Math 111! Name:_!!Key!__________________________________________________________________Grp#!________!GrpAct – Week 4B! ! ! ! Polynomial)Functions)1) ) ) )))))))))))))))))))))Sects!4.1!&!4.2!)Prerequisite)Skills)Key)Terms)Learning)Objectives)• factoring!• locating!roots/zeros/x-intercepts!• determining!domain/range!• interpreting !fa cto red !form ! !• polynomial!• degree!• leading!coefficient!• factor!!• multiplicity! !• end!behavior!• local!max/mins!- !Model!with!polynomial!function!- !Find!the!roots!of!polynomials!in!factored!form!- !Sketch!graphs!of!polynomials!in!factored!form!- !Identify!multiplicity!o f!p o lyn o mial!roots!in !f ac to re d !fo r m !!!!!!and!in!graphical!form!- !Determine!end!behavior!from!the!polynomial!formula!- !Construct!formulas!for!polynomials!given!in!graphical!form!!1. On!a!hike!in!the!woods!you!stumble!upon!a!patch!of!beautiful!blackberries.!You!very!much!want!to!pick!some!to!take!home ,!but!h ave !no !con taine r!with !wh ich !to!tran spo rt!the!b erries.!Yo u !do!h ave !a!piec e!of!8.5 !inch !x!11!inch !cardstock,!a!pair!o f !sc is s o rs !a n d !ta pe .!In!a!mo m en t!of!b rillianc e ,!yo u !re aliz e !th a t!if!y ou!cut!a!square!from !each !co rne r!!of!the!cardstock,!it!can!be!fold e d !and!taped!into !an !o p e n !b ox !(s ee !th e !d ia gra m!at!right ).!Wanting!to!take!home!as!much!as!possible,!your!goal!is!to!cut!out!the!square!so!your!box!will!have!the!maximum!possible!volume.!!Let!the!square!cut!from!each!corner!have!side!length!x"(mea sure d!in !inch es). !a. Imagin e !c u tting!squa re s !o f!va ry in g!size!from!each!corner.!Measure!the!side!lengths!and!calculate!the!volume!of!the!box,!V."Record!your!data!in!the!tab le !b elo w.!!!!!!!!!!b. State!the!function!that!will!mode l!th e!vo lum e!o f!the!b ox!in!te rm s!of!V(x)!and !its!do m ain ."!V(x)!=!!!!!x(11− 2x )(8.5− 2x )!! ! ! !!Domain:! [0,!4.25]! ! ! ! ! !" "Length'of'Cut'(inches))1!2!4.25!6!Volume'(cubic'inche s)''58.5)63)0))N/A)x"x"Define)the)Following)Terms)!Polynomial!Function!!!!Degre e!of!a!Polynomial!!!Leading!Coefficient!of!a!Polynomia l!2. The!model!that!we!created!for!V!in!proble m!1!of!the !p re vio u s!p a g e!is !a !p oly n o m ia l!function.!In!orde r!to!talk !precisely!about!polynomial!functions,!we!need!some!vocabulary.!!a. !Often!times!when!w e!en co un ter!p olyn om ia ls,!th e y!a re !in !st an d a rd !fo rm,!anxn+ ... + a1x + a0.!Give!the!formula!for!V!from!problem!1!in!standard!form.!! ! ! V(x)!=!!!x(11− 2x )(8.5− 2x )!!b. What!is!the!degree!of!V?!__3____________!!c. What!is!the!leading!coefficient!of!V?!__4_____________!!3. The!graph!of!V"is!given!(labe l!eac h!ax is,!inc lu d e!u n its ).!!a. On!what!interval(s)!is!V!increasing? !! In!the!dom a in!![0,1.5]!!of!con text,!!or!!(-inf!1.5]!U![ 4.9 ,!in f)!!if!we!are!no t!c on s id er ing !th e !co n te xt !b. On!what!interval(s)!is!V"decreasing?!!!!!!!!! [1.5,!4.25]!!c. Estimate!any!local!maxima!of!V,"and!identify!where!those!occur.!!! local!ma x!o f!ap p ro x.!!68 !!in3!!at!!x"="1.5!!d. Estimate!any!local!minima!of!V,!and !id e n t ify !where!t h o s e !occur.!!!! There!are!no!local!mins!in!the!context!domain.!(If!we!a re !n o t!c o n sid e r in g!t h e!c o n t ex t!t h en!the!domain!there!is!a!L-min!of!approx..-7!at!x!=!4.9)!!4. In!addit io n !to !s ta n d a rd !fo r m,!it!can!be !u s ef u l!to !work!w ith !polynomials!in!factored!form.!The!function!V!that!we!h ave !bee n!using!can!be!written!in!factored!form!as!V (x) = 4x(x − 4.25)(x − 5.5).!!a. What!value(s)!of!the!input!x!will!make!the!output!V"=!0?!" x"="0"or"4.25"(or"5.5)"!b. Explain!how!to!find!the!x-interce p ts !o f!a!p o ly n omial!wh en !g ive n !its !fo rmula!in!fa ct or ed !fo rm.!!c. Does!x!=!5.5!make!sense!in!the!context!of!the!problem!described!on!page!1?!Why!or!why!not?!!! No,!because!it!is!not!possible!to!cut!a!square!of!length!5.5!inches!out!of!all!4!corners!!!d. Your!original!goal!in!this!problem!was!to!find!the!value!of!x!that!gives!the!largest!p ossib le!valu e!for!V.!!What!is!that!value,!and!h ow !large !of!a!vo lum e!ca n!yo u !ma ke !for!you r!bo x?!!! ! ! The!maximum!volume!occurs!at!x=approx.!1.5!in,!and!has!a!volume!of!approx.!68!in3!Length"of"squa re"cut -out"(inches) "Volume"(in3)"5. Lets!compare!the!growth!rate!of!polynomials.!!For!example,!g(x)!=!x6!and!h(x)!=!x4!have!the!same!end!behavior!(for!both!functions!the!graph!rises!to!the!left!and!the!right),!but!does!on e!grow!faster!than!the!other?!!!!a. Consider!the!degree!5!polynomialf (x) = 2x5+ 3x4+ 4 x3+ 2x2+ 6x.!We!wa n t!to !inv es tiga te !h ow!this!polynomial!grows!as!x!becomes!large!and!positive.!!Please!fill!in!the!last!column!of!the!table.!!x'2x5)3x4)4x3)2x2)6x'f(x))=))2x5+3x4+4x3+2x2+6x'%'of'f(x)'that'comes'from'the'leading'term:'!1!2!3!4!2!6!17!2/17=0.117!11.7%!10!200,000!30,000!4000!200!60!234,260!200000/234260!85.4%!100!20,000,000,000!300,000,000!4,000,000!20,000!600!20,304,020,600!98.5%!1000!2,000,000,000,000,000!3,000,000,000,000!4,000,000,000!2,000,000!6000!2,003,004,002,006,000!99.85%!!As!x!becomes!large,!you!should!see!that!the!highest!powered!term!becomes!much!more!significant!than!the!other!terms.!We!sa y!tha t!the!h ighe st!po we red !term !dominates!the!po lyno m ial!as!x"gets!really!large.!!!!b. Now!lets!look!more!at!g(x)!=!x6!and!h(x)!=!x4,!and!co mpare!the ir!grow th !rate.!Fill!ou t!the!ta ble!b elo w .!!x'h(x))=)x4)g(x))=)x6)1!1!1!10!10,000!1,000,000!100!100,000,000!1,000,000,000,000!1000!1,000,000,000,000!1,000,000,000,000,000,000!!Which!function!grows!faster!as!x!gets!large?!!!g(x) =