Indices. Indices. Curriculum Ready ACMNA: 209, 210, 212,
|
|
|
- Sandra Jefferson
- 5 years ago
- Views:
Transcription
1 Inis Inis Curriulum Ry ACMNA: 09, 0,, 6
2
3 Inis INDICES Inis is t plurl or inx. An inx is us to writ prouts o numrs or pronumrls sily. For xmpl is tully sortr wy o writin #. T is t inx. Anotr wor or inx is xponnt. Answr ts qustions, or workin trou t ptr. I us to tink: Wt r inis? (Wt r xponnts?) How r inis us to writ vry lr n vry smll numrs? Wt is sintii nottion? Answr ts qustions, tr workin trou t ptr. But now I tink: Wt r inis? (Wt r xponnts?) How r inis us to writ vry lr n vry smll numrs? Wt is sintii nottion? Wt o I know now tt I in't know or? Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
4 Inis Bsis Inx Nottion Inx nottion is us to writ prout o numr wit itsl in n sir wy. For xmpl: Inx or xponnt or powr # # # 6 Bsi numrl So wn multiplyin numr, sy 6, y itsl 00 tims, it s sir to writ 6 00 inst o 6# 6# 6 #...(00 tims). Bs I t inx is w usully mk it invisil so w writ 7 inst o 7. Ain n Sutrtin wit Inis Two xprssions in inx orm r lik trms i ty v t sm s AND ty v t sm inx. Cn simplii (lik trms) Cnnot simplii (unlik trms) Sm inx Dirnt inis + ^ Sm s Dirnt s Simpliyin lik trms Lik trms Lik trms Lik trms roup totr ^ ^ + + Simpliy lik trms ^6 + ^ + J 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
5 Inis Bsis Multiplition wit Inis Consir t prout # ^ # # ^ # # +. In t prout, t s is t sm n t inis v n totr. In nrl you n pply t ormul or multiplition wit inis: m n m+ n # Sm s A inis Multiplyin trms wit inis 7 + t # 9t # 9 # t # t 6t 6t Coiints multipli sprtly Coiints multipli sprtly REMEMBER A oiint is t numr or t vril in n xprssion. E. T oiint o x is. + + pq # pq # # p # p # q # q p q pq Sm ss roup totr Division wit Inis I w ivi ' # # # # # # m # # m 7 # # 7# 7 7# 7# 7 In t ivision, t son inx () s n sutrt rom t irst (). In nrl, pply tis ormul or ivision wit inis: m n m n ' Sm s Sutrt inis Diviin trms wit inis Coiints ivi sprtly 7 7 0y ' y ^0 ' ^y ' y y y 7 Coiints ivi sprtly 6 6 ' 7 ^ ' 7^ ' ^ ' ^ ' 6 Sm ss roup totr Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
6 Inis Qustions Bsis. Writ in xpn orm: 6 9 ^ ` j ` j x xy i. Intiy t s, inx n si numrl o o t ollowin: 0 Bs Bs Bs Inx Inx Inx Bsi numrl Bsi numrl Bsi numrl ` j Bs ^ Bs ^ Bs Inx Inx Inx Bsi numrl Bsi numrl Bsi numrl ^ 6 Bs Bs Inx Inx Bsi numrl Bsi numrl. Intiy t ollowin vlus: Wt is t si numrl o to t inx? Wt is t s o n xprssion wit inx n si numrl 6? Wt woul t inx o n xprssion i t si numrl is 8 n t s is? Wt is t inx in n xprssion wit s 7 n si numrl? J 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
7 Inis Qustions Bsis. Us lik trms to simpliy t ollowin in inx orm (i possil): + + ^ ^ ^6 + ^ ^ ^ + ^ +. Fin t ollowin prouts in simplst inx orm: 6 # # # 7 ^ # ^ ^ # ^ ` j # ` # j ` j q # q 7 x # y# y # x 0 # i 8 6# wv# w # v Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
8 Inis Qustions Bsis 6. Fin t ollowin prouts in simplst inx orm: 8 ' 8 7 ' 7 6 ^ ' ^ 9 m ' m 9 rs ' rs ' 9 x y 7 6xy i 6 ' 8 J 6 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
9 Inis Knowin Mor Risin Inis to Inis W us wt w know out multiplyin trms wit inis, to in rul or risin inis to inis. Consir or xmpl: 8 ^ # # Noti tt t s rmins t sm n w in t prout o t inis. In nrl, pply t ormul ^ m n mn Sm s Multiply Inis Risin inis to inis ^ # 0 x x x ^ # Mor Inx Lws I prout or rtion is ris to n inx, tn t inx pplis to trm. Brkts wit inis ^ m m m m n m m m ^x x pq ^ p ^q 7x 6pq # 8 6pq x x ` j x 8 ^ m 6 9 Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC 7
10 Inis Knowin Mor T Zro Inx m 0 m m m Tis mns 0 wn! 0 Anytin ris to n inx o 0 is. T zro inx ^ ` ^+ x ^ j ^6 + ^ ^ 0 # 7 # 7 7 Frtionl Inis Lt s try iur out wt to o wn t inis r rtions, su s For xmpl, onsir : 6 or p 7. Usin t inx lw or multiplition w n sy ^ Fin t squr root o ot sis to otin ^ Simpliy y nllin t inx o wit t squr root For ny, w n sy is t squr root o t numr. In t sm wy is t u root o. Bsilly, or ny n, n is t n t root o. Alwys us t ormul n n Frtionl inis ^ x # 8 # x x J 8 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
11 Inis Knowin Mor Ntiv Inis Lt s s i w n us wt w know to iur out ow to us ntiv inis su s, or x For xmpl: Ntiv inis 0 ^. ` Sin 0 Usin t ivision o inis lw Aorin to t zro inx lw In nrl, or ntiv inis w us t ormul: n n Ntiv inis xmpls 6 t m t t x x ` j ` j 9 ^ ^ ^ In tis xmpl, t minus () is inlu in t inx. In tis xmpl, it is NOT inlu in t inx. Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC 9
