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1 , ~.rjf) )('\.. 1,,0-- Math III Pledged_----=-+ ---'l\...--m~\r A square piece ofcardboard with each side 24 inches long has a square cut out at each corner. The sides are then turned up to form an open box. Find the side ofthe cut-out square that will produce a box ofmaximum volume. "X==? V=-(~ ~ -d. X'ry- V= ;> "z'(r ~ - X') 2)( is I'YI a.:x: V1= ~~ [(I :;1.-)(lYIJ + X.~ (J2-XJ(-I)] vi: l-l(12-)<)[i~-"x-.2)lj \J I::: 1-{ (1.:2 - )<.j ( I ~ - 3 X) \ + 1-, + i X= \:1. -x.== t-j o 4 \2 :JL( 0T@I@ 6= 1-/!;) min rr fyll1~ oc.,\.~~. A rock is thrown upward from the top ofan 80 foot high building. After 2 seconds the '" \~ rock reaches its maximum height (v(2) = 0) and then falls to the ground. What is the ~ c.,e; terminal velocity? (a(t) = -32 ftlsec) ~ V (t:) = - 3;;fi. + C. V (;Z') = - 3::1 (;2) -t e. == 0 V(t)=-3;n.-+. ~L\ c.= ~CV~ -{,(t)= -IL,-L'l. +L,4-t: + ~O -\tt,(i:-'l.- L\t - SJ ~ O~~ C-l::-"SJ(i::+\) -1::= S sec ~ w-he;~5 /~C

2 x~ \\f'.\ j. Evaluate the following and ~ate a graphical interpretation for each: 4),..--" ( ( r-- \ a) li~arctan - -== l~jl, \\I 'AO II 0 -Jlj x~o X ~.J n~ ) ~ ~ Ix+ll ~ -C~_:t:--') := - I c) 11m "------'- X'~-,- (X-t-i) x~-l x+ 1 3 I X -2 d) lim.l ::: x~oo 4 _ x 2 ~ (X-t\') =- \ )(~- i + (X:~) qo-p ~ (j'wyy'(l I " x x:::-~ e) lm- x~-4- X + 4

3 (J9 fb){j (i)' ),J.. A large museum display case is in the ~hape of a rectangular box with a square base and a volume of 216,,,W8F8 feet~he front is a special glass that costs $40 per square foot; the bottom, top, and back are of wood that costs $20 per square foot; the two sides are glass but not as expensive as the front and costs $10 per square foot. (12 points) a) Find the dimensions of the least expensive case. h - ;( 1~ r?... - X \j == ~2...t-. = ~\ \.0 ft'?/ C. :::$ 40(xh) +~~ )(' + t.io6chj... $ I ~ (:L'1-h) 1.- -rrov-rt:: bo-tfom.,. bo.cl't SideS -:z. -f-dp h ':L (!.= YOXh+40X1-:l0~h+c;:lO"h = BOX +40X C(X) -= 80*(~\~\ +yox '2. == 80.;1I~,,-' +40 X 2 X~' ), _ 3- \~ ~~ c.t(",) ~ _ go.~\\o,,-?+<goi<. == - '8'O,:2ljq 4- ~OX) h - t,:l x:2. - ~Ol~\lp ~<6'OX3 ~ 0 => 1t6:A'= /~I~\ b :::=-> X ~ (P-t-+ X _ + ~ 'ro -;-;-;;) S\~~ or <D~N@) &~~r~ b) What is this cost? ~= 4goxh+4'yo)(.:1. ::- $~ocs/p)+~l{ol3/p) C. ~ 3~ X $. )~ 0 = L4 J.f3 ;) 0 \

4 ~ 'ti'.f\ 0;(;\ "Along sheet of metal, 18 inches wide is to be turned up at both sides to make a horizontal gutten 1\ with vertical sides (90 degree angles). How many inches should be turned up at each side for maximum carrying capacity for this gutter. '" U)C. ox ~d.. - Ig-~~ r=-=-:-- 1DAr -= " I go.. ;;2.){ rn o.jle, X t A =- X(I S'-~~) =- \ ~x - ~'X:1- A I ex)::: I g -1-/1- J-f-x., :;;-/ ~ V+ " - ex = J-j.5 Y o G) 4.5.s 'f't\f\ the following limits: a) li~ Arc tan('!') :=. -\ -;ll' \ x-+o x ~ ~..L '= -<::>0 )(-:1)0- X b) lim 9 - X2 _ \", x-+2+ 2X2-8 - \+ 00 L I\J: ~blj ~~ c) lill!,x-9, "'" ~ ~,:=~- \ ~"",~X-9_ )(~ot- -~~o ~ d) lim 4X_~4 ~ "" ~ ~ - 'f.. _ :=.~ 00] x-+-«> 3 - x \ )(' ~... ()O 0 JI X~ ~ - \ "-i-~ /I.'" _\ e) lim ~cos X ) ~\- ()C ] x-+-+«> ex ~ t&?>.< - 0 e')( '" ~- t)o

5 \,..J ~~/ \ I '\ 1 i l../ A \ Math 111 Quiz 6 & 7 PLEDGED \. \.' \ \ When you sign the pledged line above you state that you have neaher given nor I received help on this quiz from a person, book, or notes; that you have not observed anyone else obtaining unauthorized material; and that you have fmished this quiz in one sitting. Starting time and date Ending time and date SHOW ALL WORK ON THIS PAPER! Note.' -FI(-%") ~o F'(o): I f(x) =. -Jr -Jr Slnx,-- < x < 2 4 -Jr Arctan x, - < x < 0 4 x,o < x <2 _I) 12

6 ~. Sketch the graph of Y - 3x 4 4x (15 points) 32 2 y'= /;)X--/dX y"= 36X-~Jjx Points where the tangent line is horizontal (0 1 I ') (I) 0 ) Inflection points ( OJ t) (J r~h ),, Intervals of increase X_>-----'I Intervals ofupward concavity X~ 0 ~C:R X"7 ~ Sketch: (Make your sketch big enough and clearly defined in shape. Label all important aspects.) (O} \J ~)i> " f.\~ }.., /\ ~i \ 0 l~ J? Y1 ( J)61 \-l:ltj rf" ;>X II l~x(3x-2j ~ I ~ 0 ~ I ~)<.2(:X_I) j x~o,,:=9= }<.=o "f.::: I 3,.+ T ~\.+ 9~ 9 o@ \ ~ "(1 \ I o S\~\l 0) 3 ~0 D\;~

7 Math III You know the rules. This represents my work: ~ Sketch the graph ofthe following and find the area between the function and the x-axis from x = -3 to x = o. f(x) = X 2 +x-2 +I(X) ;::. :;;;2X -I- I x-= -5:. (5 X::=.-:t. X== I (- -k /-~) (-;).)0\ (\)0) 0== (/<-I-;L)cX-0 Ve-trt 6'/

8 2 4 2 r; t.. 2) I 7 J '3 a)!(x -1)x dx =' :/-x. - x ri. =- '7x - 3" x _ I = -I I~ ~ - J + t -~ - I ;~ - ~ : I ~ ~ - f" ~WJ ~ b) f (X + CSC 2 x)dx % --,,- -rrl.. Lf o - ~ '/co + CPt. Ii

9 f) f sin-ix ~x 8)j~ I X

10 3 l) X -f I dx -, X + X + II, Prove}he following using mathematical induction: (15 points) }: 4i = 2n(n+l) 1-1 I /) Far Y\ -= \: ~ Lll = Lj (I') - 4 L ~,

11 dx: dy: I ;}.S.. Area: _-----:l~_= Evaluate: (3 points each, 6 total) a) f S e ex dx - e)< 4 X _ 4': + 4 (el<_ :2.J~

12 . Prove the following using Mathematical Induction: 8 2 (2i -1) = n Clearly state each step. (20 points) Ln ;=1 10 For *'he -h "s;t +e'\(y"\) L:= I, 2 ~ (~L -I) = :26)- \ = I 0/f\-c1 (\) == I I L=i 'v2 '~0 Ass~ L:(~l- ~ ~ ~<- ~-~ L==-/ ~) t (,;h' -I) + [;2(R+I) --IJ ~ -it + [2( R+I)-Q. L :::= I odd ~ Yl.Q/{..i ib((yl -to ~t)l~~ b) ~ (;JL _ /) - ~2-1 ;;)-i2 + :2 - ) oj~ - \2-+\ Q) ~ C~L - () == ~"l--+;2-1c -\- I ~ (\'2+ I); (2-r I l=1 oj~ca.. ~ 15 fr~ n ~~O&) C~(?L-') =-n 2 ~~