The Handbook of Essential Mathematics

Transcription

1 Fo Pulic Relese: Distiutio Ulimited The Ai Foce Resech Lotoy The Hdook of Essetil Mthemtics Fomuls, Pocesses, d Tles Plus Applictios i Pesol Fice X Y Y XY Y X X XY X Y X XY Y Compiltio d Epltios: Joh C. Spks Editos: Dold D. Gegoy d Vicet R. Mille Fo Puic Relese: Distiutio Ulimited

2 The Hdook of Essetil Mthemtics Ai Foce Pulictio 6 Fo Pulic Relese: Distiutio Ulimited

3 Fowd Wight-Ptteso Ai Foce Bse WPAFB hs ejoyed legthy d distiguished histoy of sevig the Gete-Dyto commuity i viety of wys. Oe of these wys is though the WPAFB Eductiol Outech EO Pogm, fo which the Ai Foce Resech Lotoy AFRL is poud d cotiuous suppote, povidig oth techicl epetise fom ove pcticig scietists d egiees d ogoig esouces fo the vious pogms sposoed y the WPAFB Eductiol Outech. The missio of the WPAFB EO pogm is To fom leig pteships with the K- eductiol commuity i ode to icese studet weess d ecitemet i ll fields of mth, sciece, vitio, d eospce; ultimtely developig ou tio s futue scietific d techicl wokfoce. I suppot of this missio, the WPAFB EO spies to e the est oe-stop esouce fo ecougemet d ehcemet of K- sciece, mth d techology eductio thoughout the Uited Sttes Ai Foce. It is i this spiit tht AFRL offes The Hdook of Essetil Mthemtics, compedium of mthemticl fomuls d othe useful techicl ifomtio tht will well seve oth studets d teches like fom ely gdes though ely college. It is ou sicee hope tht you will use this esouce to eithe futhe you ow eductio o the eductio of those futue scietists d egiees so vitl to pesevig ou cheished Ameic feedoms. LESTER MCFAWN, SES Eecutive Diecto Ai Foce Resech Lotoy 3

4 Itoductio Fomuls! They seem to e the e of evey egiig mthemtics studet who hs yet to elize tht fomuls e out stuctue d eltioship d ot out memoiztio. Gted, fomuls hve to e memoized; fo, it is ptly though memoiztio tht we evetully ecome ucosciously competet : tue mste of ou skill, pcticig it i lmost effotless, utomtic sese. I mthemtics, eig ucosciously competet mes we hve msteed the udelyig lgeic lguge to the sme degee tht we hve msteed ou tive togue. Kowig fomuls d udestdig the esoig ehid them popels oe towds the od to mthemticl fluecy, so essetil i ou mode high-tech society. The Hdook of Essetil Mthemtics cotis thee mjo sectios. Sectio I, Fomuls, cotis most of the mthemticl fomuls tht peso would epect to ecoute though the secod ye of college egdless of mjo. I dditio, thee e fomuls ely see i such compiltios, icluded s mthemticl tet fo the iquisitive. Sectio I lso icludes select mthemticl pocesses, such s the pocess fo solvig lie equtio i oe ukow, with suppotig emples. Sectio II, Tles, icludes oth pue mth tles d physicl-sciece tles, useful i viety of disciples gig fom physics to usig. As i Sectio I, some tles e icluded just to utue cuiosity i spiit of fu. I Sectios I d II, ech fomul d tle is eumeted fo esy efel. Sectio III, Applictios i Pesol Fice, is smll tetook withi ook whee the lguge of lge is pplied to tht eveydy ficil wold ffectig ll of us thoughout ou lives fom ith to deth. Note: The ide of comiig mthemtics fomuls with ficil pplictios is ot oigil i tht my fthe hd simil type ook s Pudue egieeig studet i the ely 93s. I would like to tke this oppotuity to thk M. Al Gimoe Chim of the Deptmet of Mthemtics, Sicli Commuity College, Dyto, Ohio fo povidig equiedmemoiztio fomul lists fo Sicli mthemtics couses fom which the fomul compiltio ws ptilly uilt. Joh C. Spks Mch 6 4

5 Dedictio The Hdook of Essetil Mthemtics is dedicted to ll Ai Foce fmilies O Icus I ide high... With whoosh to my ck Ad o wid to my fce, Folded hds I quiet est Wtchig...O Icus... The clouds glide y, Thei fields f elow Of gold-illumed sow, Ple yellow, tquil moo To my ight Eveig sky. Ad Wight...O Icus... Mde it so Silveed chiot stekig O togues of fie lepig Ad I will soo e sleepig Aove you dems... August : Joh C. Spks th Aivesy of Poweed Flight

6 Tle of Cotets Sectio I: Fomuls with Select Pocesses Ide to Pocesses Pge 6. Alge 3.. Wht is Vile? 3.. Field Aioms 4.3. Divisiility Tests 5.4. Sutctio, Divisio, Siged Numes 6.5. Rules fo Fctios 8.6. Ptil Fctios 9.7. Rules fo Epoets.8. Rules fo Rdicls.9. Fcto Fomuls.. Lws of Equlity 4.. Lws of Iequlity 6.. Ode of Opetios 7.3. Thee Meigs of Equls 7.4. The Seve Petheses Rules 8.5. Rules fo Logithms 3.6. Comple Numes 3.7. Wht is Fuctio? 3.8. Fuctio Alge Qudtic Equtios & Fuctios 34.. Cdo s Cuic Solutio 36.. Theoy of Polyomil Equtios 37.. Detemits d Cme s Rule Biomil Theoem 4.4. Aithmetic Seies 4.5. Geometic Seies 4.6. Boole Alge 4.7. Vitio Fomuls 43 6

7 Tle of Cotets cot. Clssicl d Alytic Geomety 44.. The Pllel Postultes 44.. Agles d Lies Tigles Coguet Tigles Simil Tigles Pl Figues Solid Figues Pythgoe Theoem 5.9. Heo s Fomul 5.. Golde Rtio 53.. Distce d Lie Fomuls 54.. Fomuls fo Coic Sectios Coic Sectios 3. Tigoomety Bsic Defiitios: Fuctios & Iveses Fudmetl Defiitio-Bsed Idetities Pythgoe Idetities Negtive Agle Idetities Sum d Diffeece Idetities Doule Agle Idetities Hlf Agle Idetities Geel Tigle Fomuls Ac d Secto Fomuls Degee/Rdi Reltioship Additio of Sie d Cosie Pol Fom of Comple Numes Rectgul to Pol Coodites Tigoometic Vlues fom Right Tigles Elemety Vecto Alge Bsic Defiitios d Popeties Dot Poducts Coss Poducts Lie d Ple Equtios Miscelleous Vecto Equtios 67 7

8 Tle of Cotets cot 5. Elemety Clculus Wht is Limit? Wht is Diffeetil? Bsic Diffeetitio Rules Tscedetl Diffeetitio Bsic Atidiffeetitio Rules Tscedetl Atidiffeetitio Lies d Appoimtio Itepettio of Defiite Itegl Fudmetl Theoem of Clculus Geometic Itegl Fomuls Select Elemety Diffeetil Equtios Lplce Tsfom: Geel Popeties Lplce Tsfom: Specific Tsfoms Moey d Fice Wht is Iteest? Simple Iteest Compoud d Cotiuous Iteest Effective Iteest Rtes Peset-to-Futue Vlue Fomuls Peset Vlue of Futue Deposit Stem Peset Vlue of with Iitil Lump Sum Peset Vlue of Cotiuous Types o Retiemet Svigs Accouts Lo Amotiztio Auity Fomuls Mkup d Mkdow Clculus of Fice Poility d Sttistics Poility Fomuls Bsic Cocepts of Sttistics Mesues of Cetl Tedecy Mesues of Dispesio Smplig Distiutio of the Me Smplig Distiutio of the Popotio 9 8

