# TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

Size: px
Start display at page:

Transcription

1 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS UNIT-I PARTIAL DIFFERENTIAL EQUATIONS PART-A. Elimit th ritrry ott & from = ( + )(y + ) = ( + )(y + ) Diff prtilly w.r.to & y hr p & q p = (y + ) ; q = ( + ) y (y + ) = p/ ; ( + ) = q/y = (p/)(q/y) 4y = pq. Form th PDE y limitig th ritrry futio from = f(y) = f(y), Diff prtilly w.r.to & y hr p & q p = f ( y). y q = f ( y). p/q = y/ p qy = 3. Form th PDE y limitig th ott d from = + y = + y, Diff. w.r. t. d y hr p & q p = - ; q = y - p ; q y p q y p qy 4. Fid th omplt itgrl of p + q =pq Put p =, q =, p + q =pq += = - Th omplt itgrl i = + 5. Fid th olutio of p q y + = +y+ ----() i th rquird olutio giv p q -----() put p=, q = i () ( ) ( ) y

2 6. Fid th Grl olutio of p t + q ty = t. d dy d t t y t ot d ot y dy ot d tk ot d ot y dy ot y dy ot d log i log i y log log i y log i log i i y i y i i i y, i y i 7. Elimit th ritrry futio f from f f d form th PDE. p f q f y ; ( ) p py q q y 8. Fid th qutio of th pl who tr li o th -i Grl form of th phr qutio i Whr r i ott. From () +(-) p= () y +(-) q = (3) From () d (3) p q 9. Elimit th ritrry ott y p ; q r (), Tht i py -q = whih i rquird PDE. y d form th PDE. p qy p q. Fid th igulr itgrl of p qy pq Th omplt olutio i y ; y ; y ( y) ( ) y ( y. ) y y y y y

3 . Fid th grl olutio of p+qy= Th uiliry qutio i From d dy y y. dy d y y o implifyig Thrfor, y d dy d Itgrtig w gt i grl olutio.. Fid th grl olutio of p +qy = d dy d Th uiliry qutio i d dy From y dy d Alo y Thrfor 3. Solv Auiliry qutio i 4. Solv 5. Solv. log = log y + log Itgrtig w gt y Itgrtig w gt y, y y D DD 3D i grl olutio. m m 3, 3 Th olutio i f y f y 3 D 4DD 3D Auiliry qutio i m m, m, m 3 m 4m 3, 3 Th CF i CF f y f y 3 PI D 4DD 3D Put D, D Domitor =. PI D 4D Z=CF + PI f y f y 3 D 3DD 4D Auiliry qutio i C.F i = f(y + 4) + f(y - ) m m, m = 4, m = - m 3m 4, 4 m m, m, m 3

4 PI D 3DD 4D Fid P.I D 4DD 4D PI D 4DD 4D Put D, D PART-B. Solv PI D D p y q 6 y. Solv y p y q y 3. Solv m y l ly m 4. Solv 3 4y p 4 q y 3 5. Solv p yq 6. Solv y p yq 7. Solv y p q 8. Solv y p q 9. Solv. Solv. Solv. Solv 3. Solv 4. Solv 5. Solv 3 y D DD D 3 i(3 ) o o y D DD 6D y o D DD 3D y 6 D 6DD 5D ih y y D 4DD 4D 3 3 D D D DD D o( ) 6. Solv 7. Solv p qy p q p qy p q 8. Solv p q 9. Solv. Solv (i) ( p q ) ( p q ) (ii) ( p q )

5 UNIT-II FOURIER SERIES PART-A. Dfi R.M.S vlu. If lt f() futio dfid i th itrvl (, ), th th R.M.S vlu of f() i dfid y y f ( ) d. Stt Prvl Thorm. Lt f() priodi futio with priod l dfid i th itrvl (, +l). l o f ( ) d l 4 Whr, & r Fourir ott o 3. Dfi priodi futio with mpl. If futio f() tifi th oditio tht f( + T) = f(), th w y f() i priodi futio with th priod T. Empl:- i) Si, o r priodi futio with priod ii) t i r priodi futio with priod. 4. Stt Dirihlt oditio. (i) f() i igl vlud priodi d wll dfid pt poily t Fiit umr of poit. (ii) f () h t mot fiit umr of fiit diotiuou d o ifiit Diotiuou. (iii) f () h t mot fiit umr of mim d miim. 5. Stt Eulr formul. I (, l) o f o i l whr f ( ) d o l f ( )o d l f ( )i d 6. Writ Fourir ott formul for f() i th itrvl (, ) o f ( ) d f ( )o d f ( )i d

6 7. I th Fourir pio of, f() = i (-π, π ), fid th vlu of,. Si f(-)=f() th f() i v futio. H = 8. If f() = 3 i π < < π, fid th ott trm of it Fourir ri. Giv f() = 3 f(-) = (- ) 3 = - 3 = - f() H f() i odd futio Th rquird ott trm of th Fourir ri = o = 9. Wht r th ott trm d th offiit of o i th Fourir Epio f() = 3 i π < < π Giv f() = 3 f(-) = - - (- ) 3 = - [ - 3 ] = - f() H f() i odd futio Th rquird ott trm of th Fourir ri = =. Fid th vlu of for f() = ++ i (, ) o f ( ) d 3 ( ) d (i)fid i th pio of Fourir ri i (, ) (ii)fid i th pio of i Fourir ri i (, ) (i) Giv f() = f(-) = = f() H f() i v futio I th Fourir ri = (ii) Giv f() = i f(-) = (-)i(-) = i = f() H f() i v futio I th Fourir ri =. Oti th i ri for f l / l l / l Giv Giv f f Fourir i ri i l / l l / l l / l l / l f i l

