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  1. 1. axis of parabola .3. J ‘ I-‘om! dismntc FV " n Ln. -nglh of Imus rectum (LR) = 4.1 I Gencral Fquations: :1. Axis Vcnjcznl A s’ - 1); »: s Eyi I-‘-~o b. Axis llorizonxlnl . Cy-‘ + Dx o | Z_v 4 r»' = 0 ll . ‘t: md: |rd I-Zqualimrsz ‘ zn. wncx at V ( II. R) l. (X -- h)1= -ln( y - k) . opens upward 2. (x -- h)’ = -4a(_v — k) . opens downward 3. (y - kM)’= 4:: (Mx - It). opens to the right -I. (_' K): ’- -In (x M. opens lo the left b. vcmrx an origin : V ( O. 0) I. x’ 4ay . opens upward x’ - -day . opens do'nvnrd V’ -" -lax . opens to the righl . . ’ ‘ 4. y‘ ' -«lax . opens Iutlu: left 2. 3 o " ‘ o1'; a‘pI. ~wl| icl1 moves soélhnl the gum of Ike distances from two fixed points, c‘a1le‘d ‘fO¢i. i5‘0¢2fl3.13"1.: _ V‘
  2. 2. "Vt * IKV. ' F. V, + | -‘N1 ,1’ Bl“ Flvl ‘ F3‘/3 Thmlbve; NW: :gu‘; "|; ‘._/ i: PM = v. v, -= In _. eu; . ° two" I General equation: Ax‘+Cy’+Dx+Ey+[= .-.0 When: A at C and ( A and C have the S'. lmL' sign) ll Standard Equation: A. Center at (h. k) x — 3 '— k)’ l- (‘a-'') + (Lb'! ‘~ 1 . MA: Horizontal 2. ("_b; _“)3+ (>_'_-J03: 1 , MA : Vertical . 3. B. Center at ( 0, 0) l. x 3 3 _ 7 + _‘lZ: T- 1 . MA 2 Horizontal 2. x 1 3 _ b: + -2? — l , MA : Vertical Relations of constants: a’ = b’ + c’ Semi - Major Axis = CV. = CV1 = a Seml- Minor Axis = CB , =CB3 = b C = CF, = CF; MA = Major Axis = V. V, = 2a ma = minor axis = B. B, = 2b LR '= 2b’ / a Eccentricity e = cl 3 d = distance of the directrix from the center = a I e Z°. °°. ".°‘E": "‘! '°'! ":" . '£K3Qt| -A ( 9 > 1) I I ' is the locus , of a point which moves so that the differen _ o
  3. 3. l i l i ‘ Vs‘ lime emistntt length _“, ‘ PD ll me: ll i Ll'ti. lltt‘H ' ~C_ l”) - l} l U help - and C has: opposite »t: _'h» ll St. nxl. ird i‘orin A. Center oi l_ h, k) 2. _ (mi-h): =i - b- a B. Center at (0,0) . l. "V ‘H a‘ b‘ y: Y: 7 - '~w= l . . 5'. ‘ b- Vote . Relations ofconstants : a‘ ~r bi = c :4 l 2 SEMl—TA= CV. =CV, =a 3 SEMl—CA= CB. =CB3=b -1 TA = V.V3=2a 5. CA= B.B; =2b 6. FiF3=2C 7 L. R. =2b’/ a. 8 e = e la > I 9. d= a/e l0. Equations of as mptote: TA: Horizontal . TA : Vertical . l'/ lluri/ .mtl; tl . TA 2 Vertical , a>bora= bora<b a. y -k j_— ( x — h) . for horizontal TA il b, y~k = ji‘) (3: — h). for vertical TA LEGEND: TA --- Transverse axis CA Conjugate axis
