day33 - D \s\~ - or Di:sc... \ j == ' 1\,~X- ~ €/LL ' T\...

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Unformatted text preview: D \s\~ - or Di:sc... \ j == ' 1\,~X- ~ €/LL ' T\ \"J. iIJ '+-tu... ~ ~cLe... i V= +k ' i-k " i ~e e.OY\e S 3 \j = <: ~ ~rV'\ ~ TX:J.4 -r~ (;:;?S -1-- 2-")4 3 o / DDF 3 II To -h~d v~~ \j = -- 'i A.e.v~ .· 1\~ [ Ro<! - ~;J +an'A ~ d lsc.. yne{-ho& ---t--- ~ I I I (2ecu:t: n~'nt -to \ e..ft: + 0 p -t-n bo+torY\ Lt \) = 'il~ [~S - L;:js - IOU + '1-)Jc:l¥­ o V= lIr a ~~o(S) l\s ~::.-\ I ( '1-~'3 , ':J. [x ~- (-/1J ~ · x- [r-2. c 3 3 : X3)r I \ (-r 3 I<. 5 _r "3) C rj *. 3) = 1 1' r3 m ost e J\rors: C Clrelc.ss / ) F="or'j e ti1 A~ C. t- .J.) \\ R e-su..b" J +o\~ei:::b~j .-1::0 pur X bo..cI< iY'\to t -he. o .r.sw e..-r u~~ +he. Q I'\S£....)et'" i n "u.."} 3 .) V Is-h-'(G~~5 e.. \ ,o's ( l i We : - (c~+b) == - a.-b b(A-t-wrl ' fi~ - o..+-b) Lon QOY'Y1r 1e;h''''5 -the. c..h.o..lr. i£.A.le. (I.L= 5 if) 3)<. 2 ­ du =Cos(3x2.1 · ~ .) ~d:h"j P roble'rY's .' / .0-) S-ta. 1t I n l .(3x):;,ee.'i(3XJdX ej cd) d.) S X X~l'oX+~5 ~)( 3X~~'f ctx. j ~ / _ ~~ I..A./'f­ F \no. -the. QVeK~e... +CXJ;: X sin-x, ~ 3~ ~ sin 3(i)ctv S oLve: a YeA' vo..lu..e. [0) ~rrJ +Or ~~ _ _ XQ.()c(u't)(x2+~)'/2. I cJ..,¥ ~3 J L/~ r -l 'y \o\ +(~) : },"(x) = e .)(t--SinX , . t'(o)=Y; +(0)==-3 hs / a.) - ;iff I ~ t Un 5 (3 X) + * -tan 3(3 X) i - ~ b) ! i(, + rx)Sf.J_ !1 ( t-tiX)3/;J.+ Q. 5 3 c.j - 3 C.os d) e.') . -'\ 0 1') (1) +- Cos 3 ( i) -f- c.. ~ -L-n I'~ + (xt-3)1 + ~ 4rctcw-(X: 3) +c.. - 3((- Xz)'(:2-+- (( _ X2-)3/:z +- -L f fF ~ Probl~yY\S :±t ~ ( FrO'fY' ~o..st ?J u.. i L -ZC5 / -te s f:s ) /. S h 0 uJ - tho..t + h e. m e.a.n V0.. \ v...e ----rh eore. J YY'\ +0 \ Lni-9r-o.-l s Q pflieS -to f (x):::: x + l over [0,;;2]. Q) G Ive - the cOY\dit-lOY\(s) thaJ-- Mu-st b e m et . 'bJ r ,'y\ol +he.. Qve-rCNje. voLu..e. e ) F \'",ol ~ d) 5 He-tc.h - the. £j r~h o f -the. ~n.c-bon ~ s how !:he. re.c.to..Y'\<jle C jivin'j The. aVeA-o-5 e volu...c J ~ '( e . e ) E,xf lo..in ' the rneo..Y\ \ ('\~ o f t fae m ean a nd !:he vo.lu-e Va !c.Ae. The.oretYvl +0'1 Tn-l:-e.<jro..ls. ~.) So \ v e .' ~::: dJi- f a)') 2.(d':j 11 C OS w:­ 3 '-lx:5 3~ [voJu..CAte .' b~ · lnd.u.chO/'\ : ~n > 3 r \ I n=: ::{, 3) · - · p roie.cti\e. 'I S s hot S+r-o.\~ht u .pwa.d 0 *0... 4.) ? rove.. 5~ A '\o.:\:.e at l q.iPrn/se.c - h--om t'he.~ d t.(, S m eJex-s 0-') -+- b') c') h i ~h 0 ..( t ) 0+ 0 - bu....i\dinq =- q, S? rt)/sec. ~ W hcd--"tS ~e. ~y-e.ode.st 4--.e\~-l: -.LdJ A t .-J r: w l\oi veJ.ow.~ w ; t i t- h it -t:'he. !jro(Al"d? 30 points ~---+1t------- Math 111 P ledged_--.tdf-­ ..... Quiz 11 You may use the reference sheet for this quiz but you may not use any other notes or text or person or calculator. Your pledge indicates that you took this quiz in accordance with the understood rules. ~1. S how that the Mean Value Theorem 'for Integrals applies to f (x) ( / ( 1 0 points) = x 3 + l over [0,2]. (a) Give the condition(s) that must be met. ~ (~)::: X'3+-\ \s c..OY'\~Y'\.