7 l f ( )i d l l l i d ( l )i d l l l l l l o i o i l l () l l l( l ) l l ( ) l l l l l o l i l o l l i l l i 4li l f Fourir ri i 4l i i l 3. If f() i odd futio i ll,. Wht r th vlu of & If f() i odd futio, o =, = 4. I th Epio f() = Fourir ri i (-. ) fid th vlu of Giv f() = f(-) = - = = f() H f() i v futio o d 5. Fid hlf rg oi ri of f() =, i o d o i i d () Fourir ri i o o f o o

8 6. Fid th RMS vlu of f() =, Giv f() = R.M.S vlu l y f ( ) d d l PART-B. Epd f( ) (, ) Fourir ri d h ddu tht (, ) Fid th Fourir ri for f() = i (-. ) d lo prov tht (i) (ii) Epd f() = o Fourir ri i (-. ). 4. Fid oi ri for f() = i (-, ) u Prvl idtity to Show tht Epd f() = i Fourir ri i (, ) 6. Epd f() = Fourir ri i (-. ) d ddu to 7. If f( ), (,) i, (, ) Fid th Fourir ri up to od hrmoi Fid th Fourir ri d h ddu tht X Y Fid th Fourir ri up to third hrmoi X π/3 π/3 π 4π/3 5π/3 π F() Fid th Fourir pio of f ( ) ( ) i (, ) d H ddu tht Fid Fourir ri to rprt f ( ) with priod 3 i th rg (,3). Fid th Fourir ri of f i (, ).

9 3. Fid th Fourir ri for f i (, ) i (, ) d h.t Fid th th hlf rg i ri for f i th itrvl (, ) d ddu tht Oti th hlf rg oi ri for f i (,) d lo ddu tht Fid th Fourir ri for f() = i (-. ) d lo prov tht UNIT-III APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS PART-A u u. Clify th Prtil Diffrtil Equtio i) u u hr A=,B=,&C=- B - 4AC=-4()(-)=4>. Th Prtil Diffrtil Equtio i hyproli. u u u. Clify th Prtil Diffrtil Equtio y y u u u y y B -4AC=-4()()=>. hr A=,B=,&C= Th Prtil Diffrtil Equtio i hyproli. 3. Clify th followig od ordr Prtil Diffrtil qutio u u u u y hr A=,B=,&C= B -4AC=-4()()=-4<. Th Prtil Diffrtil Equtio i Ellipti. 4. Clify th followig od ordr Prtil Diffrtil qutio u u u u u y 4 u u u 4 6 u 8 u y hr A= 4,B =4, & C = B -4AC =6-4(4)() =. Th Prtil Diffrtil Equtio i Proli. 5. Clify th followig od ordr Prtil Diffrtil qutio i) y u yu u u 3u ii) y yy y u u u u y 7 i) Proli ii) Hyproli (If y = ) iii)ellipti (If y my +v or v) u u u u y

10 6. I th wv qutio y y t wht do td for? y y T hr t m T-Tio d m- M 7. I o dimiol ht qutio ut = α u wht do α td for? - u u t = k i lld diffuivity of th ut Whr k Thrml odutivity - Dity Spifi ht 8. Stt y two lw whih r umd to driv o dimiol ht qutio i) Ht flow from highr to lowr tmp ii) Th rt t whih ht flow ro y r i proportiol to th r d to th tmprtur grdit orml to th urv. Thi ott of proportiolity i kow th odutivity of th mtril. It i kow Fourir lw of ht odutio 9. A tightly trthd trig of lgth i ftd t oth d. Th midpoit of th trig i dipld to dit d rld from rt i thi poitio. Writ th iitil oditio. (i) y(, t) = (ii) y(,t) = y (iii) t t (iv) y(, ) = ( ). Wht r th poil olutio of o dimiol Wv qutio? Th poil olutio r y(,t) = (A + B - ) (C t + D - t ) y(,t) = (A o + B i )( C o t + D i t) y(,t) = (A + B) ( Ct + D). Wht r th poil olutio of o dimiol hd flow qutio? Th poil olutio r u(, t) ( A B ) C u(, t) ( Ao Bi ) C u(, t) ( A B) C t. Stt Fourir lw of ht odutio Q u ka t (th rt t whih ht flow ro r A t dit from o d of r i proportiol to tmprtur grdit) Q=Qutity of ht flowig k Thrml odutivity, A=r of ro tio ; u =Tmprtur grdit

11 3. Wht r th poil olutio of two dimiol hd flow qutio? Th poil olutio r u(, y) ( A B )( C o y Di y) y y u(, y) ( Ao Bi )( C D ) u(, y) ( A B)( Cy D) 4. Th tdy tt tmprtur ditriutio i oidrd i qur plt with id =,y =, = d y =. Th dg y = i kpt t ott tmprtur T d th thr dg r iultd. Th m tt i otiud uqutly. Epr th prolm mthmtilly. U(,y) =, U(,y) =,U(,) =, U(,) = T. PART-B. A tightly trthd trig with fid d poit = d = l i iitilly t rt i it quilirium poitio. If it i t virtig givig h poit vloity 3 (l-). Fid th diplmt.. A trig i trthd d ftd to two poit d prt. Motio i trtd y diplig th trig ito th form y = K(l- ) from whih it i rld t tim t =. Fid th diplmt t y poit of th trig. 3. A trig of lgth l i ftd t oth d. Th midpoit of trig i tk to hight d th rld from rt i tht poitio. Fid th diplmt of th trig. 4. A tightly trthd trig with fid d poit = d = l i iitilly t rt i it poitio giv 3 y y(, ) = y i. If it i rld from rt fid th diplmt. l 5. A trig i trthd tw two fid poit t dit l prt d poit of th trig r < < l giv iitil vloiti whr V. Fid th diplmt. ( l ) < < l 6. Driv ll poil olutio of o dimiol wv qutio. Driv ll poil olutio of o dimiol ht qutio. Driv ll poil olutio of two dimiol ht qutio. 7. A rod 3 m log h it d A d B kpt t o C d 8 o C, rptivly util tdy tt oditio prvil. Th tmprtur t h d i th rdud to o C d kpt o. Fid th rultig tmprtur u(, t) tkig =. 8. A rtgulr plt with iultd urf i 8 m wid o log omprd to it width tht ita r m log, with iultd id h it d A & B kpt t o C d 4 o C rptivly util th tdy tt oditio prvil. Th tmprtur t A i uddly rid to 5 o C d B i lowrd to o C. Fid th uqut tmprtur futio u(, t). 9. A rtgulr plt with iultd urf i 8 m wid d o log omprd to it width tht it my oidrd ifiit plt. If th tmprtur log hort dg y = i u (,) = i 8, <<8 Whil two dg = d = 8 wll th othr hort dg r kpt t o C. Fid th tdy tt tmprtur.. A rtgulr plt with iultd urf i m wid o log omprd to it width tht it my oidrd ifiit plt. If th tmprtur log hort dg y = i giv y 5 u d ll othr thr dg r kpt t o C. Fid th tdy tt ( ) 5 tmprtur t y poit of th plt.

12 UNIT-IV FOURIER TRANSFORMS PART-A. Stt Fourir Itgrl Thorm. If f( ) i pi wi otiuouly diffrtil d olutly o, th, i( t) f ( ) f t dt d.. Stt d prov Modultio thorm. F f o F F Proof: i F f o f o d i i i f d i( ) i( ) f d f d F F F f o F F 3. Stt Prvl Idtity. If F i Fourir trform of f, th F d f d 4. Stt Covolutio thorm. Th Fourir trform of Covolutio of f d g i th produt of thir Fourir trform. F f g F G 5. Stt d prov Chg of l of proprty. If F F f, th F f F i F f f d i t dt f t ; whr t, F f F

13 d 6. Prov tht if F[f()] = F() th F f ( ) ( i) F( ) d i F f d, Diff w.r.t tim d i F f i d d i f ( i) ( ) d d i ( ) F f d () i d d i ( i) F ( ) f d d d d F f i F 7. Solv for f() from th itgrl qutio f ( )o d f ( )o d F f f o d F f f ( ) F f o d o d o d o d,

14 8. Fid th ompl Fourir Trform of f( ) i F f f d ; i F f d (o ii ) d i (o ) d i [U v d odd proprty od trm om ro] 9. Fid th ompl Fourir Trform of f( ) i F f f d i d ; (o i i ) d i o i ( ( ii ) d () i o i [U v d odd proprty firt trm om ro]. Writ Fourir Trform pir. If f( ) i dfid i,, th it Fourir trform i dfid i F f d

15 If F i Fourir trform of f, th t vry poit of Cotiuity of f, w hv i f F d.. Fid th Fourir oi Trform of f() = - F f f o d F o d F. Fid th Fourir Trform of i F f f d f( ) im,, im i i m d d othrwi i m i m i m 3. Fid th Fourir i Trform of. o d i m i m F f f i d i d F 4. Fid th Fourir i trform of F f f i d F i d F i d

16 5. Fid th Fourir oi trform of F f f o d F o d o d o d 4 4, 6. Fid th Fourir i trform of f( ) F f f i d F f f i d f i d o i d o PART-B. Fid th Fourir Trform of f( ) if if d h o i 3 ddu tht (i) o 3 d 6. Fid th Fourir Trform of 3.Fid th Fourir Trform of i) i d ii) i f( ) f( ) d i o (ii) 3 d 5 i o. h S.T 3 d if if d h vlut if 4. Fid Fourir Trform of f( ) d h vlut if 4 i) i d ii) i 4 d

17 5. Evlut i) d d ii) 6. Evlut i) d d ii) d 7. Evlut () 4 () t dt t 4 t 9 i ; wh o 8. (i)fid th Fourir i trform of f( ) ; wh o ; wh o (ii) Fid th Fourir oi trform of f( ) ; wh 9. (i) Show tht Fourir trform i (ii)oti Fourir oi Trform of d h fid Fourir i Trform. (i) Solv for f() from th itgrl qutio (ii) Solv for f() from th itgrl qutio. Fid Fourir i Trform of f ( )o d, t f ( )i t d, t, > d h ddu tht, t i d. (i)fid FS & F (ii) Fid FS & F UNIT-V Z- TRANSFORMS PART-A. Dfi Z trform Lt {f()} qu dfid for =,, d f() = for < th it Z Trform i dfid Z f ( ) F f ( ) (Two idd trform) Z f ( ) F f ( ) (O idd trform). Fid th Z Trform of Z f f Z ()... Z

18 3. Fid th Z Trform of Z f f Z 4. Fid th Z Trform of d Z Z Z d, y th proprty, d d ( ) ( ) Stt Iitil & Fil vlu thorm o Z Trform Iitil Vlu Thorm If Z [f ()] = F () th f () = lim F ( ) Fil Vlu Thorm If Z [f ()] = F () th lim f ( ) lim( ) F( ) 6. Stt ovolutio thorm of Z- Trform. Z[f()] = F() d Z[g()] = G() th Z{f()*g()} = F() G() 7. Fid Z Trform of Z f f Z Fid Z Trform of o d i W kow tht Z f f

19 Z o Z i o o Similrly 9. Fid Z Trform of =>> Z i o o o Z o i i =>> o Z Z f f log log log. Fid Z Trform of! Z f f Z!! 3...!!! 3!

20 . Fid Z Trform of Z f f Z ( ) log log. Stt d prov Firt hiftig thorm t Sttmt: If Z f t F, th Z f () t F Proof: Z f ( t) f ( T ) t T A f(t) i futio dfid for dirt vlu of t, whr t = T, th th Z-trform i Z f ( t) f ( T ) F( ). t T T Z f ( t) f ( T ) F( ) 3. Dfi uit impul futio d uit tp futio. T Th uit mpl qu i dfid follow: for ( ) for Th uit tp qu i dfid follow: u ( ) for for 4. Fid Z Trform of t Z t T T T Z T [Uig Firt hiftig thorm]

21 5. Fid Z Trform of Z t t Z t t Z t T T T T T T [Uig Firt hiftig thorm] t 6. Fid Z Trform of Z o t o Z o t Z o t T o T T ot t T T T ot [Uig Firt hiftig thorm] 7. Fid Z Trform of t T Z Lt f (t) = t, y od iftig thorm ( t T) Z Z f t T F f ( ) ( ) () T 8. Fid Z Trform of Z i t T T T Lt f (t) = it, y od iftig thorm Z i( t T) Z f ( t T) F( ) f () i t i t t t o o. Fid Z trform of Z f f Z Z Z

22 PART-B. Fid (i) Z. Fid (i) Z 3. Fid (i) Z ( )(4 ) 8 (ii) Z (ii) Z ( )( ) ( ) (ii) Z ( ) ( )(4 ) 8 y ovolutio thorm. y ovolutio thorm ( )( 3) y ovolutio thorm 4. (i ) Stt d prov Iitil & Fil vlu thorm. (ii) Stt d prov Sod hiftig thorm 3 5. Fid th Z trform of (i) (ii) ( )( ) ( )( ) 6. (i) Fid Z ( 4) y ridu. (ii) Fid th ivr Z trform of 7. Fid (i) Z 8. (i)fid th Z trform of (ii) Fid (ii) Z f( )! ( ) 7 H fid y prtil frtio. Z d ( )! Z d lo fid th vlu of i( ) d o( ).! 9. Solv y 6y 9y with y & y. Solv y 4y 4y y() =,y() =. Solv y 3y 4y, giv y() 3& y (). Solv y 3 3y y, y 4, y & y 8, 3. Fid Z o & Z i d lo fid Z o & Z i Z. ( )!

### TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS UNIT-I PARTIAL DIFFERENTIAL EQUATIONS PART-A. Elimit th ritrry ott & from = ( + )(y + ) Awr: = ( + )(y + ) Diff prtilly w.r.to & y hr p & q y p = (y + ) ;

### Chapter 3 Fourier Series Representation of Periodic Signals

Chptr Fourir Sris Rprsttio of Priodic Sigls If ritrry sigl x(t or x[] is xprssd s lir comitio of som sic sigls th rspos of LI systm coms th sum of th idividul rsposs of thos sic sigls Such sic sigl must:

### 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

### 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.

### 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

### MM1. Introduction to State-Space Method

MM Itroductio to Stt-Spc Mthod Wht tt-pc thod? How to gt th tt-pc dcriptio? 3 Proprty Alyi Bd o SS Modl Rdig Mtril: FC: p469-49 C: p- /4/8 Modr Cotrol Wht th SttS tt-spc Mthod? I th tt-pc thod th dyic

### x, x, e are not periodic. Properties of periodic function: 1. For any integer n,

Chpr Fourir Sri, Igrl, d Tror. Fourir Sri A uio i lld priodi i hr i o poiiv ur p uh h p, p i lld priod o R i,, r priodi uio.,, r o priodi. Propri o priodi uio:. For y igr, p. I d g hv priod p, h h g lo

### Fourier Transform Methods for Partial Differential Equations

Itrtiol Jourl o Prtil Dirtil Equtio d Applitio,, Vol, No 3, -57 Avill oli t http://puipuom/ijpd//3/ Si d Edutio Pulihig DOI:69/ijpd--3- Fourir Trorm Mthod or Prtil Dirtil Equtio Nol Tu Ngro * Dprtmt o

### 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

### 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

### [ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is

Discrt-tim ourir Trsform Rviw or discrt-tim priodic sigl x with priod, th ourir sris rprsttio is x + < > < > x, Rviw or discrt-tim LTI systm with priodic iput sigl, y H ( ) < > < > x H r rfrrd to s th

### MA6351-TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS SUBJECT NOTES. Department of Mathematics FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY

MA635-TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS SUBJECT NOTES Deprtmet of Mthemtics FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY MADURAI 65, Tmildu, Idi Bsic Formule DIFFERENTIATION &INTEGRATION

### Quantum Mechanics & Spectroscopy Prof. Jason Goodpaster. Problem Set #2 ANSWER KEY (5 questions, 10 points)

Chm 5 Problm St # ANSWER KEY 5 qustios, poits Qutum Mchics & Spctroscopy Prof. Jso Goodpstr Du ridy, b. 6 S th lst pgs for possibly usful costts, qutios d itgrls. Ths will lso b icludd o our futur ms..

### Chapter 3 Higher Order Linear ODEs

ht High Od i ODEs. Hoogous i ODEs A li qutio: is lld ohoogous. is lld hoogous. Tho. Sus d ostt ultils of solutios of o so o itvl I gi solutios of o I. Dfiitio. futios lld lil iddt o so itvl I if th qutio

### On Gaussian Distribution

Prpr b Çğt C MTU ltril gi. Dpt. 30 Sprig 0089 oumt vrio. Gui itributio i i ollow O Gui Ditributio π Th utio i lrl poitiv vlu. Bor llig thi utio probbilit it utio w houl h whthr th r ur th urv i qul to

### 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

### How much air is required by the people in this lecture theatre during this lecture?

3 NTEGRATON tgrtio is us to swr qustios rltig to Ar Volum Totl qutity such s: Wht is th wig r of Boig 747? How much will this yr projct cost? How much wtr os this rsrvoir hol? How much ir is rquir y th

### Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.

Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%

### Section 5.1/5.2: Areas and Distances the Definite Integral

Scto./.: Ars d Dstcs th Dt Itgrl Sgm Notto Prctc HW rom Stwrt Ttook ot to hd p. #,, 9 p. 6 #,, 9- odd, - odd Th sum o trms,,, s wrtt s, whr th d o summto Empl : Fd th sum. Soluto: Th Dt Itgrl Suppos w

### IIT JEE MATHS MATRICES AND DETERMINANTS

IIT JEE MTHS MTRICES ND DETERMINNTS THIRUMURUGN.K PGT Mths IIT Trir 978757 Pg. Lt = 5, th () =, = () = -, = () =, = - (d) = -, = -. Lt sw smmtri mtri of odd th quls () () () - (d) o of ths. Th vlu of th

### PREPARATORY MATHEMATICS FOR ENGINEERS

CIVE 690 This qusti ppr csists f 6 pritd pgs, ch f which is idtifid by th Cd Numbr CIVE690 FORMULA SHEET ATTACHED UNIVERSITY OF LEEDS Jury 008 Emiti fr th dgr f BEg/ MEg Civil Egirig PREPARATORY MATHEMATICS

### INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..

### Lectures 2 & 3 - Population ecology mathematics refresher

Lcturs & - Poultio cology mthmtics rrshr To s th mov ito vloig oultio mols, th olloig mthmtics crisht is suli I i out r mthmtics ttook! Eots logrithms i i q q q q q q ( tims) / c c c c ) ( ) ( Clculus

### Q.28 Q.29 Q.30. Q.31 Evaluate: ( log x ) Q.32 Evaluate: ( ) Q.33. Q.34 Evaluate: Q.35 Q.36 Q.37 Q.38 Q.39 Q.40 Q.41 Q.42. Q.43 Evaluate : ( x 2) Q.

LASS XII Q Evlut : Q sc Evlut c Q Evlut: ( ) Q Evlut: Q5 α Evlut: α Q Evlut: Q7 Evlut: { t (t sc )} / Q8 Evlut : ( )( ) Q9 Evlut: Q0 Evlut: Q Evlut : ( ) ( ) Q Evlut : / ( ) Q Evlut: / ( ) Q Evlut : )

TWO MARKS WITH ANSWER MA65/TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS REGULATION: UNIT I PARTIAL DIFFERENTIAL EQUATIONS Formtio o rti dirti utio Sigur itgr -- Soutio o tdrd t o irt ordr rti dirti utio

### Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals

Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion

### Vtusolution.in FOURIER SERIES. Dr.A.T.Eswara Professor and Head Department of Mathematics P.E.S.College of Engineering Mandya

