Study Guides (248,518)
Canada (121,606)
Mathematics (559)
MAT237Y1 (48)
Final

MAT 237 Exam Guide Part 2

161 Pages
265 Views
Unlock Document

Department
Mathematics
Course
MAT237Y1
Professor
Robert Brym
Semester
Fall

Description
a tuples bear numbers each Lis an element of R (xi ER) R se Use ase lengths distances x LASER is called the or lengti Distance between Li is (from abe) a tuples bear numbers each Lis an element of R (xi ER) R se Use ase lengths distances x LASER is called the or lengti Distance between Li is (from abe)MAT冈 イー-----xeyellxlllla-plcase ーーIrialtex-4e보t4n-tylellelltlly .Ery. こ46 2叫に415. s-U4tua-Ilz/ltlliAll Cass 4:4-4-4)-is 24.以川ーーーー):4k-axame xt3)+-A(ema)-rexa uxVE.4^ ニ2j-t-k-트14.-3,a a subsetof R_b4 Piving.conditions 一unions Up, zedle-if. xe4ーーーーーーーーーーー χchahif.ed-and-zee Sat Differences Ae AAA c AUR2 MAT 冈 イー ----- xeyellxlllla - plcase ーー Irialtex - 4e 보 t4n - tylellelltlly .Ery . こ 46 2 叫 に 415. s - U4tua - Ilz / ltlliAll Cass 4 : 4-4-4 ) -is 24. 以川 ーーーー ) : 4k - axame xt3 ) + - A ( ema ) -rexa uxVE.4 ^ ニ 2j - t - k- 트 14. - 3 , a a subsetof R_b4 Piving.conditions 一 unions Up , zedle - if . xe4 ー ー ー ー ー ー ー ー ー ー ー χchahif.ed - and - zee Sat Differences Ae AAA c AUR2MAT EN 0m2. Ex/ A UA nktion a Line: p.NER fixed Exy Find the line belman... I.3.2). and 4. Plane TnR t,s ER tut SV pin, R are fixed In I-DL calculus, ha can deine one-sidad detvrdives. In dimensions, this is a proAem,sone awoid it. To say Whitwer a function red on a set ucR is LhR want u to "surround" ev Det A soi NCR is a neghbourhood a point zeR if also say that x is an i N non) MAT EN 0m2. Ex/ A UA nktion a Line: p.NER fixed Exy Find the line belman... I.3.2). and 4. Plane TnR t,s ER tut SV pin, R are fixed In I-DL calculus, ha can deine one-sidad detvrdives. In dimensions, this is a proAem,sone awoid it. To say Whitwer a function red on a set ucR is LhR want u to "surround" ev Det A soi NCR is a neghbourhood a point zeR if also say that x is an i N non)MAT A subset UCRn s open if it is a neighbourhood each. That rs is open if a neigh THM Open bolls are open Ex The open interval as b Lab as xs b Des. A swbeat ACRA Te cloegd iff its com,plement is open ACR closed. RAA open Ex Compeanads open bolts are closc VE Ex Closed intervals ahe closed. A Car bl CR Suppose xEAT the either xEtoo ay o xe cb, en A 133 closed u V CR are open then the union is LEMMA open the viewsartion is spen THM If A, R are closed, then AUR, 2 A AB are closed MAT A subset UCRn s open if it is a neighbourhood each. That rs is open if a neigh THM Open bolls are open Ex The open interval as b Lab as xs b Des. A swbeat ACRA Te cloegd iff its com,plement is open ACR closed. RAA open Ex Compeanads open bolts are closc VE Ex Closed intervals ahe closed. A Car bl CR Suppose xEAT the either xEtoo ay o xe cb, en A 133 closed u V CR are open then the union is LEMMA open the viewsartion is spen THM If A, R are closed, then AUR, 2 A AB are closedMAT The Di uCR is pen a neighbourhood of each of rts points its complement TisoRn Lt. If u, V CBN are open, then uUV is pen,and unvisafen. On HMW, shaA that. nute unions of open set awe s not a LP. If A BCR are desed AUB dosed AnB ceeeed i. closad for. LEN closed. Ex/ The amptM sg s the sd which contains no elenents, R is closed (Rty is closed They are open closed! XER is a limit point a set ACR if every nechbourimod N20, leek at P THM ACR is clused it curioins all its limit points limit has to be in A bourhood 6f some point z EA s a hivat po A A Ex is a linn pant not i I ACR. tine claswe st A is smallest closed set that MAT The Di uCR is pen a neighbourhood of each of rts points its complement TisoRn Lt. If u, V CBN are open, then uUV is pen,and unvisafen. On HMW, shaA that. nute unions of open set awe s not a LP. If A BCR are desed AUB dosed AnB ceeeed i. closad for. LEN closed. Ex/ The amptM sg s the sd which contains no elenents, R is closed (Rty is closed They are open closed! XER is a limit point a set ACR if every nechbourimod N20, leek at P THM ACR is clused it curioins all its limit points limit has to be in A bourhood 6f some point z EA s a hivat po A A Ex is a linn pant not i