  4. 4. DIFFER E. 'TI. -l nogv FORMUl. »tS l 3 (Cl 0. ulierec lS. lll'COli$l. llll . 3. 9’ (xi 1 d 3. 5’ (u“l nu""iL1 d l 4. — » ‘ll . M d “M udx d @ Q1 5. g (2, dx V V‘ . l 6 5‘ ((tl): ‘ 2/ ti d“ - _a- - a-n Q 7 d(u, .) um, d‘ g u. ». 9.! 9. dx (a) a(na)dx Q _ VdU+ dv 10‘ dx (ul—U(; a lnuggl d . =lfl ll. ‘-jx(ll'lU) U dx 9 , =1 9.“ '1 dx (l°‘; .uU) u(l°Sb€)dx l3 9- (sinu)= cosui‘—‘ ' dx dx l4. gx (cosu)= -sinugxfl l5. gx (tanu)= seczug’-:3 I6, fix (cotu)= -csczuilafl I7. 5 (secu)= secutanugi X dx I8. ix (cscu)= -cscucotu% tl I9. tl . 'l) “ d ti d 7| -n tl d d 27. d 9 23. d 29. 9 d 30. d d 3:. 9 32. Q d 33. 9 d Q 34. d d 35. E Q 36' dx - RU, ‘ ". ltL um u) ‘ll ~ ll‘ d‘ l-lit. mt. ti) L "J V‘! ~ ll‘ d" (ntc l. lll ii) I -‘l-V" 1 u d . [ J talc Lul u; , L. “ l t u‘ d. _ ‘ _ 1 (. _ll: (art sec u) d‘ usu’ - 1 (‘NC csc u) -' 1 d—” Nu‘ - l d‘ E (sinh u) = cos h u . (cosh u) = sinh u % (tanlt u) = scch‘ u ‘£3 (coth u) = - csch‘ u (such u) = - sech u tanh u 3-‘? (csch u) = — csch u coth u 3;" I (arc sinh u) = (arc cosh u) = ‘T ‘(arc tanh u) = ’I‘l_‘? '—‘ % (arc coth u) = —T| _—u! "‘ . | Q (are sech u) = ""‘u rfl _ u_ dx 412 (are csclt U): u l+u1 ax . (U>'l . ,(u: <l) , (u: >|) _(o<I: <ll _(ux0l
  5. 5. MAXIMUM AND MINIMUM VALUES y - axis Max. Value of f(x) ' Min. PL 4- Min. Value off(x) x v axis Note: 1. At point of inflection y‘ = 0 2. At critical point y’ = 0 l. P. (lnflection Point) = a point where the sense of concavity of the curve changes. I. THE FIRST DERIVATIVE TEST (F. D. T. ) A. ) The function y = f(x) is a maximum for x = a if, y‘ > 0 for x < a y‘ = 0 for x = a y‘ < 0 for x > a B. ) The function y = f(x) is a minimum for x = a if, y' < 0 for x < a —y' = 0 for x = a y‘ > 0 for x > a ll. THE SECOND DERlVATlVE TEST (8. D. T. ) A. ) The function y = f(x) is a maximum at x = a. if f'(a) = 0 and f‘(a) < 0. B. ) The function y = f(x) is a minimum at x = a, if f'(a) = 0 and f'(a) > 0- Note that it f'(a) = O or if f'(a) does not exist then the S. D. T. fails. then it is the time to use F. D. T POINT OF lNFLECTlON TEST (P. l. T. ) 1.) ll f'(a) = 0 and if f“(x) § 0 for x < a and rm = § 0 for X , 3; men Y = f(x) has a point of inflection at x = a. 2_) [f f'(a) = o and if f"(a) it 0; then y = f(x) has a point of inflection at x - a. V ‘ Note: y = f(x) has a point of inflection at x = a it t’(3) = 0 Of f'(a) d°°5 "°* “'5 -
  6. 6. IR. I III§1‘'IS I0 I I 16 I7 I8 I9 i. .. Ii. l., i. - “INK, t“iw‘i ‘ U ‘A I “‘ ““ ““— ' ' ~3'. cv. ‘II. v -l II -I I I V" . I, ‘ J III‘. ILJ . .‘iitI l Ic' ifii ~‘ t I ‘J. IILI ~ ‘I - I in . i In III - III -‘I I2i I| 'l'iC£I. IIltlli ht mm Inn u Ju ‘ *. I II - { Icm u du »” HI) I: * L‘ I! aiIut. IiI;3‘>«. 'ci: 'u ms Il ‘ K Icot u du = In s. ii I) - Q . '.ci: iidu= lni~ci: ti~i; mii: ~t. ' Its; ll d. i ~ In I cs; Ii — not Ii I‘ C — - ‘in ICSC U ’ cot u) I C Isi: »;"u du = tan u * L" II: s:' u du v— - cut u -~ ( Iscc u lzxn U du v‘ set: U ' C Ii: scuccitidu= -cscu+C Va‘ II‘ rl I-1-@—v—= l iri; ian‘—‘ I-c u'Il' | 3 du I U I r, -—-! -'——zircsi: c- +C uu-.1 J 31 Isinh u du -'~ cosh u 4' Icosli :2 du ~ ‘. lIl it > C I54.