Math20c Midterm 2 - Spring 2007

# Math20c Midterm 2 - Spring 2007 -...

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Unformatted text preview: _____________________________——————————-——-———-——- \ CLLP w lk-“Iath 20C Calculus NIidtel'm Exam II 1\-*'Iarcl1 2, 2007 umo: —__— Lm’mrc lluur: Sew-lion Hour: (iuhlvlimrs for tle lost: 0 N0 hunks. notes. or (‘illt'lllHl-[H'H 2111-! allowed. 0 You may lam-'9 answnrs in H)’llll)()llt' form. likv unless ilu-y sinlplifv I‘urtilml'. mm v’ﬁ = m” = 1.“ 01' (-()5(;5:,,x.:1) : —\/j/2. I [.an l.l1v lew proviclurl. ll’ lu‘c‘vssnry. write "Ht't‘ nlllm' sink-5" and ("outimur \\'(_}1'l{i115.§ m1 Ilu‘ hack of tllv sumo sheet. a Circle your ﬁnal answers when rvlm'nm. 0 Show all 5101'»; i1] _\'t')111‘.~;c1111l.i(')11f~5 mul lil'dlif‘ your renaming clear. Allan-«rm with no ("xplar1:11:1011 will l't‘L'OthP Im vrmlil. {‘\'(‘]1 il‘ Ilu‘y are ('fil‘l'thL. a You llilVI‘ Si} 111i111.1l.(:s. — M \Q (l) (a) Mau'l] 1.11(\.f0]luwit1g equations (i)—[\'i) with the graphs (I)—{\'I). Yum Iluml 1101 explain your {‘lmimas. T110 ﬁrst. mu‘ has 1mm (lvuit'led for you. (1} H3"- .U} = Hil12;1'+ in? II (ii) ffn‘. y) = t1}.~+(:-;i1'l('d::"‘3 + '93)) (iii) f(.-:‘. y) = sin(.r) siu(:jy) (i\') f[.:'. y) = (135(3J') t‘ns[y) MPH (v) f(.r- y) = 5i11(I-I'l + Jul) |H (vi) ﬂux 3;) : .Ir’uyl' I (‘3) Suppose. that. if (u. e) 2% (U. U): (o. e). . z f(u..¢.) {07 mm (a) Is ﬂu. 1.?) continuous at. ((i. {21) ‘? Clearly explain your answer. 1. - "u UK Slﬂ‘v U2. um ,.._,t.————. : _ (_ j (Yrtﬂ'7f0ﬂll U 'I V1 U‘erv'l ( “at” UFE‘O V170 L'H'r’i {J'Il/{L\/ : ﬂ! - _ C) '>' ,I.n'.—-£-u,-u\ U1 a} V1 ’30 a "L I ‘— a. 1 g‘iani-O m-lv am 03-414»?— 1 , ._.._._-—--' "1 \_J—- 1 9 Oi % \ L": l 92x] SQLLIQZL “HM” H 4 J . lol- chW-lv __ OS. l‘u‘r’l Upde A {v.on-‘rwrm \I‘ Ye": ‘t-T 'lg p¢-.-1~1m«0w\S (1)) Suppose further that u. = “(1‘) and e = (9(3‘) are eoutinumls functions off. for all 3‘ E (i) \Vrite down the chain rule that. would allow one to eonnmte % in terms of f“, {I‘m 11.; and :3. You need not. eonnulte any derivative. [9% (1 4p explicitly: .9 6le o m‘ﬁu‘Ll-f +*Cv 'l' Val q/ \V '1 1 "'f '"l’ (ii) Use the two tables of values given l1elou-' to ealeulute % f=| Page 2 (:3) 81113111150 f(:1:.y) : :r2 + 1,12. where (:1; y) E R2. The surface 2 = ftp-'11) is a. [Il‘ru‘alJUthIL (2-1) One of the two eontour 111aps below is for n paraboloid and [the other 11 cone. Circle the one. that corresponds to a paralmloid. (1)) Let P : [2. L 5) is a point on the surface .2 = f(:1:, 37;). Find a unit. vet-tor 1? for which the dimer-101121] derivative Dag/(P) is the largest. possible. \ a “v w "" ?:C2r‘t3 224(xw1) r?‘ 173‘ : 2x 1:14 : 27 . . -—3 ____ V¥£21l32<q ‘2') Dx-lil?)- v4 (133-11 \ 611+ 9 i U; ‘3'. TEE-5L.le 7 (C) Consider all possible lines that. are tangent. to the surface 2 : [(3:31) at. P. One of these lines has 1.110 largest. possible shupe. \-'\-"l1a.t. is the magnitude of this slope "5‘ lV-H1A7lzlé. \$33) -: {If .-—' ‘ He "‘ kl : \20 [all Find the equation of [.110 plane tangent 10 r: = f(:1:. y) al. P(‘2. 1. a= 2.x "ELF-2‘1 l1}.11=O FKY :' 2/ Fun? 5 Z 2;?er +?x(1117(x -23 + 1:1, ('21 I\ W") : S +L1Cxe23+20443 12:5—‘r lee\$+ :L,‘ Pagefj’ : 3* 4x+1.1,(-IO 2mg)“. zviag (:1) (a) For each (if thc i'hllr.)\\-'i11g sets. (1011.01'11'1iuo \\-'11(-‘.lzher it. is (‘losed and Whetln-zl' it. is bounded. (i) {(:1:.g) :12 + y? >1} / I Closetl'3Q‘1’e; .I Boumlecl‘l’ (Yes \ . m)ummzyzm Closed? Bomulm‘ll’ (Yes 511:1)?“ *‘ ‘41"?‘441 -+ x7ﬁ29<x1 “1542/ L +2 (1)) Suppose f(:.{:. y) = (:1: — H" + (y — 1)2 + (::--+- “)3. 27(L_2_7I 4 2w w'lux 43(‘4 (i) Find all critical points of f(:r:. y). xleu-n-+ch+w> 1.. = 2044) + ti-WI) {if £chz+2°ﬂ fxqtl Q.” 5 ’2,+7/" “l \$1,:o xxx-2+1X +1”! 17.. :L1><+7v;~7- “Milk/l +1y Rx +7»: : Ll” Qw=0=1w1+lﬂbl waxﬂ— 1%4‘411‘4J)’ O _ 111,1 4-1? "2- x i?! le+ 25v»?! CL (0%;54-JZIK ‘51": 3‘ ’ / (ii) Classiij' each of _\_-'0111‘ critical points as li'x-al maxilnllni. local mini- mum 01' saddle point. -. "L D ('39:? = kH-‘wriw )0 31x24 >0 l a '1 ‘2. = «1-01 w “T =1“ Ll ' 1 109a};- / (53 ‘5; mm'uwamw‘l (iii) Find the. shortest. (listm'uro from the point (1.1.—1) to the plain-r. :1‘+'y— z: 1. ?= man“. in 431-41“: +131 PEN43141144114m-ill‘zx:2_ x_\ Razor—13 2xr2=o 2 :2 Hal ,5 ?\f:2(\4-—1) 3k;*2‘-O 91—»ZL14'3 121-5110 52 = '2 ‘ Page 2; Use the Il'l()I'-h(')('l of Lagrange Illultipliel's to ﬁnd the maxin'mm and minimum values of the functiml '3? .l'(:r.-'. y) = 3'2 * '9" subject to 1.110. constraint. ;I:“+y : kw) QUHH 9(‘11‘43: )(1— ‘41— :x = u // c3): = 12,} , t-Z ./ "1 7 9%“ “I V¥=AU3 - / 5 2—} / x14 trt—‘l / ____ “3357‘? ‘7 /'1_? ‘x-SO Lfg.‘ E£U:D _: i —__ﬁ' ' M 43(0 ,ii) : ﬂ! «bu-Hi}Emu-“4'1 ‘ £04.49) T I ﬁ-waxirmnm Page: 5 ((5) Note: You would have to have earned full credit 011 problmn 2 to qualify for any portion of this extra credit. Cullﬁitlt‘l' again llw ﬁlm-lion deﬁned in l'n‘olﬁeln ‘2: 11‘ (u. r} ,i (U. n): u. if (u. r) = (0.0). 15 _/'(u. t') rlii'i‘m‘onl.iuhlo Ht. (1. l) 2’ At ((1.0) '? Clearly and fully nxpluiu using Lin-3 not ion of dill]?r‘vHHrm-EMy. HH- r') = v r—['“ “is Y€§ 112— CUmf) Ix; Ourﬁh‘ﬂuolﬁ\ 0“- V) h j l -\ I \.1-\-{\v'\I'L‘. I Q P\ \J‘MPY} '2' [Rf ' '3 gywww’} L L , L- \ I 2 ‘ k1 g'w'ij V ; 11 Ink“ :_.______:L ‘ 2 2- f’riyp fu‘ F‘UFIIIIIIH Slwclt [1) Sulm.‘ surfm-m: _f(.:’. y) = I: y; i5 :1 Hmldlt‘; fly. y} = my": — I” L»; .‘1 Illunkl‘j.‘ 5;|¢lt|ll.‘: yr) — 1H — .r'“) — 3;: has :I [muk a] UL”. Iii). {'3} f is mntinunus :lt (nub) if lim j'(::r.y) = j'(u.r‘;). Hay} —- [n.h'] (3) ' .2 t f; AI 4.- _."N A”: d: r: L rI'J‘ * L, dy. \vliil'll is 11m smut). as L{.:') -- ﬂu} I'MHJ' u} [for singlv \‘nriahltﬂ .) . . L(.:'.y} _!'(u.h) -+ {—IUJJJHJ‘ — n) + ﬁ{a.h)(y — h}. r).r d}; (-l) '11; [iml critical points. :40le f_,. {I and f,_, - I:l HilllllllIllll;!t_>ll:-i|}' for .r. y. I fJ‘J' Jr“; “2r .v'ry u (u) If D } [l and flrlrmJi} )— U. l.|10|1f(_u.b) is :1 load minimum. (:3) Suppose“ that [n.h} is u r:riIi(':ll [mint nl‘f. :uul lvl D = . Then {a} If D “:3 (I and {HULL} < (J. Hum ﬂail) is a low} maximum. in} If D < (J. Hum _f{r:.h] E51101 u Int'ul maximqu or minimum. ((5) Suppose : = f:'_.r.y']. .r : .11!) um} 31 y”). d: r'an’J‘ I Hf u'y E—Erh‘ +Fydf' {T} Huppusv : .'-: f(.:'._t;1. .r = 413.?) and Ir; 2 Maui). f): H; HJ' I r"): Hy m _ m :1.- ' @05- i): _ H: Um ('1: fly . — -— -+- — —. EN EL:- r'H (if; (H ('8) Suppose! F(.r.y. .2} -- II and ‘ = fi.r._r,r}. Thou. r'): F, fir I H: F.” 0—H _ 3’7“: (E!) Suppnst' {T = {ah} is a unit \‘(u-mr. . 4! £)|]}"_['_ . Elli—[RI WM : Vﬂr. y} ‘ 11 {ill} Sulw vf{.r.y] ,\ Tszyl fur .r. y. (‘nmpuiv f{.r'.y] at. all snluliuns to ﬁnd mux and min. ...
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Math20c Midterm 2 - Spring 2007 -...

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