I ACR. tine claswe st A is smallest closed set thatMAT SBT THM A s the set if ou limit poi is not in a closed F2 is not a limit A Ex. The closure of an open boll is a bale Take o consider ex ER where os take c desa enough to then-St-c ESE Exy R take A is largest open set contorned in A MM int A is the set all inte Nor points st A this cant be A if nt sa limit A ond a limit point The bosundaMA st A demoted DAY, Ts the set stall boww dory pirds of A MAT SBT THM A s the set if ou limit poi is not in a closed F2 is not a limit A Ex. The closure of an open boll is a bale Take o consider ex ER where os take c desa enough to then-St-c ESE Exy R take A is largest open set contorned in A MM int A is the set all inte Nor points st A this cant be A if nt sa limit A ond a limit point The bosundaMA st A demoted DAY, Ts the set stall boww dory pirds of AMAT sl sequence ausians to each natural numbers n an object an. Exp A. an hihai doab t mean for a sequence unverges a At L n R For every E20, there is NEN st dan E for n N For D, let N than an L LEO Con O, O E Assume At A. Naut to show B closed MAT sl sequence ausians to each natural numbers n an object an. Exp A. an hihai doab t mean for a sequence unverges a At L n R For every E20, there is NEN st dan E for n N For D, let N than an L LEO Con O, O E Assume At A. Naut to show B closedfunction as elements one set to elemanis CO,DO) Functions f L non-linen XE Gions. LER is the limit f as x approaches as 3.13 Ve o orin SIWAA function as elements one set to elemanis CO,DO) Functions f L non-linen XE Gions. LER is the limit f as x approaches as 3.13 Ve o orin SIWAAMAT IHM Let Sc R S R then Ex Pf Suppose want to show. Li Lee S. Non say xe S and o lix-alls S io,of Fot am Let Core) is condi May 17 where MAT IHM Let Sc R S R then Ex Pf Suppose want to show. Li Lee S. Non say xe S and o lix-alls S io,of Fot am Let Core) is condi May 17 whereMAT loll alls THM Let SCR f S R the llowing are eguvalent in verging to a li Ca prove the contrapostiive (not (not D Suppose f not cont a For some o and all Then, a Le is the oncle radius N MAT loll alls THM Let SCR f S R the llowing are eguvalent in verging to a li Ca prove the contrapostiive (not (not D Suppose f not cont a For some o and all Then, a Le is the oncle radius NMAT CRA 21- Am AC 2. the innade of A under Ce e') exp (ti, exp (R)s lo oo). under f is the set IA) xeu Exl THM IRA, the fo OWN are equivalent bood of f ca CN exercise, si R following au is conti wuows. the De-image a anu closed szt under f closed Some O MAT CRA 21- Am AC 2. the innade of A under Ce e') exp (ti, exp (R)s lo oo). under f is the set IA) xeu Exl THM IRA, the fo OWN are equivalent bood of f ca CN exercise, si R following au is conti wuows. the De-image a anu closed szt under f closed Some OMAT S31 exercise. LS Continuous deine forsy Dot is upper bound for -co, S La.... BLIS L, So is Loco is the least upper bound US Aare Ext is an uppe bound pose y is an upper bound and 4s L then and so- y is an upper- R has bound has a upper bound. CR is bounded if FM ER is bounded ux is bounded closed balls ane bawodod ACR is compact if it is both closed bounded Ex Co, is compact In general, any desod iniwwal the form Ca,bl bER impac Exp Cersed bolls are mpa MAT S31 exercise. LS Continuous deine forsy Dot is upper bound for -co, S La.... BLIS L, So is Loco is the least upper bound US Aare Ext is an uppe bound pose y is an upper bound and 4s L then and so- y is an upper- R has bound has a upper bound. CR is bounded if FM ER is bounded ux is bounded closed balls ane bawodod ACR is compact if it is both closed bounded Ex Co, is compact In general, any desod iniwwal the form Ca,bl bER impac Exp Cersed bolls are mpaMAT Amazing IHM H k SAn is compact, and f k- ig conf. Conn Ex, A path a The imoage any such path also campaa. Comrade Carde. Ramads. Neither boundedness nor closedness is eneugh K is boundsd THM LA non compact subset kaR Aasa maximum element. That is, yek, st. xsy Y DLE NOTE This is not true for non-compact sets (0,l) has co mar, element.. k CR compact K. is Goda k has an upper baud. -Show. supK a limit. uppet bound z z is an This commadiat