Review Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2

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

1985 AP Calculus BC: Section I

985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b

+ x. x 2x. 12. dx. 24. dx + 1)

INTEGRATION of FUNCTION of ONE VARIABLE INDEFINITE INTEGRAL Fidig th idfiit itgrals Rductio to basic itgrals, usig th rul f ( ) f ( ) d =... ( ). ( )d. d. d ( ). d. d. d 7. d 8. d 9. d. d. d. d 9. d 9.

Chapter At each point (x, y) on the curve, y satisfies the condition

Chaptr 6. At ach poit (, y) o th curv, y satisfis th coditio d y 6; th li y = 5 is tagt to th curv at th poit whr =. I Erciss -6, valuat th itgral ivolvig si ad cosi.. cos si. si 5 cos 5. si cos 5. cos

Taylor and Maclaurin Series

Taylor ad Maclauri Sris Taylor ad Maclauri Sris Thory sctio which dals with th followig topics: - Th Sigma otatio for summatio. - Dfiitio of Taylor sris. - Commo Maclauri sris. - Taylor sris ad Itrval

Option 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges.

Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt

PURE MATHEMATICS A-LEVEL PAPER 1

-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series

Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris

Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net

Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.

Fooling Newton s Method a) Find a formula for the Newton sequence, and verify that it converges to a nonzero of f. A Stirling-like Inequality

Foolig Nwto s Mthod a Fid a formla for th Nwto sqc, ad vrify that it covrgs to a ozro of f. ( si si + cos 4 4 3 4 8 8 bt f. b Fid a formla for f ( ad dtrmi its bhavior as. f ( cos si + as A Stirlig-li

Probability & Statistics,

Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said

COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS II

COLLECTION OF SUPPLEMENTARY PROBLEMS I. CHAPTER 6 --- Trscdtl Fuctios CALCULUS II A. FROM CALCULUS BY J. STEWART:. ( How is th umbr dfid? ( Wht is pproimt vlu for? (c ) Sktch th grph of th turl potil fuctios.

Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120

Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,

Calculus & analytic geometry

Calculus & aalytic gomtry B Sc MATHEMATICS Admissio owards IV SEMESTER CORE COURSE UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITYPO, MALAPPURAM, KERALA, INDIA 67 65 5 School of Distac

1973 AP Calculus BC: Section I

97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f

Student s Printed Name:

Studt s Pritd Nam: Istructor: CUID: Sctio: Istructios: You ar ot prmittd to us a calculator o ay portio of this tst. You ar ot allowd to us a txtbook, ots, cll pho, computr, or ay othr tchology o ay portio

NET/JRF, GATE, IIT JAM, JEST, TIFR

Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Mathmatical Physics JEST-6 Q. Giv th coditio φ, th solutio of th quatio ψ φ φ is giv by k. kφ kφ lφ kφ lφ (a) ψ (b) ψ kφ (c) ψ

Chapter (8) Estimation and Confedence Intervals Examples

Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit

H2 Mathematics Arithmetic & Geometric Series ( )

H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic

z 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z

Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist

Chapter Taylor Theorem Revisited

Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o

ln x = n e = 20 (nearest integer)

H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77

An Introduction to Asymptotic Expansions

A Itroductio to Asmptotic Expasios R. Shaar Subramaia Asmptotic xpasios ar usd i aalsis to dscrib th bhavior of a fuctio i a limitig situatio. Wh a fuctio ( x, dpds o a small paramtr, ad th solutio of

Statistics 3858 : Likelihood Ratio for Exponential Distribution

Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai

On the approximation of the constant of Napier

Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of

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

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

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

STIRLING'S 1 FORMULA AND ITS APPLICATION

MAT-KOL (Baja Luka) XXIV ()(08) 57-64 http://wwwimviblorg/dmbl/dmblhtm DOI: 075/МК80057A ISSN 0354-6969 (o) ISSN 986-588 (o) STIRLING'S FORMULA AND ITS APPLICATION Šfkt Arslaagić Sarajvo B&H Abstract:

