b a 2 ((g(x))2 (f(x)) 2 dx

Transcription

1 Clc II Fll 005 MATH Nme: T3 Istructios: Write swers to problems o seprte pper. You my NOT use clcultors or y electroic devices or otes of y kid. Ech st rred problem is extr credit d ech is worth 5 poits. (These re just more problems, but hrder. They re worth fewer poits so tht you re ot uduly tempted.) Lods of poits re possible o the test, but the highest grde tht I will wrd is 5 poits.. (6 poits) Write the formuls for x d ȳ, where ( x, ȳ) is the cetroid of ple regio lyig betwee two curves y = f(x) d y = g(x) with f(x) < g(x) d x b. x = x(g(x) f(x) dx, ȳ = (g(x) f(x) dx ((g(x)) (f(x)) dx (g(x) f(x) dx. (8 poits) Fid the cetroid of the regio bouded by the grph of the curve y = x d the positive x- d y-xes. Evlute completely d simplify. For extr 4 poits, check your swer by showig explicitly tht the poit lies i the regio. (Tht does ot me to drw grph, lthough tht would t hurt.) The curve itersects the positive xes t the poits (0, ) d (, 0) (too esy!) so x = 0 x( x ) dx 0 ( x ) dx = / /4 /3 = 3 8. Likewise, ȳ = 0 ( x ) dx /3 + /5 /3 = 5. ( 0 x ) dx = Lookig t the grph, it mkes sese tht these re both less th hlf, o? To check the swer for resobleess, we c verify tht the cetroid lies withig the regio, by verifyig it lies below the curve. (This is ot geerlly true, of course! A cetroid of the form give i problem # eed ot lie below the top curve. C you give exmple? I our cse it is true, but why?) It is esy to verify tht /5 < (3/8), which is ll tht my questio mouts to.

2 3. (8 poits) Recll tht we proved tht the cetroid of the semidisk of rdius r (the regio cotied i the semicircle of rdius r) lies 4r uits bove its bse. Fid the cetroid of the 3π segmet of the disk of rdius r t height h, s show (shded) i the figure below. (Set it up.) h r For some reso, the h d r o the digrm were corrupted to i d s i the pritig. Nobody sked bout, so I ll ssume it ws otherwise self-expoltory. I ll use r = d rescle it whe I m doe. I ll let the circle be the uit circle (cetered t (0, 0). It is cler tht x = 0, so we oly eed ȳ. If you drw lie segmets from the ceter of the disk to the poits o the circle t the bse of the disk segmet, you ll see right trigles d you c clculte tht tht x rus from h to h. The regio is bouded by the fuctios g(x) = x d f(x) = h. Tht does it: h h (( x ) h )dx ȳ = h ( x h)dx h Yes, you c use symmetry to cle it up. 4. (5 poits) Express the umber.34 = s rtio of itegers. Slp some 9 s: the swer is + 34/99 = 3/99. Use geometric series or other trick.

3 5. (5 poits) Stte the Mootoic Covergece Theorem. (I clled it the Big M o Cmpus Theorem.) Look it up. 6. (8 poits) Defie the sequece { } s follows. Let = π d for, defie Prove tht the sequece is coverget. + = ( + 5). There is exmple worked i the text tht is similr. Bsiclly, ech suggessive elemet i the sequece is the vrge of the previous oe with 5. Sice the sequece strts t π, we expect the umbers to icrese to 5. We c prove this without fidig formul for the th term (lthough tht is t hrd) by showig the sequece is bouded bove (by 5) d icresig. We use iductio first to show the sequece is bdd. Clerly, = π < 5, so the bse cse is true. Now suppose tht 5. The + = ( + 5) (5 + 5) (by the iductive hypothesis) = 5. So we hve proved tht 5 for ll, i.e., tht { } is bdd bove by 5. Showig { } is icresig is ow esy: + = ( + 5) = (5 ) > (0) = 0, (sice < 5) hece +. The result is proved. 7. (5 poits) Write the formul for the sum of the geometric series, c p c c k, where < c <. k=p 3