9 Tle of Cotets cot Sectio II: Tles. Numeicl 94.. Fctos of Iteges though Pime Numes less th Rom Numel d Aic Equivlets Nie Elemety Memoy Numes Ameic Nmes fo Lge Numes Selected Mgic Sques Thitee-y-Thitee Multiplictio Tle.8. The Rdom Digits of PI.9. Stdd Noml Distiutio 3.. Two-Sided Studet s t Distiutio 4.. Dte d Dy of Ye 5. Physicl Scieces 6.. Covesio Fctos i Allied Helth 6.. Medicl Aevitios i Allied Helth 7.3. Wid Chill Tle 8.4. Het Ide Tle 8.5. Tempetue Covesio Fomuls 9.6. Uit Covesio Tle 9.7. Popeties of Eth d Moo.8. Metic System 3.9. Bitish System 4 Sectio III: Applictios i Pesol Fice. The Alge of Iteest 8.. Wht is Iteest? 8.. Simple Iteest.3. Compoud Iteest.4. Cotiuous Iteest 4.5. Effective Iteest Rte 9 9

10 Tle of Cotets cot. The Alge of the Nest Egg 35.. Peset d Futue Vlue 35.. Gowth of Iitil Lump Sum Deposit Gowth of Deposit Stem 4.4. The Two Gowth Mechisms i Cocet Summy 5 3. The Alge of Cosume Det Lo Amotiztio You Home Motgge C Los d Leses The Auity s Motgge i Revese The Clculus of Fice Jco Beoulli s Diffeetil Equtio Diffeetils d Iteest Rte Beoulli d Moey Applictios 9 Appedices A. Geek Alphet B. Mthemticl Symols C. My Most Used Fomuls 4

11 Sectio I Fomuls with Select Pocesses

12 Ide to Pocesses Pocess Whee i Sectio I. Comple Rtioliztio Pocess.8.. Qudtic Tiomil Fctoig Pocess Lie Equtio Solutio Pocess.. 4. Lie Iequlity Solutio Pocess Ode of Opetios.. 6. Ode of Opetios with Petheses Rules Logithmic Simplifictio Pocess Comple Nume Multiplictio Comple Nume Divisio.6.9. Pocess of Costuctig Ivese Fuctios.8.6. Qudtic Equtios y Fomul.9.3. Qudtic Equtios y Fctoig Cdo s Cuic Solutio Pocess Cme s Rule, Two-y-Two System Cme s Rule, Thee-y-Thee System Removl of y Tem i Coic Sectios The Lie Fist-Ode Diffeetil Equtio Medi Clcultio 7.3.6

13 . Alge.. Wht is Vile? I the fll of 96, I fist ecouteed the moste clled i my high-school feshm lge clss. The lette is still moste to my, whose el tue hs ee cofused y such wods s vile d ukow: pehps the most hoifyig desciptio of eve iveted! Actully, is vey esily udestood i tems of lguge metpho. I Eglish, we hve oth pope ous d poous whee oth e distict d diffeet pts of speech. Pope ous e specific pesos, plces, o thigs such s Joh, Ohio, d Toyot. Poous e ospecific pesos o etities such s he, she, o it. To see how the cocept of poous d ous pplies to lge, we fist emie ithmetic, which c e thought of s pecise lguge of qutifictio hvig fou ctio ves, ve of eig, d pletho of pope ous. The fou ctio ves e dditio, sutctio, multiplictio, d divisio deoted espectively y,,,. The ve of eig is clled equls o is, 3 deoted y. Specific umes such s, 3. 45, 3 5, 3 769,.4563, 45,, seve s the ithmeticl equivlet to pope ous i Eglish. So, wht is? is meely ospecific ume, the mthemticl equivlet to poou i Eglish. Eglish poous getly epd ou cpility to descie d ifom i geel fshio. Hece, poous dd icesed fleiility to the Eglish lguge. Likewise, mthemticl poous such s, y, z, see Appedi B fo list of symols used i this ook getly epd ou cpility to qutify i geel fshio y ddig fleiility to ou lguge of ithmetic. Aithmetic, with the dditio of, y, z d othe mthemticl poous s ew pt of speech, is clled lge. I Summy: Alge c e defied s geelized ithmetic tht is much moe poweful d fleile th stdd ithmetic. The icesed cpility of lge ove ithmetic is due to the iclusio of the mthemticl poou d its ssocites y, z, etc. A moe use-fiedly me fo vile o ukow is poume. 3

14 .. Field Aioms The field ioms decee the fudmetl opetig popeties of the el ume system d povide the sis fo ll dvced opetig popeties i mthemtics. Let, & c e y thee el umes poumes. The field ioms e s follows. Popeties Additio Multiplictio. Closue is uique el ume is uique el ume Commuttive Associtive c c c c Idetity Ivese Distiutive o Likig Popety Tsitivity Note: c c & c c > & > c > c < & < c < c e ltete epesettios of 4

15 .3. Divisiility Tests Diviso Coditio Tht Mkes it So The lst digit is,,4,6, o 8 3 The sum of the digits is divisile y 3 4 The lst two digits e divisile y 4 5 The lst digit is o 5 6 The ume is divisile y oth d 3 The ume fomed y ddig five times the lst 7 digit to the ume defied y the emiig digits is divisile y 7** 8 The lst thee digits e divisile y 8 9 The sum of the digits is divisile y 9 The lst digit is divides the ume fomed y sutctig two times the lst digit fom the emiig digits** The ume is divisile y oth 3 d divides the ume fomed y ddig fou times the lst digit to the emiig digits** 4 The ume is divisile y oth d 7 5 The ume is divisile y oth 3 d divides the ume fomed y sutctig five times the lst digit fom the emiig digits** 9 9 divides the ume fomed y ddig two times the lst digit to the emiig digits** 3 3 divides the ume fomed y ddig seve times the lst digit to the emiig digits** 9 9 divides the ume fomed y ddig thee times the lst digit to the emiig digits** 3 divides the ume fomed y sutctig 3 thee times the lst digit fom the emiig digits** 37 divides the ume fomed y sutctig 37 eleve times the lst digit fom the emiig digits** **These tests e itetive tests i tht you cotiue to cycle though the pocess util ume is fomed tht c e esily divided y the diviso i questio. 5

16 .4. Sutctio, Divisio, Siged Numes.4.. Defiitios: Sutctio: Divisio:.4.. Altete epesettio of :.4.3. Divisio Popeties of Zeo Zeo i umeto: Zeo i deomito: is udefied Zeo i oth: is udefied.4.4. Demosttio tht divisio-y-zeo is udefied c c fo ll el umes c, the c fo ll el umes, If lgeic impossiility.4.5. Demosttio tht ttempted divisio-y-zeo leds to eoeous esults. y Let ; the multiplyig oth sides y gives y y y y y y y y Dividig oth sides y y whee y gives y y y y. The lst equlity is flse sttemet. 6

17 .4.6. Siged Nume Multiplictio:.4.7. Tle fo Multiplictio of Siged Numes: the itlicized wods i the ody of the tle idicte the esultig sig of the ssocited poduct. Multiplictio of Sig of Sig of Plus Mius Plus Plus Mius Mius Mius Plus.4.8. Demosttio of the lgeic esoleess of the lws of multiplictio fo siged umes. I oth colums, oth the middle d ightmost umes decese i the epected logicl fshio

