where i is the index of summation, a i is the ith term of the sum, and the upper and lower bounds of summation are n and 1.

Transcription

1 Chpter Itegrtio. Are Use sigm ottio to write d evlute sum. Uderstd the cocept o re. Approimte the re o ple regio. Fid the re o ple regio usig limits. Sigm Nottio I the precedig sectio, ou studied tidieretitio. I this sectio, ou will look urther ito prolem itroduced i Sectio. tht o idig the re o regio i the ple. At irst glce, these two ides m seem urelted, ut ou will discover i Sectio. tht the re closel relted etremel importt theorem clled the Fudmetl Theorem o Clculus. This sectio egis itroducig cocise ottio or sums. This ottio is clled sigm ottio ecuse it uses the uppercse Greek letter sigm, writte s. Sigm Nottio The sum o terms,,,..., is writte s i... where i is the ide o summtio, i is the ith term o the sum, d the upper d lower ouds o summtio re d. REMARK The upper d lower ouds must e costt with respect to the ide o summtio. However, the lower oud does t hve to e. A iteger less th or equl to the upper oud is legitimte. Emples o Sigm Nottio FOR FURTHER INFORMATION For geometric iterprettio o summtio ormuls, see the rticle Lookig t k k d k k Geometricll Eric Heglom i Mthemtics Techer. To view this rticle, go to MthArticles.com... c. d. e.. i i i0 7 j 7 j j k... j k i... From prts () d (), otice tht the sme sum c e represeted i dieret ws usig sigm ottio. Although vrile c e used s the ide o summtio, i, j, d k re ote used. Notice i Emple tht the ide o summtio does ot pper i the terms o the epded sum. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

2 . Are THE SUM OF THE FIRST 00 INTEGERS A techer o Crl Friedrich Guss (777 ) sked him to dd ll the itegers rom to 00.Whe Guss retured with the correct swer ter ol ew momets, the techer could ol look t him i stouded silece.this is wht Guss did: This is geerlized Theorem., Propert, where 00 i t The properties o summtio show elow c e derived usig the Associtive d Commuttive Properties o Additio d the Distriutive Propert o Additio over Multiplictio. (I the irst propert, k is costt.).. k i k i The et theorem lists some useul ormuls or sums o powers. THEOREM. Summtio Formuls. c c, c is costt.. i. Evlutig Sum i ± i i ± i i i A proo o this theorem is give i Appedi A. See LrsoClculus.com or Bruce Edwrds s video o this proo. i Evlute or 0, 00, 000, d 0,000. Solutio i i i Fctor the costt out o sum. Write s two sums. Appl Theorem.. Simpli. Simpli. Now ou c evlute the sum sustitutig the pproprite vlues o, s show i the tle elow ,000 i I the tle, ote tht the sum ppers to pproch limit s icreses. Although the discussio o limits t iiit i Sectio. pplies to vrile, where c e rel umer, m o the sme results hold true or limits ivolvig the vrile, where is restricted to positive iteger vlues. So, to id the limit o s pproches iiit, ou c write lim lim lim 0. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

3 Chpter Itegrtio Are I Euclide geometr, the simplest tpe o ple regio is rectgle. Although people ote s tht the ormul or the re o rectgle is A h it is ctull more proper to s tht this is the deiitio o the re o rectgle. From this deiitio, ou c develop ormuls or the res o m other ple regios. For emple, to determie the re o trigle, ou c orm rectgle whose re is twice tht o the trigle, s show i Figure.. Oce ou kow how to id the re o trigle, ou c determie the re o polgo sudividig the polgo ito trigulr regios, s show i Figure.. h Trigle: A h Figure. Prllelogrm Hego Polgo Figure. ARCHIMEDES (7 B.C.) Archimedes used the method o ehustio to derive ormuls or the res o ellipses, prolic segmets, d sectors o spirl. He is cosidered to hve ee the gretest pplied mthemtici o tiquit. See LrsoClculus.com to red more o this iogrph. Fidig the res o regios other th polgos is more diicult. The ciet Greeks were le to determie ormuls or the res o some geerl regios (pricipll those ouded coics) the ehustio method. The clerest descriptio o this method ws give Archimedes. Essetill, the method is limitig process i which the re is squeezed etwee two polgos oe iscried i the regio d oe circumscried out the regio. For istce, i Figure.7, the re o circulr regio is pproimted -sided iscried polgo d -sided circumscried polgo. For ech vlue o, the re o the iscried polgo is less th the re o the circle, d the re o the circumscried polgo is greter th the re o the circle. Moreover, s icreses, the res o oth polgos ecome etter d etter pproimtios o the re o the circle. FOR FURTHER INFORMATION For ltertive developmet o the ormul or the re o circle, see the rticle Proo Without Words: Are o Disk is R Russell J Hedel i Mthemtics Mgzie. To view this rticle, go to MthArticles.com. = The ehustio method or idig the re o circulr regio Figure.7 = A process tht is similr to tht used Archimedes to determie the re o ple regio is used i the remiig emples i this sectio. Mr Evs Picture Lirr Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