4 8. ( poits ech) Determie whether ech of the followig sequeces { } coverges or erges for the give. If it coverges, fid its limit. If it erges to +, sy so. If it erges to, sy so. If it erges i some other wy, sy how. No credit for erges or coverges, but o pelties for icorrect swers. If you c prove your results firly rigorously, sve tht for extr credit problem E, but write it o seprte pge. () = ( + ) + to + = ( ) cov to 0 5 (c) = cov to 0 l( + ) ( ) (d) = rct cov to π/4 + (e) = ( + ) + cov to (f) = ( + )! (g) = cos ( ) (h) = 3 3 cov to 0 (i) = 3 3 to (j) = 3 3 cov to 0 cov to 0 cov to 9. ( poits for ech correct swer, for ech icorrect swer, o pelty for blks) Determie whether ech of the followig sequeces is evetully icresig, evetully resig or ot evetully mootoic. If you c prove your results firly rigorously, sve tht for extr credit problem F o seprte pge. () = e + = e (c) = 3 + (d) = (e) = ( ) 5 (f) = + ic ( ) (g) = rct + (h) = ( + )! ic (i) = cos ( ) ic (j) = (k) = + ev ic l( + ) 3 + (l) = + ev 0. (4 poits ech) Evlute the sum of ech of the followig series (ll re coverget). Be midful of the lower limits i the sums. () =3 = 5 4 = 5/48 + = 5/ (c) (d) = ( ) 3 = 3/4 ( + 3 ) =0 7 OOPS! Does t coverge!. (5 poits) Stte the hypotheses d coclusio of the theorem we cll the Itegrl Test for covergece of series. Look it up. 4

5 . (5 poits) Use the itegrl test to ide the covergece or ergece of the series = (l ) (l l ). (First of ll, the lower limit ws met to be.) This ws doe i clss: the fuctio f(x) = /(x(l x)(l l x)) is resig to 0; let u = l l x... the itegrl erges, so the series erges. 3. (5 poits) Stte the hypotheses d coclusio of the theorem we cll the Limit Compriso Test for covergece of series. Look it up. 4. (5 poits) Use the limit compriso test to ide the covergece or ergece of the series = Compre it with /, which coverges. ( ) (5 poits) Stte the hypotheses d coclusio of the theorem we cll the Altertig Series Test for covergece of series. Look it up. 6. ( poit for ech correct swer, for ech icorrect swer, o pelty for blks) Determie whether ech of the followig series is coverget or erget. If you c justify your results with resoble clrity d brevity, sve tht for extr credit problem J o seprte pge. () (c) (d) (e) (f) (g) (h) = = = = = e + e 3 + ( ) 3 + cov cov 3 cov 3 = ( ) rct + = = ( + )! si ( ) cov cov (i) (j) (k) (l) (m) () (o) (p) = = = = + si 3 + rcsi cov cos ( ) ( + ) + 3 l (l ) = = = cov ( ) + l ( ) = cov cov cov cov 5

6 Extrs Feel free to do these o the bck of the previous pge or elsewhere. Just tell me where to look. A. ( ) Write the pproximtio to the re uder the curve y = x 3 betwee x = 0 d x = by pplyig the trpezoid rule with 5 equl subitervls. B. ( ) Stte d prove the theorem of Pppus. C. ( ) Fid the volume of the solid of revolutio obtied by rottig the regio described i problem # bout () the x-xis; the y-xis; (c) the lie x = 4; (d) the lie y =. D. ( ) Sme s the previous problem, but bout the lie () the lie x = /; the lie y = x. E. ( ) For up to two poits ech, prove five of your swers i problem #8. F. ( ) For up to two poits ech, prove five of your swers i problem #9. G. ( ) Discuss the covergece or ergece of these sequeces. () { }. { } l 5 l 3 { ( (c) + 5 ) } 3 H. ( ) Fid the vlue of the followig repetig cotiued frctio. [3;, 3,, 3,, 3,...] = I. ( ) We write our umbers i bse-0, but wht if we were members of rce with 6 digits o ech hd? Aside from the musicl implictios, we d probbly be usig bse- umber system. Fid the exct frctio, expressed s rtiol umber (i.e., rtio of itegers), tht is represeted by the bse- umber 0.ABBBBBBB, where A d B re is the bse- digits for te d eleve, respectively. 6

7 J. ( ) For up to poit ech, briefly justify your swers i problem #6. K. ( ) Discuss the covergece or ergece of these series. () = = si π + l + L. ( ) Use mthemticl iductio to prove tht k = k= ( + )( + ). 6 M. ( ) Ask questio you wish I hd sked d swer it. Poits my vry. Offer void where prohibited by lw. 7