18 .5. Rules fo Fctios c Let d e fctios with d d. d c d c c c c c.5.. Fctiol Equlity: d c.5.. Fctiol Equivlecy:.5.3. Additio like deomitos:.5.4. Additio ulike deomitos: c d c d c d d d d Note: d is the commo deomito.5.5. Sutctio like deomitos:.5.6. Sutctio ulike deomitos: c d c d.5.7. Multiplictio:.5.8. Divisio:.5.9. Divisio missig qutity:.5.. Reductio of Comple Fctio:.5.. Plcemet of Sig: c c c c c d d d d c c d d c d d c d c c c c c c d d c d c c 8

19 9.6. Ptil Fctios Let P e polyomil epessio with degee less th the degee of the fctoed deomito s show..6.. Two Distict Lie Fctos: B A P The umetos B A, e give y P B P A,.6.. Thee Distict Lie Fctos: c C B A c P The umetos C B A,, e give y,, c c c P C c P B c P A.6.3. N Distict Lie Fctos: i i i i i A P with i j j j i i i P A

20 .7. Rules fo Epoets.7.. Additio: m m.7.. Sutctio: m m.7.3. Multiplictio: m m.7.4. Distiuted ove Simple Poduct:.7.5. Distiuted ove Comple Poduct: p m p m.7.6. Distiuted ove Simple Quotiet:.7.7. Distiuted ove Comple Quotiet: p m p m.7.8. Defiitio of Negtive Epoet:.7.9. Defiitio of Rdicl Epessio:.7.. Defiitio whe No Epoet is Peset:.7.. Defiitio of Zeo Epoet:.7.. Demosttio of the lgeic esoleess of the defiitios fo d vi successive divisios y. Notice the powe deceses y with ech divisio [] [] []

21 .8. Rules fo Rdicls.8.. Bsic Defiitios: d.8.. m Comple Rdicl:.8.3. m m m Associtive:.8.4. Simple Poduct:.8.5. Simple Quotiet:.8.6. Comple Poduct: m m m.8.7. Comple Quotiet: m m.8.8. m m Nestig:.8.9. Rtiolizig Numeto fo > m :.8.. Rtiolizig Deomito fo > m :.8.. Comple Rtioliztio Pocess: c c c c c c c m m m m m Numeto: c c c.8.. Defiitio of Sud Pis: If ± the the ssocited sud is give y is dicl epessio, m.

22 .9. Fcto Fomuls.9.. Simple Commo Fcto: c c c.9.. Gouped Commo Fcto: d c d c c c d c dc d c.9.3. Diffeece of Sques:.9.4. Epded Diffeece of Sques: c c c.9.5. Sum of Sques: i i i comple.9.6. Pefect Sque: ± ±.9.7. Geel Tiomil:.9.8. Sum of Cues: Diffeece of Cues: Diffeece of Fouths: Powe Reductio to Itege: Powe Reductio to Rdicl:.9.3. Powe Reductio to Itege plus Rdicl:

23 .9.4. Qudtic Tiomil Fctoig Pocess c Let e qudtic tiomil whee the thee coefficiets,, c e iteges. M, N such tht Step : Fid iteges M N. M N c Step : Sustitute fo, c M N c Step 3: Fcto y Goupig.9. M N c M N M N M M N M N Note: if thee e o pi of iteges M, N with oth M N d M N c the the qudtic tiomil is pime. Emple: Fcto the epessio 3 7. : MN 7 4 & M N 3 M 4, N : :

24 .. Lws of Equlity Let A B e lgeic equlity dc, D e y qutities.... Additio: A C B C... Sutctio: A C B C..3. Multiplictio: A C B C..4. Divisio: A B povided C C C..5. Epoet: A B povided is itege..6. Recipocl: povided A, B A B C D..7. Mes & Etemes: CB AD if A, B A B..8. Zeo Poduct Popety: A B A o B..9. The Cocept of Equivlecy Whe solvig equtios, the Lws of Equlity with the eceptio of..5, which poduces equtios with et o eteous solutios i dditio to those fo the oigil equtio e used to mufctue equtios tht e equivlet to the oigil equtio. Equivlet equtios e equtios tht hve ideticl solutios. Howeve, equivlet equtios e ot ideticl i ppece. The gol of y equtio-solvig pocess is to use the Lws of Equlity to cete successio of equivlet equtios whee ech equtio i the equivlecy chi is lgeiclly simple th the pecedig oe. The fil equtio i the chi should e epessio of the fom, the o-ie fom tht llows the solutio to e immeditely detemied. I tht lgeic mistkes c e mde whe poducig the equivlecy chi, the fil swe must lwys e checked i the oigil equtio. Whe usig..5, oe must check fo eteous solutios d delete them fom the solutio set.... Lie Equtio Solutio Pocess Stt with the geel fom L R whee L d R e fist-degee polyomil epessios o the left-hd side d ight-hd side of the equls sig. 4

25 Step : Usig pope lge, idepedetly comie like tems fo oth L d R Step : Use.. d.. o s-eeded sis to cete equivlet equtio of the fom. Step 3: use eithe..3 o..4 to cete the fil equivlet fom fom which the solutio is esily deduced. Step 4: Check solutio i oigil equtio. Emple: Solve 4{3[7 y 3 9] y 9} 5 y 3. : 4{3[7 y 3 9] y 9} 5 y 3 4{3[7 y 9] y 8} 5y 5 3 4{3[7 y ] y 8} 5y 8 4{y 36 y 8} 5y 8 4{3y 54} 5y 8 9y 6 5y 8 9y 7 5y 8 : 9y 7 5y 8 9y 5y 7 5y 5y 8 87y y y 9 3 :87y 9 9 y : Check the fil swe y i the oigil equtio 87 4{3[7 y 3 9] y 9} 5 y 3. 5

26 .. Lws of Iequlity Let A > B e lgeic iequlity d C e y qutity.... Additio: A C > B C... Sutctio: A C > B C C > A C > B C..3. Multiplictio: C < A C < B C A B C > > C C..4. Divisio: A B C < < C C..5. Recipocl: < povided A, B A B Simil lws hold fo A < B, A B, d A B. Whe multiplyig o dividig y egtive C, oe must evese the diectio of the oigil iequlity sig. Replcig ech side of the iequlity with its ecipocl lso eveses the diectio of the oigil iequlity...6. Lie Iequlity Solutio Pocess Stt with the geel fom L > R whee L d R e s descied i... Follow the sme fou-step pocess s tht give i.. modifyig pe the checks elow. Revese the diectio of the iequlity sig whe multiplyig o dividig oth sides of the iequlity y egtive qutity. Revese the diectio of the iequlity sig whe eplcig ech side of iequlity with its ecipocl. The fil swe will hve oe the fou foms >,, <, d. Oe must ememe tht i of the fou cses, hs ifiitely my solutios s opposed to oe solutio fo the lie equtio. 6

27 .. Ode of Opetios Step : Pefom ll powe isigs i the ode they occu fom left to ight Step : Pefom ll multiplictios d divisios i the ode they occu fom left to ight Step 3: Pefom ll dditios d sutctios i the ode they occu fom left to ight Step 4: If petheses e peset, fist pefom steps though 3 o s-eeded sis withi the iemost set of petheses util sigle ume is chieved. The pefom steps though 3 gi, o s-eeded sis fo the et level of petheses util ll petheses hve ee systemticlly emoved. Step 5: If fctio is peset, simulteously pefom steps though 4 fo the umeto d deomito, tetig ech s totlly-septe polem util sigle ume is chieved. Oce sigle umes hve ee chieved fo oth the umeto d the deomito, the fil divisio c e pefomed..3. Thee Meigs of Equls. Equls is the mthemticl equivlet of the Eglish ve is, the fudmetl ve of eig. A simple ut sutle use of equls i this fshio is.. Equls implies equivlecy of mig i tht the sme udelyig qutity is eig med i two diffeet wys. This c e illustted y the epessio 3 MMIII. Hee, the two divese symols o oth sides of the equls sig efe to the sme d ect udelyig qutity. 3. Equls sttes the poduct eithe itemedite o fil tht esults fom pocess o ctio. Fo emple, i the epessio 4, we e ddig two umes o the lefthd side of the equls sig. Hee, dditio c e viewed s pocess o ctio etwee the umes d. The esult o poduct fom this pocess o ctio is the sigle ume 4, which ppes o the ight-hd side of the equls sig. 7

