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# Calculus I MTH 251

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This 12 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 251 at Portland Community College taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/224639/mth-251-portland-community-college in Math at Portland Community College.

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Date Created: 10/19/15

Mr Simonds MTH 251 more volume gar k 130 Volume of Cylindrical Shell g y Vy pyk T hy Pym 1 kmquot Vy 71M T hyk 71M T hm npltykgt1pltm2 hm pyk My pyk My 2 pyk2pykhykAy 2 PyhyAy Please nofe The for This parficular drawing phat yk 772 Page 1 of 2 Mr Simonds MTH 251 more volume The parabolic boundary of The region below was gener39aTed by The formula x 6 y 22 Find The volume of The solid ThaT r esuITs from r39oTaTion ThaT region abouT The line y 2 In facT find The volume Twice Find The volume using shells and Then find The volume again using washer39s killii i l39 M 43th AV 39L 9 13 L M eri LYJ 1qu bl V39W39L j J9 QSLK 2 f I xLUquotl 9 5 j21 639 M ii 94 Xi1 a ill dcx agl gt4 V 115 611quot mquotg 39 W S L LIJrWIlkvmju 3 V Page 2 of 2 My SwmundsMTH 2517nmauun Nnminn Nnminn anzlinn Suppusemm 312 Fmdvhevcnueuf 3 llurt39J quot1 A N fh JW M alumnae t 43 1 d 2 Suppusemm 312 Fmdvhe va ue uf x 3 ff 1 if a Suppusemm gxx2 5x77 mem va ue uf g399 Prehmmary wurk Fmdafumum fur g x A Jquot 5 J 1 34Hf 3 HM 4 Suppuse Mm 1 mem va ue uf k2 7 z The furmum fur k t has mini to tin with this 9mm 4 i 39J L a H yr 1 0 P39212112 Mr Simonds MTH 251 notation t11 d F39d h l k6 7 In T evaueof th 5 SupposeThaT ktt 1 J Il l U K A quot J qJr to Q JKml o m I f a 6 Buu was given The formula 3x2 7 7x 1 and asked To find Buu wroTe The work shown below Buu losT half The poinTs for The problem WhaT39s The problem wiTh Buu39s answer A 5dqf 7137127771 1 3 4 Jquot om f H ltgt NW 11 4 rick i d ltHgt H m dw dillJ 4460 52 7 f93 f 972 g49 g3945and hrfgJE Find h 16 and gmm L1H 7i my591402 i HM C39glfulf l39 l l J 4Lljmil I k I J l 9mg m gammy Lllul l39l lmlj lmlbf 1 i L All 7IjVlljllll39 VL 5 J C 4j39lquot7l 3 1 S 7 Page20f2 39 Auga g Volc343 Mu gm pl2 l 7 f e wanna7 74 45 S 3 s 1 A gt I Mr Simonds MTH 251 Week 1 LecTure NoTes In a simplified model of The siTuaTion The heighT above ground fT of a ball I seconds afTer iT is dropped from a heighT of 1024 feeT is given by The formula pt 1024 7161 2 The heighT of The ball aT several Times is shown in Table 1 t 7 t The average velociTy of The ball fTs beTween Times t1 and t2 is defined To be 2 T 1 For example The average velociTy for The enTire 8 seconds iT Takes for The ball To reach The ground is p8e p0 7 0134102413 7 7102413 7 7128 s 8s70s 8s LeT39s go ahead and fill The average velociTies over The Time inTervals indicaTed in Table 2 Table 1 Heig wt of ball once dropped Table 2 Average velocities of the ball I S Flt H rptz s vm fTs 0 1024 29 88944 23993 L l 299 8809584 2993 q 7 Li 2999 880095984 7 3 880 29993 9 g 7ng 3001 879903984 33001 q C 0 L 301 8790384 31301 7 q 31 87024 8 0 3 31 q We define The insTanTaneous velociTy of The ball 3 seconds inTo The flighT To be The limiT of The average velociTy over The inTerval t3 or 3t as t approaches 3 Symbolically we wriTe This parTicular limiT as shown below LeT39s go ahead and indicaTe our besT guess for The value of This limiT including uniT v3lim pt l3 tgt3 Page 1 of 8 Mr Simonds MTH 251 Week 1 LecTure NoTes LeT39s go ahead and make Tables from which we can guess The value of some generic limiTs LeT39s fill Table 3 wiTh values from which we can sin57rt infer The value of 11m HZ s1n37rt our conclusion using a compleTe maThemaTical LeT39s sTaTe sTaTernenT 92 S39 a 3 S39 I 13 f39 v2 Table 3 y LeT39s fill Table 4 wiTh values from which we can sinx infer The value of lim LeT39s sTaTe our Xgt07 conclusion compleTe maThemaTical sTaTernenT i39m xao using a sui 39 Table 4 l M S mixl gtlt LeT39s fill Table 5 shows values of The funcTion fxx2x1 The value of lim xgt71 A lab ll quot b n WhaT can we deduce abouT hLlVJ39I39q LDJ La J uro MA at 39I L L Ll Table 4 yx 2mm Page 2 of 8 Mr Simonds MTH 251 Week 1 LecTure NoTes Figure 1 shows The funcTion y gt LeT39s sTaTe each of The following values g71 3 lim gt 3 312ng 3 Figure l gltsgt 5 glt2gt l Eirglglt l Jiritglttgt l 13570 l gltrgt l lim gt J raf e For each of The following limiTs leT39s sTaTe wheTher or noT The limiT exisTs and eiTher sTaTe The value of The limiT or sTaTe The reason ThaT The limiT does noT exisT tr 5 m 16 limgt 1imgt lim gtand 1imgt H3 H71 ram ram 1 4y I 5394 439 9 L0 va N1 kY a A Ill36 5 1 Jlr39V Jul r H 39 L in 4 quotI kenjla 391 J 515 I Dt UT 90quot I L Q aj 5hhh1u L SabK r 1in 4 7 new 39 L H L39a Lyn3 41111 Huff AMEM710 quot q a Vk UI 1 1395 LLYR 392 LA I L Page 3 of 8 quot lo I lII J I 3 I n xil39lJ kaa In Ark 39m 3 4 L443 Mr Simonds MTH 251 Week 1 LecTure NoTes Every funcTion formula has an associaTed formula called The difference guoTienT The difference quoTienT is a very imporTanT formula in The field of dI39ffer39enra cacuLs39 h 7 The difference quoTienT for The funcTion y x is The formula We adapT The difference formula quoTienT To The name and independenT variable of The given funcTion For example we39d say ThaT The difference quoTienT of The funcTion y 379 is The g9 h 7 g9 h formula LeT39s find and simplify The difference quoTienT formula for The funcTion pt1024716t2 Please pay aTTenTion To The formaT and sTeps used in This example I expecT you To use The same formaT and for you To show equivalenT sTeps when working problems like These 39 39 wk lol lL Li52 Ilu clquot ne paw 7C i f f l l quot1 i lu Gan mum flownu L L ogt1 Jlt12 RL1l v1 u E M 3HkI4A 7RL L 324l 4H k la 32 LA X Jl LL L750 Page 4 of 8 Mr Simonds MTH 251 Week 1 LecTure NoTes Table 6 shows The average velociTies we cacuaTed on page 1 for The heighT funcTion pt 1024 7161 2 On page 4 we deTerrnined ThaT The difference quoTienT for This funcTion is t h 7 t W 42461 LeT39s go ahead and evaluaTe This difference quoTienT when t 3 and 11 has The values indicaTed in Table 7 Table 6 Average velociTies of The ball Table 7 Difference quoTienT values t 1 tun s vcve fTs tandh 23 1 R 293 7944 t3 21701 7 if 2993 79584 t3 117001 g5 1 29993 795984 t 3 h 7 0001 if 7 6 33001 796016 t 3 h 0001 If L o 1 3301 79616 t3 h001 quot 71 1 331 7976 t3 h01 quotH C LeT39s pltlipl3 73 On page 1 we defined The insTanTaneous velociTy aT t3 To be v31i redefine This velociTy in Terms of The difference quoTienT gcru LL it Iquotquot j lw J VJ w W OM Ce 1 0 0 0 k L J 913 2 L 4 f lt 3 l a o la 6 W P Page5of8 P lt 3 04 M0 H l H 30 Mr Simonds MTH 251 Week 1 Lec rure NoTes Le r39s find The difference quoTienT for The func riong x x73 I LL J Hum Pawn 75w jm xi M amp 3lt j VLjx xk 3 X 3 J A xl k x4 i Lit 3 xHs xS X 3 XH L L 7328 33 A 3L Qx k 3x 3 29 ltL 3 Kx 31 3 we lx 3X 77 l Ls o Page 6 of 8 3 Mr Simonds MTH 251 Week 1 Lecture Notes Figure 2 shows the function yqx7x2 5xi3 along with the secent line connecting the fixed point 2q2 to the variable point 2 hq2 h Lets stcite each of the expressions requested below the figure in terms of q and h X 3 x IVl This is the point 2q2 This is the point 2 hq2 h 1 395 MI H L On The run between the 2 points is x r x L The rise between the 2 points is y2 r y Thesiopeofthesecentiineis e L Page 7 ME Mr Simonds MTH 251 Week 1 LecTure NoTes SomeThing39s amok wiTh each of The following limiT Tables STaTe The amokness for each Table Please noTe ThaT some of The errors seem like wriTing errors buT are acTually maThemaTical errors as in misidenTificaTion of variables or lack of definiTion of variables 5x15 I 9 T whaTs amok 5x15 Table 8 was creaTed To find lim 3 Table 8 lim x 3 x 3 Two Things are amok wiTh Table 8 DTL uklu o l x Tvu 39I rl Juka I 3l sin2x whaT39s amok Sin2x Table 9 y Table 9 was creaTed To find 111 lL willquot 5 LltHUalv39b l39 l luau CandiAh hfm WT al Milt3939 Ju39 s rem mij 3ll l n 74 2 4 t Table 10 was creaTed To find lim whaT39s amok tarz39 t 7 2 v tautquot on I 12 7 Table 10 y 1 03 Table 11 was creaTed To find li1g1lt4t 2 whaT39s amok j J 439J Ifo quot39lj I Hal 4 31 lump 3 39lA39fl l 43 l L urruLJ39 quot dquot Cvsl lquot I lt 4 l U j 5 4 Table 12 was creaTed To find lim whaT39s amok xgt039 17 x Table 12 fxjx In Lc 08ng 391 x x 1L y A o 79 18 78 3 l M db 799 198 7999 1998 Page 8 of 8

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