the fact thai z is the least upper k closed f KCR ns com then f has a max. on K. Tha MAT Amazing IHM H k SAn is compact, and f k- ig conf. Conn Ex, A path a The imoage any such path also campaa. Comrade Carde. Ramads. Neither boundedness nor closedness is eneugh K is boundsd THM LA non compact subset kaR Aasa maximum element. That is, yek, st. xsy Y DLE NOTE This is not true for non-compact sets (0,l) has co mar, element.. k CR compact K. is Goda k has an upper baud. -Show. supK a limit. uppet bound z z is an This commadiat the fact thai z is the least upper k closed f KCR ns com then f has a max. on K. ThakCR is compact, empty f: k- R is continuous roo has a minimum maximum on k a has maximum n the Ne Is C o st. WIS2 f A is coorooted if it bas no discuorections. -Check Ulu up PCB A check unvnA Suppose. E unv. mpossible2 ExeNasa SCQ.Ts dwsconozatod if it has air least two points Ex is connected. such connection cant exist comecled..- sat Imagina a disconnection Cu,v) (u,v s not a das Coonaution. A ixi is connected kCR is compact, empty f: k- R is continuous roo has a minimum maximum on k a has maximum n the Ne Is C o st. WIS2 f A is coorooted if it bas no discuorections. -Check Ulu up PCB A check unvnA Suppose. E unv. mpossible2 ExeNasa SCQ.Ts dwsconozatod if it has air least two points Ex is connected. such connection cant exist comecled..- sat Imagina a disconnection Cu,v) (u,v s not a das Coonaution. A ixi is connectedMAT Conn Suppose. Curv) is a disconnection EV un I closed, since unIL VOI is the intersection two closed n I is bounded, as it is: a Subsze 5 bounded set I and Con empty asumpfion) so these is a maximum zeunI. EV If EV so fats tro some open-and z Eu., But z is the maximum. of No swch dusconection can exist. I IS con the contrapositiv Assume fIA) is duscoonaated and ove that A for f et (uiV) be a is continuous, u is open Cu ,V) is a disconnection A Check. disconnection Ulin Vin A so t MAT Conn Suppose. Curv) is a disconnection EV un I closed, since unIL VOI is the intersection two closed n I is bounded, as it is: a Subsze 5 bounded set I and Con empty asumpfion) so these is a maximum zeunI. EV If EV so fats tro some open-and z Eu., But z is the maximum. of No swch dusconection can exist. I IS con the contrapositiv Assume fIA) is duscoonaated and ove that A for f et (uiV) be a is continuous, u is open Cu ,V) is a disconnection A Check. disconnection Ulin Vin A so tMAT 37 Similarly, VLMA be ve shown (u V) is a disconnection for A. Exp Paths the fom Lo have conceded Exeroke Show thas the textbook connected and de are any two points x u EA there is a continuous function A THM If ACRA is e connected. it is connected Assume A is atenuise conneoRd, but is disconnetect that is it has a disconnection (uv There is a path A st V ane open (u,v) is a disconnection of (to, s coone tool Exi Open balls are arc wise cono Rad from y to z. The same function to show that closed balls RT is a civis nn and are the only clopen closed- i open) upsets of Suppose Than (A, A) is a disconnection is en since A is closed A AC ART Value THM ALCA is connectaci r: A R is continuous so for some ae A fc 20 fo some beA MAT 37 Similarly, VLMA be ve shown (u V) is a disconnection for A. Exp Paths the fom Lo have conceded Exeroke Show thas the textbook connected and de are any two points x u EA there is a continuous function A THM If ACRA is e connected. it is connected Assume A is atenuise conneoRd, but is disconnetect that is it has a disconnection (uv There is a path A st V ane open (u,v) is a disconnection of (to, s coone tool Exi Open balls are arc wise cono Rad from y to z. The same function to show that closed balls RT is a civis nn and are the only clopen closed- i open) upsets of Suppose Than (A, A) is a disconnection is en since A is closed A AC ART Value THM ALCA is connectaci r: A R is continuous so for some ae A fc 20 fo some beAMAT 3 Assume the hypotheses hold but fx o for xEA. Then, f CA) is duscoone (a) f(A) nu EXA Show that the function f: R xy- kas- a zero is connected is comtiwnows. TKerefoNe, but f has a zero. B are conne EAnb then u P, is If Ai connected EI, and REAL VieI UAE