28 .4. The Seve Petheses Rules.4.. Cosecutive pocessig sigs,,, e septed y petheses..4.. Thee o moe cosecutive pocessig sigs e septed y ested pethesis whee the ightmost sig will e i the iemost set of petheses Nested petheses e typiclly witte usig the vious cketig symols to fcilitte edig The ightmost pocessig sig d the ume to the immedite ight of the ightmost sig e oth eclosed withi the sme set of petheses Petheses my eclose siged o usiged ume y itself ut eve sig y itself Moe th oe ume c e witte iside set of petheses depedig o wht pt of the ovell pocess is emphsized Whe petheses septe umes with o iteveig multiplictio sig, multiplictio is udestood. The sme is tue if just oe plus o mius sig septes the two umes d the petheses eclose oth the ightmost ume d the septig sig Demosttig the Seve Bsic Petheses Rules 5 : Popely witte s 5..4., : Popely witte s 5..4., : Popely witte s 5 [ ]..4. thu 4 5 : Icoect pe : Coect pe.4.,.4.4, : Does ot eed petheses to chieve septio sice the 5 seves the sme pupose. Ay use of petheses would e optiol 5 : The optiol petheses, though ot eeded, emphsize the egtive5 pe : The optiol petheses emphsize the fct tht the fil outcome is egtive pe.4.5,.4.6 8

29 4 : The mdtoy petheses idicte tht 4 is multiplyig. Without the petheses, the epessio would e popely ed s the sigle ume 4, : The mdtoy petheses idicte tht 7 is multiplyig 5. Without the iteveig petheses, the epessio is popely ed s the diffeece 7 5, : The mdtoy petheses idicte tht 3 is multiplyig 5. The epessio 3 5 lso sigifies the sme, Demosttio of Use of Ode-of-Opetios with Petheses Rules to Reduce Rtiol Epessio { 8} { 8} [8 8]

30 .5. Rules fo Logithms.5.. Defiitio of Logithm to Bse > : y log if d oly if y.5.. Logithm of the Sme Bse: log.5.3. Logithm of Oe: log.5.4. Logithm of the Bse to Powe: log p p log p.5.5. Bse to the Logithm: p.5.6. Nottio fo Logithm Bse : Log log.5.7. Nottio fo Logithm Bse e : l log log N.5.8. Chge of Bse Fomul: log N log.5.9. Poduct: log MN log N log M M.5.. Quotiet: log log M log N N p.5.. Powe: log N plog N.5.. Logithmic Simplifictio Pocess m A B Let X, the p C m A B log X log p C m p log X log A B log C log X log log X log m p A log B log C A mlog B p log C Note: The use of logithms tsfoms comple lgeic epessios whee poducts ecome sums, quotiets ecome diffeeces, d epoets ecome coefficiets, mkig the mipultio of these epessios esie i some istces. e 3

31 .6. Comple Numes.6.. Defiitio of the imgiy uit i : i is defied to e the solutio to the equtio..6.. Popeties of the imgiy uit i : i i i.6.3. Defiitio of Comple Nume: Numes of the fom i whee, e el umes.6.4. Defiitio of Comple Cojugte: i i.6.5. Defiitio of Comple Modulus: i.6.6. Additio: i c di c d i.6.7. Sutctio: i c di c d i.6.8. Pocess of Comple Nume Multiplictio i c di c d c i di c d c i d c d d c i.6.9. Pocess of Comple Nume Divisio i c di i c di c di c di i c di c di c di c d c d i c d c d c d i c d c d 3

32 .7. Wht is Fuctio? The mthemticl cocept clled fuctio is foudtiol to the study of highe mthemtics. With this sttemet i mid, let us defie i wokig sese the wod fuctio: A fuctio is y pocess whee umeicl iput is tsfomed ito umeicl output with the opetig estictio tht ech uique iput must led to oe d oly oe output. Fuctio Nme f Iput Side Pocessig Rule f Output Side The ove figue is digm of the geel fuctio pocess fo fuctio med f. Fuctio mes e usully lowe-cse lettes, f, g, h, etc. Whe mthemtici sys, let f e fuctio, the etie iput-output pocess stt to fiish comes ito discussio. If two diffeet fuctio mes e eig used i oe discussio, the two diffeet fuctios e eig discussed, ofte i tems of thei eltioship to ech othe. The vile is the idepedet o iput vile; it is idepedet ecuse y specific iput vlue c e feely chose. Oce specific iput vlue is chose, the fuctio the pocesses the iput vlue vi the pocessig ule i ode to cete the output vile f, lso clled the depedet vile sice the vlue of f is etiely detemied y the ctio of the pocessig ule upo. Notice tht the comple symol f eifoces the fct tht output vlues e ceted y diect ctio of the fuctio pocess f upo the idepedet vile. Sometimes, simple y will e used to epeset the output vile f whe it is well udestood tht fuctio pocess is ideed i plce. Two moe defiitios e oted. The set of ll possile iput vlues fo fuctio f is clled the domi d is deoted y the symol Df. The set of ll possile output vlues is clled the ge d is deoted y Rf. 3

33 .8. Fuctio Alge Let f d g e fuctios, d let f e the ivese fo f.8.. Ivese Popety: f f f [ ] [ f ].8.. Additio/Sutctio: f ± g f ± g.8.3. Multiplictio: f g f g f f g g.8.4. Divisio: ; g g f f f o g f [ g ].8.5. Compositio: g o f g[ f ].8.6. Pocess fo Costuctig Ivese Fuctios Step : Stt with f f, the pocess equlity tht must e i plce fo ivese fuctio to eist. Step : Replce f with y to fom the equlity f y. Step 3: Solve fo y i tems of. The esultig y is f. Step 4: Veify y the popety f f f f Demosttio of.8.6: Fid f fo f 3. : f f : y 3 : y y 4 : f : f f f f

34 .9. Qudtic Equtios & Fuctios.9.. Defiitio d Discussio A complete qudtic equtio i stdd fom edy-to-esolved is equtio hvig the lgeic stuctue c whee,, c. If eithe o c, the qudtic equtio is clled icomplete. If, the qudtic equtio educes to lie equtio. All qudtic equtios hve ectly two solutios if comple solutios e llowed. Solutios e otied y eithe fctoig o y use of the qudtic fomul. If, withi the cotet of pticul polem comple solutios e ot dmissile, qudtic equtios c hve up to two el solutios. As with ll el-wold pplictios, the ume of dmissile solutios depeds o cotet..9.. Qudtic Fomul with Developmet: c c 4 ± ± c 4c 4 4c 4 4c.9.3. Solutio of Qudtic Equtios y Fomul To solve qudtic equtio usig the qudtic fomul the moe poweful of two commo methods fo solvig qudtic equtios pply the followig fou steps. Step : Rewite the qudtic equtio so it mtches the stdd fom c. 34

