2018 MAΘ National Convention Mu Individual Solutions ( ) ( ) + + +

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Dominick Edgar Warren
5 years ago
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1 8 MΘ Natioal ovetio Mu Idividual Solutios b a f + f + + f ) (... ) ( l ( ) l ( ) l ( 7) l ( ) ) l ( 8) ) a( ) cos + si ( ) a' ( ) cos ( ) a" ( ) si ( ) ) For a fuctio to be differetiable at c, it must be defied at c ad f ( ) f ( ) + c c ad f ' f '. These values must also be fiite. This does ot occur at for l, but does for + c c the others. Therefore, the aswer is I, III, ad IV ol. ) For the first it: The secod it: For the third it, this graph has a oblique asmptote that is a lie with positive slope, so: Therefore, two of the its eist ad are fiite. 5) fter verifig that the coditios for the MVT for derivatives are met: f ( b) f ( a) f '( c) c c c b a 6) fter verifig that the coditios for the MVT for itegrals are met: ( ) ( ) b a f c f d ( ) c c c 8c c, c 9 The largest value of c that satisfies the MVT for itegrals is sice it must be o the ope iterval (-,) 7) Solvig for ields:. However, the absolute value smbols makes the itegratio tougher. To itegrate, simplif the itegral as d d

2 8 MΘ Natioal ovetio Mu Idividual Solutios Notice here that whe, so d d l ( l ( 6) + ) l ( 6) 8) + e cos ( ) e at the poit,. d d d d + + e cos ( ) e si ( ) + e cos ( ) e si ( ) d d d d Now, b pluggig i the values of ad : + d e si esi d e cos + ecos e ) g ( ) 7 g '( ) g '( ) ecause the slope of the taget lie is, the slope of the perpedicular lie is ) This revolutio forms a Ellipsoid ) ( ) Note for this problem l hopital s rule is aother method of solvig. ) hoice is ol true if this graph is odd with respect to the origi, ad the problem does ot specif if it is. f " for all. ) hoice is alwas true because ' f ad hoice is ot true. The iverse is a fuctio because the origial graph passes the horizotal lie test. hoice is ot applicable to this situatio because there is o value of where the secod derivative is. + e d e d e e. ) Usig differetials, Newto s method is used to approimate zeros for various fuctios: f 5 f ( ) + f '( )( ) ' f Now, usig the iitial guess: ( cos ( si )) ( ) rea d d 5) 6) si cos ta si cos d d. This itegral ca be computed usig u-substitutio with u : si ( ) u u du 6

3 8 MΘ Natioal ovetio Mu Idividual Solutios 7) First, work with the cube ad fid the rate of chage of the mai diagoal. Let the legth of the side of the cube be s. dv ds ds ds V s s 5 5 dt dt dt dt 5 Usig the Pthagorea theorem, the mai diagoal of a cube is s so d ds dt dt However, is the diameter of the circumscribed sphere, so the rate of chage of the radius is 8) To fid the iterval of covergece, use the ratio test: ( ) + + ( ) Now create a preiar iterval to evetuall isolate : ( ) fter testig the edpoits for divergece (or covergece) b pluggig them ito the series, the iterval remais ope eded. 9) I polar coordiates, it is difficult to represet straight lies with equatios. To mediate this, taget lies are writte usig cartesia coordiates. This requires coversios. r si r cos d dr si d d d d d dr cos d d r + r cos r si at. pluggig i all the respective values: 6 ( ) cos ( ) r' ( ) 6si ( ) + d d 7 ) Use the MVT for derivatives i such a wa that f '( ) L' ( ) f ( a) f b b a The poit of tagec is therefore ( 7, ). Use slope itercept form with iformatio acquired to fid the equatio of L( ) ad compute its -itercept itercept:,

4 8 MΘ Natioal ovetio Mu Idividual Solutios ) Use trig idetities. Oce it is simplified, use u-substitutio. 5 6 sec ( ) 6 sec ( ) d ( ) 6sec( ) sec( ) ta ( ) d 6sec( ) ta ( ) sec( ) d u sec( ) 6 sec( ) ta ( ) sec( ) du d sec( ) ta ( ) 6 sec sec d d u du Itegrate ormall ad udo the substitutio. Never forget the costat of itegratio o idefiite itegrals. u + sec + ) Whe chagig the order of itegratio, the goal is to rewrite the bouds so that o ot chage the itegral itself, as that would lead to a differet aswer, so is out. s writte, the bouds are ad 6 The area of itegratio is show. Istead of represetig the graphs i terms of, put them i terms of to get the horizotal bouds, ad itegrate verticall from to 8. Rewrite the bouds as 8 ad. Fiall, to keep everthig cosistet, itegrate with respect to first, ad et. f, is easier to itegrate. Fial aswer: d d d d ) Euler s method uses a differetial equatio to approimate values if the atiderivative of a fuctio is tough to compute, but ca be applied to eas fuctios as well.

5 8 MΘ Natioal ovetio Mu Idividual Solutios ) Let be the amout of rope that is pulled up. t the bottom of the hole,, ad at the top of the hole 8. s the rope is pulled, there is 8 feet of rope remaiig. Now, usig Hooke s Law: 8 W F d Force i this case is the weight of the sstem (rope ad briefcase) as the rope is pulled. Sice the rope has weight, the weight of the sstem chages, ad this gives us a itegrable fuctio with respect to. F ( ) ( 8 ) W ( 8 ) + 5 d 7 d 8,8 foot-pouds 5) hoice is alwas true. If the product of + a is writte as a sum of logs, use the it compariso test to prove that it coverges. This works because a approaches as approaches ifiit. 6) ( + a) l ( + a) Make the compariso: hoice is ol true if, which is ot stated. a b ( + ) ( + a ) l l a hoice is ot true. If the ratio test ields a value of, the outcome is either diverget, coverget, or coditioall coverget, but we do ot kow which uless aother test is used. hoice is false because the harmoic series is diverget but the alteratig harmoic series is coverget. f ( ) ( + ) f ( + ) ( + ) + 7) This problem is t tough per-se, but it is trick to the ee. Make sure ou correctl represet the factorials ad the epoet o the e term. ( f 8 ) ( 8) -[ E 8 ( 8) 9! 8 8 e ] 8) There are a couple differet was to prove that the area of polgos with a give circumradius (c) approaches the area of a circle with radius r. The method of ehaustio states that accout that 6 9) sih ( ) si 6 6. c : ( ) e e e e e e e e sih Use the quadratic formula to solve for e : e l sih a csi c Now fid the derivative of the atural log: d d + otiued o the et page Usig this iformatio, use itegratio b parts to compute the itegral: Takig ito

6 8 MΘ Natioal ovetio Mu Idividual Solutios sih ( ) sih ( ) d d. The via u-substitutio: + u + sih ( ) sih ( ) ( u ) du sih ( ) d d u + u du d simplifig the itegral: ( ) sih sih d u du 6 u ( ) sih u u 6 sih ( ) u 6 u 6 ( ) u( u ) sih 6 The, b resubstitutig u + : sih Fiall: ( ) + ( + ) sih d 9 + ( ) sih sih ( ) d + 9 ) The imagiar umber i acts as a costat, ad the derivative is take with respect to : i! i + i f ( ) l l f '( ) f '( ) + + i ( + ) + swer Ke:! ) ) ) ) 5) 6) 7) 8) 9) ) ) ) ) ) 5) 6) 7) 8) 9) ) ) ) ) ) 5) 6) 7) 8) 9) )

Solutions to quizzes Math Spring 2007

Solutions to quizzes Math Spring 2007 to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x