A Review of Complex Arithmetic

/0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd

( A) ( B) ( C) ( D) ( E)

d Smsr Fial Exam Worksh x 5x.( NC)If f ( ) d + 7, h 4 f ( ) d is 9x + x 5 6 ( B) ( C) 0 7 ( E) divrg +. (NC) Th ifii sris ak has h parial sum S ( ) for. k Wha is h sum of h sris a? ( B) 0 ( C) ( E) divrgs

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)

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

In exercises 1 and 2, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers _

Chapter 9 Curve I eercises ad, (a) write the repeatig decimal as a geometric series ad (b) write its sum as the ratio of two itegers _.9.976 Distace A ball is dropped from a height of 8 meters. Each time

Washington State University

he 3 Ktics ad Ractor Dsig Sprg, 00 Washgto Stat Uivrsity Dpartmt of hmical Egrg Richard L. Zollars Exam # You will hav o hour (60 muts) to complt this xam which cosists of four (4) problms. You may us

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

Chapter 9 Infinite Series

Sctio 9. 77. Cotiud d + d + C Ar lim b lim b b b + b b lim + b b lim + b b 6. () d (b) lim b b d (c) Not tht d c b foud by prts: d ( ) ( ) d + C. b Ar b b lim d lim b b b b lim ( b + ). b dy 7. () π dy

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=

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.

SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3

SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos

e to approximate (using 4

Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial

امتحانات الشهادة الثانوية العامة فرع: العلوم العامة

وزارة التربية والتعليم العالي المديرية العامة للتربية دائرة االمتحانات امتحانات الشهادة الثانوية العامة فرع: العلوم العامة االسم: الرقم: مسابقة في مادة الرياضيات المدة أربع ساعات عدد المسائل: ست مالحظة:

Discrete Fourier Transform (DFT)

Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial

Calculus II exam 1 6/18/07 All problems are worth 10 points unless otherwise noted. Show all analytic work.

9.-0 Calculus II exam 6/8/07 All problems are worth 0 poits uless otherwise oted. Show all aalytic work.. (5 poits) Prove that the area eclosed i the circle. f( x) = x +, 0 x. Use the approximate the area

An Introduction to Asymptotic Expansions

A Itroductio to Asmptotic Expasios R. Shaar Subramaia Dpartmt o Chmical ad Biomolcular Egirig Clarso Uivrsit Asmptotic xpasios ar usd i aalsis to dscrib th bhavior o a uctio i a limitig situatio. Wh a

Law of large numbers

Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs

Calculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.)

Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics

1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute

Math, Calculus II Fial Eam Solutios. 5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute 4 d. The check your aswer usig the Evaluatio Theorem. ) ) Solutio: I this itegral,

Hadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms

Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of

CDS 101: Lecture 5.1 Reachability and State Space Feedback

CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray ad Hido Mabuchi 5 Octobr 4 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls

Calculus 2 Test File Fall 2013

Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to

Calculus II - Problem Drill 21: Power Series, Taylor and Maclaurin Polynomial Series

Calculus II - Problem Drill : Power Series, Taylor ad Maclauri Polyomial Series Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully () Work the problems o paper as 3 4 3 4. Fill

A Simple Proof that e is Irrational

Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural

07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n

07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l

MAT 182: Calculus II Test on Chapter 9: Sequences and Infinite Series Take-Home Portion Solutions

MAT 8: Clculus II Tst o Chptr 9: qucs d Ifiit ris T-Hom Portio olutios. l l l l 0 0 L'Hôpitl's Rul 0 . Bgi by computig svrl prtil sums to dvlop pttr: 6 7 8 7 6 6 9 9 99 99 Th squc of prtil sums is s follows:,,,,,

Example 2. Find the upper bound for the remainder for the approximation from Example 1.

Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

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

CDS 101: Lecture 5.1 Reachability and State Space Feedback

CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray 7 Octobr 3 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls Dscrib th dsig o

Quiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.

Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where

2 ) 5. (a) (1)(3) + (1)(2) = 5 (b) {area of shaded region in Fig. 24b} < 5

Odd Aswers: Chapter Four Cotemporary Calculus PROBLEM ANSWERS Chapter Four Sectio 4.. (a) ()() + (8)(4) = 5 (b) ()() ()(8) = 76. bh + b(h h) = bh + bh bh = b ( h + H ) 5. (a) ()() + ()() = 5 (b) {area

National Quali cations

Ntiol Quli ctios AH07 X77/77/ Mthmtics FRIDAY, 5 MAY 9:00 AM :00 NOON Totl mrks 00 Attmpt ALL qustios. You my us clcultor. Full crdit will b giv oly to solutios which coti pproprit workig. Stt th uits

Problem Value Score Earned No/Wrong Rec -3 Total

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING ECE6 Fall Quiz # Writt Eam Novmr, NAME: Solutio Kys GT Usram: LAST FIRST.g., gtiit Rcitatio Sctio: Circl t dat & tim w your Rcitatio

Calculus 2 Test File Spring Test #1

Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad

Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1

Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida

page 11 equation (1.2-10c), break the bar over the right side in the middle

I. Corrctios Lst Updtd: Ju 00 Complx Vrils with Applictios, 3 rd ditio, A. Dvid Wusch First Pritig. A ook ought for My 007 will proly first pritig With Thks to Christi Hos of Swd pg qutio (.-0c), rk th

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.

Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si

ENGI 3424 Appendix Formulæ Page A-01

ENGI 344 Appdix Formulæ g A-0 ENGI 344 Egirig Mthmtics ossibilitis or your Formul Shts You my slct itms rom this documt or plcmt o your ormul shts. Howvr, dsigig your ow ormul sht c b vlubl rvisio xrcis

2.29 Numerical Fluid Mechanics Spring 2015 Lecture 12

REVIEW Lctur 11: Numrical Fluid Mchaics Sprig 2015 Lctur 12 Fiit Diffrcs basd Polyomial approximatios Obtai polyomial (i gral u-qually spacd), th diffrtiat as dd Nwto s itrpolatig polyomial formulas Triagular

Taylor Series (BC Only)

Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor

Lectures 9 IIR Systems: First Order System

EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work

National Quali cations

PRINT COPY OF BRAILLE Ntiol Quli ctios AH08 X747/77/ Mthmtics THURSDAY, MAY INSTRUCTIONS TO CANDIDATES Cdidts should tr thir surm, form(s), dt of birth, Scottish cdidt umbr d th m d Lvl of th subjct t

SIGNALS AND LINEAR SYSTEMS UNIT-1 SIGNALS

SIGNALS AND LINEAR SYSTEMS UNIT- SIGNALS. Dfi a sigal. A sigal is a fuctio of o or mor idpdt variabls which cotais som iformatio. Eg: Radio sigal, TV sigal, Tlpho sigal, tc.. Dfi systm. A systm is a st

ASSERTION AND REASON

ASSERTION AND REASON Som qustios (Assrtio Rso typ) r giv low. Ech qustio cotis Sttmt (Assrtio) d Sttmt (Rso). Ech qustio hs choics (A), (B), (C) d (D) out of which ONLY ONE is corrct. So slct th corrct

LECTURE 13 Filling the bands. Occupancy of Available Energy Levels

LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad

WBJEEM MATHEMATICS. Q.No. μ β γ δ 56 C A C B

WBJEEM - MATHEMATICS Q.No. μ β γ δ C A C B B A C C A B C A B B D B 5 A C A C 6 A A C C 7 B A B D 8 C B B C 9 A C A A C C A B B A C A B D A C D A A B C B A A 5 C A C B 6 A C D C 7 B A C A 8 A A A A 9 A

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

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,

Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative

3 Show in each case that there is a root of the given equation in the given interval. a x 3 = 12 4

C Worksheet A Show i each case that there is a root of the equatio f() = 0 i the give iterval a f() = + 7 (, ) f() = 5 cos (05, ) c f() = e + + 5 ( 6, 5) d f() = 4 5 + (, ) e f() = l (4 ) + (04, 05) f

2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005

Mark Schm 67 Ju 5 GENERAL INSTRUCTIONS Marks i th mark schm ar plicitly dsigatd as M, A, B, E or G. M marks ("mthod" ar for a attmpt to us a corrct mthod (ot mrly for statig th mthod. A marks ("accuracy"

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)

Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages

f x x c x c x c... x c...

CALCULUS BC WORKSHEET ON POWER SERIES. Derive the Taylor series formula by fillig i the blaks below. 4 5 Let f a a c a c a c a4 c a5 c a c What happes to this series if we let = c? f c so a Now differetiate

DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1

DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.

SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad

MATH 129 FINAL EXAM REVIEW PACKET (Revised Spring 2008)

MATH 9 FINAL EXAM REVIEW PACKET (Revised Sprig 8) The followig questios ca be used as a review for Math 9. These questios are ot actual samples of questios that will appear o the fial exam, but they will

EC1305 SIGNALS & SYSTEMS

EC35 SIGNALS & SYSTES DEPT/ YEAR/ SE: IT/ III/ V PREPARED BY: s. S. TENOZI/ Lcturr/ECE SYLLABUS UNIT I CLASSIFICATION OF SIGNALS AND SYSTES Cotiuous Tim Sigals (CT Sigals Discrt Tim Sigals (DT Sigals Stp

Math 113 Exam 3 Practice

Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This

Math 113 Exam 4 Practice

Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

.8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

Taylor Polynomials and Approximations - Classwork

Taylor Polyomials ad Approimatios - Classwork Suppose you were asked to id si 37 o. You have o calculator other tha oe that ca do simple additio, subtractio, multiplicatio, or divisio. Fareched\ Not really.

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

On a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G.

O a problm of J. d Graaf coctd with algbras of uboudd oprators d Bruij, N.G. Publishd: 01/01/1984 Documt Vrsio Publishr s PDF, also kow as Vrsio of Rcord (icluds fial pag, issu ad volum umbrs) Plas chck

Math 122 Test 3 - Review 1

I. Sequeces ad Series Math Test 3 - Review A) Sequeces Fid the limit of the followig sequeces:. a = +. a = l 3. a = π 4 4. a = ta( ) 5. a = + 6. a = + 3 B) Geometric ad Telescopig Series For the followig

Physics of the Interstellar and Intergalactic Medium

PYA0 Sior Sophistr Physics of th Itrstllar ad Itrgalactic Mdium Lctur 7: II gios Dr Graham M. arpr School of Physics, TCD Follow-up radig for this ad t lctur Chaptr 5: Dyso ad Williams (lss dtaild) Chaptr

Mathematics Extension 1

016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved

APPENDIX: STATISTICAL TOOLS

I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.

Ordinary Differential Equations

Basi Nomlatur MAE 0 all 005 Egirig Aalsis Ltur Nots o: Ordiar Diffrtial Equatios Author: Profssor Albrt Y. Tog Tpist: Sakurako Takahashi Cosidr a gral O. D. E. with t as th idpdt variabl, ad th dpdt variabl.

Taylor Series Mixed Exercise 6

Taylor Series Mied Eercise Let f() cot ad a f(a) f () ( cos ec ) + cot f (a) f''() cotcosec + ( cosec ) f''(a) f'''() ( cosec cot cosec ) + cotcosec Substitutig ito the Taylor series epasio gives f() +

Chapter 4 - The Fourier Series

M. J. Robrts - 8/8/4 Chaptr 4 - Th Fourir Sris Slctd Solutios (I this solutio maual, th symbol,, is usd for priodic covolutio bcaus th prfrrd symbol which appars i th txt is ot i th fot slctio of th word