12 Inis Knowin Mor Mor Frtionl Inis Up until now w v only work wit rtionl inis wit numrtor lik or. As ll mtmtiins know, ts r not t only typ o rtions. It is importnt to lrn ow to us rtionl inis wos numrtors r not or xmpl: Usin t risin n inx to n inx lw, w in ^ ^ In nrl, t ormul or rtionl inis is: m m n n m ^ ^n ^ m n n m Mor rtionl inis 7 ^7 ^ ^ xy 6 ^ 8^00xy B 8 ^ 00 xy ^0xy 000xy 6 9 ` 8 j ` 8 j 6 J 0 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
13 Inis Qustions Knowin Mor. Us t lw or risin inis to inis to rwrit t ollowin in simplst inx orm: ^ 6 ^ ^ 0 ^ x ^ 6 ^. Us inx lws to rwrit t ollowin in simplst inx orm: ^x ^t ^xy ^p q 0 ^ 7 ^u ^ 7 xy ^ i ^ j ^pqr 7. Us inx lws to rwrit t ollowin in simplst inx orm: ` j x y o x m y x y o ` j xy m y Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
14 Inis Qustions Knowin Mor. Us t zro inx lw to simpliy t ollowin: ^6 0 ^ 0 0 ^ 0 ^# # i 0 ^ 7 0 j 6p 0 k 0 # x l # x 0. Simpliy t ollowin xprssions usin t lw or ntiv inis: 0 ^ ^ p 7 i 6p j ^6p k 7 0x ' 0x l pq' 0pq 6. Us rtionl inis to writ t ollowin in sur orm: 7 0 n 6 6 q 7 x mn 6 mn 6 ^ i 7 J 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
15 Inis Qustions Knowin Mor 7. Writ t ollowin in inx nottion: 8 x 0 0 x y x 8. Writ t ollowin in simplst positiv inx nottion: ^ 8 ` j w # w ^7 # ^7 6 p ' p 7 x # x y ' y ^ pq 7 0 i 0 x # ` # 0 # x j j ^6 ' 6 Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
16 Inis Usin Our Knowl Ar, Volum n Sur Ar Inx nottion n ppli to msurmnt, to in xprssions or ts msurmnts or to solv prolms. Fin n xprssion or trinl wit t ollowin imnsions: x An xprssion or t r o tis trinl is x Ar # ^Bs# ^Hit # x# x # # # x# x x ` ^ j Tis is t pln or swimmin pool: y 6y y I t volum o t swimmin pool nnot x Stp : Dtrmin n xprssion or t volum: Volum ^lnt# ^rt# ^it 6y# y# y y Stp : St t xprssion qul to 768 n solv or y: 768m wt is t mximum vlu or y? y 768 y y 6 y 6 Divi ot sis y Fin t u root o ot sis J 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
17 Inis Qustions Usin Our Knowl. A rpt ns to put own on st or n upomin onrt. Tr is only 90 m o rpt vill or t st wi s t ollowin sp: Fin t vlu o x to nsur ll t rpt is us. x x x. A nw uilin is in uilt ntirly out o lss (inluin t roo, xluin t loor). T sp o t uilin is isply low. I tr is only 6 60m o lss vill, tn wt is t mximum prmissil vlu o p to nsur tr is nou lss or t ntir uilin? p p p Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
18 Inis Qustions Usin Our Knowl. In n ropln, t pssnr in s imnsions: k k 6k k Bor lit, t in must ill wit oxyn. Writ n xprssion or t volum o oxyn rquir or ropln in trms o k. I prtiulr irlin rquirs omin volum o m o oxyn or 0 roplns, tn solv or k. J 6 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
19 Inis Usin Our Knowl Sintii Nottion or Lr Numrs (Grtr Tn ) Sintii nottion pplis inx nottion to lp writ vry lr n vry smll numrs in n sy wy. For xmpl, # # Som xmpls r: 000 # # # Tis my sm silly or numrs tis smll ut it is rtinly lpul or lr numrs su s # 0 0 or # 0 00 Not tt is not writtn s #. Sintii nottion rquirs t irst numr to twn 0 n 0 (so on o,,,..., 9). Sintii nottion or lr numrs For lr numrs (rtr tn ) t inx o 0 is t numr o iits twn t irst iit n t iml point. 7 6 First iit # 0 7 Diml point Btwm n 0 Sintii Nottion or Smll Numrs (Lss Tn ) Sintii nottion is writtn s (numr twn 0 n 0) (inx o 0), t only irn or smll numrs is tt t inx is ntiv. Som xmpls r: # # 0 ^7. # 0# 0.7 # Ain t nswr is not writtn s 7 # 0 sin t irst numr is rquir to twn 0 n 0. Sintii nottion or smll numrs For smll numrs (lss tn ) t inx is t ntiv vlu o t mount o iits tr t iml point up to n inluin t irst nonzro iit. Diml point # 0 First nonzro iit Btwm n 0 Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC 7
20 Inis Usin Our Knowl Sintii Nottion Mks Clultions Esir Usin sintii nottion # ' ^. # 0 # ^. # #. # 0 # # 0 0. # 0 ^0. # 0 # # 0 (Rmmr t irst numr must twn n 0) ^. # 0 ' ^# 0 8. # 0 # 0 8. # # 0 0. # 0 ^# 0 # 0 # (Rmmr t irst numr must twn n 0) # ' ^. # 0 # ^# 0. # # 0 # # # 0 (Rmmr t irst numr must twn n 0) 9 ^. # 0 ' ^9# 0. # 0 9 9# 0. # ^9 0.# # 0 0.# 0 ^# 0 # 0 # 0 (Rmmr t irst numr must twn n 0) # 0 ^96 ^0 8 8 # 0 ^. # 0 # 0. # 0 (Rmmr t irst numr must twn n 0) J 8 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
