Recall from the previous lecture: Extreme Value Theorem Suppose a real-valued function f( x1,, x n. ) is continuous on a closed and bounded

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1 Multvrle Clculus Lecture # Notes I ths lecture we cotue the dscusso of extrem of fuctos of severl vrles. I prtculr, we ll dscuss protocol for fdg extrem ouded rego, we ll look t some pplctos ecoomcs, d we ll ddress how to hdle multple costrts. We ll lso get strted o the de of tegrto of fucto. Recll from the prevous lecture: Extreme Vlue Theorem Suppose rel-vlued fucto f( x,, x ) s cotuous o closed d ouded dom R. The ths fucto must tt ts mxmum d mmum vlues somewhere wth ths dom. [Note: You c thk of closed set s oe for whch y coverget sequece ths set hs ts lmt the set. I prtculr, ths mes tht the oudry of the set must e cluded.] For fucto f( x,, x ) of severl vrles, extrem wll occur t: () Sttory pots (pots where f =0,.e. where ll of the prtl dervtves of f vsh); or () No-dfferetle pots (pots where the gve fucto s ot dfferetle); or (c) O the oudry of the dom. The sttory pots d o-dfferetle pots re crtcl pots. The lyss of typcl prolem usully s dvded to two steps: () seekg crtcl pots the teror of the rego (though they could occur o the oudry), d () seekg ddtol cddtes for extrem o the oudry of the rego usg the Method of Lgrge Multplers. gg ggg f = λ g gx (,, x) = c Assocted wth ths theorem s the followg protocol for detfyg extreme of fuctos ouded rego: () Frst seek sttory pots the teror of the ouded dom (whch mght possly occur o the oudry of the dom) s well s other crtcl pots, e.g. pots of o-dfferetlty; the () Use the Method of Lgrge Multplers to detfy cddtes for extrem o the oudry of the dom (ssumg the oudry s descred y level sets of dfferetle fucto). If the oudry cossts of severl dstct peces, you wll lso hve to clude where those peces tersect s possle loctos for extrem. It must lso e sd tht we ofte wsh to fd extrem of fuctos uouded doms. I ths cse we proceed s ove, ut determg whch pots yeld mxm or mm my requre some ddtol rgumet. Ofte, these methods come dow to detfyg short lst of cddtes for extrem d evluto of the gve fucto to determe whch yeld reltve d solute mxm d mm. Exmple: Fd the mxmum d mmum vlues of the fucto ouded y the curves 4 y = x d y = 4. Soluto: () We frst seek teror crtcl pots: These occur where fx = x+ = 0 ( xy, ) = (, ). You c pply the fy = y 4= 0 d ervtve Test, f you wsh, to see tht ths wll yeld locl mmum. f( xy, ) = x + y + x 4y+ 5 the rego () We ext exme the oudry for possle extrem. Alog the top edge ( y = 4 ) we c just susttute to get f = x + x+ 5 d sc Clculus methods gve crtcl pot where x + = 0,.e. x =. So the pot (, 4) s cddte for mxmum or mmum. For the other oudry ( 4 y = x ) we could lso susttute, ut f we use the Method of Revsed Jury 7, 07