is connectRd... SA SA SA. D--- and each Ai is both compose and conceatok -connect&d f R R differentiable at aER if the limit DeO If T C is an open interval, I-R is differendiade ael if th limit vector the point Ct The linit in each comporoud is a T goes through 3-D space tEr In the last example, elo MAT 3 Assume the hypotheses hold but fx o for xEA. Then, f CA) is duscoone (a) f(A) nu EXA Show that the function f: R xy- kas- a zero is connected is comtiwnows. TKerefoNe, but f has a zero. B are conne EAnb then u P, is If Ai connected EI, and REAL VieI UAE is connectRd... SA SA SA. D--- and each Ai is both compose and conceatok -connect&d f R R differentiable at aER if the limit DeO If T C is an open interval, I-R is differendiade ael if th limit vector the point Ct The linit in each comporoud is a T goes through 3-D space tEr In the last example, eloMAT Cap What about R-- R Car differen Tate one variable a time open, u- R the jth at a (ai, az, an is the limit Also derotod and even Exl CompuR the Martial derivatives f R R Xe ..Ex/ Portal daHvohves may exist, even when the function is not continuous exist (exerowse) Ott At Koio O O What is the total" dewiodive of a function? nonsense, cart divide by XER RER CON CR be econ f nu-R and ae u there is CER c the gradiant of V- V MAT Cap What about R-- R Car differen Tate one variable a time open, u- R the jth at a (ai, az, an is the limit Also derotod and even Exl CompuR the Martial derivatives f R R Xe ..Ex/ Portal daHvohves may exist, even when the function is not continuous exist (exerowse) Ott At Koio O O What is the total" dewiodive of a function? nonsense, cart divide by XER RER CON CR be econ f nu-R and ae u there is CER c the gradiant of V- VwewAld icatio se +c.ex. is sort, at a in the u is COO 3 Sa f is height tertan at This mans if were at a Cs,L and walking in the headma us (Est) mg'd be nalking down hill at and a fan ca) attu as-C-tu. Cetu at two tw you can see TO, O (a) an) Ca ,a o) an) Ca laura aitt attei e, whete cs cm) then (c So, if fis diff. at a then the gradewit is avon by wewAld icatio se +c.ex. is sort, at a in the u is COO 3 Sa f is height tertan at This mans if were at a Cs,L and walking in the headma us (Est) mg'd be nalking down hill at and a fan ca) attu as-C-tu. Cetu at two tw you can see TO, O (a) an) Ca ,a o) an) Ca laura aitt attei e, whete cs cm) then (c So, if fis diff. at a then the gradewit is avon bydel ecland.continuousin-a-ei3kloounxxcl -Just paove-for-ua Without loss taeneralit4r-de(Gd) CothaAdshXLr.consider tw.tu CuLfcut a)) |Nant to-show-fis d ft. -heknon,- 卉-蓊 ane.coof. all llvlS8 hill show Nghethat -re,0) -le%yte)(Cz土5sーレ't eLexl Similarly (ztu) s 4(uvilt ilvl): Ellvil ーーナTiviī tftV)_ftoh-evkE Partial-Darius Total Deriv Partial Daris - -what about-By-sly? Ideas approxiot-ann-linear function bu a linoaroe Dog A lino at ap is a funation T: R^→R"st.f EX-4ER Recall linear map Tpr-Rn ean bo hop esnted bu- an-mon-natrix 4 denato the-space ofall-lineat mnapsーーーーーー et yj"ED Recau. If Te L (RnD") and Sel RaN) and is- RocalRensTel L the and SelR:00-andin epresented.b4th-mxn-matix-A,Sb4 the lem the ken matrix P, than SeTE / (R" R) .isrepesenad bl 并 del ecland.continuousin - a - ei3kloounxxcl -Just paove - for - ua Without loss taeneralit4r - de ( Gd ) CothaAdshXLr.consider tw.tu CuLfcut a ) ) | Nant to - show - fis d ft . -heknon , - 卉 - 蓊 ane.coof . all llvlS8 hill show Nghethat -re , 0 ) -le % yte ) ( Cz 土 5s ー レ ' t eLexl Similarly ( ztu ) s 4 ( uvilt ilvl ) : Ellvil ーーナ Tiviī tftV ) _ftoh - evkE Partial - Darius Total Deriv Partial Daris - -what about - By - sly ? Ideas approxiot - ann - linear function bu a linoaroe Dog A lino at ap is a funation T : R ^ → R " st.f EX - 4ER Recall linear map Tpr - Rn ean bo hop esnted bu- an - mon - natrix 4 denato the - space ofall - lineat mnaps ーーーーーー et yj " ED Recau . If Te L ( RnD " ) and Sel RaN ) and is- RocalRensTel L the and SelR : 00 - andin epresented.b4th - mxn - matix - A , Sb4 the lem the ken matrix P , than SeTE / ( R " R ) .isrepesenad bl 并MAT T is called the EMchet desauttve of f ot a E Ts at a then d May THMa 2dcR open, f R and the al desvatives R