21 Inis Qustions Usin Our Knowl. Rprsnt t ollowin lr numrs usin sintii nottion: Rprsnt t ollowin smll numrs usin sintii nottion: Rprsnt t ollowin numrs in sintii nottion: Twlv tousnts Two million, tr unr tousn Four unr n twnty millionts Nin illion, iv unr n sixty svn million 7. Rwrit t ollowin in norml numrs: 0. # # # # # # 0 Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC 9
22 Inis Qustions Usin Our Knowl 8. Evlut t ollowin prouts usin sintii nottion: 7 8 ^# 0 # ^7# 0 ^. # 0 # ^8. # 0 7 ^ ^ 6. 0 ^. 0 ^. # 0 #. # 0 # # # 0 ^ ^. ^6. # 0 #. # 0 7 6# 0 # ^# 0 9. Evlut t ollowin quotints usin sintii nottion: 8 7 ^6# 0 ' ^# 0 ^96. # 0 ' ^. # 0 8. # 0 '. # 0 ^ ^ ^7. # 0 ' ^6# 0 ^6. # 0 ' ^8# 0 6 ^. # 0 ' ^9# 0 J 0 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
23 Inis Qustions Usin Our Knowl 0. Evlut t ollowin usin sintii nottion: 6 # 0 00# # 0 6 # ^ # 8 0 ^ #. Evlut t ollowin usin sintii nottion: 0 000# ' ' 0.6 Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
24 Inis Tinkin Mor Comprin Sintii Nottion Arrn t ollowin numrs in snin orr:. # # 0. # # 0 8. # 0 Stp : Arrn y inx 6.7 # 0. # 0. # 0. # # 0 8 Stp : Arrn y numr in inx 6.7 # 0. # 0. # 0. # # 0 8 Sm inx Arrn t ollowin numrs in snin orr: 9. # 0 8. # 0 7. # 0 6. # 0 8. # 0 Stp : Arrn y inx 7. # 0.9 # 0 8. # 0 8. # 0 6. # 0 Stp : Arrn y numr in inx 7. # 0 8. # 0.9 # 0 8. # 0 6. # 0 Sm inx Prixs or Inis o 0 W ll know tt tr r 000 m in km. T prix kilo tully mns 000 or 0. So wn w sy kilomtr w mn # 0 mtrs, or wn w sy kilorms w mn 0 # rms. T prix milli in millimtrs mns 0 (or 0.00). So wn w sy istn is mm, w mn # 0 m (or 0.00 m). Som otr prixs r: Smll numr Lr numr Prix Inx Arvition pio 0 p nno 0 9 n miro 0 6 μ milli 0 m to 0 kilo 0 k m 0 6 M i 0 9 G tr 0 T Wn w sy o tryt (T), tis mns 0 # yts o mmory. J 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
25 Inis Qustions Tinkin Mor. Arrn t ollowin in snin orr: # 0. # 0 7 7# 0 6.# 0 7.# 0 6 # 0.# 0.# 0 # 0 # 0 7.# 0 6# 0 7.# 0 # 0. Complt t ollowin tl: Msurmnt Prix Inx Smll or Lr Numr Expn Form 00 M M 0 6 Lr nm Nno 97 μm 60 p 0 0 Lr 00 m C m T mss o t Ert is stimt to.98 0 # k. Writ tis vlu in trrms (T). T istn rom t Ert to t Sun is stimt t.96 0 # m. Writ tis in Gimtrs (Gm). T r o n ltron is # Couloms (C). Writ tis in pioouloms (pc). Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
26 Inis Tinkin Evn Mor Finin Missin Trms Fin in o t ollowin xmpls: Finin t missin trm in prout To otin y itsl, ivi ot sis y vrytin otr tn : 7 # # 7 7 Finin t missin trm in prout To otin y itsl, ivi ot sis y vrytin otr tn : # vw vw # vw vw vw vw v w vw Finin t missin numrtor in quotint To otin y itsl, multiply ot sis y t nomintor unr : 6 # 6 # 6 # 8 J 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
27 Inis Tinkin Evn Mor Finin Missin Trms Finin t missin nomintor in quotint To otin y itsl, swp t nomintor wit t trm on t otr si o t quls sin: Finin t missin vlu in sur To otin y itsl, ris ot sis to t inx wi will nl t sur wy: ^ ^ 6 8 Finin t missin vlu in n inx To otin y itsl, in t squr root o ot sis to nl t squr wy (ssum is positiv). ^ ^ (Rmmr t squr root is positiv y onvntion) Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
28 Inis Qustions Tinkin Evn Mor. Fin t vlu o in t ollowin prouts: r # r # mn # mn # 8 xyz 0yzx # pqr # 6p qr # # # # J 6 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
29 Inis Qustions Tinkin Evn Mor. Fin t vlu o in t ollowin quotints: 0y 6 y 6 9tu tu 6pq 6pq ' xy z ' 6x y. Fin t vlu o in t ollowin: xy z 8 9 ^ 6p q ^ 6m n p Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC 7
30 Inis Qustions Tinkin Evn Mor. Qustions to tink out: Explin t irn twn ^# n #. Wt is t irn twn ^ 0 n 0? Wt is t irn twn 6 n 0 6 #? For ny numr, os 0 xist? I so tn wt is its vlu, i not tn wy not? Dos 0 xist? I so tn wt is its vlu, i not tn wy not? 0 Is tr vlu or so tt x # x? Do 6 0 n 6 0 v irnt vlus? Wt r t vlus o x n y i ^ 8 6? y x J 8 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
31 Inis Answrs Bsis: Bsis:. #. 6 6# 6# 6# 6 9 9# 9# 9 ^ # # ` j # # # # Bs Inx Bsi numrl Bs Inx 6 Bsi numrl 6 ` j ` # # # j ` j ` j ` j. 6 x x# x ( so s is ) xy x# x# x# y# y# y (Inx is ) i # # # # # (Inx is ). Bs 0. ^ ^ Inx Bsi numrl 0 Bs Inx ^ ^ + ^6 ^ + 7^ ^ Bsi numrl 9 Bs Inx Bsi numrl 6 Bs. 9 ^ 0 9 ` j xy 8 ^ 7 q 0 0 Inx Bsi numrl 8 i 8wv 6 Bs Inx Bsi numrl 9 Bs Inx ^ 7 7 ` j rs 9xy Bsi numrl i 8 Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC 9
32 Inis Answrs Knowin Mor: x 0 x 8 t xy p 0 q 0 u Knowin Mor: i 6 7 n 7 q 6 x m n 6 mn ^ x y 7. 8 x i j p q r 0. x y x y 8 6 x y 9 x x x y 6 9 w 7. p y 0 x i i 0 x j j 6 k l Usin Our Knowl:. 00. x 7m 7 7. p 9m 6 7 p 6. Totlvolum k m 8k i k 6 j p 6p x l pq.. # # 0 J 0 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