35 Step : Idetify the two coefficiets d costt tem,,& c. Step 3: Apply the fomul d solve. Step 4: Check you swes i the oigil equtio Solutio Discimito: 4c 4c > two el solutios 4c oe el solutio of multiplicity two two comple cojugtes solutios 4c <.9.5. Solutio whe & : c c.9.6. Solutio of Qudtic Equtios y Fctoig To solve qudtic equtio usig the fctoig method, pply the followig fou steps. Step : Rewite the qudtic equtio i stdd fom Step : Fcto the left-hd side ito two lie fctos usig the qudtic tiomil fctoig pocess.9.4. Step 3: Set ech lie fcto equl to zeo d solve. Step 4: Check swes i the oigil equtio Note: Use the qudtic fomul whe qudtic equtio cot e fctoed o is hd to fcto Qudtic-i-Fom Equtio: U U c whee U is lgeic epessio of vyig compleity Defiitio of Qudtic Fuctio: 4c f c Ais of Symmety fo Qudtic Fuctio: 4c.9.. Vete fo Qudtic Fuctio:, 4 35

36 .. Cdo s Cuic Solutio d 3 Let c e cuic equtio witte i stdd fom with Step : Set y. Afte this sustitutio, the ove cuic 3 3 c ecomes y py q whee p d 3 c q d Step : Defie u & v such tht y u v d p 3uv 3 Step 3: Sustitute fo y & p i the equtio y py q p This leds to u qu, which is qudtici-fom i u 7 3. Step 4: Use the qudtic fomul.9.3 to solve fo u q 4 3 q 3 7 p 3 u Step 5: Solve fo u 3 u & v whee q q 4 7 p v to oti 3u 3 p q 3 & v q 4 7 p 3 Step 6: Solve fo whee y 3 u v 3 36

37 .. Theoy of Polyomil Equtios... Let P e polyomil witte i stdd fom. The Eight Bsic Theoems... Fudmetl Theoem of Alge: Evey polyomil P of degee N hs t lest oe solutio fo which P. This solutio my e el o comple i.e. hs the fom i.... Numes Theoem fo Roots d Tuig Poits: If P is polyomil of degee N, the the equtio P hs up to N el solutios o oots. The equtio P hs ectly N oots if oe couts comple solutios of the fom i. Lstly, the gph of P will hve up to N tuig poits which icludes oth eltive mim d miim...3. Rel Root Theoem: If P is of odd degee hvig ll el coefficiets, the P hs t lest oe el oot...4. Rtiol Root Theoem: If P hs ll itege coefficiets, the y tiol oots fo the equtio P p must hve the fom q whee p is fcto of the costt coefficiet d q is fcto of the led coefficiet. Note: This esult is used to fom tiol-oot possiility list...5. Comple Cojugte Pi Root Theoem: Suppose P hs ll el coefficiets. If i is oot fo P with P i, the P i...6. Itiol Sud Pi Root Theoem: Suppose P hs ll tiol coefficiets. If is oot fo P with P, the P. 37

38 ..7. Remide Theoem: If P is divided y c, the the emide R is equl to P c. Note: this esult is etesively used to evlute give polyomil P t vious vlues of...8. Fcto Theoem: If c is y ume with P c, the c is fcto of P. This mes P c Q whee Q is ew, educed polyomil hvig degee oe less th P. The covese P c Q P c is lso tue. The Fou Advced Theoems..9. Root Loctio Theoem: Let, e itevl o the is with P P <. The thee is vlue, such tht P.... Root Boudig Theoem: Divide P y d to oti P d Q R. Cse d > : If oth R d ll the coefficiets of Q e positive, the P hs o oot > d. Cse d < : If the oots of Q ltete i sig with the emide R i syc t the ed the P hs o oot < d. Note: Coefficiets of zeo c e couted eithe s positive o egtive which eve wy helps i the susequet detemitio.... Desctes Rule of Sigs: Age P i stdd ode s show i the title. The ume of positive el solutios equls the ume of coefficiet sig vitios o tht ume decesed y eve ume. Likewise, the ume of egtive el solutios equls the ume of coefficiet sig vitios i o tht ume decesed y eve ume. P... Tuig Poit Theoem: Let polyomil P hve degee N. The the ume of tuig poits fo polyomil P c ot eceed N. 38

39 .. Detemits d Cme s Rule... Two y Two Detemit Epsio: d c c d... Thee y Thee Detemit Epsio: c e f d f d e d e f c h i g i g h g h i ei fh di fg c dh eg ei hf fg di cdh ceg..3. Cme s Rule fo Two-y-Two Lie System Give y e c dy f with D c d The e f d d D y e c f D..4. Cme s Rule fo Thee-y-Thee Lie System Give y cz j c d ey fz k with D d e f g hy iz l g h i The j c j c k e f d k f l h i g l i, y, z D D d g e h D j k l 39

40 D..5. Solutio Types i i D D, D i i D i, D i hs ifiite solutios D i, D i hs uique solutio D i, D i hs o solutio i.3. Biomil Theoem Let d e positive iteges with..3.. Defiitio of!:!...,.3.. Specil Fctoils:! d!!.3.3. Comitoil Symol:!!.3.4. Summtio Symols:... i 3 4 i i k k k k 3... i k.3.5. Biomil Theoem: i i i.3.6. Sum of Biomil Coefficiets whe : i i i i.3.7. Fomul fo the th Tem: i 4

41 :.3.8. Pscl s Tigle fo Aithmetic Seies.4.. Defiitio: S i i whee is the commo icemet.4.. Summtio Fomul fo S : S [ ].5. Geometic Seies i.5.. Defiitio: G whee is the commo tio i.5.. Summtio Fomul fo G : G i G G i G i G i i i i i i.5.3. Ifiite Sum Povided < < : i i i 4

42 .6. Boole Alge I the followig tles, the popositios o Flse F..6.. Elemety Tuth Tle: p & q e eithe Tue T d : o : egtio ~: implies, p q ~ p ~ q p q p q p q p q T T F F T T T T T F F T F T F F F T T F F T F F F F T T T F T T.6.. Tuth Tle fo the Eclusive O p q p e q T T F T F T F T T F F F e :.6.3. Modus Poes: Let p q & p T. The, q T Chi Rule: Let p q & q. The p T Modus Tolles: Let p q & q F. The ~ q ~ p T Fllcy of Affimig the Cosequet: Let p q & q T.The q p F Fllcy of Deyig the Atecedet: Let p q & p F. The ~ p ~ q F Disjuctive Syllogism fo the Eclusive O: p Let p e q T & q F. The T 4

43 .6.9. Demosttio tht the Eglish doule-egtive i the slg epessio I do t got oe ctully ffims the opposite of wht is iteded. Step Phse Commet I do ot hve y The oigil popositio p s iteded I do hve oe Assume p T I do ot hve oe Negtio of p : ~ p F 3 I do t hve oe Pope cotcted fom of 3: ~ p F 4 I do t got oe Slg vesio of 3 5 I hve some Logicl cosequece of 3: ~ p F ~ ~ p T.7. Vitio o Popotiolity Fomuls.7.. Diect: y k k.7.. Ivese: y.7.3. Joit: z ky k.7.4. Ivese Joit: z y.7.5. Diect to Powe: y k k.7.6. Ivese to Powe: y 43

44 . Geomety.. The Pllel Postultes... Let poit eside outside give lie. The thee is ectly oe lie pssig though the poit pllel to the give lie.... Let poit eside outside give lie. The thee is ectly oe lie pssig though the poit pepedicul to the give lie...3. Two lies oth pllel to thid lie e pllel to ech othe...4. If tsvese lie itesects two pllel lies, the coespodig gles i the figues so fomed e coguet...5. If tsvese lie itesects two lies d mkes coguet, coespodig gles i the figues so fomed, the the two oigil lies e pllel... Agles d Lies α β α β α β 8 α β 9... Complimety Agles: Two gles α, β with α β 9. 44