cominuow, then f is differentiate and Assume A is not compact a) Shoon Ef: st. fCA) rs not bounded FLA continmeus f is diff and vfizy If an whit vector, st f is diff a a, then same di Nection Det. Let nicR be open, feu-R. Then f is T2dfa is FNéchet deutvative... Tofa is thematrix) If c Ts the arodeut of fat a than EN- R) so Tedfa is the FNéchet de Nivative. MAT T is called the EMchet desauttve of f ot a E Ts at a then d May THMa 2dcR open, f R and the al desvatives R cominuow, then f is differentiate and Assume A is not compact a) Shoon Ef: st. fCA) rs not bounded FLA continmeus f is diff and vfizy If an whit vector, st f is diff a a, then same di Nection Det. Let nicR be open, feu-R. Then f is T2dfa is FNéchet deutvative... Tofa is thematrix) If c Ts the arodeut of fat a than EN- R) so Tedfa is the FNéchet de Nivative.MAT before..... Then L Suppose usR.6pen, f Rm, and f has a Frichet. Tm) so Tux) TTHM ueR is, and the partial ONe continuous then f is daffoNeviobe at a and a is ven by (A any dimension MAT before..... Then L Suppose usR.6pen, f Rm, and f has a Frichet. Tm) so Tux) TTHM ueR is, and the partial ONe continuous then f is daffoNeviobe at a and a is ven by (A any dimensionMAT257 2-142,zzay ) 区→N (LA VIAD → (ear-c":") 22._一語---- u.vwDEP(x.421. 01f -2. 之一-O- R") is a linear map the no-n T is - Danaark : uThListha maximum q.condanaous ーーー让-lTal_00.the compact-sek. o-Sel--thou. ITI :5 Avi+35y," t 7w nd isacuatvector ーシーT(Tax)7囟干χ_ Exorcise : SES-Jthat.l_l. that l--l is indeed a parm on LSLR). LA-N), e --in Paticular-ll St Tilsl SltILTIL THM. (the Chain-Rule 14 UcBy,.VeNAN open-f:1d크V-3三V_R._ diff.nt aev,then at a and 并ーーーwithout-les. of 0, sod SeLLER", st. →壑 ST Titfit)-setto 4js-diff. at o,-soaTe LATN)- st qgup-Tyl-a- -we-wast, toshcw_1o S is the Br aka LivwitT5xll sdal4 fa止巧alalife-TSzl Show Otudhto 2010 MAT257 2-142 , zzay ) 区 → N ( LA VIAD → ( ear - c " : " ) 22._ 一 語 ---- u.vwDEP ( x.421 . 01f -2 . 之一 -O- R " ) is a linear map the no - n T is - Danaark : uThListha maximum q.condanaous ーーー 让 -lTal_00.the compact - sek . o - Sel -- thou . ITI : 5 Avi + 35y , " t 7w nd isacuatvector ーシー T ( Tax ) 7 囟干 χ_ Exorcise : SES - Jthat.l_l . that l -- l is indeed a parm on LSLR ) . LA - N ) , e --in Paticular - ll St Tilsl SltILTIL THM . ( the Chain - Rule 14 UcBy , .VeNAN open - f : 1d 크 V - 3 三 V_R._ diff.nt aev , then at a and 并 ーーー without - les . of 0 , sod SeLLER " , st . → 壑 ST Titfit ) -setto 4js - diff . at o , -soaTe LATN ) - st qgup - Tyl - a- -we - wast , toshcw_1o S is the Br aka LivwitT5xll sdal4 fa 止 巧 alalife - TSzl Show Otudhto 2010MATS히 .3baal-O to nds-to bounded Doesn't-Lagrk a 6-Afcall be zero for al Since·ll fco-Slo) l-lo-oll 0% ll 011 Then,-M c . ItlS1,then Let-E>0 sinco 盔STdēilaa1_Tyll:0 co o S20 sta-llfull_n-for-llzllsS Asume Ssiar-toビル主11 < S 119(f(x)-エfu) that-the-linet is zero. If2dsR -isdiff at 7(t)e or :R→Disdi of tand ー并ー. DefencDfree-Dr Ex Lut Let-ret)。 ( Qt-Lt2e) (Tw)。(sint -(st-i) cost-I) (sine (近ー! ) cost,-l)-(→ st.2t ) 골을 뚫 등 H-붚솔 ttel Donut Skop-cw: entl4 LAW l b cf flour to make 30, donuis Everytime the priceofflourincreasingb4-194b-themiseriyiramer ePrice of dA andthou sell Do less How is their buthePrico flour fuantatut 千donaisseldーーー MATS 히 .3baal - O to nds - to bounded Does n't - Lagrk a 6 - Afcall be zero for al Since · ll fco - Slo ) l - lo - oll 0 % ll 011 Then , -M c . ItlS1 , then Let - E > 0 sinco 盔 STdēilaa1_Tyll : 0 co o S20 sta - llfull_n - for - llzllsS Asume Ssiar - to ビル 主 11 < S 119 ( f ( x ) - エ fu ) that - the - linet is zero . If2dsR -isdiff at 7 ( t ) e or : R → Disdi of tand ー 并ー . DefencDfree - Dr Ex Lut Let - ret ) 。 ( Qt - Lt2e ) ( Tw ) 。 ( sint - ( st - i ) cost - I ) ( sine ( 近 ー ! ) cost , -l ) - ( → st.2t ) 골 을 뚫 등 H- 붚솔 ttel Donut Skop - cw : entl4 LAW l b cf flour to make 30 , donuis Everytime the priceofflourincreasingb4-194b - themiseriyiramer ePrice of dA andthou sell Do less How is their buthePrico flour fuantatut 千 donaisseld ーーーMAT costs So e-teer 55-LOO-a -Ex Perceived Hummdax Humidex H a o- a Ca T h ose functions cloud covet c suniokt LLO How does OT oc. N MAT costs So e-teer 55-LOO-a -Ex Perceived Hummdax Humidex H a o- a Ca T h ose functions cloud covet c suniokt LLO How does OT oc. Nel set st f for the ev It c so function Can wate y as unction a Look at vative o only the the level rs bad ction least locally We Can wrfte as functi 2 say that locally, the level sze f th Note A uction f hrte as FC to,by a and el set st f for the ev It c so function Can wate y as unction a Look at vative o only the the level rs bad ction least locally We Can wrfte as functi 2 say that locally, the level sze f th Note A uction f hrte as FC to,by a andMAT 20 st. Fix Such an x, then so, F is st cheating (L, loth) Intact, f Rs (see textbook Bu the chain Rule What about Solv a system of equations imp citu? Det SCR s- tive f is the set graph and th --Smt m a,b a SR SR a and a ci function ERM v aka. MAT 20 st. Fix Such an x, then so, F is st cheating (L, loth) Intact, f Rs (see textbook Bu the chain Rule What about Solv a system of equations imp citu? Det SCR s- tive f is the set graph and th --Smt m a,b a SR SR a and a ci function ERM v aka.MAT an s to be --3-at-O- and inveHabe and the TFT Phe 4 B 220 f Ca,b R is catinuous on la, bl and diffeNeut Table. on a,b) then 3CEla,by st b-a then xel) Then Y is differentiable on Ca,b) and Then Y -Co, R is cont. and diff on Co, by the Chain Rule By the I-D MT Cay (ba) SO MAT an s to be --3-at-O- and inveHabe and the TFT Phe 4 B 220 f Ca,b R is catinuous on la, bl and diffeNeut Table. on a,b) then 3CEla,by st b-a then xel) Then Y is differentiable on Ca,b) and Then Y -Co, R is cont. and diff on Co, by the Chain Rule By the I-D MT Cay (ba) SOMAT between a Buppose SSR is a set except Rossibly at the endpoints a Then, As is cost. on Coil and duft on Co then MAT between a Buppose SSR is a set except Rossibly at the endpoints a Then, As is cost. on Coil and duft on Co thenMAT 4 Le abERA and let L be the line szoment betwzen them Suppose SER is a set contai L and f S IR immens on L and diff on ports.) Then 3 CEL st Ex Let Then tot every... VER sboru that 3 CERN st is diff. Since sin and zH llxl Take VER. If then guv Sin Then, by the MVT, 3 c bzkween a st v- o) CO O Sino Then, 3 c betwann a o b at Look at n Cordlany If abER, L is the line. gen Athena s diff on L u is a set containing L f u-re and M on L than M Mball Co MAT 4 Le abERA and let L be the line szoment betwzen them Suppose SER is a set contai L and f S IR immens on L and diff on ports.) Then 3 CEL st Ex Let Then tot every... VER sboru that 3 CERN st is diff. Since sin and zH llxl Take VER. If then guv Sin Then, by the MVT, 3 c bzkween a st v- o) CO O Sino Then, 3 c betwann a o b at Look at n Cordlany If abER, L is the line. gen Athena s diff on L u is a set containing L f u-re and M on L than M Mball CoMAT A, BCR are convex, then An Lis comex hMT is convex for a o, b x very Convex set is aron ise connected ected 3 connected s diff and vfo20 Vx Then f rs constant on u lab To do HNS riaprowdy, well use MVT. Apply MNT, 3 Ste Ca art) st att Qtt) Ca lab) Switch ra the roles MAT A, BCR are convex, then An Lis comex hMT is convex for a o, b x very Convex set is aron ise connected ected 3 connected s diff and vfo20 Vx Then f rs constant on u lab To do HNS riaprowdy, well use MVT. Apply MNT, 3 Ste Ca art) st att Qtt) Ca lab) Switch ra the rolesn, x reru are all infinitely sin, cos exist and are Como) Ex lst order Ond ordet 3rd order ade) If all of its partial to Higher Order Partial DeNvatives and the Chain Rale e fuay) inhere xiy are functions sit. TOS Hee TOS WTA Ext Palos Coordinotes. t sine TOM sine. cose. sin0 n, x reru are all infinitely sin, cos exist and are Como) Ex lst order Ond ordet 3rd order ade) If all of its partial to Higher Order Partial DeNvatives and the Chain Rale e fuay) inhere xiy are functions sit. TOS Hee TOS WTA Ext Palos Coordinotes. t sine TOM sine. cose. sin0MAT ST Macan: Multi-index Nato tion outti-index is an n-tuple con- n-k is MAT ST Macan: Multi-index Nato tion outti-index is an n-tuple con- n-k isMAT June. non neoptiva integers, ER If f aR--R is Ch then Comm Can also have constants. ER ,they Cc,, c Consider Apply the mWFromval THM Lamma suppose abeN and Lis the is CT Then, fo to