33 Inis Answrs Usin Our Knowl: Tinkin Mor:. 0. # # # 0 0. # 0 0 #,. 0 6 #, #,.# 0 7,. # # 0 7. # 0. 0 #, 0 #, 0 #,.# 0, # 0 0 # # # #, 7. 0 #,, 6# 0, # # 0. # 0 0. # # # # # 0. # 0. # # # 0 6. # 0. # 0 7. # 0. # 0 8. # 0 9 # 0 # 0 # 0 6. # 0 8 8# 0 6 6# 0. Expn Form Smll or Lr Numr Msurmnt Prix Inx 00 M M 0 6 Lr nm Nno 0 9 Smll m 97 μm miro 0 6 Smll m 60 p pio 0 Smll m to 0 Lr 00 m M C m 0 6 Lr C nm nno 0 9 Smll m 0. 8# # 0 # 0. # # # 0 T 96. # 0 Gm pc #. 6. # 0 0 # 0. # 0. # 0 Mtltis 00% 00% Inis P Lrnin J SERIES TOPIC
34 Inis Answrs Tinkin Evn Mor: Tinkin Evn Mor:. r mn 6 x z pqr. Ys, tr is vlu or tt mks 0 x # x. T vlu is 7x 6 No, t two trms o not v irnt vlus; ty r ot qul to. x n y. 0y tu 8 8 6p 9 q 0 9xyz 7. 6x yz 7 9 p q 7 m np. T irst is uin (risin to t inx o ) t prout o n, tt is. T son is tkin t prout o n u, tt is 6 9 #. T irst is risin t prout o n to t inx o zro, tt is ^. 0 T son is multiplyin y 0, tt is 0 #. T irst is sort n wy o writin # # # # #, wi is qul to 096. T son is sort n wy o writin # 0# 0# 0# 0# 0# 0, wi is qul to W n usully writ ny sur lik n s n 0 But i w o tis wit w woul t 0 n so on. Sin w n t ivi numr 0 y zro it tlls us tt os not xist J 00% Inis SERIES TOPIC Mtltis 00% P Lrnin
35
36 Inis
S i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA.
S i m p l i y i n g A l g r SIMPLIFYING ALGEBRA www.mthltis.o.nz Simpliying SIMPLIFYING Algr ALGEBRA Algr is mthmtis with mor thn just numrs. Numrs hv ix vlu, ut lgr introus vrils whos vlus n hng. Ths
OpenMx Matrices and Operators
OpnMx Mtris n Oprtors Sr Mln Mtris: t uilin loks Mny typs? Dnots r lmnt mxmtrix( typ= Zro", nrow=, nol=, nm="" ) mxmtrix( typ= Unit", nrow=, nol=, nm="" ) mxmtrix( typ= Int", nrow=, nol=, nm="" ) mxmtrix(
# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.
How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0
Tangram Fractions Overview: Students will analyze standard and nonstandard
ACTIVITY 1 Mtrils: Stunt opis o tnrm mstrs trnsprnis o tnrm mstrs sissors PROCEDURE Skills: Dsriin n nmin polyons Stuyin onrun Comprin rtions Tnrm Frtions Ovrviw: Stunts will nlyz stnr n nonstnr tnrms
Decimals DECIMALS.
Dimls DECIMALS www.mthltis.o.uk ow os it work? Solutions Dimls P qustions Pl vlu o imls 0 000 00 000 0 000 00 0 000 00 0 000 00 0 000 tnths or 0 thousnths or 000 hunrths or 00 hunrths or 00 0 tn thousnths
1 Introduction to Modulo 7 Arithmetic
1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w
learning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms
rp loritms lrnin ojtivs loritms your sotwr systm sotwr rwr lrn wt rps r in mtmtil trms lrn ow to rprsnt rps in omputrs lrn out typil rp loritms wy rps? intuitivly, rp is orm y vrtis n s twn vrtis rps r
UNCORRECTED SAMPLE PAGES
Numrs n surs Aritmti is t stuy o numrs n oprtions on tm. Tis sort ptr rviws wol numrs, intrs, rtionl numrs n rl numrs, wit prtiulr ttntion to t ritmti o surs n tir pproximtions. Most o tis mtril will milir
MAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017
MAT3707/201/1/2017 Tutoril lttr 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Smstr 1 Dprtmnt o Mtmtil Sins SOLUTIONS TO ASSIGNMENT 01 BARCODE Din tomorrow. univrsity o sout ri SOLUTIONS TO ASSIGNMENT
H SERIES. Decimals. Decimals. Curriculum Ready ACMNA: 103, 128, 129, 130, 152, 154,
Dimls H SERIES Dimls Curriulum Ry ACMNA: 0, 8, 9, 0,,, www.mthltis.om Copyriht 009 P Lrnin. All rihts rsrv. First ition print 009 in Austrli. A tlou ror or this ook is vill rom P Lrnin Lt. ISBN 978--98--9
d e c b a d c b a d e c b a a c a d c c e b
FLAT PEYOTE STITCH Bin y mkin stoppr -- sw trou n pull it lon t tr until it is out 6 rom t n. Sw trou t in witout splittin t tr. You soul l to sli it up n own t tr ut it will sty in pl wn lt lon. Evn-Count
SAMPLE PAGES. Primary. Primary Maths Basics Series THE SUBTRACTION BOOK. A progression of subtraction skills. written by Jillian Cockings
PAGES Primry Primry Mths Bsis Sris THE SUBTRACTION BOOK A prorssion o sutrtion skills writtn y Jillin Cokins INTRODUCTION This ook is intn to hlp sur th mthmtil onpt o sutrtion in hilrn o ll s. Th mstry
Grade 7/8 Math Circles March 4/5, Graph Theory I- Solutions
ulty o Mtmtis Wtrloo, Ontrio N ntr or ution in Mtmtis n omputin r / Mt irls Mr /, 0 rp Tory - Solutions * inits lln qustion. Tr t ollowin wlks on t rp low. or on, stt wtr it is pt? ow o you know? () n
Cycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!
Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik
Seven-Segment Display Driver
7-Smnt Disply Drivr, Ron s in 7-Smnt Disply Drivr, Ron s in Prolm 62. 00 0 0 00 0000 000 00 000 0 000 00 0 00 00 0 0 0 000 00 0 00 BCD Diits in inry Dsin Drivr Loi 4 inputs, 7 outputs 7 mps, h with 6 on
ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS
C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h
In which direction do compass needles always align? Why?
AQA Trloy Unt 6.7 Mntsm n Eltromntsm - Hr 1 Complt t p ll: Mnt or s typ o or n t s stronst t t o t mnt. Tr r two typs o mnt pol: n. Wrt wt woul ppn twn t pols n o t mnt ntrtons low: Drw t mnt l lns on
Complete Solutions for MATH 3012 Quiz 2, October 25, 2011, WTT
Complt Solutions or MATH 012 Quiz 2, Otor 25, 2011, WTT Not. T nswrs ivn r r mor omplt tn is xpt on n tul xm. It is intn tt t mor omprnsiv solutions prsnt r will vlul to stunts in stuyin or t inl xm. In
EE1000 Project 4 Digital Volt Meter
Ovrviw EE1000 Projt 4 Diitl Volt Mtr In this projt, w mk vi tht n msur volts in th rn o 0 to 4 Volts with on iit o ury. Th input is n nlo volt n th output is sinl 7-smnt iit tht tlls us wht tht input s
, each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management
nrl tr T is init st o on or mor nos suh tht thr is on sint no r, ll th root o T, n th rminin nos r prtition into n isjoint susts T, T,, T n, h o whih is tr, n whos roots r, r,, r n, rsptivly, r hilrn o
Physics 222 Midterm, Form: A
Pysis 222 Mitrm, Form: A Nm: Dt: Hr r som usul onstnts. 1 4πɛ 0 = 9 10 9 Nm 2 /C 2 µ0 4π = 1 10 7 tsl s/c = 1.6 10 19 C Qustions 1 5: A ipol onsistin o two r point-lik prtils wit q = 1 µc, sprt y istn
QUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt o Computr n Inormtion Sins CSCI 2710 (Trno) Disrt Struturs TEST or Sprin Smstr, 2005 R this or strtin! This tst is los ook
(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,
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
Weighted Graphs. Weighted graphs may be either directed or undirected.
1 In mny ppltons, o rp s n ssot numrl vlu, ll wt. Usully, t wts r nonntv ntrs. Wt rps my tr rt or unrt. T wt o n s otn rrr to s t "ost" o t. In ppltons, t wt my msur o t lnt o rout, t pty o ln, t nry rqur
Walk Like a Mathematician Learning Task:
Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics
5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees
/1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our
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
The University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008
Th Univrsity o Syny MATH2969/2069 Grph Thory Tutoril 5 (Wk 12) Solutions 2008 1. (i) Lt G th isonnt plnr grph shown. Drw its ul G, n th ul o th ul (G ). (ii) Show tht i G is isonnt plnr grph, thn G is
Designing A Concrete Arch Bridge
This is th mous Shwnh ri in Switzrln, sin y Rort Millrt in 1933. It spns 37.4 mtrs (122 t) n ws sin usin th sm rphil mths tht will monstrt in this lsson. To pro with this lsson, lik on th Nxt utton hr
8Algebraic UNCORRECTED SAMPLE PAGES. techniques. What you will learn. Australian curriculum. Chapter 8A 8B 8C 8D 8E 8F
8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K Chptr Wht you will lrn 8Algri thniqus Epning inomil prouts Prt squrs n irn o prt squrs Ftorising lgri prssions Ftorising th irn o two squrs Ftoristion y grouping Ftorising
Present state Next state Q + M N
Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I
Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.
