Exam 3 solution

Exam 3 solution - ES .3 Lj is s z E ii a E a. % ; § 3 i g...

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Unformatted text preview: ES .3 Lj is s z E ii a E a. % ; § 3 i g i E a ,3 E i5 m mwmwmmz 31A MA 223 ' Exam 3 Spring 2009 1. (14 pts). The first derivative of a certain function is f'(x) = x3(2x~ 3)2(x+ l)5(x~ 7) . Find all the critical numbers and determine which (if any) critical numbers give relative y maxima or relative minima. Find the x—values only, you do not need to find the y» ’ Values. - r a , ‘ / a 3; § n 7-“ g K, 1 fm)» 0 3 X (2% .3) (KT 3) Mr?) 3 we we gram! eastern X; I? @V K a W} W w?!" a k (1 f ‘1" if] QM "if a; ‘ are iaiia max ‘ Y 2. (12 pts). The first derivative of an unknown function. is f '(x)== x(x- 1)2 . Find the intervals on which this unknown function f (x) is concave upward and downward. Also find the inflection point(s) (if any). I” y 1‘ V, 7L9”; it” Was am- t): W? a» Mia X r y . m V V 3 Xmm nigh} “it in a a «I we i {am mg 4 iw , r Concave upward: i hm i fiéfiwfi “K _ a « I 3 f fi g ' Concave downward: iii”? a "wag a 2 W” a“: g fa mi a w {my 7%? f MA 223 Exam 3 Spring 2009 3. (12 pts). Sketch a possible graph of a function that has all of the following properties fie X W 7%)”; We we MM.WWWWMMMN “9m _ p y g . $3.4m ‘ ‘43 H I if w E W55 fi 0 f (x)> 0when—3<x<-landwhenx>1§l:3;_izwm “‘“‘:;,;.*i“‘§‘"i \ $ 0 fl(x) > 0 when x< Oandwhen x> 3 0 f'(x)< 0 when 0< x< 3 ‘ “:3 '3 l 0 f"(x)< 0 when x< ~3andwhen ~1< x<1 x2 4. (12 pts). Find the absolute maximum and absolute minimum values of f ( x) = mi on - x the interval —0.5 S xs 1. Give both the x- and y- of each absolute max/min. The first x(x+ 2) ‘ (3H1)2 derivative is provided: f’( x) = /\ «fine; tyJiQW’ fikvf, 2r“ w . t a, ,1: ’5? i am if!” i . : away? aim afgema f 12%;}; y: as: ' «ti 1 r, ‘L’N \, p30,; wet“ , ms « s _ 3, w» Iggy g 33 £3 'ttra m w 5,4 ‘, wamfimwmw'cmamu MA 223 Exam 3 Spring 2009 5. (15 pts). A farmer wishes to enclose a rectangular pasture with 320 feet of fence. What dimensions give the maximum area if a) the fence is on all four sides of the pasture? b) the pasture lies along a river so the fence is only required on threevsitles? H Bazaar u“ .«ewmtmuaww Wem‘,m\w W. 7:? w as: X «if, , ‘ ,. ‘ “ m t “a 3 Nail» singer f .N {K}; Na} w, «5;: 4;: :1 v r _ . . . 6.’ (15 pts). A manufacturer has been selling lamps at $6 apiece, and at this pawns? v 1 ms r r consumers have been buying 3,000 ms per month. The manufacturer wishesto raise} Q? a; A \ f? r \I «- the price and estimates that for each $1 increase in the price, 1,000 fewer lamps will be “ages 55* 24’"? t W £13 sold each month. The manufacturer can produce the lamps at a cost of $4 per lamp. A is; Mr what price should the manufacturer sell the lamps to generate the greatest possible 35o » M E; profit? ‘5 y? "I. W»: 3%? é, {:5 g M «ha “a I, g g; y p H ‘33 f if: " ' ~ Ni 0 Wage War has; eg 35% 53 my“ ' ‘ it ever; ' g} j?’ '3» i 7; Wt; 39%? m {was {54 W i; fieee w mama gm , . < a 5' , ,. " ,Va is? if mgaegglgfig m “gm? ii} ‘ r ‘ V ‘ y , . Egypw ,. i‘ 1, ,~. W, 3. » ‘ '2 h :3 231%?“ if ‘“ WQQX w figma 2% @Viiteefifi‘” afieaenf'xh wean w fiéwa G “Magi? w” w I» ~ ‘ "re -, i is {gig ‘ 53"" {:1} in: 7:: gap www.mmmmmmemwmm :rcu mmnmmmmva-V :3 g s g Mawmammfimmwnww‘10mewyxummmanmmmzmm nmmwmwmmmwmw mmmwmwmzmv MA 223 Exam 3 Spring 2009 x 7. (20 pts). Sketch the function f(") z and corgplete the missing information in x“ awe a mg a; the fouowmg Skew}! gllide. Note that the first and second derivatives are provided; ‘ I = 1'35 ” =fl2x~4 f(x) (x+1)3 and 1 (36+ 1)4. Sketch Guide 1. Domain: Xe“? 2. Intercepts ' «3;, W a. y-intercept: y= 32> $139)} away €15“ b. x—intercept: x= @ 3. Asymptotes 3 a .1...“ a: v * ~ r - 5"; L" ' we 1 w » - . = i 5‘ Maw!»- fl’O W H Engfifl a. Honzontal. y 0 Hi. , {3 a h b. Vertical: x= awe-Q v ' MQMWQJ t ‘ a a. “ ’w‘m 4. Increasing/Decreasing Intervals Mg“)? £3 M; a. Increasing: -1<x< 1 fl { ,, b. Decreasing: x< -1 andx>1 )3 gym; “W T: gr ,5 4. ‘ W . . mmmmmfi, 5. Relative max/mm «‘1 g _ ‘ w r g If “Ewe,” m a w} flaw/j 1%“ a- Relative max: :5} E ’ 3% f £3“st Frag/5%” Z?7a $13. Relative min: K ‘1 “.3 t r 1‘ gym f W11? X; Concave upward/downward intervals ffifl‘v “‘ a} k/ 1 W3 W “a w W J 4/07” M a. Concave upward: x> 2 5"} WWW mm m ‘ 4 s a mag a :2 "g ‘ b. Concave downward: x< —1 and -1 < x < 2 7. Inflection points: i 1' A»??? "‘ a .‘0 We.» ., a j a 5,? 8. Sketch the graph on next page. ' r Spring 2009 W p Exam 3 3 2 2 éfloe§§§e§§§§ 339%.S3§x§.§2fi ...
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Exam 3 solution - ES .3 Lj is s z E ii a E a. % ; § 3 i g...

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