IR is Ck and dti Let c b a whoe mejo Claim this Induction hdds. Buppose MAT June. non neoptiva integers, ER If f aR--R is Ch then Comm Can also have constants. ER ,they Cc,, c Consider Apply the mWFromval THM Lamma suppose abeN and Lis the is CT Then, fo to IR is Ck and dti Let c b a whoe mejo Claim this Induction hdds. BupposeMAT chain Mule ey induction, we proved the dam b- a at b Land a, XEL, then 3 cel a st X-ras Exl Applu Toutor's THM to exp.. on the interval. Do, 1 For each n 3 ca e lo st expun" exp rand x-a) l Since Cielo If IER is opan interval, f I R s cs and ael is fuar o s o then f has a local maximum inuoub- ---------i Apply Taylor's THM. Nith nei such (C a local Let UCR be If a, Eu are st. the line L between. at x is contanodinu than 3. CEL MAT chain Mule ey induction, we proved the dam b- a at b Land a, XEL, then 3 cel a st X-ras Exl Applu Toutor's THM to exp.. on the interval. Do, 1 For each n 3 ca e lo st expun" exp rand x-a) l Since Cielo If IER is opan interval, f I R s cs and ael is fuar o s o then f has a local maximum inuoub- ---------i Apply Taylor's THM. Nith nei such (C a local Let UCR be If a, Eu are st. the line L between. at x is contanodinu than 3. CELshowed that these m Xm-am If we take Taylors THM nitth neo, 3esL st. MVT What if lol Co, O Let v Ex-ay so Also Then the VtVi Oz an an where ai calls m m motNx A and v IS. a column vector. v.Av Nun showed that these m Xm-am If we take Taylors THM nitth neo, 3esL st. MVT What if lol Co, O Let v Ex-ay so Also Then the VtVi Oz an an where ai calls m m motNx A and v IS. a column vector. v.Av NunMAT Det If.ucR is open, R is c and ae then the a is the morax Hessat a we showed n is that. CEL. -a) bliss than Then for et be as in Tautors THM. Then st for CEesia) we have that This implie in particular, that MAT Det If.ucR is open, R is c and ae then the a is the morax Hessat a we showed n is that. CEL. -a) bliss than Then for et be as in Tautors THM. Then st for CEesia) we have that This implie in particular, thatMATS Ex Coosider the curve SCA dejined bui Find the line to S ugh the point coo. the S can be inritten with y LOAD line is given bu the taseert plane to the sphere as. radius 3 DZ Near. the sphere is quenas a graph of a C n function f: R 2,2) So the taraent plane is given the set F R is C continuously O On MATS Ex Coosider the curve SCA dejined bui Find the line to S ugh the point coo. the S can be inritten with y LOAD line is given bu the taseert plane to the sphere as. radius 3 DZ Near. the sphere is quenas a graph of a C n function f: R 2,2) So the taraent plane is given the set F R is C continuously O OnMAT these is a neobbourhood CS st. a) As this holds fo all vectors urne have a is a citi Taylor's THM hithin L at a CHtical Note The Hessian minix a Lmxm matNx. in modrix A is symmetric Then it is diagonalizable and Nts engenvalues are real. rthegonal SPT where P is o and where the a Nethe egenvalnes A deAnte if all of its eiaenvalues ane nagaliva. non- 000 nods it has at least one posieve and one oeaotive MAT these is a neobbourhood CS st. a) As this holds fo all vectors urne have a is a citi Taylor's THM hithin L at a CHtical Note The Hessian minix a Lmxm matNx. in modrix A is symmetric Then it is diagonalizable and Nts engenvalues are real. rthegonal SPT where P is o and where the a Nethe egenvalnes A deAnte if all of its eiaenvalues ane nagaliva. non- 000 nods it has at least one posieve and one oeaotiveMAT Ex J o Lhas eigenvolna and 2. ositive o 3 go the matrix is positive definite. Eigenvalues is posrtive detinxte S o positive sem (bit pot pasti dginite) O O nte Card 5 O O TO -3 Note A is an -A is an eigen veR ,vro, alt. A Ts pasitve detincte iff. A is negative nite A Ts pas. A is Sam DCE P is invertible v o et Pax then as Anvr Nav. bloe 20 and Aivit o fot ton SSWAN is as eigenvolve A Tlus means zigenvalves are positive MAT Ex J o Lhas eigenvolna and 2. ositive o 3 go the matrix is positive definite. Eigenvalues is posrtive detinxte S o positive sem (bit pot pasti dginite) O O nte Card 5 O O TO -3 Note A is an -A is an eigen veR ,vro, alt. A Ts pasitve detincte iff. A is negative nite A Ts pas. A