4 The Are o Ple Regio. Are 7 Recll rom Sectio. tht the origis o clculus re coected to two clssic prolems: the tget lie prolem d the re prolem. Emple egis the ivestigtio o the re prolem. Approimtig the Are o Ple Regio () = + 0 () The re o the prolic regio is greter th the re o the rectgles. Use the ive rectgles i Figure.() d () to id two pproimtios o the re o the regio lig etwee the grph o d the -is etwee 0 d. Solutio. The right edpoits o the ive itervls re i Right edpoits where i,,,,. The width o ech rectgle is, d the height o ech rectgle c e otied evlutig t the right edpoit o ech itervl. 0,,,,,,,,, 0 () = + Evlute t the right edpoits o these itervls. The sum o the res o the ive rectgles is 0 () The re o the prolic regio is less th the re o the rectgles. Figure. Height Width i i.. Becuse ech o the ive rectgles lies iside the prolic regio, ou c coclude tht the re o the prolic regio is greter th... The let edpoits o the ive itervls re i Let edpoits where i,,,,. The width o ech rectgle is, d the height o ech rectgle c e otied evlutig t the let edpoit o ech itervl. So, the sum is Height Width i i 0.0. Becuse the prolic regio lies withi the uio o the ive rectgulr regios, ou c coclude tht the re o the prolic regio is less th.0. B comiig the results i prts () d (), ou c coclude tht. < Are o regio <.0. B icresig the umer o rectgles used i Emple, ou c oti closer d closer pproimtios o the re o the regio. For istce, usig rectgles o width ech, ou c coclude tht 7.7 < Are o regio < 7.9. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

5 Chpter Itegrtio The regio uder curve Figure.9 (m i ) Δ The itervl, is divided ito suitervls o width. Figure.0 (M i ) Upper d Lower Sums The procedure used i Emple c e geerlized s ollows. Cosider ple regio ouded ove the grph o oegtive, cotiuous uctio s show i Figure.9. The regio is ouded elow the -is, d the let d right oudries o the regio re the verticl lies d. To pproimte the re o the regio, egi sudividig the itervl, ito suitervls, ech o width s show i Figure.0. The edpoits o the itervls re Becuse is cotiuous, the Etreme Vlue Theorem gurtees the eistece o miimum d mimum vlue o i ech suitervl. m i Miimum vlue o i ith suitervl M i Mimum vlue o i ith suitervl Net, deie iscried rectgle lig iside the ith suregio d circumscried rectgle etedig outside the ith suregio. The height o the ith iscried rectgle is m i d the height o the ith circumscried rectgle is M i. For ech i, the re o the iscried rectgle is less th or equl to the re o the circumscried rectgle. 0 0 < < <... <. Are o iscried rectgle The sum o the res o the iscried rectgles is clled lower sum, d the sum o the res o the circumscried rectgles is clled upper sum. Lower sum s Upper sum S m i M i Are o iscried rectgles Are o circumscried rectgles From Figure., ou c see tht the lower sum s is less th or equl to the upper sum S. Moreover, the ctul re o the regio lies etwee these two sums. s Are o regio S m i M i Are o circumscried rectgle = () = () s() = () S() Are o iscried rectgles Are o regio Are o circumscried is less th re o regio. rectgles is greter th re o regio. Figure. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

6 . Are 9 Iscried rectgles Circumscried rectgles Figure. () = () = Fidig Upper d Lower Sums or Regio Fid the upper d lower sums or the regio ouded the grph o d the -is etwee 0 d. Solutio To egi, prtitio the itervl 0, ito suitervls, ech o width Figure. shows the edpoits o the suitervls d severl iscried d circumscried rectgles. Becuse is icresig o the itervl 0,, the miimum vlue o ech suitervl occurs t the let edpoit, d the mimum vlue occurs t the right edpoit. Let Edpoits m i 0 i i Usig the let edpoits, the lower sum is s i i i. Lower sum Usig the right edpoits, the upper sum is S i i i i m i M i i 0.. i Upper sum Right Edpoits M i 0 i i Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

7 0 Chpter Itegrtio Eplortio For the regio give i Emple, evlute the lower sum s d the upper sum S or 0, 00, d 000. Use our results to determie the re o the regio. Emple illustrtes some importt thigs out lower d upper sums. First, otice tht or vlue o, the lower sum is less th (or equl to) the upper sum. Secod, the dierece etwee these two sums lesses s icreses. I ct, whe ou tke the limits s, oth the lower sum d the upper sum pproch d s < S lim s lim lim S lim Lower sum limit Upper sum limit The et theorem shows tht the equivlece o the limits (s ) o the upper d lower sums is ot mere coicidece. It is true or ll uctios tht re cotiuous d oegtive o the closed itervl,. The proo o this theorem is est let to course i dvced clculus.. THEOREM. Limits o the Lower d Upper Sums Let e cotiuous d oegtive o the itervl,. The limits s o oth the lower d upper sums eist d re equl to ech other. Tht is, lim s lim m i lim M i lim S where d m i d M i re the miimum d mimum vlues o o the suitervl. I Theorem., the sme limit is ttied or oth the miimum vlue m i d the mimum vlue M i. So, it ollows rom the Squeeze Theorem (Theorem.) tht the choice o i the ith suitervl does ot ect the limit. This mes tht ou re ree to choose ritrr -vlue i the ith suitervl, s show i the deiitio o the re o regio i the ple. Deiitio o the Are o Regio i the Ple Let e cotiuous d oegtive o the itervl,. (See Figure..) The re o the regio ouded the grph o, the -is, d the verticl lies d is Are lim c i where c i i d. i c i i (c i ) The width o the ith suitervl is i. Figure. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

8 . Are Fidig Are the Limit Deiitio (, ) () = (0, 0) The re o the regio ouded the grph o, the -is, 0, d is. Figure. Fid the re o the regio ouded the grph, the -is, d the verticl lies 0 d, s show i Figure.. Solutio Begi otig tht is cotiuous d oegtive o the itervl 0,. Net, prtitio the itervl 0, ito suitervls, ech o width. Accordig to the deiitio o re, ou c choose -vlue i the ith suitervl. For this emple, the right edpoits c i i re coveiet. Are lim c i lim i lim i lim lim Right edpoits: c i i The re o the regio is. () = Fidig Are the Limit Deiitio See LrsoClculus.com or iterctive versio o this tpe o emple. Fid the re o the regio ouded the grph o, the -is, d the verticl lies d, s show i Figure.. Solutio Note tht the uctio is cotiuous d oegtive o the itervl,. So, egi prtitioig the itervl ito suitervls, ech o width. Choosig the right edpoit c i i i Right edpoits The re o the regio ouded the grph o, the -is,, d is. Figure. o ech suitervl, ou oti Are lim c i lim i lim i i lim i i lim. The re o the regio is. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

9 Chpter Itegrtio The et emple looks t regio tht is ouded the -is (rther th the -is). A Regio Bouded the -is (, ) () = (0, 0) The re o the regio ouded the grph o d the -is or 0 is. Figure. Fid the re o the regio ouded the grph o d the -is or 0, s show i Figure.. Solutio Whe is cotiuous, oegtive uctio o, ou c still use the sme sic procedure show i Emples d. Begi prtitioig the itervl 0, ito suitervls, ech o width. The, usig the upper edpoits c i i, ou oti Are lim c i lim i lim i lim lim Upper edpoits: c i i. The re o the regio is. REMARK You will ler out other pproimtio methods i Sectio.. Oe o the methods, the Trpezoidl Rule, is similr to the Midpoit Rule. I Emples,, d 7, c i is chose to e vlue tht is coveiet or clcultig the limit. Becuse ech limit gives the ect re or c i, there is o eed to id vlues tht give good pproimtios whe is smll. For pproimtio, however, ou should tr to id vlue o c i tht gives good pproimtio o the re o the ith suregio. I geerl, good vlue to choose is the midpoit o the itervl, c i i, d ppl the Midpoit Rule. Are i. Midpoit Rule Approimtig Are with the Midpoit Rule () = si c π c π c π c π The re o the regio ouded the grph o si d the -is or 0 is out.0. Figure.7 Use the Midpoit Rule with to pproimte the re o the regio ouded the grph o si d the -is or 0, s show i Figure.7. Solutio For,. The midpoits o the suregios re show elow. c c So, the re is pproimted Are 0 which is out.0. c i si c i c c 7 si si 7 si si Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

10 . Are. Eercises See ClcCht.com or tutoril help d worked-out solutios to odd-umered eercises. Fidig Sum I Eercises, id the sum. Use the summtio cpilities o grphig utilit to veri our result.. i... k. c. Usig Sigm Nottio ottio to write the sum k0 k... I Eercises 7, use sigm Evlutig Sum I Eercises 0, use the properties o summtio d Theorem. to evlute the sum. Use the summtio cpilities o grphig utilit to veri our result i i. 9. ii 0. 9 k k j j i i i 0 i i i Evlutig Sum I Eercises, use the summtio ormuls to rewrite the epressio without the summtio ottio. Use the result to id the sums or 0, 00, 000, d 0,000. Approimtig the Are o Ple Regio I Eercises 0, use let d right edpoits d the give umer o rectgles to id two pproimtios o the re o the regio etwee the grph o the uctio d the -is over the give itervl.., 0,, rectgles. 9,,, rectgles 7. g,,, rectgles. g,,, rectgles 9. cos, 0, rectgles 0. g si, 0,, rectgles Usig Upper d Lower Sums I Eercises d, oud the re o the shded regio pproimtig the upper d lower sums. Use rectgles o width... Fidig Upper d Lower Sums or Regio I Eercises, use upper d lower sums to pproimte the re o the regio usig the give umer o suitervls (o equl width)....., i.. kk.. k 7j j i i Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

11 Chpter Itegrtio Fidig Limit I Eercises 7, id ormul or the sum o terms. Use the ormul to id the limit s. i 7. lim. 9. lim 0. i. lim. i. Numericl Resoig Cosider trigle o re ouded the grphs o, 0, d. () Sketch the regio. () Divide the itervl 0, ito suitervls o equl width d show tht the edpoits re (c) Show tht s (d) Show tht S (e) Complete the tle. () Show tht lim s lim S.. Numericl Resoig Cosider trpezoid o re ouded the grphs o, 0,, d. () Sketch the regio. () Divide the itervl, ito suitervls o equl width d show tht the edpoits re (c) Show tht s (d) Show tht S (e) Complete the tle. () Show tht lim i. lim i 0 < <... < <. i s S < <... < <. i. i s S s lim S. lim i lim i Fidig Are the Limit Deiitio I Eercises, use the limit process to id the re o the regio ouded the grph o the uctio d the -is over the give itervl. Sketch the regio.., 0,., 7., 0,., 9.,, 0.,. 7, [,.,.,,., Fidig Are the Limit Deiitio I Eercises 0, use the limit process to id the re o the regio ouded the grph o the uctio d the -is over the give -itervl. Sketch the regio Approimtig Are with the Midpoit Rule I Eercises, use the Midpoit Rule with to pproimte the re o the regio ouded the grph o the uctio d the -is over the give itervl...., g,,, g, h,,, t, 0 0 0,. cos, 0, 0, 0, WRITING ABOUT CONCEPTS, 0,, 0,, Approimtio I Eercises d, determie which vlue est pproimtes the re o the regio etwee the -is d the grph o the uctio over the give itervl. (Mke our selectio o the sis o sketch o the regio, ot perormig clcultios.).., () () (c) 0 (d) (e) si, 0, 0, () () (c) (d) (e) 7. Upper d Lower Sums I our ow words d usig pproprite igures, descrie the methods o upper sums d lower sums i pproimtig the re o regio.. Are o Regio i the Ple Give the deiitio o the re o regio i the ple. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

12 . Are 9. Grphicl Resoig Cosider the regio ouded the grphs o ), 0,, d 0, s show i the igure. To prit elrged cop o the grph, go to MthGrphs.com. () Redrw the igure, d complete d shde the rectgles represetig the lower sum whe. Fid this lower sum. () Redrw the igure, d complete d shde the rectgles represetig the upper sum whe. Fid this upper sum. (c) Redrw the igure, d complete d shde the rectgles whose heights re determied the uctiol vlues t the midpoit o ech suitervl whe. Fid this sum usig the Midpoit Rule. (d) Veri the ollowig ormuls or pproimtig the re o the regio usig suitervls o equl width. Lower sum: Upper sum: Midpoit Rule: s S M i i i (e) Use grphig utilit to crete tle o vlues o s, S, d M or,, 0, 00, d 00. () Epli wh s icreses d S decreses or icresig vlues o, s show i the tle i prt (e). 70. HOW DO YOU SEE IT? The uctio show i the grph elow is icresig o the itervl,. The itervl will e divided ito suitervls. True or Flse? I Eercises 7 d 7, determie whether the sttemet is true or lse. I it is lse, epli wh or give emple tht shows it is lse. 7. The sum o the irst positive itegers is. 7. I is cotiuous d oegtive o,, the the limits s o its lower sum s d upper sum S oth eist d re equl. 7. Writig Use the igure to write short prgrph epliig wh the ormul... is vlid or ll positive itegers. Figure or 7 Figure or 7 7. Grphicl Resoig Cosider -sided regulr polgo iscried i circle o rdius r. Joi the vertices o the polgo to the ceter o the circle, ormig cogruet trigles (see igure). () Determie the cetrl gle i terms o. () Show tht the re o ech trigle is r si. (c) Let A e the sum o the res o the trigles. Fid lim A. 7. Buildig Blocks A child plces cuic uildig locks i row to orm the se o trigulr desig (see igure). Ech successive row cotis two ewer locks th the precedig row. Fid ormul or the umer o locks used i the desig. (Hit: The umer o uildig locks i the desig depeds o whether is odd or eve.) θ is eve. 7. Proo Prove ech ormul mthemticl iductio. (You m eed to review the method o proo iductio rom preclculus tet.) () i () i () Wht re the let edpoits o the irst d lst suitervls? () Wht re the right edpoits o the irst two suitervls? (c) Whe usig the right edpoits, do the rectgles lie ove or elow the grph o the uctio? (d) Wht c ou coclude out the heights o the rectgles whe the uctio is costt o the give itervl? PUTNAM EXAM CHALLENGE 77. A drt, throw t rdom, hits squre trget. Assumig tht two prts o the trget o equl re re equll likel to e hit, id the proilit tht the poit hit is erer to the ceter th to edge. Write our swer i the orm cd, where,, c, d d re itegers. This prolem ws composed the Committee o the Putm Prize Competitio. The Mthemticl Associtio o Americ. All rights reserved. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

13 A Aswers to Odd-Numered Eercises Sectio. (pge ) c i 9.. i j j ,00.. 0: S. 0: S.9 00: S.0 00: S : S : S ,000: S.000 0,000: S < Are o regio < < Are o regio < < Are o regio <.. The re o the shded regio lls etwee. squre uits d. squre uits.. A S 0.7. A S 0.7 A s 0. A s lim 9. lim. lim. () () 0 9. A. A 0 0. A. A 7. A 9. A (c) s i (d) S i i (e) You c use the lie ouded d. The sum o the res o the iscried rectgles i the igure elow is the lower sum. The sum o the res o the circumscried rectgles i the igure elow is the upper sum. s S () lim i ; lim i. A 7. A 7 The rectgles i the irst grph do ot coti ll o the re o the regio, d the rectgles i the secod grph cover more th the re o the regio. The ect vlue o the re lies etwee these two sums. Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.

14 Aswers to Odd-Numered Eercises A9 9. () () (c) s S (d) Proo (e) M s S M () Becuse is icresig uctio, s is lws icresig d S is lws decresig. 7. True 7. Suppose there re rows d colums. The strs o the let totl..., s do the strs o the right. There re strs i totl. So, d For odd, locks; For eve, locks 77. Putm Prolem B, 99 π Copright 0 Cegge Lerig. All Rights Reserved. M ot e copied, sced, or duplicted, i whole or i prt. Due to electroic rights, some third prt cotet m e suppressed rom the ebook d/or echpter(s). Editoril review hs deemed tht suppressed cotet does ot mterill ect the overll lerig eperiece. Cegge Lerig reserves the right to remove dditiol cotet t time i susequet rights restrictios require it.