LECTURE NOTES OF ENGINEERING MATHEMATICS III Su Cod: MAT) Vtusoutio.i COURSE CONTENT ) Numric Aysis ) Fourir Sris ) Fourir Trsforms & Z-trsforms ) Prti Diffrti Equtios 5) Lir Agr 6) Ccuus of Vritios Tt

### UNIT I FOURIER SERIES T

UNIT I FOURIER SERIES PROBLEM : Th urig mom T o h crkh o m gi i giv or ri o vu o h crk g dgr 6 9 5 8 T 5 897 785 599 66 Epd T i ri o i. Souio: L T = i + i + i +, Sic h ir d vu o T r rpd gc o T T i T i

### Chapter 6 Perturbation theory

Ct 6 Ptutio to 6. Ti-iddt odgt tutio to i o tutio sst is giv to fid solutios of λ ' ; : iltoi of si stt : igvlus of : otool igfutios of ; δ ii Rlig-Södig tutio to ' λ..6. ; : gl iltoi ': tutio λ : sll

### APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS

Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M ISSN 5-353 APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS S.M.Khirnr, R.M.Pi*, J.N.Slun** Dprtmnt o Mthmti Mhrhtr

### Linear 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

### 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

### Integration by Guessing

Itgrtio y Gussig Th computtios i two stdrd itgrtio tchiqus, Sustitutio d Itgrtio y Prts, c strmlid y th Itgrtio y Gussig pproch. This mthod cosists of thr stps: Guss, Diffrtit to chck th guss, d th Adjust

### 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

### IX. Ordinary Differential Equations

IX. Orir Diffrtil Equtios A iffrtil qutio is qutio tht iclus t lst o rivtiv of uow fuctio. Ths qutios m iclu th uow fuctio s wll s ow fuctios of th sm vribl. Th rivtiv m b of orr thr m b svrl rivtivs prst.

### BRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I

EXERCISE I t Q. d Q. 6 6 cos si Q. Q.6 d d Q. d Q. Itegrte cos t d by the substitutio z = + e d e Q.7 cos. l cos si d d Q. cos si si si si b cos Q.9 d Q. si b cos Q. si( ) si( ) d ( ) Q. d cot d d Q. (si

### Rectangular Waveguides

Rtgulr Wvguids Wvguids tt://www.tllguid.o/wvguidlirit.tl Uss To rdu ttutio loss ig rquis ig owr C ort ol ov rti rquis Ats s ig-ss iltr Norll irulr or rtgulr W will ssu losslss rtgulr tt://www..surr..u/prsol/d.jris/wguid.tl

### (2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

. DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,

### Consider serial transmission. In Proakis notation, we receive

5..3 Dciio-Dirctd Pha Trackig [P 6..4] 5.-1 Trackr commoly work o radom data igal (plu oi), o th kow-igal modl do ot apply. W till kow much about th tructur o th igal, though, ad w ca xploit it. Coidr

### New Advanced Higher Mathematics: Formulae

Advcd High Mthmtics Nw Advcd High Mthmtics: Fomul G (G): Fomul you must mmois i od to pss Advcd High mths s thy ot o th fomul sht. Am (A): Ths fomul giv o th fomul sht. ut it will still usful fo you to

### 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:,,,,,

### COMSACO INC. NORFOLK, VA 23502

YMOL 9. / 9. / 9. / 9. YMOL 9. / 9. OT:. THI RIG VLOP ROM MIL--/ MIL-TL-H, TYP II, L ITH VITIO OLLO:. UPO RULT O HOK TTIG, HOK MOUT (ITM, HT ) HV IR ROM 0.0 THIK TO 0.090 THIK LLO Y MIL-TL-H, PRGRPH...

### Ordinary 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

### Problem Session (3) for Chapter 4 Signal Modeling

Pobm Sssio fo Cht Sig Modig Soutios to Pobms....5. d... Fid th Pdé oimtio of scod-od to sig tht is giv by [... ] T i.. d so o. I oth wods usig oimtio of th fom b b b H fid th cofficits b b b d. Soutio

### Section 3: Antiderivatives of Formulas

Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin

### DEPARTMENT OF MATHEMATICS BIT, MESRA, RANCHI MA2201 Advanced Engg. Mathematics Session: SP/ 2017

DEARMEN OF MAEMAICS BI, MESRA, RANCI MA Advad Egg. Mathatis Sssio: S/ 7 MODULE I. Cosidr th two futios f utorial Sht No. -- ad g o th itrval [,] a Show that thir Wroskia W f, g vaishs idtially. b Show

### Frequency Response & Digital Filters

Frquy Rspos & Digital Filtrs S Wogsa Dpt. of Cotrol Systms ad Istrumtatio Egirig, KUTT Today s goals Frquy rspos aalysis of digital filtrs LTI Digital Filtrs Digital filtr rprstatios ad struturs Idal filtrs

### Calculus Cheat Sheet. ( x) Relationship between the limit and one-sided limits. lim f ( x ) Does Not Exist

Clulus Cht Sht Limits Dfiitios Pris Dfiitio : W sy lim f L if Limit t Ifiity : W sy lim f L if w for vry ε > 0 thr is δ > 0 suh tht mk f ( ) s los to L s w wt y whvr 0 < < δ th f L < ε. tkig lrg ough positiv.

### I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o

I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o u l d a l w a y s b e t a k e n, i n c l u d f o l

### a f(x)dx is divergent.

Mth 250 Exm 2 review. Thursdy Mrh 5. Brig TI 30 lultor but NO NOTES. Emphsis o setios 5.5, 6., 6.2, 6.3, 3.7, 6.6, 8., 8.2, 8.3, prt of 8.4; HW- 2; Q-. Kow for trig futios tht 0.707 2/2 d 0.866 3/2. From

### ROUTH-HURWITZ CRITERION

Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7 ROUTH-HURWITZ CRITERION The Routh-Hurwitz riterio i lytil proedure for determiig

### Chapter #2 EEE Subsea Control and Communication Systems

EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio

### TOPIC 5: INTEGRATION

TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function

### The z-transform. Dept. of Electronics Eng. -1- DH26029 Signals and Systems

0 Th -Trsform Dpt. of Elctroics Eg. -- DH609 Sigls d Systms 0. Th -Trsform Lplc trsform - for cotios tim sigl/systm -trsform - for discrt tim sigl/systm 0. Th -trsform For ipt y H H h with ω rl i.. DTFT

### Right Angle Trigonometry

Righ gl Trigoomry I. si Fs d Dfiiios. Righ gl gl msurig 90. Srigh gl gl msurig 80. u gl gl msurig w 0 d 90 4. omplmry gls wo gls whos sum is 90 5. Supplmry gls wo gls whos sum is 80 6. Righ rigl rigl wih

### AE57/AC51/AT57 SIGNALS AND SYSTEMS DECEMBER 2012

AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER Q. Drmi powr d rgy of h followig igl j i ii =A co iii = Solio: i E P I I l jw l I d jw d d Powr i fii, i i powr igl ii =A cow E P I co w d / co l I I l d wd d Powr

### Trigonometric Formula

MhScop g of 9 FORMULAE SHEET If h lik blow r o-fucioig ihr Sv hi fil o your hrd driv (o h rm lf of h br bov hi pg for viwig off li or ju coll dow h pg. [] Trigoomry formul. [] Tbl of uful rigoomric vlu.

### DFT: Discrete Fourier Transform

: Discrt Fourir Trasform Cogruc (Itgr modulo m) I this sctio, all lttrs stad for itgrs. gcd m, = th gratst commo divisor of ad m Lt d = gcd(,m) All th liar combiatios r s m of ad m ar multils of d. a b

### P a g e 5 1 of R e p o r t P B 4 / 0 9

P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e

### Further Results on Pair Sum Graphs

Applid Mathmatis, 0,, 67-75 http://dx.doi.org/0.46/am.0.04 Publishd Oli Marh 0 (http://www.sirp.org/joural/am) Furthr Rsults o Pair Sum Graphs Raja Poraj, Jyaraj Vijaya Xavir Parthipa, Rukhmoi Kala Dpartmt

### (HELD ON 22nd MAY SUNDAY 2016) MATHEMATICS CODE - 2 [PAPER -2]

QUESTION PAPER WITH SOLUTION OF JEE ADVANCED - 6 7. Lt P (HELD ON d MAY SUNDAY 6) FEEL THE POWER OF OUR KNOWLEDGE & EXPERIENCE Our Top clss IITi fculty tm promiss to giv you uthtic swr ky which will b

### CSE303 - Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata

CSE303 - Introduction to th Thory of Computing Smpl Solutions for Exrciss on Finit Automt Exrcis 2.1.1 A dtrministic finit utomton M ccpts th mpty string (i.., L(M)) if nd only if its initil stt is finl

### Lectures 5-8: Fourier Series

cturs 5-8: Fourir Sris PHY6 Rfrcs Jord & Smith Ch.6, Bos Ch.7, Kryszig Ch. Som fu jv pplt dmostrtios r vilbl o th wb. Try puttig Fourir sris pplt ito Googl d lookig t th sits from jhu, Flstd d Mths Oli

### SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11

UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum

### 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

### 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,

### EE Control Systems LECTURE 11

Updtd: Tudy, Octor 8, EE 434 - Cotrol Sytm LECTUE Copyright FL Lwi 999 All right rrvd BEEFTS OF FEEBACK Fdc i uivrl cocpt tht ppr i turl ytm, itrctio of pci, d iologicl ytm icludig th ic cll d mucl cotrol

### Weighted graphs -- reminder. Data Structures LECTURE 15. Shortest paths algorithms. Example: weighted graph. Two basic properties of shortest paths

Dt Strutur LECTURE Shortt pth lgorithm Proprti of hortt pth Bllmn-For lgorithm Dijktr lgorithm Chptr in th txtook (pp ). Wight grph -- rminr A wight grph i grph in whih g hv wight (ot) w(v i, v j ) >.

### b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?

MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth

### Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.

Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,

### Signals & Systems - Chapter 3

.EgrCS.cm, i Sigls d Sysms pg 9 Sigls & Sysms - Chpr S. Ciuus-im pridic sigl is rl vlud d hs fudml prid 8. h zr Furir sris cfficis r -, - *. Eprss i h m. cs A φ Slui: 8cs cs 8 8si cs si cs Eulrs Apply

### Fast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation

Algorithm Desig d Alsis Victor Admchi CS 5-45 Sprig 4 Lecture 3 J 7, 4 Cregie Mello Uiversit Outlie Fst Fourier Trsform ) Legedre s Iterpoltio ) Vdermode Mtri 3) Roots of Uit 4) Polomil Evlutio Guss (777

### UNIT V: Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Dr. V. Valliammal Department of Applied Mathematics Sri Venkateswara College of Engineering

UNIT V: -TRANSFORMS AND DIFFERENCE EQUATIONS D. V. Vllimml Deptmet of Applied Mthemtics Si Vektesw College of Egieeig TOPICS:. -Tsfoms Elemet popeties.. Ivese -Tsfom usig ptil fctios d esidues. Covolutio

### Physics 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

### 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

### The Z transform techniques

h Z trnfor tchniqu h Z trnfor h th rol in dicrt yt tht th Lplc trnfor h in nlyi of continuou yt. h Z trnfor i th principl nlyticl tool for ingl-loop dicrt-ti yt. h Z trnfor h Z trnfor i to dicrt-ti yt

### 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

### 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

### CLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC

CLSS XI ur I CHPTER.6. Proofs d Simpl pplictios of si d cosi formul Lt C b trigl. y gl w m t gl btw t sids d C wic lis btw 0 d 80. T gls d C r similrly dfid. T sids, C d C opposit to t vrtics C, d will

### ( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition

Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of

### Model of the multi-level laser

Modl of th multilvl lsr Trih Dih Chi Fulty of Physis, Collg of turl Sis, oi tiol Uivrsity Tr Mh ug, Dih u Kho Fulty of Physis, Vih Uivrsity Astrt. Th lsr hrtristis dpd o th rgylvl digrm. A rsol rgylvl

### Digital Signal Processing, Fall 2006

Digital Sigal Procssig, Fall 6 Lctur 9: Th Discrt Fourir Trasfor Zhg-Hua Ta Dpartt of Elctroic Systs Aalborg Uivrsity, Dar zt@o.aau.d Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Cours at a glac MM Discrt-ti

### MAS221 Analysis, Semester 2 Exercises

MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)

### 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

### VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS

Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of

### k m The reason that his is very useful can be seen by examining the Taylor series expansion of some potential V(x) about a minimum point:

roic Oscilltor Pottil W r ow goig to stuy solutios to t TIS for vry usful ottil tt of t roic oscilltor. I clssicl cics tis is quivlt to t block srig robl or tt of t ulu (for sll oscilltios bot of wic r

### Helping every little saver

Spt th diffc d cut hw u c fid I c spt thigs! Hlpig v littl sv Hw d u p i? I ch Just pp it f u chs. T fid u lcl ch just visit s.c.uk/ch If u pig i chqu, it c tk ud 4 wkig ds t cl Ov th ph Just cll Tlph

### Math 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.

Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right

### Chapter 7 Infinite Series

MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2

### Eigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):

Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )

### HIGHER ORDER DIFFERENTIAL EQUATIONS

Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution

### CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018

CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs

### /99 \$10.00 (c) 1999 IEEE

P t Hw Itt C Syt S 999 P t Hw Itt C Syt S - 999 A Nw Atv C At At Cu M Syt Y ZHANG Ittut Py P S, Uvty Tuu, I 0-87, J Att I t, w tv t t u yt x wt y tty, t wt tv w (LBSB) t. T w t t x t tty t uy ; tt, t x

### Introduction to Medical Imaging. Lecture 4: Fourier Theory = = ( ) 2sin(2 ) Introduction

Introduction Introduction to Mdical aging Lctur 4: Fourir Thory Thory dvlopd by Josph Fourir (768-83) Th Fourir transform of a signal s() yilds its frquncy spctrum S(k) Klaus Mullr s() forward transform

### Emil Olteanu-The plane rotation operator as a matrix function THE PLANE ROTATION OPERATOR AS A MATRIX FUNCTION. by Emil Olteanu

Emil Oltu-Th pl rottio oprtor s mtri fuctio THE PLNE ROTTON OPERTOR S MTRX UNTON b Emil Oltu bstrct ormlism i mthmtics c offr m simplifictios, but it is istrumt which should b crfull trtd s it c sil crt

### Classical Theory of Fourier Series : Demystified and Generalised VIVEK V. RANE. The Institute of Science, 15, Madam Cama Road, Mumbai

Clssil Thoy o Foi Sis : Dmystii Glis VIVEK V RANE Th Istitt o Si 5 Mm Cm Ro Mmbi-4 3 -mil ss : v_v_@yhoooi Abstt : Fo Rim itgbl tio o itvl o poit thi w i Foi Sis t th poit o th itvl big ot how wh th tio

### 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

### EEE 303: Signals and Linear Systems

33: Sigls d Lir Sysms Orhogoliy bw wo sigls L us pproim fucio f () by fucio () ovr irvl : f ( ) = c( ); h rror i pproimio is, () = f() c () h rgy of rror sigl ovr h irvl [, ] is, { }{ } = f () c () d =