45 ... Supplemety Agles: Two gles α, β with α β Lie Sum of Agles: The sum of the two gles α, β fomed whe stight lie is itesected y lie segmet is equl to Acute Agle: A gle less th..5. Right Agle: A gle ectly equl to Otuse Agle: A gle gete th 9.3. Tigles γ α β γ 8 α c β.3.. Tigul Sum of Agles: The sum of the thee iteio gles α, β, γ i y tigle is equl to Acute Tigle: A tigle whee ll thee iteio glesα, β, γ e cute.3.3. Right Tigle: A tigle whee oe iteio gle fom the tid α, β, γ is equl to Otuse Tigle: A tigle whee oe iteio gle fom the tid α, β, γ is gete th Sclee Tigle: A tigle whee o two of the thee side-legths,, c e equl to othe.3.6. Isosceles Tigle: A tigle whee ectly two of the side-legths,, c e equl to ech othe.3.7. Equiltel Tigle: A tigle whee ll thee sidelegths,, c e ideticl c o ll thee gles α, β, γ e equl with α β γ 6 45

46 .3.8. Coguet Tigles: Two tigles e coguet equl if they hve ideticl iteio gles d side-legths Simil Tigles: Two tigles e simil if they hve ideticl iteio gles..3.. Icluded Agle: The gle tht is etwee two give sides.3.. Opposite Agle: The gle opposite give side.3.. Icluded Side: The side tht is etwee two give gles.3.3. Opposite Side: The side opposite give gle.4. Coguet Tigles Give the coguet two tigles s show elow γ e ω d α c β φ.4.. Side-Agle-Side SAS: If y two side-legths d the icluded gle e ideticl, the the two tigles e coguet. Emple: & α & c e & φ & f.4.. Agle-Side-Agle ASA: If y two gles d the icluded side e ideticl, the the two tigles e coguet. Emple: α & c & β φ & f & ϕ.4.3. Side-Side-Side SSS: If the thee side-legths e ideticl, the the tigles e coguet. Emple: & c & e & f & d.4.4. Thee Attiutes Ideticl: If y thee ttiutes side-legths d gles e equl with t lest oe ttiute eig side-legth, the the two tigles e coguet. These othe cses e of the fom Agle-Agle-Side AAS o Side-Side-Agle SSA. Emple SSA: & & β e & d & ϕ Emple AAS: & & & & d f α β φ ϕ ϕ 46

47 .5. Simil Tigles Give the two simil tigles s show elow ω γ e d α β c φ ϕ f.5.. Miiml Coditio fo Simility: If y two gles e ideticl AA, the the tigles e simil. Suppose The α φ & β ϕ α β γ 8 α 8 β γ 8 & φ ϕ ϖ 8 ϕ ϖ φ.5.. Rtio lws fo Simil Tigles: Give simil c tigles s show ove, the e f d.6. Pl Figues A is the pl e, P is the peimete, is the ume of sides..6.. Degee Sum of Iteio Agles i Geel Polygo: D 8 [ ] D 54 6 D Sque: A s : P 4s, s is the legth of side s 47

48 .6.3. Rectgle: A h : P h, & h e the se d height h.6.4. Tigle: A h, & h e the se d ltitude h.6.5. Pllelogm: A h, & h e the se d ltitude h.6.6. Tpezoid: A B h, B & e the two pllel ses d h is the ltitude h B π π.6.7. Cicle: A : P whee is the dius, o P πd whee d, the dimete Ellipse: A π ; & e the hlf legths of the mjo & mio es 48

49 .7. Solid Figues A is totl sufce e, V is the volume.7.. Cue: A 6s V s 3 :, s is the legth of side s π Sphee: A 4 : V, is the dius 3 π.7.3. Cylide: A π πl : V π l, & l the dius d legth l e.7.4. Coe: A π πt : V π h, & t & h e the dius, slt height, d ltitude 3 h t 49

50 .7.5. Pymid sque se: A s st : V 3 s h, s & t & h e the side, slt height, d ltitude s h t.8. Pythgoe Theoem.8.. Sttemet: Let ight tigle ABC hve oe side AC of legth, secod side AB of legth y, d hypoteuse log side BC of legth z. The z y B y A z C.8.. Tditiol Algeic Poof: Costuct ig sque y igig togethe fou coguet ight tigles. z y 5

51 The e of the ig sque is give y A y A z 4. Equtig: y y z 4 z y, o equivletly y y y y z y z y Visul Pe-Algeic Pythgoe Poof: The ide is to oseve tht the two five-sided iegul polygos o eithe side of the dotted lie hve equivlet es. Tkig wy thee coguet ight tigles fom ech e leds to the desied Pythgoe equlity Pythgoe Tiples: Positive iteges, such tht L M N L M, N.8.5. Geetig Fomuls fo Pythgoe tiples: Let m, with m > > e iteges. The, N m, d M m L m 5

52 5.9. Heo s Fomul Let c s e the semi-peimete of geel tigle d A e the itel e eclosed y the sme..9.. Heo s Fomul: c s s s s A.9.. Deivtio Usig Pythgoe Theoem: : Cete two equtios fo the ukows h d : : c h E h E : Sutct E fom E d solve fo c c c c c : 3 Sustitute the vlue fo ito E c c h : 4 Solve fo h [ ] 4 4 c c c h c h c

53 53 [ ] { } [ ] { } [ ] { }[ ] { } { }{ }{ }{ } c c c c c h c c c h c c c c c h : 5 Solve fo e usig ch A. { }{ }{ }{ } { }{ }{ }{ } 6 4 c c c c A c c c c c c A : 6 Sustitute c s d simplify. } { c s s s s A s s s c s A s s s c s A.. Golde Rtio... Defiitio: Let p e the semi-peimete of ectgle whose se d height e i the popotio show, defiig the Golde Rtioφ. Solvig fo leds to 68. φ. φ

54 ... Golde Tigles: Tigles whose sides e popotioed to the Golde Rtio. Two emples e show elow. B B 36 φ 8 C 7 A C A D.. Distce d Lie Fomuls Let, y d, y e two poits whee >.... -D Distce Fomul: D y y... 3-D Distce Fomul: Fo the poits, y, z d, y,, z y y z y y D z..3. Midpoit Fomul:, Lie Fomuls y y..4. Slope of Lie: m..5. Poit/Slope Fom: y y m..6. Geel Fom: A By C..7. Slope/Itecept Fom: y m whee, m d, e the d y Itecepts: 54

55 y..8. Itecept/Itecept Fom: whee, d, e the d y itecepts..9. Slope Reltioship etwee two Pllel Lies L d L hvig slopes m d m : m m... Slope Reltioship etwee two Pepedicul Lies L d L hvig slopes m d m : m m... Slope of Lie Pepedicul to Lie of Slope m : m.. Fomuls fo Coic Sectios... Geel: A By Cy D Ey F... Cicle of Rdius Ceteed t h, k : h y k..3. Ellipse Ceteed t h y k h, k : I If >, the two foci e o the lie y k d e give y h c, k & h c, k whee. II If >, the two foci e o the lie h d e give y h, k c & h, k c whee c...4. Hypeol Ceteed t h, k : h y k y k h o h I Whe is to the left of the mius sig, the two foci e o the lie y k d e give y h c, k & h c, k whee c. c 55