Divided. diamonds. Mimic the look of facets in a bracelet that s deceptively deep RIGHT-ANGLE WEAVE. designed by Peggy Brinkman Matteliano
RIGHT-ANGLE WEAVE Dv mons Mm t look o ts n rlt tt s ptvly p sn y Py Brnkmn Mttlno Dv your mons nto trnls o two or our olors. FCT-SCON0216_BNB66 2012 Klm Pulsn Co. Ts mtrl my not rprou n ny orm wtout prmsson
The University of Sydney MATH 2009
T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n
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. {, } {, } {, } {, } {, } {,
Math 166 Week in Review 2 Sections 1.1b, 1.2, 1.3, & 1.4
Mt 166 WIR, Sprin 2012, Bnjmin urisp Mt 166 Wk in Rviw 2 Stions 1.1, 1.2, 1.3, & 1.4 1. S t pproprit rions in Vnn irm tt orrspon to o t ollowin sts. () (B ) B () ( ) B B () (B ) B 1 Mt 166 WIR, Sprin 2012,
Lecture 20: Minimum Spanning Trees (CLRS 23)
Ltur 0: Mnmum Spnnn Trs (CLRS 3) Jun, 00 Grps Lst tm w n (wt) rps (unrt/rt) n ntrou s rp voulry (vrtx,, r, pt, onnt omponnts,... ) W lso suss jny lst n jny mtrx rprsntton W wll us jny lst rprsntton unlss
16.unified Introduction to Computers and Programming. SOLUTIONS to Examination 4/30/04 9:05am - 10:00am
16.unii Introution to Computrs n Prormmin SOLUTIONS to Exmintion /30/0 9:05m - 10:00m Pro. I. Kristin Lunqvist Sprin 00 Grin Stion: Qustion 1 (5) Qustion (15) Qustion 3 (10) Qustion (35) Qustion 5 (10)
Constructive Geometric Constraint Solving
Construtiv Gomtri Constrint Solving Antoni Soto i Rir Dprtmnt Llngutgs i Sistms Inormàtis Univrsitt Politèni Ctluny Brlon, Sptmr 2002 CGCS p.1/37 Prliminris CGCS p.2/37 Gomtri onstrint prolm C 2 D L BC
Binomials and Pascal s Triangle
Binomils n Psl s Tringl Binomils n Psl s Tringl Curriulum R AC: 0, 0, 08 ACS: 00 www.mthltis.om Binomils n Psl s Tringl Bsis 0. Intif th prts of th polnomil: 8. (i) Th gr. Th gr is. (Sin is th highst
STRUCTURAL GENERAL NOTES
UILIN OS: SIN LOS: RUTURL NRL NOTS NRL NOTS: US ROUP: - SSMLY USS INTN OR PRTIIPTION IN OR VIWIN OUTOOR TIVITIS PR MIIN UILIN O STION. SSONL. T UNTION O TIS ILITY IS NOT OR QUIPP OR OUPNY URIN WINTR/ TIN
QUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt of Computr n Informtion Sins CSCI 710 (Trnoff) Disrt Struturs TEST for Fll Smstr, 00 R this for strtin! This tst is los ook
Outline. Binary Tree
Outlin Similrity Srh Th Binry Brnh Distn Nikolus Austn nikolus.ustn@s..t Dpt. o Computr Sins Univrsity o Slzur http://rsrh.uni-slzur.t 1 Binry Brnh Distn Binry Rprsnttion o Tr Binry Brnhs Lowr Boun or
Edge-Triggered D Flip-flop. Formal Analysis. Fundamental-Mode Sequential Circuits. D latch: How do flip-flops work?
E-Trir D Flip-Flop Funamntal-Mo Squntial Ciruits PR A How o lip-lops work? How to analys aviour o lip-lops? R How to sin unamntal-mo iruits? Funamntal mo rstrition - only on input an an at a tim; iruit
4.1 Interval Scheduling. Chapter 4. Greedy Algorithms. Interval Scheduling: Greedy Algorithms. Interval Scheduling. Interval scheduling.
Cptr 4 4 Intrvl Suln Gry Alortms Sls y Kvn Wyn Copyrt 005 Prson-Ason Wsly All rts rsrv Intrvl Suln Intrvl Suln: Gry Alortms Intrvl suln! Jo strts t s n nss t! Two os omptl ty on't ovrlp! Gol: n mxmum sust
Exam 1 Solution. CS 542 Advanced Data Structures and Algorithms 2/14/2013
CS Avn Dt Struturs n Algorithms Exm Solution Jon Turnr //. ( points) Suppos you r givn grph G=(V,E) with g wights w() n minimum spnning tr T o G. Now, suppos nw g {u,v} is to G. Dsri (in wors) mtho or
SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan
LOCUS 58 SOLVED EXAMPLES Empl Lt F n F th foci of n llips with ccntricit. For n point P on th llips, prov tht tn PF F tn PF F Assum th llips to, n lt P th point (, sin ). P(, sin ) F F F = (-, 0) F = (,
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
Outline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)
Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall
Hvn lps o so o t posslts or solutons o lnr systs, w ov to tos o nn ts solutons. T s w sll us s to try to sply t syst y lntn so o t vrls n so ts qutons. Tus, w rr to t to s lnton. T prry oprton nvolv s
Multipoint Alternate Marking method for passive and hybrid performance monitoring
Multipoint Altrnt Mrkin mtho or pssiv n hyri prormn monitorin rt-iool-ippm-multipoint-lt-mrk-00 Pru, Jul 2017, IETF 99 Giuspp Fiool (Tlom Itli) Muro Coilio (Tlom Itli) Amo Spio (Politnio i Torino) Riro
Module graph.py. 1 Introduction. 2 Graph basics. 3 Module graph.py. 3.1 Objects. CS 231 Naomi Nishimura
Moul grph.py CS 231 Nomi Nishimur 1 Introution Just lik th Python list n th Python itionry provi wys of storing, ssing, n moifying t, grph n viw s wy of storing, ssing, n moifying t. Bus Python os not
Module 2 Motion Instructions
Moul 2 Motion Instrutions CAUTION: Bor you strt this xprimnt, unrstn tht you r xpt to ollow irtions EXPLICITLY! Tk your tim n r th irtions or h stp n or h prt o th xprimnt. You will rquir to ntr t in prtiulr
3 a 15a 6 b 21a 5 c 30a 6 d 12a 9. e 125a 8 f 36a 12 g 90a 13 h 56a a 6a b 5 c 3a 4 d 6a 4. e 10a 4 f 8a 2 g 5a 4 h 12a 2
Answrs Cptr Workin wit surs Eris A Surs 8 6 6 8 6 6 6 + 6 8 6 6 6 6 + 8 + + 8 6 9 6 m ( + m Cptr Simpliin prssions usin t lws o inis Eris A Inis 9 i j 6 k 8 l 6 6 i j k l 6 6 9 8 6 9 6 6 6 8 8 6 9 8 8
BASIC CAGE DETAILS SHOWN 3D MODEL: PSM ASY INNER WALL TABS ARE COINED OVER BASE AND COVER FOR RIGIDITY SPRING FINGERS CLOSED TOP
MO: PSM SY SI TIS SOWN SPRIN INRS OS TOP INNR W TS R OIN OVR S N OVR OR RIIITY. R TURS US WIT OPTION T SINS. R (UNOMPRSS) RR S OPTION (S T ON ST ) IMNSIONS O INNR SIN TO UNTION WIT QU SM ORM-TOR (zqsp+)
CS September 2018
Loil los Distriut Systms 06. Loil los Assin squn numrs to msss All ooprtin prosss n r on orr o vnts vs. physil los: rport tim o y Assum no ntrl tim sour Eh systm mintins its own lol lo No totl orrin o
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
CONVERTING UNITS. Converting Units PASSPORT
CONVERTING UNITS PASSPORT www.mthltis.om.u This ook shows how to writ th sm vlu using smllr or lrgr units o msurmnt. Mny nint ivilistions msur lngths/istns y rlting thm to rtin oy prts. Invstigt ths trms
Garnir Polynomial and their Properties
Univrsity of Cliforni, Dvis Dprtmnt of Mthmtis Grnir Polynomil n thir Proprtis Author: Yu Wng Suprvisor: Prof. Gorsky Eugny My 8, 07 Grnir Polynomil n thir Proprtis Yu Wng mil: uywng@uvis.u. In this ppr,
CS 103 BFS Alorithm. Mark Redekopp
CS 3 BFS Aloritm Mrk Rkopp Brt-First Sr (BFS) HIGHLIGHTED ALGORITHM 3 Pt Plnnin W'v sn BFS in t ontxt o inin t sortst pt trou mz? S?? 4 Pt Plnnin W xplor t 4 niors s on irtion 3 3 3 S 3 3 3 3 3 F I you
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
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
Minimum Spanning Trees
Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst
Designing A Uniformly Loaded Arch Or Cable
Dsinin A Unirmy Ar Or C T pr wit tis ssn, i n t Nxt uttn r r t t tp ny p. Wn yu r n wit tis ssn, i n t Cntnts uttn r r t t tp ny p t rturn t t ist ssns. Tis is t Mx Eyt Bri in Stuttrt, Grmny, sin y Si
Using the Printable Sticker Function. Using the Edit Screen. Computer. Tablet. ScanNCutCanvas
SnNCutCnvs Using th Printl Stikr Funtion On-o--kin stikrs n sily rt y using your inkjt printr n th Dirt Cut untion o th SnNCut mhin. For inormtion on si oprtions o th SnNCutCnvs, rr to th Hlp. To viw th
Experiment # 3 Introduction to Digital Logic Simulation and Xilinx Schematic Editor
EE2L - Introution to Diitl Ciruits Exprimnt # 3 Exprimnt # 3 Introution to Diitl Loi Simultion n Xilinx Smti Eitor. Synopsis: Tis l introus CAD tool (Computr Ai Dsin tool) ll Xilinx Smti Eitor, wi is us
UNCORRECTED SAMPLE PAGES 4-1. Naming fractions KEY IDEAS. 1 Each shape represents ONE whole. a i ii. b i ii
- Nming frtions Chptr Frtions Eh shp rprsnts ONE whol. i ii Wht frtion is shdd? Writ s frtion nd in words. Wht frtion is not shdd? Writ s frtion nd in words. i ii i ii Writ s mny diffrnt frtions s you
BASIC CAGE DETAILS D C SHOWN CLOSED TOP SPRING FINGERS INNER WALL TABS ARE COINED OVER BASE AND COVER FOR RIGIDITY
SI TIS SOWN OS TOP SPRIN INRS INNR W TS R OIN OVR S N OVR OR RIIITY. R IMNSIONS O INNR SIN TO UNTION WIT QU SM ORM-TOR (zqsp+) TRNSIVR. R. RR S OPTION (S T ON ST ) TURS US WIT OPTION T SINS. R (INSI TO
CSC Design and Analysis of Algorithms. Example: Change-Making Problem
CSC 801- Dsign n Anlysis of Algorithms Ltur 11 Gry Thniqu Exmpl: Chng-Mking Prolm Givn unlimit mounts of oins of nomintions 1 > > m, giv hng for mount n with th lst numr of oins Exmpl: 1 = 25, 2 =10, =
Nefertiti. Echoes of. Regal components evoke visions of the past MULTIPLE STITCHES. designed by Helena Tang-Lim
MULTIPLE STITCHES Nrtiti Ehos o Rgl omponnts vok visions o th pst sign y Hln Tng-Lim Us vrity o stiths to rt this rgl yt wrl sign. Prt sping llows squr s to mk roun omponnts tht rp utiully. FCT-SC-030617-07
An undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V
Unirt Grphs An unirt grph G = (V, E) V st o vrtis E st o unorr gs (v,w) whr v, w in V USE: to mol symmtri rltionships twn ntitis vrtis v n w r jnt i thr is n g (v,w) [or (w,v)] th g (v,w) is inint upon
Chem 104A, Fall 2016, Midterm 1 Key
hm 104A, ll 2016, Mitrm 1 Ky 1) onstruct microstt tl for p 4 configurtion. Pls numrt th ms n ml for ch lctron in ch microstt in th tl. (Us th formt ml m s. Tht is spin -½ lctron in n s oritl woul writtn
COMPLEXITY OF COUNTING PLANAR TILINGS BY TWO BARS
OMPLXITY O OUNTING PLNR TILINGS Y TWO RS KYL MYR strt. W show tht th prolm o trmining th numr o wys o tiling plnr igur with horizontl n vrtil r is #P-omplt. W uil o o th rsults o uquir, Nivt, Rmil, n Roson
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 : )
0.1. Exercise 1: the distances between four points in a graph
Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 pg 1 Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 u: W, 3 My 2017, in lss or y mil (grinr@umn.u) or lss S th wsit or rlvnt mtril. Rsults provn in th nots, or in
CSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp
CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140
Polygons POLYGONS.
Polgons PLYGNS www.mthltis.o.uk ow os it work? Solutions Polgons Pg qustions Polgons Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon f g h Polgon Not polgon Polgon Not polgon Polgon
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..
Why the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.