2 Lgrge Multplers we frst rewrte ths costrt s gxy (, ) = 4y x = 0. The optmlty codto gves: x+ = λ ( x) x+ x = x + = 4xy + x xy = y 4 = λ (4) y 4 4 Comg ths wth the costrt 4 y = x we get x =, so x = d y =,.e. the pot (,). The oudres meet t the pots ( 4,4) d (4,4) d these must lso e cosdered s cddtes for possle extrem. Evluto d comprso gves, s we strt wth the sttory pot d the work our wy sequetlly roud the oudry: Cddte pot Vlue Notes (, ) f (, ) = 0 solute mmum t ths teror sttory pot ( 4,4) f ( 4, 4) = reltve mxmum t ths oudry pot (, 4) f (, 4) = 4 ether mxmum or mmum (4, 4) f (4, 4) = solute mxmum (,) f (,) = ether mxmum or mmum Lgrge Multpler methods Ecoomcs Perhps the most commo settg where costred optmzto s ecoutered s the feld of Ecoomcs. Ideed, you could lmost defe sgfct porto of ecoomcs y ths. Here s typcl prolem tht you mght ecouter: Prolem: Suppose producto t fctory s modeled y Pxy (, ) = 400x y 4 where x represets the umer m of uts of lor d y represets the umer of uts of cptl. [A producto model of ths form ( P = x y ) s kow s Co-ougls model.] If lor costs $0/ut d cptl costs $40/ut, fd the optml comto of lor d cptl tht yelds the gretest producto gve udget of $000. [Note: These fgures mght correspod to hourly producto.] Soluto: We wt to mxmze Pxy (, ) = 400x y 4 suject to the costrt Bxy (, ) = 0x+ 40y = 000. Extrem wll occur where: P 4 x = λbx 00x y =λ 0 y P= λ B y 00 x Py B = = = λ y x y 4 =λ 40 x 4 Geerlly, we see tht uder optml codtos Px Bx ut prce of lor = =. Tht s, uder Py By ut prce of cptl optml codtos the rto of the mrgl productvtes s equl to the rto of the ut prces. I the specfc prolem, the equto y = x represets le of optmlty depedet of whtever udget we hve wth whch to work. If we solve ths smulteously wth the gve udget 0x+ 40 x = 75x= 000, costrt we get tht ( ) Revsed Jury 7, 07

3 so x = 40 uts of lor d y = x= 45 uts of cptl. Uder these optml codtos, the mxmum 4 mx = (40, 45) = There s reso why we re usg excessve umer of sgfct fgures. To see why, let s cosder wht would hppe f we were to mrglly crese the udget. productvty wll e P P ( ) ( ) If we sted hd $00 to sped optmlly, how would thgs chge? Optmlly we would stll hve to mt the codto tht y = x, so the oly chge would e tht 75x = 00 whch would gve tht x = 40. uts of lor d y = uts of cptl. Usg these, the ew mxmum producto would e Pmx = P(40.0, 45.05) So get crese producto of P.4. If we thk of the udget B s prmeter tht c e djusted, we see tht y cresg the udget y $ we hve tht P.4. Curously, we c clculte tht the vlue of the Lgrge Multpler s B ( ) ( ) λ=.4. Ths seems ulkely to hve occurred y chce, so wht s the explto? 0 Everythg c e expled usg the Ch Rule. The method of soluto c e thought of sequetlly s: solve v L.M. B ( xb ( ), yb ( )) PxB ( ( ), yb ( )) = Pmx The Ch Rule plus susttuto of the Lgrge Multpler codtos gves tht: dp P x P y B x B y B x B y = + = λ + λ =λ +. db x B y B x B y B x B y B To mke sese of the somewht cryptc expresso sde the pretheses (where B ppers oth s prmeter d s fucto of x d y, cosder the somewht slly sequece: solve v L.M. B ( xb ( ), yb ( )) BxB ( ( ), yb ( )) = B Ths sclly sys tht f you re gve B dollrs to sped optmlly d someoe the sks you how much moey you must the sped, the swer s smply s much moey s I hve to work wth. The Ch Rules db B x B y dp B x B y the gves tht = + =, so =λ + =λ =λ. Ths expls why we foud db x B y B db x B y B P tht for smll chge udget we hve λ. B Spekg colloqully, we mght sy tht λ mesures how much more g for the uck. Recprocty: I prolem lke the prevous oe, the Lgrge Multpler codto P= λ B smply descres the fct tht uder optml codtos P d B wll e prllel.. It could just s esly hve ee expressed s B= λ P. Ths c e terpreted s syg tht the prolem of mxmzg producto suject to fxed udget s fudmetlly the sme s mmzg cost suject to fxed producto. Essetlly, effcecy s the rel pot d we c terchge the roles of wht qutty s eg optmzed d wht qutty s eg costred. Revsed Jury 7, 07

4 The Method of Lgrge Multplers wth Multple Costrts Suppose we wsh to fd the extrem of fucto f( xyz,, ) suject to two costrts: gxyz (,, ) = c d hxyz (,, ) = c. Let s further ssume tht the surfces represet y oth costrts re smooth surfces d tht they tersect smooth curve,.e. curve tht c e prmeterzed wth dfferetle compoet fuctos. It s worth otg tht poorly-posed prolem the costrts mght ot eve e comptle. Eve more sutle, the two costrts mght oly tersect tgetlly, so we wll further ssume tht they hve cle tersecto or trsverse tersecto whch the orml vectors to these costrt surfce re ever prllel log ther tersecto. Uder these codtos, f r () t = xt (), yt (), zt () prmeterzes the tersecto curve, the t y extremum the fucto f( r ( t)) = f( xt ( ), yt ( ), zt ( )) would hve sttory pot. By the Ch Rule, ths mes tht d d [ f( r( t)) ] = f v= 0 where v s the velocty (tget) vector to the tersecto curve. Ths mes tht dt f v t extremum. But we kow tht ggg g d h re perpedculr t ths pot to the respectve costrt surfces. As log s we kow tht ggg g d h re ot prllel, they wll sp the ple ggg ggg ggg perpedculr to the vector v, so we must therefore e le to express f =λ g+λ h for some sclrs (Lgrge Multplers) λ d λ. Together wth the two costrts, ths s our ehced Method of Lgrge ggg ggg ggg f =λ g+λ h Multplers for multple costrts: gxyz (,, ) = c. Note tht ths yelds totl of 5 equtos the 5 hxyz (,, ) = c xyzλ,,,, λ. It my ot e esy, ut these c prcple e solved. ukows { } Ths method c, uder del codtos, e exteded to fucto of vrles wth m costrts,.e. to fd g( x,, x ) = c the extrem of f( x,, x ) suject to the costrts we would solve the equtos: gm( x,, x) = c m ggg ggg gggg f =λ g+ +λm g m g( x,, x ) = c ( + m equtos the + m ukows { x,, x, λ,, λm}. gm( x,, x) = cm Itegrto of fuctos of severl vrles over regos Itegrto s relly out mesuremet specfclly the mesuremet of ggregte mout of somethg. Recll the des ehd tegrto of fucto of oe vrle over tervl,.e. f ( x ) dx, the defte tegrl. The motvtg exmple ws most lkely fdg the re of rego uder the grph d ove the horzotl xs of postve fucto f( x ) defed o tervl [., ] The method utlzed to clculte ths ws the Method of Rem Sums. Ths cossts of four steps: R 4 Revsed Jury 7, 07

5 ) Prtto the dom: Choose ddtol pots { } x = x x e the wdth of the -th tervl. x so tht = x0 < < x < x < < x = d let ) Approxmte wht s to e mesured: Choose smple pot c ech tervl, d pproxmte the re of the -th rectgle s A f( c ) x. ) Sum: Add up these pproxmte res to get pproxmto of the totl re,.e. A= A f( c ) x. = = 4) Lmt: Refe the prtto y sertg pots so tht the wdth of the lrgest tervl gets progressvely smller d fd the lmt of the pproxmte totl re (f t exsts) s these wdths ecome rtrrly smll. Tht s, f we deote the mesh of the prtto y = mx ( x ), we defe lm f ( c) x f ( x) dx 0 = = f ths lmt exsts depedet of y choces. If ths lmt exsts we sy tht the fucto f( x ) s (Rem) tegrle over the tervl [., ] It s prove (or should e prove) sgle-vrle Clculus tht f f( x ) s cotuous or pecewse-cotuous o the tervl [, ] wth oly jump dscotutes, the f( x ) wll e tegrle. It must e oted tht the clculto of defte tegrls oe-vrle Clculus s mde much smpler v the pplcto of the Fudmetl Theorem of Clculus. Oce the de of the defte tegrl s estlshed, t s smple to oserve tht there s relly o reso to ssume f( x ) to e postve o the tervl [., ] The defto of f ( x ) dx stll mkes sese, ut must e terpreted the s sged re where regos ove the horzotl xs re couted s postve re d regos elow the horzotl xs re couted s egtve re. The tegrl the mesures the sum of these vlues. More sgfctly, the defte tegrl does t hve to e terpreted or costructed s re t ll. Here re two other smple exmples of some sgfcce: () Suppose r of mterl of vrle desty s locted such wy tht t occupes the tervl [., ] If σ ( x) mesures the desty t ech pot the tervl (perhps uts such s grms per cetmeter), the sme Method of Rem Sums would gve m σ( c ) x for the pproxmte mss of the -th tervl d σ( x) dx would the gve the totl mss of ths r of mterl. If the desty mesures electrc chrge desty, the the tegrl wll mesure the totl chrge. Ideed, f σ ( x) mesures the desty of y qutty whtsoever ths tervl, the tegrl σ( x) dx wll mesure the totl mout of ths qutty. () Suppose tht over the tme tervl t oject moves sptlly (perhps log strght le) such wy tht ts velocty t y tme t s gve y fucto vt (). Over smll tme tervl, ths oject would e dsplced y mout s vt ( ) t,.e. rte tmes tme gves dstce, d the totl dsplcemet (cludg oth postve d egtve dsplcemets, depedg upo whe the velocty s postve or egtve) would e pproxmtely vt ( ) t, so the lmt we terpret v () t dt s the totl = dsplcemet of ths movg oject. 5 Revsed Jury 7, 07

6 How mght we thk out the de of the tegrl of fucto of two (or more) vrles? We c proceed y logy t lest to get strted. Suppose f( xy, ) s fucto of two vrles defed over (closed d ouded) dom. If ths fucto hs postve vlues ths dom, we c cosder ts grph z= f( xy, ) over ths dom d, y logy, try to fd wy of mesurg the volume uder ths grph (d ove the xy-ple d wth the vertcl curt wll lyg ove the oudry of the dom ). Proceedg s efore usg the Method of Rem Sums, we would: ) Prtto the dom: Ths tme we would hve to chop up the two-dmesol dom to uformly smll peces perhps mly smll rectgulr peces, ut ot ecessrly. Let A deote the re of the -th pece d defe the mesh = mx dmeter( ). of the prtto y ( ) ) Approxmte wht s to e mesured: Choose smple pot ( x, y ) ech pece, d pproxmte the volume of the vertcl shft ove ths pece y V f( x, y ) A. ) Sum: Add up these pproxmte volumes to get pproxmto of the totl volume,.e. V = V f( x, y ) A. = = 4) Lmt: Refe the prtto such wy tht the mesh of the prtto mx ( dmeter( )) = ecomes rtrrly smll. We the defe lm f( x, y) A f( x, y) da 0 = = y choces. We would the sy tht ths fucto s tegrle over ths dom d we refer to f ( x, y ) da s the (doule) tegrl of the fucto f( xy, ) over the dom. f ths lmt exsts depedet of The procedure ove s perfectly dole for computer ut, s ws the cse sgle vrle Clculus whe we used the Fudmetl Theorem of Clculus s lterte wy to clculte tegrls, we wll wt to dscover smple wys to clculte such multple tegrl (whe possle) wthout the eed of computer. Oce g, t s ot relly ecessry to ssume tht f( xy, ) s postve everywhere the dom. The defto of f ( x, y ) da stll mkes sese, ut must e terpreted the s sged volume where regos ove the horzotl xy-ple re couted s postve volume d regos elow the horzotl xy-ple re couted s egtve volume. The tegrl the mesures the sum of these vlues. The tegrl of fucto of two vrles over dom does t hve to mesure volume or sged volume. Suppose tht σ ( xy, ) mesures the desty of oject tht occupes dom the xy-ple (perhps mesured uts of grms per squre cetmeter). If we were to cut up the dom to uformly smll peces, we mght the pproxmte the mss of ech smll pece s m σ( x, y) A d proceed s efore. We would the terpret the tegrl s the totl mss of ths oject (ofte referred to s lm),.e. mss( ) = s( x, y) da σ x y da wll the gve. If the desty mesures electrcl chrge, the tegrl (, ) 6 Revsed Jury 7, 07

7 the totl chrge (summg oth postve d egtvely chrged regos to produce the et chrge). If σ ( xy, ) mesures populto desty rego, the σ( x, y) da wll gve the totl populto the rego. Perhps the smplest mesuremet we mght wt to cosder would e just the totl re of rego. I ths cse, ths volves oly summg up the res of dvdul peces, d the lmt we get smply tht Are( ) = da. I the ext lecture, we ll descre rge of ddtol pplctos of multple tegrls d develop techques for clcultg them. We ll lso get strted o trple tegrls. Notes y Roert Wters d Reée Chpm 7 Revsed Jury 7, 07