-ch’ u du — IZIIIII Ii +C I csch’ u du = -coili u vC Iscch u I; inI1 u L. Iu = -SL‘CI1 u +C Icsch u coil: udu = - csch u + C I . dU . — . _I in| I"—“)o('_iI‘Ii: >;i: II’-J‘ 23 3.3 du _| pt . I — - I I , .. 1; )vL_| IlIx. I I II}-II’ 2;; H u-. ‘l I _—A‘—’——— = lii(II'*Iu‘HI' I’C ‘Iu'~a' ‘I . iIlII'Ii-iiiii. |,. )ll II‘_l- 'I_II‘ I41‘-, -i; LIL~, ._ _13=Il~w ~~ (III-Iillii: ~i'i~2iIiii~. Li IlL‘lL‘ m iI‘Ii. I II . In: IiI: i»i: cg, ati-I; lEl'L“_’I'I I ii '~ ml}. ifboih in and n . iic L'L'Ii utlii. -I I. » L‘L’Yl ll * 1 il ('lIlL'rVl>L‘ Nut: II’ | ‘ll or II i: qii: II: I IIIQ Iipplj. Iii: ii Ittilc II the Incttt-rt III — Inn. ’ in liiii IIg' I. rizplzicc Ihc ploducl Ill utiiizh lIlI l. b'd<' hi I'l, iNli . -Rli, S ll’ RIC I. -. ‘(il '3 . -l~‘. t iii il". |)E . II 'i II ‘J. I Using a vertical rectangular clciiiciii A: y= I( / Izgtxi U — upper I. - Itmci | I.'= I}u", ‘|Id I. r= I I)t. '}II‘l I i. ,. — j_ I tIl—g(III‘I‘ H [mug i“. ,.l, .mi, .| | (cci. iI_uiIl. Ir I IL‘III‘| II I 'IiIiII"q A III I 5-Ixiiir d) . '_¥‘ l: 'I I. ’ -‘I < I‘ lilw , in.11I-ic llIIlI Lx, Iift : I_ - , I L. ‘ llgui ! .iId
  7. 7. ‘in~ | ' ~ ‘ L Nlr ( . mrmn. "_‘ 3* 1»: . lU| ':N 'll. (r v :1» ' r ‘0|. l‘u: m'sm | |)()FRl>I‘0LLr|10 '“ ‘ W--J= v=v= ’»? ~.'! l'.1c1m-«1 Hw'« ‘ ' ‘ Hvcmn -v -lIm. m»mn. ‘.. .‘. _V_ ‘‘ “ “ "W-: l.1rn1|| ‘¢, mm M _ 5 v . HIE ‘(1'-‘ . xu. u. I.‘. ,¢4, , .,‘m h‘ “ _’ “ IN mun J‘ Zruyh dr . v ‘ 2.13 121 dr . P. ‘l'| 'l‘5 I'lIE()| Ui. l I-‘ml Prupuuhun Ihu nnrldcc L! IL‘A -‘f n. 'rJ, ;1., ;. ., N. ,_‘ H ‘ nflhc gv: ucr. mug mu hunt» the ulruulzlcrclLy‘-1'1|. ‘.1u‘ . lhu ccnlnud ul Lhu ;1n. ‘, prmudcd lhc . .m H! u~. .". ;m : x the ; _:L-ncnunng .4’; f H / ) . (1 2 RE! Melhods of Finding Ihc Volumes of Solid of Rtvolutiuu uhcrc K. as lhu: lcnglh ufnn; I Disk l: Ihod Second Proposition The Kxlumc ufxln; wiIdrcw. »1x. w- u L'k: “:J| um- gcucrzuing awn luncx mu cnrcunufcrcmc nt lhv: um: dc~. v=hcd M the Rulcs 1 The nus o! 'm(; monxs: np.1r of the boundary M’ Ih»: p| ;1m: ;nc. I ccnlmld uflhn: .lI'| I(1,[1|‘U‘| dCdlhC. I: ) nt lc| |uliHI* doc» 4 vi u. u zlu 7 fhc ch: mL'n1 chuscn must bc ; to lhc axis ul mumon gene-mung awn . ‘ } c. 'L" v—_. c-—»q2nc) ‘ _ nr '~l. 'Ix"/ > (T1-nlrnid of: Plan: Area I-'ormulu M » I , M av = as an M . j,. M = —' n . V X J‘r d “h¢; ..- , and : an: courdnnulcs 01 NW r~‘| “""d *“ . " - hat clcmcnl ll Ring ur “usher Method N-U-‘"5" V _ . , . - ~ I flh ' ccnumd «:1 . m.. u Rules 1 1h«; nus nfrotzmon Is not . pan ufllu boundary 0|’ V and 3‘ are Looldlna Ch 0 k the Inn: an. -.1 2 '| ';l, lo. ' clcmcm choscn mus! be L In the . m: ul'n-I: -lmn | |IIIslr'. IlinH
  8. 8. Area of Common Polar Cufyc‘ Ccnlrokd on Solid of Rcvululion Fotmulu vx- Ix, dV V? '=I>. dV SHELL “V § DISK WASHER . ‘. l. .1‘ . . ‘.
