (worth 8 marks). Th papr smd accssibl
DSP-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
are given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
AerE 344: Undergraduate Aerodynamics and Propulsion Laboratory. Lab Instructions
ArE 344: Undrgraduat Arodynamics and ropulsion Laboratory Lab Instructions Lab #08: Visualization of th Shock Wavs in a Suprsonic Jt by using Schlirn tchniqu Instructor: Dr. Hui Hu Dpartmnt of Arospac
Complex 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
Kingdom of Saudi Arabia. The National Commission for Academic Accreditation & Assessment
ATTACHMENT 2 () Cours Spcifications Kingdom of Saudi Arabia Th National Commission for Acadmic Accrditation & Assssmnt Cours Spcifications (CS) Cours Spcifications Institution Dat Univrsity of Dammam 13/3/2014
Problem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
Elements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
Thomas 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
Differential 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
Hydrogen 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
CHAPTER 1. Introductory Concepts Elements of Vector Analysis Newton s Laws Units The basis of Newtonian Mechanics D Alembert s Principle
CHPTER 1 Introductory Concpts Elmnts of Vctor nalysis Nwton s Laws Units Th basis of Nwtonian Mchanics D lmbrt s Principl 1 Scinc of Mchanics: It is concrnd with th motion of matrial bodis. odis hav diffrnt
The 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
u 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
7' The growth of yeast, a microscopic fungus used to make bread, in a test tube can be
N Sction A: Pur Mathmatics 55 marks] / Th rgion R is boundd by th curv y, th -ais, and th lins = V - +7 and = m, whr m >. Find th volum gnratd whn R is rotatd through right angls about th -ais, laving
AS 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
Integration by Parts
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'(
Brief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
A. 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,
1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
Thomas 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
AP Calculus Multiple-Choice Question Collection connect to college success
AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos
Ewald s Method Revisited: Rapidly Convergent Series Representations of Certain Green s Functions. Vassilis G. Papanicolaou 1
wald s Mthod Rvisitd: Rapidly Convrgnt Sris Rprsntations of Crtain Grn s Functions Vassilis G. Papanicolaou 1 Suggstd Running Had: wald s Mthod Rvisitd Complt Mailing Addrss of Contact Author for offic
Recall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
Mathematics 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
Homotopy perturbation technique
Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,
VSMN30 FINITA ELEMENTMETODEN - DUGGA
VSMN3 FINITA ELEMENTMETODEN - DUGGA 1-11-6 kl. 8.-1. Maximum points: 4, Rquird points to pass: Assistanc: CALFEM manual and calculator Problm 1 ( 8p ) 8 7 6 5 y 4 1. m x 1 3 1. m Th isotropic two-dimnsional
(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
Function Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
4.2 Design of Sections for Flexure
4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt
Chapter 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
Probability 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
That is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx
MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of