\.A.ov..S ( 01 'J.l l rol~Y'lol'V\ 'I~ o 'JV\ (b) Find the Average value, A(f). ?(~ ~ 1 ((~ ;). -0 f j.'+\) o [ 4 =~ 4)( 4+~\=: l­ 0 =~(t..)=0 (c) Find c. ?\(~l:: f (e.,') =- e-3 + \ ::: 3 Ow'f'd-- ~ e ,"3.: d.) \c..::.~ 1 0 ~ ~ ~ 2. (d) Sketch the graph o f the functi n; show the rectangle giving the average value ~ 4 and the value for c. G )(X"3+\)dv.t = -tt')< +1<\ o 0 r :L ='~ /dTeCLJ u... ------~-+~~~--------~x -\ B·~=-~ A-r"-e.a.. o~ c... CA)..,Y-V t""e..do..n<3\ e... (e) Explain the meaning o f the Mean Value Theorem for Integrals . . A"u~ \J~ ~ ~ ,e w id.+h C.jJ..:C V e. a t \V \ d e. d. "? <::J \0 o....-s e.. I 5 Ct."(" e o. \J.. Y"\ d e K' v J i olth. Ptv-e.o... o k C ls 0.... e...o- r e- c.;t~ '('\.cl u- -e-h c=... lA.. b Q.SG C-LL rv C- • -<..0 '-t::tt ~ 5~ ~ 3. Evaluate: (4 points each) & a) flnvfx dx l A.= X 1 ~~=~\:.O k S'i cl.u. = . .l....l- cD+ ~ ~~ ~ dJ.A. = ..LdJt 'I. .J.. n ;l~ e., -~ v . .:;;'' \o ~ b) f ex x dx e'x - 6e + 9 5 ~~ SCe'e.- O)"J.. ')( 1 .....1 l ' U..J't _ -~ -n . lA- ~ ~ '/?-- hf - 0 .... L -\ - LL +- . ....------\---­ C. -=- \ - --e!-3 ~ (l ' -­ ~-----~2 c) f o x ~1- 4x4 LA. "" ? . '). dx olu:=- t ~ u..'f... 4 eL.A.:::: "1. d.J...t.- J -( I 1..\ ~ -r d,..u.... ::. J... k c..s \ n I - U. ::t '-\ ..L ri"·n:..s \ Y\ (~ - ~ fu c. OS ;. I r (u.J 0 Y\ *(~- ~lOl =\~ \ lOJ n points) Prove the following by induction: 3 ~ 3n, n =2, 3, . . . Show all step s clearly . q ' i t.o i s -h-u~ :f) A ss lA..h\e -:3 ~ >,.. 3 ~ f-o, ~ >- 2. ; <E> h -u. ~ u.) 3'3~? ,;).3R --?> 3'R+1 ~ q R (m~ . ~ '3) 'c) q R ~, 3 (l'-< +\\ - 7 q R ~ 3\Q. +3 \ S --\-y\Ae., -Fu'{" '"* > ';l sll"\ee. ~-t< ~ 3 ( s LA.'o 3 ~ Go+!" s ~d.. es) Q.) Ihex-e.fo"e) 3 ~ -t \ > q ~ >,... 3(~ + \) ~ 0... " - b ..... cA) B 'j ~",si-nVe ~ro~) 3 ~+\ ~ 3( ~-t-I) A-nd 3 ~3 n f;J~ Y\ =~) 3.; , I) t=o, \ \=1. 3 ? 3 t~) --+ t. l ~ (~gh. ~ 9.8m I A projectile is shot straight upwa rd a t a rate o f 19.6 Use aCt) sec 2 • m/~c from the t op o f a buildi ng 24.5 meters Answer the following: (8 points) a) When will the projectile reach the greatest height? V ( - t) = 0 - Cf. rg t + f l, fp q ,gt = , =- 0 0.. Ct") =- 9.g ""/s eG . . =- f, <? -t -I- J 9. (p V(i:) .5 ( t ) :: - 1.(. ?t 19.&' -t = ~ =l~ seC!] '1,8' b) W hat is t he greatest height? (~)"J.-\- \ Cf. [P l;J.) + ;> ~ , t> :=:-19.lL>+ 3C1.:2..+~Y~S S (:l.') -= - 4,9 -:~l 1 Y ¥\ c) When will the projectile reach the ground? s et) = 0 -:. - L\ . 9t'-t \9 , [Pi: + ~4. s o = -4.Q Lt~ L-\t - sJ t t-s)(c-+ I ) = 0 t 5 se.S) -t = - / se.c.. L =:! d) At what velocity will it hit the ground? V (S)=: -~g(5 -= - ;;< ct, +1'1,/P t-/ NIl -Se C. ~~ :,.., '1:(, t f- .J. t.f.~ ...
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