  9. 9. ' ' ; 'e“; :l: u%l; tCm; bl| Iltega: f the pure substance in a tank is equal to the as the tank. “ . ILLUSTRATION: Initially. t = 0 Rs Incoming Cu stirrer Let S = amount of substance in the tank at anytime. I S9 = amount of substance in the tank at time t = 0 § = rate of change ofS dt V = volume of mixture or solution at anytime, 1 V0 = original volume of mixture or solution C; = concentration of incoming solution } C2 = concentration of outgoing solution R. = rate of inflow volume R1 = rate of outflow Tnit_tir_t1E- weight unit vo ume Note: lt'R; > R; , "R; = R2 , lfR. < R; , V > V, ———-> it will overflow ‘V = 'V. , ———> constant V < V. , ——> it will be emptied later Formulas: dS 3- = rate of Sufingfi -- rate of S. .,, . = CIR! - C2R2 ’ ll 3! S in general at anytime t , C = --I . _¢-§_ L__ — V Vo+(Ri '''R2)‘ ([3 RIS Va -t-(R, -R, )t V = V, ,+ (R. — R2)! "“= CtRi" flherefore. dt . alum: rm ‘R: (V. S. ) « int. am, (L. D. E. ) lfcrcncc of the rate at ws to the tank and the rate at which the pure substance flows out of R2 0 outgo'n c ' E I
  10. 10. SECOND ORDER DIFFERISNTIAL l". UATlONS° d2' Liv 21—~—'. ~ +b ' my :0 dx‘ dx where a. b. and c are constant and a if 0. I Let D ‘ — tlx aD: _ + bl)_' + c_' : 0 tan’ « bl) cl)‘ -0 Auxiliary equation: by factoring am‘ -* bin t c = 0 ‘ —b i x/ b3 — 4ac by Q. E. F. . m = ~——— 2a CASE I: When the roots nil and mg are real and unequal. The general solution is. '_ = Clem. ‘ +C2e| fll CASE 2: When the roots mi and mg are real and equal. The general solution is. y_ _: C‘elYI +C2xelVlV 0|’ ID‘ y= (C, +C1x)e CASE 3: When the roots mi and mg are imaginary and unequal. , bijl4ac—b2 = Aim 221 then the general solution is, y = c"‘ (C. cosl3x +. C_, sin Bx) m:
  11. 11. .-ft| ilI| t‘llt‘ . le: m t. ». M. ): . .+. . ; .-O . l = L "' " “H Mctlinnt Is the mttldlu aluc ilL‘ll all data are arranged in increasing or decreasing order Mode : ls the value that occurs most frequently. Range: R-. utge= lvlaxunum value—-Minimum value Va rinnce: The variance ofa set ofnumbers is defined by v 3‘ _ 3 O. ___ , _ gxnl SQ where Ti -1’ A. M. Standard Deviation: Std. Deviation = OT Std. Deviation = Fundamental Principle: It‘ an event can happen in any one of n. ways. and if when this has occurred anotl_1'er fiver‘! !! '| ‘1nnPPC" "1 any one of n, ways. then the number of ways in which both events can happen tn the spect re or c . I . In general for k events, (17 I n. I ll; ' fl). .,rfly Permutation (P): - grouping nfthings in a definite order. To pennute :5 set of things means to arrange them in a definite order I. Permutation ofn different elements taken r at n time is P - --—"' n I" (n - r)'. Note: .. I‘. = P(n. r) '-' P. "
  12. 12. V t- . .. : ‘. ._l’. sirletIieai. g,ls. e. the no. of pennutntion tiilienzm I _ Itmtitei way: 05- hi. at. ca. be. eh I (I or by l-‘iiiidaniental Principle 3 - 2 - 6 2. Permutation oiii different elements taken all (r - n) at a time is, . l’. . = ii! Illustration: For a, b. c, the no. of pennutation taken all at a time is, J’; = = 6 Another way, abc. acb. bac. bca, cba. cab = 6 or by Fundamental Principle, 3 --2 - I - 6 3. Permutations of n elements some of which are alike is, npa-I: i : ln —(n — 5)]! s! where s is the number of times the element is repeated in the set. Illustration: , For letter a, a, c the number of permutation taken all nt l 3i p = IL. = _= = 3 s! 2! Another way, aac, aca, caa'= 3 4. Permutation ofn elements not all dlflerent talten all at a time is. n! ‘Dr! D)! I13! . ... .l'l| .i P: Where n. , H}, n, . n. — number of elements whichare alike n - total no. of elements in a given set 5. Theorem on Partitioning: _ . . _ _ I The number of ways of partitioning a set of n objects into r eel elements in the second cell and so forth is, H 1'. ’ _ ll! I'll! I'l; !.. ... n,i ii. , I11, . ... . n, where n = it, + n; '.