Why th Juntion Tr lgorithm? Th Juntion Tr lgorithm hris Willims 1 Shool of Informtis, Univrsity of Einurgh Otor 2009 Th JT is gnrl-purpos lgorithm for omputing (onitionl) mrginls on grphs. It os this y
a b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...
Progrssiv Printing T.M. CPITLS g 4½+ Th sy, fun (n FR!) wy to tch cpitl lttrs. ook : C o - For Kinrgrtn or First Gr (not for pr-school). - Tchs tht cpitl lttrs mk th sm souns s th littl lttrs. - Tchs th
CSI35 Chapter 11 Review
1. Which of th grphs r trs? c f c g f c x y f z p q r 1 1. Which of th grphs r trs? c f c g f c x y f z p q r . Answr th qustions out th following tr 1) Which vrtx is th root of c th tr? ) wht is th hight
Planar Upward Drawings
C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th
this is called an indeterninateformof-oior.fi?afleleitns derivatives can now differentiable and give 0 on on open interval containing I agree to.
hl sidd r L Hospitl s Rul 11/7/18 Pronouncd Loh mtims splld Non p t mtims w wnt vlut limit ii m itn ) but irst indtrnintmori?lltns indtrmint t inn gl in which cs th clld n i 9kt ti not ncssrily snsign
Problem solving by search
Prolm solving y srh Tomáš voo Dprtmnt o Cyrntis, Vision or Roots n Autonomous ystms Mrh 5, 208 / 3 Outlin rh prolm. tt sp grphs. rh trs. trtgis, whih tr rnhs to hoos? trtgy/algorithm proprtis? Progrmming
Combinatorial Optimization
Cominoril Opimizion Prolm : oluion. Suppo impl unir rp mor n on minimum pnnin r. Cn Prim lorim (or Krukl lorim) u o in ll o m? Explin wy or wy no, n iv n xmpl. Soluion. Y, Prim lorim (or Krukl lorim) n
UNCORRECTED SAMPLE PAGES. Length, area, surface 5area and volume. Online resources. What you will learn
Onlin rsours Auto-mrk hptr pr-tst Vio monstrtions o ll work xmpls Intrtiv wigts Intrtiv wlkthroughs Downlol HOTshts Ass to ll HOTmths Austrlin Curriulum ourss Ass to th HOTmths gms lirry Lngth, r, sur
DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski
Dut with Dimons Brlt DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Lsli Roglski Photo y Anrw Wirth Supruo DUETS TM from BSmith rt olor shifting fft tht mks your work tk on lif of its own s you mov! This
Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1
Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd
DFA Minimization. DFA minimization: the idea. Not in Sipser. Background: Questions: Assignments: Previously: Today: Then:
Assinmnts: DFA Minimiztion CMPU 24 Lnu Tory n Computtion Fll 28 Assinmnt 3 out toy. Prviously: Computtionl mols or t rulr lnus: DFAs, NFAs, rulr xprssions. Toy: How o w in t miniml DFA or lnu? Tis is t
CONTINUITY AND DIFFERENTIABILITY
MCD CONTINUITY AND DIFFERENTIABILITY NCERT Solvd mpls upto th sction 5 (Introduction) nd 5 (Continuity) : Empl : Chck th continuity of th function f givn by f() = + t = Empl : Emin whthr th function f
Solutions to Homework 5
Solutions to Homwork 5 Pro. Silvia Frnánz Disrt Mathmatis Math 53A, Fall 2008. [3.4 #] (a) Thr ar x olor hois or vrtx an x or ah o th othr thr vrtis. So th hromati polynomial is P (G, x) =x (x ) 3. ()
COMP108 Algorithmic Foundations
Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht
VLSI Testing Assignment 2
1. 5-vlu D-clculus trut tbl or t XOR unction: XOR 0 1 X D ~D 0 0 1 X D ~D 1 1 0 X ~D D X X X X X X D D ~D X 0 1 ~D ~D D X 1 0 Tbl 1: 5-vlu D-clculus Trut Tbl or t XOR Function Sinc 2-input XOR t wors s
Graphs. Graphs. Graphs: Basic Terminology. Directed Graphs. Dr Papalaskari 1
CSC 00 Disrt Struturs : Introuon to Grph Thory Grphs Grphs CSC 00 Disrt Struturs Villnov Univrsity Grphs r isrt struturs onsisng o vrs n gs tht onnt ths vrs. Grphs n us to mol: omputr systms/ntworks mthml
A Simple Code Generator. Code generation Algorithm. Register and Address Descriptors. Example 3/31/2008. Code Generation
A Simpl Co Gnrtor Co Gnrtion Chptr 8 II Gnrt o for singl si lok How to us rgistrs? In most mhin rhitturs, som or ll of th oprnsmust in rgistrs Rgistrs mk goo tmporris Hol vlus tht r omput in on si lok
1 I ( can / can t ) read a book, but I ( can / can t ) read a map. 2 They ( can / can t ) swim, but they ( can / can t ) pitch a tent.
Avntur mp Writ t nms o t rtrs Flip Flo Hnn Mri Tom Cirl I lik ( ply / plyin ) sor, ut I ( on t / osn t ) lik mpin H osn t lik ( wt / wtin ) TV H ( lik / liks ) rin Ty r ( Brzil / Britis ) Ty r ( liv /
CS150 Sp 98 R. Newton & K. Pister 1
Outin Cok Synronous Finit- Mins Lst tim: Introution to numr systms: sin/mnitu, ons ompmnt, twos ompmnt Rviw o ts, ip ops, ountrs Tis tur: Rviw Ts & Trnsition Dirms Impmnttion Usin D Fip-Fops Min Equivn
A prefix word in each of these sentences is incorrect. Rewrite the prefix words correctly.
Spring Trm 2 Cirl th possssiv pronoun: Th hilrn wr thrill tht th i tht h n hosn or th inl prout ws thirs. Rwrit this sntn with th vril phrs t th ginning. Don t orgt omm! Thr wsn t on pi o pizz lt tr th
CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review
rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht