zojt ( lying (12-15 N 0 long height the ground required to of IZM?

Transcription

1 Quiz 3 Solutions

2 A How 4) I chain lying on ground is 15 m mass is 100 kg one end of chain to a height much work is long required to of IZM? and its raise IxFyfq@fnmwefhtpometeriYfmkt98mk20j9trymWwkfd220jIlrx ) dx (12 15 N 0 zojt ( 144 J

3 A If Its 2 bottom swimming pool is 20ft wide and 40ft long is an inclined plane shallow end a having depth of 3ft and deep end 9ft is pool full of water estimate hydrostatic fore on shallow end deep end one of sides and bottom of pool yzoftd 40 weight of water : ftpt#jfg azse#t (a) Shallow end :# : aftereffect F xdx 5625 lbs

4 FX# (b) Deep end i EEmmhE#EdII Note that length of sheet does not affect hydrostatic force F x dx lbs * msnie#ejfge*xo * 3 bṭo ) 9 +ge : tii T amitotic lbs

5 lb # (d) 40ft Boggan 10 *E #tx Area * a IE#FekMBtI 4309 DX pressure F 962 Fx 20 DX 625

6 0052 If 3 A spring has a natural length of 20cm a 25N force is required to keep it stretched to a length of 30cm how much work Is required to stretch it from 20cm to 25cm? ran tuuuuu # klllllle <10 1) K 250 dx F f254 /o # N

7 Find 4 antroid of region and yx parabola yx2e bounded by line tw#xx2dxix3glotztztuxtafxlxx4dxeeiieiixe/eetkefie : 31 's 3k ) Centnid : It ty )

8 For 5 each condition below give sequence { an } with desired that satisfies your sequence an example of a property and justify given property (a) Converges to 0 { n't nh;a±o (b) Converges but not to 0 { if fit 11 (c) Diverges banded { th } nhlngtn does not exist it E an ( 1 ) I 1 for all n (d) Diverges unbounded nhjng ante or { than } nhjmaazn nhjg 2n fly an ft knit a So { an } diverges and nlghg an is note or a

9 Justify 0 bn Both 6 Classify each statement as never true alwayssometimes true your claim true or (a) The sequence { an } converges to zero and Ian converges : an th nbjmo an ' 0 but [ nt diverges Sonnette bn lbl series n't nhjgbn sequence { an } converges to 1 Ean converges 0 and [ converges and Never true : If bin an L series [ an n d diverges by divergence test (c) sequence { an + b } satisfies fn*m* Anthon nhjng an + nhgzbn Sometimes true : If { an } and { bn } converge statement is true given using Limit laws Take an and { bn } but antbn HY C IT so fig diverges an + So of all n { an } nhjgbn doesn't exist nhjgantbn 0

10 (d) The series [ an and Ebn Ean bn diverges converge and Never true : According to Unit laws Zan and Zbn converge Ean bn and converges if n Zan bn Ian 2 bn

11 Find a 7 value of c such that fence 10 1 n The series Zen n ' is a geometric series with first term ec and common ratio ec So ni d [ em 11 C ' e c e ec # c klifl c

12 8 Determine if following series converge or diverge If a series Ia ) IF Converges determine what It converges to I n + F 2 anti n :EI) 2 An + Bn + B +1 ±ln 4 # ) sn ' TEN nhjmasn nhjmx 2 2 series IE II converges and sum is 2 lb ) lnln ) Since nhjnalnln ) series EE lnln ) diverges by divergence test

13 Since (c) no A I EE 'ite±l This series is geometric common ratio It series will converge with first term and and 1 1<1 tes ±

14 Thus 9 Determine if following series are absolutely convergent in conditionally convergent Etn# or divergent ( 2 ) him Tz doesn't exist so by n * test of divergence IT f diverges a r: t an (b) [ An and note lankan > 0 Consider bn nt > 0 fig E;a thus by t converges LCT I > o 'F sinaie converges IEot n is absolutely A comment tnh;yty %l fan ftp?ihe;tintit' series nhjng n t is absolutely convergent 0 < 1

15 ln bnlntz hln test this so (d) n 1 First consider FE/tn diverges by p ( p± i) series If # is not absolutely convergent Note that E th for all n and nhjmy 0 so by alternating series test IEt'f converges in Thus %ftf lnln# is conditionally convergent Note that lnl # 1 > 0 f n > I lnlnntt ) ) ) a fhlntzthf tn# WI by # + htttlhlbth + µ#hht#4hh)+ t + lhlntil (2) + lnlnh ) + lnlnt 2) ) fnggsntnhjmjlnklthlntl ) + hlntzl So ME hln 'F ) diverges a

16 test HITE an a 2 an + lf3 ) an Consider Links /aa / hmm 1h' f l him 5 hey 4h +3 I 4 Since L iet > I and an > 0 for all n series 12 an diverges Note that End p I # ' I ( 11 Furr p diverges by If is alternating n 't E ht and fig alternating series test IF ' 0 so by converges Hong 4 converges conditionally

17 Since ' HIE fi# +7 Innt ft+h n Consider series EYE ( 1 1 EL ( ) ) is i on geometric and common ratio 1 I 1 nu absolutely convergent # with first term this series is 2 ( t ) is geometric common ratio with first term and 114 this Since above two series KII + 7 series is absolutely convergent converge we have that TIE #tiet which is sm of two absolutely convergent series and hence is absolutely convergent

18 How 0 10 many terms should be added to estimate to within 001 of its true value? Note that is alternating and tn his Henig by alternating we can bound error En by series remainder estimate IENI ± So we can solve inequality 001 a not < e n's Z nt ) 7 qqqq n We need at least 9999 terms to estimate ±Er within 001

19 Lvtuutttttnttttt Tim# (a) vn IF [ positive!nxm that go a 4) 0 < (b) lrnl 01 dxfytx decreasing and continuous leaflet F* ) arctanlx E ) no#ggetneemesyite+hjgarotanlh Smallest n such that arctanln ) E 01 lnymniegaf ) n9 does not satisfy necessary inequality but n 10 does After 10 seconds will be pie within 01 ft of total distance traveled f? # dx a F dx a

20 So if s is true distance pie will travel we know fi dx + foes 's + [ * dx s ± Taking average of bounds that pie travelled we estimate Sa ft