+ . ... ... ... . . .+ n. it lutti‘ is, a time is. s with n. elements in the first cell. n:
  13. 13. TV (1 (')‘t‘lIr l‘i~riiitit. itmm; lllr iiiiiiilscy . i| - . iuliititatinii-. (it It tllll('| I‘lll IllIjn'_| , ,, , ,, - « - _t'v‘: | Ill I ittlc i-. I‘. - tii- I)’ llltll’. tll0l'l lor the letter» ‘L l‘- 1‘ -I”-HIi'. vd Ill . i circle, P. =l+l—”-2 ByFormula, P¢= (3-l)! =2!=2-l=2 Combination (C): is a selection of things considered witliout regard to order. - grouping of things where arrangement is immaterial I. The number of combinations ofn objects taken r at a time is. _ E __ n. ' "C" Brll ‘ r! (n- r)! Note: .c, = C(n. r) = C," 2. The number of combination ol'n objects taken all (n = r) at I! time is. . ,C, .=l 7' V’) 3.The number of combinations that can be made taking successively I at a time. 2 at a time. 3 at 11 time and so on up to n at a time. C= nC| J'nC1+nCJ+ - - - ' "+riCn=2n‘l Probability (p): probabimy . . No. of favorable outcomes No. of possible outcomes Probability of Success + Probability of Failure = I . . ' . -- . . ll | ‘kl'. thcnin.1 I. Probabi| ity in Single Event. lfan event can happen in h ways and can fail in fw*i)’5 ""9 “W3 Y ' c > single trial the probability will happen is given b. V. 9 ' h -v f and the probability that it will fail is given by. t’ "I77
  14. 14. iii’ tlieni can _. _ _ ’ ‘I'll. -*‘~_ “hi” Two or more eventsate mutually exclusive urn. " mm. mm, "M. . - happen in tl given trial The probability that some one or other ofa set of mutually exclusive events Ill ll'| ii ‘II I - ' t trial is the sum of their separate probabilities of happening. I II I l " J mm c P‘pi +l>i+ . ~--1* n. llltlepentlent Events. Two or more events are said to be independent if the happening of one tines not affect the happening of the others. The probability that two or more independent events will happen is the product of their separate probabilities. p= p1‘p; ' . . . . . . . . . . . .-p, , 4. Dependent Events. Two or more events are said to be dependent if the happening of mic effects the pl0l)3l'. ttltl_' that the other will happen. If p. is the probability that an event will happen, and after it has happened ilie second will occur with probability P}, then the probability that the first event and then the second event will happen is the product p. ~ p, 5.Probabllity for Repeated Trials (Binomial Density Distribution) ‘ p = .c. p. ’ <1 - pm" where n = no. oftrials r = no. of desired successful outcome p. = probability of a successful outcome in a trial . _. 5‘
  15. 15. Coniplex Ntiiiilien; ‘ ‘ I. ‘ ' rtuislfi iv isiiilli lL[ll _ . I ” I C , . “lists I -I mi v’-I l’0(‘r of ('oiiiple ~l| ||]h('| ’'; 5' " t l | _ l" > r" tens it II t [Sill ii iii‘ H‘ qt gill , -: , JL, _,ii = we Roots of Complex Numliers: since for . in whole iiiiiiibet k. sin (U + 3(V0“ki = sin ll cos til + 7<tV0"l; i = cos H therefore Ill gciiernl lonii. , , . p l) 4 l60'_‘l_t ‘ K: %_ rulcls , ;_ n _—_ 1' ll%. "l. > ll where k = 0, when taking the first root k = 1, when taking the second root k = 2, when taking the third root k = n - 1, when taking the it"' root Trigonometrie and Hyperbolic Functions of Complex Numbers: 1 cosli ti“) = cos H 2. sinli tilt i = _isiii H 3 cos in = cosli u 4. sin in = _tsiiiliii 1 sin t. : ,i_ I = sin x cos it 1 cos x sin _i_ = sin x cosh} jjcos x siiiliy t» cos t it. ‘ l = cos x cos _i_i' 7+ sin x sinjj: —— cos x cosli_ 3F jsiii x siiilit L: iplace'l"ransfon1i: - '~ ~- -' . ':'. 'll"l tl‘ The L; ipl: ice trntisforiii ofn fiiiictioii ftti. deiioled b_ [{(lll- lb 'J¢r| ||€(l -15 -1 {“"tl|0'| Or ‘ 1" ' ‘ 9 5 ’. ‘ lk uitegml. Ft, <)= °[_[l'tll. l=, l-i-ftlte "dt. vi, ._-w i ii; ind s IS. 'ttl_ llllIlll)t. ‘I’. real or C0lllplC. Fnmitilzisz l. - . . — 5 I J; [;, ]:L' J . l:, lsiiikl| :;: TE? “Ll‘"“M'll sit ll1!'_ll ‘V. .l nil _ _ 5 1 I —» | , iiltelt [V -l 2 Lip-1: it I 3, ‘ c[—lL“5l"'l‘ 5-+1; V L“ I ‘ I IIOII-lIIlL‘t1l". ll. S V k _; L| C-, Vil_ ll L'lllll
  16. 16. * * -v -'-“I-—I: IIv1Ti’FT( ‘T Im mrmn ' - x - - I I “turn nu on l. .upl. ncc Trans-funm Theorem 1. (Lmc.1nt_ Thcorcm) L [a tm + b -_; m| ; I,, (_[{u)] ~o bcL[ “,3: Theorem 2. (Furs! Slulhn-_. 'Thcorc1nI Lie t fml LI mu . .. . Theorem 3. (Second Sluftmg Theorem; Ll Ill ~— a) nu —« 3)] = : [nu] Theorem -I (Transforms of Dem ntnc) L [r‘ ml = s J; .| rm] énm . L[t" ml = s‘. [,[rm] — is no) + run] . LIf"m1= s‘. [.mn1 — Is’ an» + snm + t"un] . [_[ f"'m] = s", [, [fm] ~ [s'“ fun + s'*-' fun + , ..+ s t‘*-‘an + t"“un] Theorem 5. (Transform of Integml) If it I) us of exponential order and at least pic«: cu'sc continuous. then .0 L lJ. ~ t'mdt]= f. [,[rm1+_f J‘ fmdt Theorem 6. 1r . [,[ rm] = rws). then I, [z rm] = -4>'(s) Theorem 7. If lnu Em exists. and if L [f(t)] = «tvtst. then LIE? I: ‘1"5’d5 t_m‘ | Inverse Laplace Transform: I It’ , ,[_ [rmy Fm . then rm= L [Fm] prox ulcd that L [fut] €. ISlS Gamma Function: _ . E «-1 4 ' h 2 - zcnt for u ‘- U The gamma function denoted by Hm I5 d¢r| "¢d bit“-E: " ":5 r‘: :‘‘: hl“): =': “°: °(l': )‘: rfl! A recursion or recurrence fonnula for the §i"‘"‘3 °“° Note I Ifn < 0 but n x -I -2. -3. Use l'(n) = '—1'——+—i1 u
  17. 17. 4.11 It ‘ -; Fnru r‘-. - t n. i » ‘“"‘" ‘V '""Trri5ivrATr(‘S . -_ -1 1 (in r- . - SWUCIICC and Series: scqucnce of Numb ‘"3 " defined ‘IS 1 succession ‘ ' - of numbers formed For c. iiiiple I 1 according to somc {Ned mic . If" lcmi V 2. -1_o. s.. |n_ [3 . I. 4 9, In 35_ H Series‘ — deliiied ‘IR tli - - - . e indicated suni ol 1 <. ,-( - - . |llLllCL‘ ot iiniiibcrs For exzunple. for the sequcnoc ; .,_ ; ,3_ mp 3“ “W °°"‘°$P0nding series is a. i 3,, ;, ,+_ . + 3,, Type of Series. I. Finite Series — the number of terms is limited. 2. Infinite Series'— the number or terms is u'n]imn°d_ Power Series: A power series in x — a has the form Ox: “£06.. (x—a)“= c.+c. (x—a)+c; (x—a)’+_ _. +c, _(_‘. .a)"+A When : i = 0. and the series become a power -nx. which is 2' C. .x"= Cn+C. x+C; x’+. . . +C. ..v€'+ Forcxample: l : . l_‘ = l+x+x+x+ General Method: for expaiidiiig 21 function in a power series in x and in tx — a) is git en belou. Note the rcqiiireiiiciit that the function an d ilS. dCl‘l'£lll’C$ of all orders iinist exist at . ‘= U or at X = :1 This In x. and cot X cannot be expanded in power of x Maclaurin’s Series — power series expansion of f(x) about x = 0. - power series in x. rm = r(o; + l"(0)x +5133 xi + ‘:59 x’ + () ‘, Li1(_>). ,. .+M 3t n‘ OI’ ' (ii) " rm : Zr 0 X ii it I'll Ta}‘l(Il"S Series - pouer series in . — a . -pmier series CVDJIIISIOII of It l about ‘ : i This series. lllCll iuclud special case (It " 0) es M; iel: iiiriii‘s sciics -is -I
  18. 18. f'(a) 2i Wit al’~ W’ 3' ril rrxirraiirrairx air txAal’i ii ii» . int. til i’ —“‘7rxsai“ no Fourier Series - is a series used to represent a periodic wave in either exponential or trigonometric form. the trigonometric form is in terms in sine and cosine functions. The series has the form, Flt) e l A, cost i A2 cos2t+A, cos3t+ -«An cosnt+B, slnt i B, sin2l A B‘ sin3t ~ i E For a particular period were the coefficient of the series are determined by means of the following equations. no so é-f§"F(t)dt A0:%r§“eri)cosnior » en es giglrrri sin in or
  19. 19. lECTOR ANALYSIS
  20. 20. '3‘. '9“‘““"W'lflltluwhkhnconipktelyspecifledwhenrheirm ' ‘I A anrtude are given. ‘ 7°"‘- ‘9'°‘~ °l“"'°*~ “WK V°l'lm¢. mug. specific heat. gravitational potential; time. . etc. V? ’ qlllflflel - which require both mlgnitude and direction in order to be cc mpletely specified. ' R” °"- "‘l°¢"! . GISPIICGIMHL momeflmm. Weight. torque. centrifugal force. electric field intensity, etc. Victor Representntion: I. Graphically: For example. 0/? length - represents the magnitude arrowhead - represents the direction b. Analyticallyz _. for ex. vector A can be represented as A or A. Note: —-K is a vector with the same magnitude as A but opposite in direction. Unit Vector — is a vector having unit magnitude. Cartesian Unit vectors i, j, It in 3D-space: X l . 4. A -im. sj+A. .-« = <Ar- A» A») i ,4“ ,4, are scalar components of A |
  21. 21. _4'LAd‘"m‘Am. vmmcmmomm°rA. gvalueot'x. yendzrespectiveiy Note: ' l. Zero or Null Vector — 2. Eqmmy or vmmi- it his zero magnitude and direction is undefined or no specific direction. I two vectors are equal if and only if their corresponding components are also equal Magnitude or length of A tn denoted by [Al or A. IAI*‘1.4,: + .4: * A1: Unit Vector in the direction of A is denoted by 6,: a 3 _. I_t_= A,i+Ai| .+/ ilk ‘ Mi .4, +A, +4, Position/ Radius Vector , — any vector starts from the origin which is usually represented by r. r= xi+yj+z| t X | r|= r‘= ‘Jx: +y' +2: Laws of Vector Algebra: If A, B. and C are vectors and m and n are scalars then. I. A+B= B+A 2_ A+(B+C)= (A+B)+C 3. mA= Am I 4. m(nA)= (mn)A 5_ (m+ n)A= mA+nA 6_ m(A+ B)= mA+mB Multiplication of Vectors: 1. nor on SCALAR PRODUCT B ' ' By definition. 9 ' I ‘l A . 3 as [Milli cost) A. , l o where 9 - smallerringle between A and B ' 0 :0 § n ‘ Important Laws (909 ""‘? d'"°')‘ i. A - B = B - A ' 2_ Aa(B-t-C) ‘ A'3+A'C . _ r—-*4-.3- Lllttdltuilltltvoutursindiedlreetionofincreuin E
  22. 22. - . » x . 3. B 5 . wheretitisasealar m ) (A mm 4. since I. Land It are orthogonal iojuioknj . ka(| X|)cos90o: o i°i-I-i-In k= (IXI)coso-t 5. ll-A= .“; ‘+‘.4;‘i *.4,kafldB= B,i+B. ,j-OBI“. Then Ac 8 = .-t, B, + 4,3, + 4,3, 6* . A'’‘ = All-41) ‘ -"1(-42) * AIM! ) " 4:) + As’ + 21;! ‘ Ml: also. ' B - B = 848.2 + B. ~(Bx) +’B. (B. ) = B. ’ + 8.3 + B. ’ = int’ 7. ll’ Ac 8 = 0, where A and B are not null vectors. then A and B are perpendicular Sample Application of DotVProduct: Scalar Projection of B unto A (S, ,.hB): lBl SM‘ 8 = |B| cos 9 = AQ but cos 0 Mum therefore. B : —.__A. B 2‘. M "-' BO (7 SPWIA ‘M ‘M A M Similarly for scalar projection of A unto B is. S. ..” A = A 00» - Note: mo“ we-: I80“ t-Scalar prnivclium gm; 3 2. 90° ( rscalar projection)
  23. 23. Vector Projection of B unto A (V, ,, 4‘ B); ll" . ,_>L'_r, C._ s '. < . “B I‘ "1’ i I‘ B W (SN >, B)(aA) "‘ (B ' OAXOA) Similarly for Vector Projection of A unto B is, v| r., «.]nI '_ (A' 65); )“ A X B n. crzoss on VECTOR PRODUCT: A By definition, A x B = lAllBlsin0 0,, 0 A B :1 where 0,, : ‘Axx B‘ '0 then AXB BrrA= —-AxB ‘A x B = lA| |B| (sin B) (l: —x—B‘| ) it- X i sin 8 important Laws(Cross Product): I. A ‘A3 = -B x A 2. Ax(B+C)= AxB+AxC 3. m(AxB)= (mA)xB= Ax(mB)= (AxB)m. where m is a scalar 4. ixi=0 ixj= k ixk= -j jxj-0 jxi= —k jxk= i kxk=0 kxi= j kXj= -i 5. mt ~ xl, i + Ag + A, k'and n = B, l + 3,} + 31k A x B = 91,13, — A, a,)i + (A, B, - A, Bi)i + (A18: - 4230" i j it ~ xi; ll) /1; B] B} B] 6. Arm. ’~0andBxB=0 Also A x 0 = = 0 0 x A ' 0 7, It’ A inB . it means A is parallel to B: then it 0 or It and we define A x B " 0
  24. 24. S“‘“l“"' -‘l‘l‘l| c-ttioit oft Toss Product‘ -'| ’°'-I M ‘-I l‘: n‘: tllclo): ram with tcctor sides A 'llld B- ‘ v . -[j to it Ag l. —ll, l3lsinti but | .-| lBl<mEl I: X 8| therefore, Ag : |. - x Bl Note: the area oftriangle with vector sides A and B is '/2 ofthe area ofparallelogram. thus, 1 A '~ '/2 iA X Bl PLANE IN 3D-SPACE: X: Po —-> (x, .. , v., . I. .) P —* (x. y. 2) x-x, ,, y— , 2-2.) NJ_Pn , thenN-P. , =0 , b, ' " . » an '' Z O , <31‘ __ ? )'+(; (), f y”}; _+yc(: _ Z? ) : 0 (Standard Equation ofa Plane) at 4 by + cz + (-cvt. .~ by. .- C2») = 0 at + by + C2 , ,1: 0 ( General Equation of a Plane) SCALAR TRIPLE PRODUCT: for ex . l. l “'3 Cmss product mu. “ be evaw-mcf, rmduct is not nccessan‘ ‘ lhe parenthesis used In scalar tn? !‘ Pm , ,_ d ' k H, dot product and the cross symbol can be interchange . I e ‘
  25. 25. .iiiiu' litipiirrini [ tn, ‘ jn g _ ' -- . cal I ‘ | - , . . t , n. ( , _B(‘“I rlllt I'rt)l| lLl. | l'(' (f. ,[; <"'“<' l3'C. «. (f-Axll ‘ ""“' -l3°r(‘ (‘-nxrx '('lt ‘ i ' -- (j A A o C 0. since A A 0 l- . - . ( _. . C I‘ 0‘ ‘Hum. A ‘ A ( '”l. l": ‘|l5 OF . I‘. -R/ l. Ll~Zl. EPlPED: st“ ll . .» C / T (F. ii — Si-wt‘ ~»t’, ‘; »/ AF‘/ ‘H tiniu B p t’ 1/5 . .+’ ’ B ' A0 tli) [B (‘its-calnr proj ofA unto B x C) lB C3 l A o 0,. ) M B. '(T * lB(l(t"BxC.1): A'BXC xi] 1‘) 4“; V. '’ IA‘ B 2 ,3, B; ‘B3 (’, C; C; VOLUME OF TETRAIIEDRON: Vol. of Pyramid 4 (Area of its base)(He-ight) Vol. ot‘ Tetrahedron = ( g A, )(Sproj ofA unto B : : C) vol. of Parallelepiped Vol. of‘l'etr; ilicdron -‘ | A 0 B x Cl 3‘l— Oi-' VECTOR TRIPLE PRODUCT: for ex. A x (B . 'C). (A x B)x C. etc. lmportzint Laws of Vector Triple Product: I. Ax(B. '(‘)v(AxB)-‘~'C , , Slmwiiig the need for parentheses in A x B x C tc avoid ambiguity. 2. A (B x (‘i (A-Cm (A-B)C «ll'~Il. (A . x. If) . ~. (3 (Av c‘)B (ll'C)A , (| s--. (')= (BC)xA L. )
  26. 26. DIFFERENTIAL OPERATOR DEL OR NEBLA ( V) V tax+]ay+kaz If ¢(x, y, 2) is a differential scalar field. Then the gradient ofcti is _ wt. aigg V¢— (5x—i+3y—i azk THE DIVERGENCE: If V (x, y, z) is a vector function with components whose first derivatives are continuous in "the domain of V, the divergence of V is given by _ . a . i v. v=(. §;. t,%+kE)-(iv. +jv, +kv, ) av, av2 avs ""“7a‘; *W*az"
  27. 27. THANK YOU GOD BLESS YOU

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