56 y k II Whe is to the left of the mius sig, the two foci e o the lie h d e give y h, k c & h, k c whee c...5. Pol with Vete t h, k d Focl Legth p : y k 4 p h o h 4 p y k I Fo y k, the focus is h p, k d the diecti is give y the lie h p. II Fo h, the focus is h, k p d the diecti is give y the lie. y k p..6. Tsfomtio Pocess fo Removl of y Tem i the Geel Coic Equtio A By Cy D Ey F : B Step : Set t θ d solve fo θ. A C Step : let cosθ y siθ y siθ y cosθ Step 3: Sustitute the vlues fo, y otied i Step ito A By Cy D Ey F. Step 4: Reduce. The fil esult should e of the fom A C y D E y F. 56

57 3. Tigoomety 3.. Bsic Defiitios of Tigoometic Fuctios & Tigoometic Ivese Fuctios z y α Let the figue ove e ight tigle with oe side of legth, secod side of legth y, d hypoteuse of legth z. The gle α is opposite the side of legth. The si tigoometic fuctios whee ech is fuctio of α e defied s follows: Z : Aity Z is Ivese whe Z is y 3... siα siα y si y α z 3... cosα cosα cos α z y y y tα tα t α cotα cotα cot α y y y z secα sec α sec α z cscα csc α csc α y y y si Note: is lso kow s csi. Likewise, the othe iveses e lso kow s ccos, ct, c cot, csec d c csc. 57

58 3.. Fudmetl Defiitio-Bsed Idetities 3... csc α si α 3... sec α cos α si α t α cos α cos α cot α si α t α cot α 3.3. Pythgoe Idetities si α cos α t α sec α cot α csc α 3.4. Negtive Agle Idetities si α si α cos α cos α t α t α cot α cot α 3.5. Sum d Diffeece Idetities α si β si α cos β cos αsi β si α β si α cos β cos αsi β cos α β cos α cos β si αsi β cos β cos α cos β si αsi β α 58

59 t α t β t α β t α t β t α t β t α β t α t β Deivtio of Fomuls fo cos β si α β : α d I the figue elow, ech coodite of the poit {cos α β,si α β } is decomposed ito two compoets usig oth defiitios fo the sie d cosie i 3... d 3... y {cos α β,si α β } si β si α siβ cosβ α y si β cos α {cos α,si α} y si αcos β β, α, cos αcos β Fom the figue, we hve cos α β cos α β cos αcos β si αsi β si α β y y. si α β si αcos β si β cos α 59

60 3.6. Doule Agle Idetities si α si α cos α α α α cos cos si α cos cos si t α t α t α α 3.7. Hlf Agle Idetities α α si ± cos α α cos α cos ± α t ± cos α cos α si α cos α cos α si α 3.8. Geel Tigle Fomuls Applicle to ll tigles: ight d o-ight z β y α θ Sum of Iteio Agles: α β θ 8 lso.3.. si α si β si θ Lw of Sies: y z 6

61 Lw of Cosies: y z z cos α y z yz cos β c z y y cos θ Ae Fomuls fo Geel Tigle: A z si α A yz si β y si θ c A Deivtio of Lw of Sies d Cosies: Let ABC e geel tigle d dop pepedicul fom the pe s show. C A β γ h y c y α B Fo the Lw of Sies we hve h : si α h si α h : si β h si β 3 : si α si β si β si α The lst equlity is esily eteded to iclude the thid gle γ. 6

62 Fo the Lw of Cosies we hve h siα. : Solve fo y d i tems of the gle α y cos α y cos α c y c cos α : Use the Pythgoe Theoem to complete the deivtio. h [ c cos α] c c ccos α ccos α c [ si α] ccos α cos α si α Simil epessios c e witte fo the emiig two sides Ac d Secto Fomuls θ s Ac Legth s : s θ Ae of Secto: A θ 3.. Degee/Rdi Reltioship 3... Bsic Covesio: 8 π dis 6

63 3... Covesio Fomuls: Fom To Multiply y Rdis Degees Degees Rdis 8 π π Additio of Sie d Cosie siθ cosθ k si θ α whee k α si o α cos 3.. Pol Fom of Comple Numes 3... i cos i si whee θ, θ T iθ θ 3... Defiitio of e : e i cosθ i siθ iπ Eule s Fmous Equlity: e iθ iθ De-Moive s Theoem: e e o [ cosθ i siθ ] cos[ θ ] i si[ θ ] θ 63

64 3..5. Pol Fom Multiplictio: iα iβ i α β e e e Pol Fom Divisio: e iα i α β e iβ e 3.3. Rectgul to Pol Coodites, y, θ cosθ, y siθ y, θ t y / 3.4. Tigoometic Vlues fom Right Tigles I the ight tigle elow, let si α si α. α The cos α t α cot α sec α csc α 64

65 4. Elemety Vecto Alge 4.. Bsic Defiitios d Popeties Let V v, v,, U u, u, e two vectos. v3 u3 U θ V 4... Sum d/o Diffeece: U ± V U ± V u ± v, u ± v, u3 ± v Scl Multiplictio: α U αu, αu, αu Negtive Vecto: U U Zeo Vecto:,, Vecto Legth: U u u u Uit Vecto Pllel tov : V V Two Pllel Vectos: V U mes thee is scl c such tht V c U 4.. Dot Poducts 4... Defiitio of Dot Poduct: U V uv u v u3v3 U V 4... Agleθ Betwee Two Vectos: cosθ U V Othogol Vectos: U V 65

66 Pojectio of U otov : [ ] cos V V U V V V V U V V V U U poj V θ 4.3. Coss Poducts Defiitio of Coss Poduct: 3 3 v v v u u u k j i V U Oiettio of V U ; Othogol to BothU dv : V U V V U U Ae of Pllelogm: θ si V U V U A Itepettio of the Tiple Scl Poduct: w w w v v v u u u W V U The tiple scl poduct is umeiclly equl to the volume of the pllelepiped t the top of the et pge U V θ V U

67 U W V 4.4. Lie d Ple Equtios Give poit P,, c Lie Pllel to P Pssig Though, y, z : If, y, z is poit o the lie, the y y z z c Ple Noml to P Pssig Though, y, z. If, y, z is poit o the ple, the,, c, y y, z z Distce D etwee poit & ple: If poit is give y, y, z d y cz d is ple, the y cz d D c 4.5. Miscelleous Vecto Equtios The Thee Diectio Cosies: v v v3 cosα,cos β,cosγ V V V Defiitio of Wok: costt foce F log the pth PQ : W F PQ poj F PQ PQ 67

68 5. Elemety Clculus 5.. Wht is Limit? Limits e foudtiol to clculus d will lwys e so. Limits led to esults uotile y lge loe. So wht is limit? A limit is umeicl tget, tget cquied d locked. Coside the epessio 7 whee is idepedet vile. The ow poits to tget o the ight, i this cse the ume 7. The vile o the left is tgetig 7 i mode smt-wepo sese. This mes is movig, movig towds tget, closig ge, d pogmmed to mege evetully with the tget. Notice tht the qutity is tue idepedet vile i tht hs ee luched d set i motio towds tget, tget tht cot escpe fom its sights. Idepedet viles usully fid themselves emedded iside lgeic o tscedetl epessio of some sot, which is eig used s pocessig ule fo fuctio. Coside the epessio 3 whee the idepedet vile is out to e set o the missio 5. Does the etie epessio 3 i tu tget umeicl vlue s 5? A wy to phse this questio usig ew type of mthemticl ottio might e t g et 3? Itepetig the ottio, we e 5 skig if the dymic output stem fom the epessio 3 tgets umeicl vlue i the mode smt-wepo sese s the eqully-dymic tgets the vlue 5. Mthemticl judgmet sys yes; the output stem tgets the vlue 7. Hece, we complete ou ew ottio s t get This epltio is esole ecept fo oe little polem: the wod tget is owhee to e foud i clculus tets. The tditiol eplcemet weighig i with 3 yes of histoy is the wod limit, which leds to the followig wokig defiitio: Wokig Defiitio: A limit is tget i the mode smt-wepo sese. I the ove emple, we will wite lim