is Sam DCE P is invertible v o et Pax then as Anvr Nav. bloe 20 and Aivit o fot ton SSWAN is as eigenvolve A Tlus means zigenvalves are positiveMAT 3 O S open, f t is Hessa Ts postive te, has a local man at a its e ane POSitive. o A summitic nxn matrix A Dail is n2aa definite Hess af is posttive de inite, f has a locol mio. ai a Exercise Check for Noactive ddinite motices that IE 3 inita ness is an open MAT 3 O S open, f t is Hessa Ts postive te, has a local man at a its e ane POSitive. o A summitic nxn matrix A Dail is n2aa definite Hess af is posttive de inite, f has a locol mio. ai a Exercise Check for Noactive ddinite motices that IE 3 inita ness is an openMAT 37 also I.OO Note that the determinant a is a continuas fundren the cells of the matix (ID fact, rtis a Mnomal....) O Oka -ark these is Ek o st. O Let E min SE for n then O end continuously on. (O AAte W-Taylor TH Here, hei has a local minimum a. a Hess Hessa Ex Also a So CL-1) is the only critical point. RSS MAT 37 also I.OO Note that the determinant a is a continuas fundren the cells of the matix (ID fact, rtis a Mnomal....) O Oka -ark these is Ek o st. O Let E min SE for n then O end continuously on. (O AAte W-Taylor TH Here, hei has a local minimum a. a Hess Hessa Ex Also a So CL-1) is the only critical point. RSSMAT BI Only cal part at Hess (A) inite. O f has neither a min a max. at Coy. COLO) rt which is rather a min called saddle THAM TiusR open, a c and Hassi does have a Let A20 be an VERA (Hess, V. Av hithoki loss generobru IVI v ninth O 3 t o st. v. Hess 20. tw cannot ha MAT BI Only cal part at Hess (A) inite. O f has neither a min a max. at Coy. COLO) rt which is rather a min called saddle THAM TiusR open, a c and Hassi does have a Let A20 be an VERA (Hess, V. Av hithoki loss generobru IVI v ninth O 3 t o st. v. Hess 20. tw cannot haMATJ3T 4G Recall.... A summetic n n matrix-A is posative semi-definite if ali its eigenvolve are non-neoptre. A semi of its eigenvalue abe non pastiva. In words,the THM abae states iff is C and has a local max. at a then Vf (a) o and Hesse is. Megafive a rogative egenvaluwe not have a local min. ata. In other nds if f is ci and has a local min ae a,then Hessat Ts.poSfive nite. For f is cat a CHfical Hoss pas. local min pas...sem max. Hess point and is indefinite, then f has a saddle. part at a D it has a postive eigenvalue. it has a so b dozs not ha Local min. at a does not has a local max..at a it has a saddle point and det (Hessaf) so. Then are the eigenvalue Hessif, SO, so one has to be regodive and the other Positive Hessf is ind inter and has a saddle it at a Mas a saddle pot at Hessa i is sevi extNemwm ot a saddle point Ex ve CHtical point, i e a ertical pant Hess MATJ3T 4G Recall.... A summetic n n matrix-A is posative semi-definite if ali its eigenvolve are non-neoptre. A semi of its eigenvalue abe non pastiva. In words,the THM abae states iff is C and has a local max. at a then Vf (a) o and Hesse is. Megafive a rogative egenvaluwe not have a local min. ata. In other nds if f is ci and has a local min ae a,then Hessat Ts.poSfive nite. For f is cat a CHfical Hoss pas. local min pas...sem max. Hess point and is indefinite, then f has a saddle. part at a D it has a postive eigenvalue. it has a so b dozs not ha Local min. at a does not has a local max..at a it has a saddle point and det (Hessaf) so. Then are the eigenvalue Hessif, SO, so one has to be regodive and the other Positive Hessf is ind inter and has a saddle it at a Mas a saddle pot at Hessa i is sevi extNemwm ot a saddle point Ex ve CHtical point, i e a ertical pant HessMAT on t Com since k in the interiot, say ot aEK "Kt then is a loca max, it nill be NOTE ak Attoa iter, the max. of f On k has to occurr at one the point looy. LO), (011 the max of f occurs act (LO) C-,o). H ne look functions on sets which are not bounded may mst have aldol miniona ot maxima Let SCR be unbounded, f 3- R... E. o, so that ln E. SLI. HM I SCR closed, unbounded and f S is CODinuaub and linn oo... has a global min., but no max MAT on t Com s
More Less

Related notes for MAT237Y1

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit