Student s Printed Name:
|
|
- Randolph Dean
- 5 years ago
- Views:
Transcription
1 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 of this tst. All dvics must b turd off ad stord away whil you ar i th tstig room. Durig this tst, ay kid of commuicatio with ay prso othr tha th istructor or a dsigatd proctor is udrstood to b a violatio of acadmic itgrity. No part of this tst may b rmovd from th xamiatio room. Rad ach qustio carfully. To rciv full crdit for th fr rspos portio of th tst, you must:. Show lgibl, logical, ad rlvat justificatio which supports your fial aswr.. Us complt ad corrct mathmatical otatio.. Iclud propr uits, if cssary.. Giv aswrs as xact valus whvr possibl. You hav 90 miuts to complt th tir tst. Do ot writ blow this li. Fr Rspos Problm Possibl Eard Fr Rspos Problm Possibl Eard. 5. a b. 7. a c.. b a (Scatro). b. 5 Fr Rspos 70. c. 5 Multipl Choic 0. d. 5 Tst Total 00 Vrsio A Pag of 8
2 Multipl Choic. Thr ar 0 multipl choic qustios. Each qustio is worth poits ad has o corrct aswr. Th multipl choic problms will cout 0% of th total grad. Circl your choic o your tst papr.. Exprss th followig sum usig sigma otatio. ( pts.) 87 a) c) k k ( ) ( k ) b) k k k ( ) ( k ) d) k k k0 k k ( ) ( k) k ( ) ( k ). Dtrmi whr th fuctio f( x ) is dcrasig if th first drivativ is ( pts.) x f ( x) x ( x ) ( x ). a), b) (, ) 0, c),0 (, ) d) (0,) Vrsio A Pag of 8
3 . ( pts.) Assum f ( x) g( x) dx 6 ad f ( x) dx. Fid g( x) dx. a) c) g( x) dx 9 b) g( x) dx d) g( x) dx 0 g( x) dx 9. Th figur shows th aras of thr rgios boudd by th graph of f ad th x-axis. ( pts.) Fid 5 f ( x) dx. a) c) 5 f ( x) dx b) 5 f ( x) dx 0 d) 5 f ( x) dx 60 5 f ( x) dx Vrsio A Pag of 8
4 5. Fid th x-valus of th absolut xtrma of ( pts.) h( x) x x 8 o th itrval [0, ]. a) absolut maximum at x, absolut miimum at x b) o absolut maximum, absolut miimum at x 0 c) absolut maximum at x, absolut miimum at x 0 d) o absolut xtrma 6. Lt f( x ) b twic diffrtiabl o th itrval (, ). Th graph of th drivativ of ( pts.) f( x) is show blow. Us it to dtrmi th itrval(s) o which th graph of f( x) is cocav up. a) (,) (5, ) b) (,5) c) (,) d) No such itrvals Vrsio A Pag of 8
5 7. ( pts.) Fid f( x ) if f ( x) x ad f. x x a) f ( x) x b) f ( x) x l x c) f ( x) x l( x) d) f ( x) x l x 8. Us rctagls to stimat th ara abov th x-axis ad udr th graph of ( pts.) f ( x) si x o th itrval [0, ] (s graph). Partitio th itrval ito two subitrvals of qual width ad valuat th fuctio at th midpoits of th subitrvals. a) 8 b) c) d) 6 Vrsio A Pag 5 of 8
6 Th graph of gx ( ) is show blow. It cosists of two straight lis. Us it to valuat th dfiit itgrals i problms 9 ad ( pts.) gx ( ) dx a) c) gx ( ) dx 9 b) gx ( ) 7 dx d) gx ( ) dx 7 gx ( ) dx 0. ( pts.) 6 g( x) dx a) c) 6 g( x) dx b) 6 g( x) dx d) 6 6 g( x) dx 7 g( x) dx Vrsio A Pag 6 of 8
7 (this pag ittioally lft blak) Vrsio A Pag 7 of 8
8 Fr Rspos. Th Fr Rspos qustios will cout 70% of th total grad. Rad ach qustio carfully. To rciv full crdit, you must show lgibl, logical, ad rlvat justificatio which supports your fial aswr. Giv aswrs as xact valus.. ( pts.) A right triagl whos hypotus is 8 mtrs log is rvolvd about o of its lgs to grat a right circular co (s figur). Fid th radius ad hight that maximizs th volum of th co. I your work you should: Stat th fuctio to b optimizd i trms of a sigl variabl. Stat th domai of th fuctio. Show all work dd to fid ad vrify th valus of r ad h that maximiz th volum. h r 8 Not: Th volum V of a right circular co is V r h V r h, whr h r 8 r 8 h V ( h) (8 h ) h (8 h h ) V h h h h V ( h) (8 h ) Solv V( h) 0 to fid critical poits ( ) (8 ) domai: 0, 8 or 0, 8 (8 h ) 0 8 h 0 h 6; h ( h oly solutio i domai) Vrify maximum volum at h 8 For a closd itrval: chck V (0) 0, V (), V 8 =0 (closd itrval mthod); or first drivativ tst or scod drivativ tst For a op itrval: First drivativ tst: V '( h) 0 o (0,) ad V '( h) 0 o (, 8), so thr is a maximum at h ; or apply scod drivativ tst V ( h) h 0 o ( 0, 8], so cocav dow at th critical poit h is a maximum. Solvig for r : r 8 ; r 8 Th maximum volum th co ca hav is cubic uits wh h m ad r m. Vrsio A Pag 8 of 8
9 Fids a volum fuctio i trms of a sigl variabl (two poits to rcogiz poits h r 8 ) Domai of volum fuctio poit Fids th critical valu poits Vrifis maximum poits Givs th volum maximizig valus for radius ad hight poit Subtract ½ poit for missig or icorrct drivativ otatio. Subtract ½ poit for ot showig wh takig squar root. Subtract poit for ot givig valus for both r ad h. Subtract ½ poit for ot icludig uits. Los poits for drawig a umbr li with + ad - ovr parts but o justificatio. Los poit if thy justify but do't say aythig about that maig it is a max. Vrsio A Pag 9 of 8
10 . (0 pts.) At a Fourth of July clbratio, a bottl rockt is fird straight upward from a picic tabl o mtr high. Th acclratio fuctio for th rockt is giv blow. Fid th vlocity ad positio fuctios for th rockt. t a( t) cos t m/s, whr t is tim sic th rockt lauchd t a( t) cost t v( t) si t C Apply v(0) si(0) C 0 0 C 0 C t v( t) si t m/s t s( t) cost t C Apply s(0) cos(0) 0 C 0 C C t s( t) cost t m Work o Problm Fids vt () up to costat poits Solvs for costat i vt () poit Stats complt vt () with uits poit Fids st () up to costat poits Solvs for costat i st () poits Stats complt st () with uits poit OK if calculat vt () ad st () without showig itgratio symbol ½ poit dductio for icorrct otatio with a maximum palty of poit for all otatio rrors Vrsio A Pag 0 of 8
11 . (0 pts.) Evaluat th limits. Us of L Hôpital s Rul must b idicatd ach tim it is usd, ithr symbolically or i words. No crdit will b awardd without supportig work. x0 x0 x0 bx b bl( bx) a. (5 pts.) lim x0 x bx b b l( bx) 0 lim x 0 L L lim lim bx b b( b) bx 0 x 0 bx b b bx b b(0) b b b b b b b ( ) ( ) ( (0)) ( ) ( ) ( ) (costat b>0) Rcogizs idtrmiat form (xplicitly or / poit implicitly) Applis L Hopital s Rul corrctly two tims poits (two poits ach applicatio) Substituts to gt fial aswr / poit Subtract ½ poit for failig to idicat us of L Hopital s Rul Subtract ½ poit for ach otatio rror with a maximum of o poit total for all otatio rrors (xcludig rrors idicatig us of L Hopital s Rul) Subtract ½ poit for statmt: aythig = a idtrmiat form Subtract ½ poit for wrog idtrmiat form b b b b Vrsio A Pag of 8
12 b. (5 pts.) lim x x ta x ta x lim x x L lim x lim x l ta x x x x ta l x x lim ta l 0 x x x l 0 lim x cot 0 lim x x csc x csc Rcogizs idtrmiat form (xplicitly) / poit Rwrits usig ad atural log fuctio / poit Limit i xpot / poit Uss log proprty to mov xpot / poit Divids by rciprocal of tagt fuctio (OK if / poit covrtd to si ad cosi) Applis L Hopital s Rul corrctly ( poit for poits umrator ad poit for domiator) Substituts to gt fial aswr / poit Subtract ½ poit for failig to idicat us of L Hopital s Rul Subtract ½ poit for ach otatio rror with a maximum of o poit total for all otatio rrors (xcludig rrors idicatig us of L Hopital s Rul) Subtract ½ poit for statmt: aythig = a idtrmiat form Subtract ½ poit for wrog idtrmiat form Max of /5 if divids by th wrog rciprocal Othr tchiqus OK o Dfi y as fuctio, atural log of both sids o Dfi y as limit i xpot, fid valu, th xpotiat at th d Vrsio A Pag of 8
13 . (0 pts.) Lt f( x) x a) (5 pts.) Dtrmi th quatio(s) of ay horizotal asymptots o th graph of f( x ). lim 0 x x y 0 lim x x 0 y Fids limit as x Equatio of horizotal asymptot as x Fids limit as x Equatio of horizotal asymptot as x poits / poit poits / poit b) (5 pts.) Dtrmi th itrvals o which f( x) is icrasig or dcrasig. B sur to show th calculatio of th first drivativ. Put your fial aswrs i th appropriat spacs blow. f( x) f( x) x x f( x) x ( ) x ( ) x x ( ) ( ) Calculats first drivativ Icrasig itrval Dcrasig itrval (OK if lft blak) poits poit poit f( x) 0 has o solutio f ( x) is dfid for all x o critical valus f ( x) 0 for all x Icrasig: (, ) Dcrasig: o itrvals Vrsio A Pag of 8
14 c) (5 pts.) Th scod drivativ of f( x ) is show blow. Us it to dtrmi th itrvals o which f( x) is cocav up or cocav dow. Put you fial aswrs i th appropriat spacs blow. f x x ( ) ''( x) x ( ) x x ( ) f( x) x ( ) f( x) 0 x x ( ) 0 x ( ) x x 0 x 0 Solvs f( x) 0 poits Cocav up itrval poit Cocav dow itrval poit f ( x) 0 for all x 0 f ( x) 0 for all x 0 Cocav Up: (,0) Cocav Dow: (0, ) d) (5 pts.) Sktch f( x) x. Show th ordrd pair (x, y) at ay poit whr f has a local xtrm or a iflctio poit. Labl all axis itrcpts. Show th quatio of ay horizotal asymptots o th graph. Horizotal asymptots (OK if is ot labld) y 0 y-itrcpt Basic shap (cocavity, icrasig, o-liar, o x- itrcpt) poits poit poits Vrsio A Pag of 8
15 5. ( pts.) Cosidr th limit blow. * lim (xi ) x, whr i th itrval [0, 5] is partitiod ito subitrvals of width * xi is th right dpoit of th i th subitrval 5 x a. ( pts.) Exprss th limit as a dfiit itgral. 5 * lim ( i ) ( ) i 0 x x x dx Itgrad Limits of itgratio (/ ach) poit poit b. (7 pts.) Usig th summatio formulas blow as dd, valuat th limit. ( ) ( )( ) ( ) c c, i, i, i 6 i i i i 5i 5 x x x x * * lim ( i ), i, i 5i 5 lim i 50i 5 lim i 50i 5 lim lim i i 50 5 lim lim i i i 50 ( ) 5 lim lim 5lim lim 5 5() 5 0 dtrmis Substituts * x i x ito f( x ) * i Summad formula i trms of i ad Uss summatio formulas to gt a xprssio o oly Evaluats limit (OK if o work show to rsolv idtrmiat form ) Fial aswr poit poit poit poits poit poit Vrsio A Pag 5 of 8
16 c. ( pts.) Evaluat th dfiit itgral by usig basic ara formulas. Iclud a sktch Or (x ) dx Ara of rctagl + ara of triagl (5)() + (5)(0) (x ) dx Ara of trapzoid Sktchs rgio Fial aswr with supportig work: rctagl + triagl OR ara of a trapzoid poit poit Vrsio A Pag 6 of 8
17 6. (5 pts.) Th air forc dcids to tst a xprimtal jt by flyig it from o military bas to aothr military bas 00 mils away. Th jt maks this trip i xactly 0 miuts. Popl complai of harig a soic boom, but th air forc dis thir jt vr brok th soud barrir of 768 mils pr hour. Us calculus to show that th jt must hav brok th soud barrir at som tim durig its flight. B sur to stat ay rlvat thorms from calculus you us i makig your coclusio. Lt st () b th distac (mils) travld by th jt at tim t (hours), whr t [0,0.5]. Th avrag vlocity for th trip is s(0.5) s(0) MPH Sic th positio fuctio is cssarily cotiuous ad diffrtiabl, th Ma Valu Thorm applis. Calculats avrag vlocity Mtios cotiuous ad diffrtiabl (/ poit ach) Coclusio that mtios th Ma Valu Thorm poits poit poits Do t d to dfi a positio fuctio; OK if just calculat avrag vlocity Do t d a complt stc fial aswr, but th argumt should b clar By th Ma Valu Thorm, thr must b som tim c (0,0.5) such that s( c) v( c) Thrfor, th jt must hav brok th soud barrir at som tim durig its flight. Vrsio A Pag 7 of 8
18 Scatro ( pt.) My Scatro: Chck to mak sur your Scatro form mts th followig critria. If ay of th itms ar NOT satisfid wh your Scatro is hadd i ad/or wh your Scatro is procssd o poit will b subtractd from your tst total. is bubbld with firm marks so that th form ca b machi rad; is ot damagd ad has o stray marks (th form ca b machi rad); has 0 bubbld i aswrs; has MATH 060 ad my sctio umbr writt at th top; has my istructor s last am writt at th top; has Tst No. writt at th top; has th corrct tst vrsio writt at th top ad bubbld i blow my XID; shows my corrct XID both writt ad bubbld i; Bubbl a zro for th ladig C i your XID. Plas rad ad sig th hoor pldg blow. O my hoor, I hav ithr giv or rcivd iappropriat or uauthorizd iformatio at ay tim bfor or durig this tst. Studt s Sigatur: Vrsio A Pag 8 of 8
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
More informationMONTGOMERY 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
More informationChapter 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
More informationPURE 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
More informationStudent s Printed Name:
Studet s Prited Name: Istructor: XID: C Sectio: No questios will be aswered durig this eam. If you cosider a questio to be ambiguous, state your assumptios i the margi ad do the best you ca to provide
More informationReview Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2
MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm
More information+ 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.
More informationChapter 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
More informationProblem 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
More informationStatistics 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
More informationOption 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
More informationTime : 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,
More informationH2 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
More informationCalculus & 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
More information1973 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
More informationProbability & 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
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationSolution to 1223 The Evil Warden.
Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud
More informationz 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
More informationDiscrete 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
More informationWorksheet: 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.
More informationWashington 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
More informationامتحانات 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:
More informationA 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
More informationTriple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling
Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga (186-1871) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical
More informationTaylor 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
More information07 - 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
More informationFooling 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
More informationChapter (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
More informationCDS 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
More information2617 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"
More informationare 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
More informationNET/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
More informationCDS 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
More informationln 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
More informationDTFT 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
More informationChapter 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
More informationLECTURE 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
More informationAP 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)
More informationCOLLECTION 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.
More informationWBJEEM 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
More informationLectures 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
More informationChapter 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
More informationPhysics 302 Exam Find the curve that passes through endpoints (0,0) and (1,1) and minimizes 1
Physis Exam 6. Fid th urv that passs through dpoits (, ad (, ad miimizs J [ y' y ]dx Solutio: Si th itgrad f dos ot dpd upo th variabl of itgratio x, w will us th sod form of Eulr s quatio: f f y' y' y
More informationMath 105: Review for Final Exam, Part II - SOLUTIONS
Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio f(x) = x 3 lx o the iterval [/e, e ]. (a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum
More informationChapter 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
More informationNational 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
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationDerivation of a Predictor of Combination #1 and the MSE for a Predictor of a Position in Two Stage Sampling with Response Error.
Drivatio of a Prdictor of Cobiatio # ad th SE for a Prdictor of a Positio i Two Stag Saplig with Rspos Error troductio Ed Stak W driv th prdictor ad its SE of a prdictor for a rado fuctio corrspodig to
More informationCALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.
AP Calculus AB Portfolio Project Multiple Choice Practice Name: CALCULUS AB SECTION I, Part A Time 60 miutes Number of questios 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directios: Solve
More informationEquation Sheet Please tear off this page and keep it with you
ECE 30L, Exam Fall 05 Equatio Sht Plas tar off this ag ad k it with you Gral Smicoductor: 0 i ( EF EFi ) kt 0 i ( EFi EF ) kt Eg i N C NV kt 0 0 V IR L, D, τ, d d τ c g L D kt I J diff D D µ * m σ (µ +
More informationSession : Plasmas in Equilibrium
Sssio : Plasmas i Equilibrium Ioizatio ad Coductio i a High-prssur Plasma A ormal gas at T < 3000 K is a good lctrical isulator, bcaus thr ar almost o fr lctros i it. For prssurs > 0.1 atm, collisio amog
More informationSOLVED 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
More informationASSERTION 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
More informationOn 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
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationpage 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
More information4 x 4, and. where x is Town Square
Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and
More informationas a derivative. 7. [3.3] On Earth, you can easily shoot a paper clip straight up into the air with a rubber band. In t sec
MATH6 Fall 8 MIDTERM II PRACTICE QUESTIONS PART I. + if
More informationMILLIKAN OIL DROP EXPERIMENT
11 Oct 18 Millika.1 MILLIKAN OIL DROP EXPERIMENT This xprimt is dsigd to show th quatizatio of lctric charg ad allow dtrmiatio of th lmtary charg,. As i Millika s origial xprimt, oil drops ar sprayd ito
More informationA 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
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationNEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES
Digst Joural of Naomatrials ad Biostructurs Vol 4, No, March 009, p 67-76 NEW VERSION OF SZEGED INDEX AND ITS COMPUTATION FOR SOME NANOTUBES A IRANMANESH a*, O KHORMALI b, I NAJAFI KHALILSARAEE c, B SOLEIMANI
More informationn n ee (for index notation practice, see if you can verify the derivation given in class)
EN24: Computatioal mthods i Structural ad Solid Mchaics Homwork 6: Noliar matrials Du Wd Oct 2, 205 School of Egirig Brow Uivrsity I this homwork you will xtd EN24FEA to solv problms for oliar matrials.
More informationGeneral 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
More informationSTIRLING'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:
More informationAPPENDIX: 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.
More informationNarayana IIT Academy
INDIA Sc: LT-IIT-SPARK Dat: 9--8 6_P Max.Mars: 86 KEY SHEET PHYSIS A 5 D 6 7 A,B 8 B,D 9 A,B A,,D A,B, A,B B, A,B 5 A 6 D 7 8 A HEMISTRY 9 A B D B B 5 A,B,,D 6 A,,D 7 B,,D 8 A,B,,D 9 A,B, A,B, A,B,,D A,B,
More informationOrdinary Differential Equations
Ordiary Diffrtial Equatio Aftr radig thi chaptr, you hould b abl to:. dfi a ordiary diffrtial quatio,. diffrtiat btw a ordiary ad partial diffrtial quatio, ad. Solv liar ordiary diffrtial quatio with fid
More informationDiscrete Fourier Transform. Nuno Vasconcelos UCSD
Discrt Fourir Trasform uo Vascoclos UCSD Liar Shift Ivariat (LSI) systms o of th most importat cocpts i liar systms thory is that of a LSI systm Dfiitio: a systm T that maps [ ito y[ is LSI if ad oly if
More information3 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
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationECE602 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
More informationName: Math 10550, Final Exam: December 15, 2007
Math 55, Fial Exam: December 5, 7 Name: Be sure that you have all pages of the test. No calculators are to be used. The exam lasts for two hours. Whe told to begi, remove this aswer sheet ad keep it uder
More informationThomas J. Osler. 1. INTRODUCTION. This paper gives another proof for the remarkable simple
5/24/5 A PROOF OF THE CONTINUED FRACTION EXPANSION OF / Thomas J Oslr INTRODUCTION This ar givs aothr roof for th rmarkabl siml cotiud fractio = 3 5 / Hr is ay ositiv umbr W us th otatio x= [ a; a, a2,
More informationThe 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!
More informationOn 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
More informationECE594I Notes set 6: Thermal Noise
C594I ots, M. odwll, copyrightd C594I Nots st 6: Thrmal Nois Mark odwll Uivrsity of Califoria, ata Barbara rodwll@c.ucsb.du 805-893-344, 805-893-36 fax frcs ad Citatios: C594I ots, M. odwll, copyrightd
More informationThe 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!
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationMTH 133 Solutions to Exam 2 November 16th, Without fully opening the exam, check that you have pages 1 through 12.
Name: Sectio: Recitatio Istructor: INSTRUCTIONS Fill i your ame, etc. o this first page. Without fully opeig the exam, check that you have pages through. Show all your work o the stadard respose questios.
More informationClass #24 Monday, April 16, φ φ φ
lass #4 Moday, April 6, 08 haptr 3: Partial Diffrtial Equatios (PDE s First of all, this sctio is vry, vry difficult. But it s also supr cool. PDE s thr is mor tha o idpdt variabl. Exampl: φ φ φ φ = 0
More informationIdeal crystal : Regulary ordered point masses connected via harmonic springs
Statistical thrmodyamics of crystals Mooatomic crystal Idal crystal : Rgulary ordrd poit masss coctd via harmoic sprigs Itratomic itractios Rprstd by th lattic forc-costat quivalt atom positios miima o
More informationMID-YEAR EXAMINATION 2018 H2 MATHEMATICS 9758/01. Paper 1 JUNE 2018
MID-YEAR EXAMINATION 08 H MATHEMATICS 9758/0 Paper JUNE 08 Additioal Materials: Writig Paper, MF6 Duratio: hours DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO READ THESE INSTRUCTIONS FIRST Write
More informationELG3150 Assignment 3
ELG350 Aigmt 3 Aigmt 3: E5.7; P5.6; P5.6; P5.9; AP5.; DP5.4 E5.7 A cotrol ytm for poitioig th had of a floppy dik driv ha th clodloop trafr fuctio 0.33( + 0.8) T ( ) ( + 0.6)( + 4 + 5) Plot th pol ad zro
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationMath 142, Final Exam. 5/2/11.
Math 4, Fial Exam 5// No otes, calculator, or text There are poits total Partial credit may be give Write your full ame i the upper right corer of page Number the pages i the upper right corer Do problem
More information(Reference: sections in Silberberg 5 th ed.)
ALE. Atomic Structur Nam HEM K. Marr Tam No. Sctio What is a atom? What is th structur of a atom? Th Modl th structur of a atom (Rfrc: sctios.4 -. i Silbrbrg 5 th d.) Th subatomic articls that chmists
More informationMaximum and Minimum Values
Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f
More informationFor example suppose we divide the interval [0,2] into 5 equal subintervals of length
Math 1206 Calculus Sec 1: Estimatig with Fiite Sums Abbreviatios: wrt with respect to! for all! there exists! therefore Def defiitio Th m Theorem sol solutio! perpedicular iff or! if ad oly if pt poit
More information(A) 0 (B) (C) (D) (E) 2.703
Class Questios 007 BC Calculus Istitute Questios for 007 BC Calculus Istitutes CALCULATOR. How may zeros does the fuctio f ( x) si ( l ( x) ) Explai how you kow. = have i the iterval (0,]? LIMITS. 00 Released
More informationMath 116 Second Midterm November 13, 2017
Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.
More informationNational 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
More information(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:
More informationLinear Algebra Existence of the determinant. Expansion according to a row.
Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit
More informationMATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.)
MATH A FINAL (7: PM VERSION) SOLUTION (Last edited December 5, 3 at 9:4pm.) Problem. (i) Give the precise defiitio of the defiite itegral usig Riema sums. (ii) Write a epressio for the defiite itegral
More informationOrdinary 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.
More informationMath 116 Practice for Exam 3
Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More information