69 5.. Wht is Diffeetil? The diffeetil cocept is oe of the two coe cocepts udelyig clculus, limits eig the othe. Wee is Scottish wod tht mes vey smll, tiy, dimiutive, o miuscule. I the cotet of clculus, wee c e used i simil fshio to help epli the cocept of diffeetil, lso clled ifiitesiml. To hve diffeetil, we fist must hve vile,, y, z etc. Oce we hve vile, sy, we c cete secody qutity d, which is clled the diffeetil of the vile. Wht ectly is this d, ed dee? The qutity d is vey smll, tiy, dimiutive, o miuscule umeicl mout whe comped to the oigil. Moeove, it is the vey smll size of d tht mkes it, y defiitio, wee. How smll? I mthemticl tems, the followig two coditios hold: d < d << d < <<. The two ove coditios stte d is smll eough to gutee tht oth its poduct d quotiet with the oigil qutity is still vey smll d much, much close to zeo th to oe the meig of the symol <<. Both iequlities imply tht d is lso vey smll whe cosideed idepedetly < d <<. Lstly, oth iequlities stte tht d >, which igs us to the followig vey impott poit: lthough vey smll, the qutity d is eve zeo. Oe c lso thik of d s the fil h i limit pocess lim whee the pocess uptly stops just shot of tget, h i effect svig the pidly vishig h fom disppeig ito olivio! Thikig of d i this fshio mkes the diffeetil pepckged o foze limit of sots. Diffeetils e desiged to e so smll tht secod-ode d highe tems ivolvig diffeetils, such s 7 d, c e totlly igoed i ssocited lgeic epessios. This fil popety distiguishes the diffeetil s topic elogig to the suject of clculus. 69

70 5.3. Bsic Diffeetitio Rules Limit Defiitio of Deivtive: f h f f ' lim h h Diffeetitio Pocess Idicto: [] Costt: [ k ] Powe:[ ], c e y epoet Coefficiet:[ f ] f ' Sum/Diffeece:[ f ± g ] f ± g Poduct:[ f g ] f g' g f ' f g f ' f g' Quotiet: g g Chi:[ f g ] f ' g g' Ivese: [ f ] f ' f Geelized Powe: [{ f } ] { f } f ' ; Agi, c e y epoet 5.4. Tscedetl Diffeetitio [l ] [log ] l [ e ] e [ ] l [si ] cos 7

71 [si ] [cos ] si [cos ] [t ] sec [t ] [sec ] sec t [sec ] 5.5. Bsic Atidiffeetitio Rules Atidiffeetitio Pocess Idicto: Costt: kd k C Coefficiet: f d f d Powe Rule fo : d C Powe Rule fo : d d l C Sum: [ g ] d f d f g d Diffeece: [ f g ] d f d g d Pts: f g d f g g f d Chi: f g g d f g C 7

72 5.5.. Geelized Powe Rule fo : [ f ] f d [ f ] C Geelized Powe Rule fo : f d l f C, f f f Geel Epoetil: e f d e C 5.6. Tscedetl Atidiffeetitio l d l C e d e C e d e C d C l cos d si C si d cos C t d l cos C cot d l si C sec d l sec t C sec t d sec C sec d t C csc d l csc cot C csc d cot C 7

73 d si C d t C d l C 5.7. Lies d Appoimtio Tget Lie t, f : y f f Noml Lie t, f : y f f Lie Appoimtio: f f f Secod Ode Appoimtio: f f f f f Newto s Itetive Fomul: f Diffeetil Equlities: y f dy f d f d f f d F d F f d 5.8. Itepettio of Defiite Itegl At lest thee itepettios e vlid fo the defiite itegl. Fist Itepettio: As pocessig symol fo fuctios, the defiite itegl f d istucts the opeto to stt the pocess y fidig F the pimy tideivtive fo f d d fiish it y evlutig the qutity F F F. This itepettio is pue pocess-to-poduct with o cotet. 73

74 Secod Itepettio: As summtio symol fo diffeetil qutities, f d sigls to the opeto tht myids of ifiitesiml qutities of the fom f d e eig cotiuously summed o the itevl [, ] with the summtio pocess sttig t d edig t. Depedig o the cotet fo give polem, such s summig e ude cuve, the diffeetil qutities f d d susequet totl c tke o viety of meigs. This mkes cotiuous summig poweful tool fo solvig el-wold polems. The fct tht cotiuous sums c lso e evluted y f d F F F is key cosequece of the Fudmetl Theoem of Clculus Thid Itepettio: The defiite itegl f d c e itepeted s poit solutio y to y eplicit diffeetil equtio hvig the geel fom dy f d : y. I this itepettio f d is fist modified y itegtig ove the vile suitevl [, z] [, ]. This leds to z y z f d F z F. Sustitutig gives the stted oudy coditio y F F d sustitutig gives y F F f d. I this cotet, the fuctio y z F z F, s uique solutio to dy f d : y, c lso e itepeted s cotiuous uig sum fom to z. 74

75 5.9. The Fudmetl Theoem of Clculus Let f d e defiite itegl epesetig cotiuous summtio pocess, d let F e such tht F f. The, f d c e evluted y the ltetive pocess f d F F F. Note: A cotiuous summtio o dditio pocess o the itevl[, ] sums millios upo millios of cosecutive, tiy qutities fom to whee ech idividul qutity hs the geel fom f d. 5.. Geometic Itegl Fomuls 5... Ae Betwee two Cuves fo f g o [, ] : A [ f g ] d 5... Ae Ude f o [, ] : A f d Volume of Revolutio out Ais Usig Disks: V π [ f ] d Volume of Revolutio out y Ais usig Shells: V π f d Ac Legth: s [ f ] d 75

76 5..6. Revolved Sufce Ae out Ais: SA π f [ f ' ] d Revolved Sufce Ae out y Ais: SA π [ f ' ] d Totl Wok with Vile Foce F o [, ] : W F d 5.. Select Odiy Diffeetil Equtios ODE dy 5... Fist Ode Lie: f y g d dy 5... Beoulli Equtio: f y g y d ODE Seple if it educes to: g y dy f d dv Fllig Body with Dg: m mg kv dt Costt Rte Gowth o Decy: dy ky : y y dt dy Logistic Gowth: k L y y : y y dt Cotiuous Piciple Gowth: dp P c : P P dt Newto s Lw i Oe Dimesio: d mv F dt Newto s Lw i Thee d Dimesios: mv F dt 76

77 Pocess fo Solvig Lie ODE Step: Let F e such tht f F Step : Fomulte the itegtig fcto F e Step 3: Multiply oth sides of g y f d dy y F e g e ye d d g e y f e d dy e F F F F F Step 4: Pefom the idefiite itegtio. [ ] F F F F F Ce d g e e y y C d g e y e 5.. Lplce Tsfom; Geel Popeties 5... Defiitio: ] [ s F dt e t f t f L st 5... Lie Opeto Popety: ] [ s G s F t g t f L Tsfom of the Deivtive:... ] [ f f s f s s F s t f L Deivtive of the Tsfom: t f t s F Tsfom of the Defiite Itegl: s s F d f L t / ] [ τ τ