Quiz-2_Fall-2005-2006_Makdisi_Jaber

Quiz-2_Fall-2005-2006_Makdisi_Jaber - ,5; Math 201 — Fall...

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Unformatted text preview: ,5; Math 201 — Fall 2005—06 ,6“ Calculus and Analytic Geometry III, sections 5—8 ‘ 2 ‘ Quiz 2, December 1 —- Duration: 1 hour De C‘ ' Mfim‘wfi nae-Gm»- 1”“: YOUR NAME: 3 ©\o LU“ 06 3h? YOUR AUB ID#= ’ngwfl, wwgg ‘5 C2 Va: (2.— WW ‘Lm , emu (Loos PLEASE CIRCLE YOUR SECTION: Section 5 Section 6 Section 7 Section 8 Recitation Tu 11 Recitation Tu 12:30 Recitation Tu 2 Recitation Tu 3:30 Ms. Jaber Ms. Jaber Ms. Jaber Professor Makdisi INSTRUCTIONS: 1. Write yourNAME and AUB ID number, and circle your SECTION above. 2. Solve the problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. Each problem is worth 12 points. 3. You may use the back of each page for scratchwork OR for solutions. There are three extra blank sheets at the end? for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY WHERE THE GRADER SHOULD CONTINUE READING. 4. Open book and notes. NO CALCULATORS ALLOWED. Turn OFF and put away any cell phones. GOOD LUCK! An overview of the'exam problems. Each problem is worth 12 'points. Take a minute to look at all the questions, THEN solve each problem on its corresponding page INSIDE the booklet. 1. Given the function f(:r,y,z) = $2 + :32 — 3;. Let S be the level set of f given by S={(m,y,z) | (cg-+22 ~—y=5}. a) (6 pts) Find the equation of the tangent plane to S at the point P0(2, 0, l). h) (6 pts) Draw a rough picture of S. (Hint: S is a. paraboloid. Your drawing should clearly indicate the 3:, y, and z axes and the coordinates of the vertex [i.e., summit] of S 3 2. Given the parametrized curve P(t) = (4379.32, 2t). a) (2 pts) Find the velocity vector 17(1). . h) (5 pts) Find the arclength of the curve between the points Q1 2 (0,0,0) and o2 = 3,4,4). . c) (5 pts) Given P I ,:1 : (é, 1, 2), we know that there exist certain constants a,b,c giving us an approximation Plt=l+m % + GATE, 1 + bAt, 2 + CAT) Find a, b, and c. 3. a) (5 pts) Make a table of values, and use it to sketch the curve C given in polar coordinates by r = l —l— cos 29. 0, lf—Ti' S x < —7r/2 b) (7 pts) Define = m, if —7r/2 g :c S 7r/2. [we extend f(.z) to be periodic 0, if 7r / 2 < x 5 7r with period 2w]. In the Fourier series f(.’B) : o0 / 2 + on cos m; + 22:1 1),, sin nr, find ONLY the coefficient ()3. 4. VV'c are given a. function f (:3, y) which satisfies: .... —. m. 2) z 10, W (112): (3,4), W .u = 5,6, v = ,2. (3,, < ) f (5,6) (1 J a) (4 pts) Find the directional derivative of f at the point 130(5, 6) in the direction of the vector i? = (3,4). b) (4 pts) Find an approximate value for f(1.02, 1.99). 8 r, c) (4 pts) Find the partial derivative — [f(82 + t‘, 3515)] . (In other words, as we have substituted 2: = 52 + t2 and y 2 3st.) (sat)=(la2) 5. Consider an elliptic disk R in the plane, given by R = {(r, y) l I2 + 2y2 3 12}. Find the maximum and minimum of the function f (x, y) = my Jr 103.; on the region R. Indicate both the maximum/ minimum values and the points where they are attained. 3:2 6. a.) (4 pts) Show that hm does NOT exist. (ma—40,0) :62 + 3/2 $4 b) (4 pts) Show that lim 2 0. (which) 3:2 + 342 €m2/2 . . - cos y 4 ts Find the vaer of hm C) ( p ) (as,y)—-(0.o> 1'2 +192 notation. Even if you have not done part (b), you may use the result from there as well 4 as the similar result limW‘waTO) : 0_ . Hint: use Taylor series and 2 1. Given the function flag, 2) = :1: + ' ; . Let S be the level set of f given by 8: {mm | +zQ—y=s}. (6 pts) Find the equation of the tangent plane to S at the point 130(2, 0, 1). 1““. Jug-ud- firth,“ is Ian/“suauql ‘l'a “Hue ‘lflnstnx' tht «’1’ “Pa. 3F —_ (21., 'i, 2,1) 5" 3 (Uri—Ll). 7.4"“) Q&,1,;)ebjul- Riva-I. e-a m J. 6;“;b G") (x-lljli-‘D '(qp-II'L.) to 6‘3 b) (6 pts) Draw a rough picture of S. (Hint: S is a paraboloid. Your drawing should clearly indicate the ac, y, and z axes and the coordinates of the vertex [i.e., summit] of S inf-3:5 c: J: frat- L3 memst an. in n 3m ‘u 3 ‘ n Lir-CMLJ "AAA-Is 1—5 3 2. Given the parametrized curve P = t2, 2t). 0 a) (2 pts) Find the velocity vector b) (5 pts) Find the arclength of the curvebetween the points (21 2 (0,0,0) and Q2 : _ Q muse-es iv tr» “gawk: 0M) 3. Q1. OrMSQ‘NJ-s *‘u lc'bwl ,t2129\|:: : m «flask aw Qt Jr» Q7. 3 “3 WM t= t. 2. =§We 2‘1——”it 2 L; + “‘ H 'L -: {:0 +049 2' S {LN-{L19 k t2) k 4L): r5 1"“ ‘1 Ctlnf' _ 9 '1- - :- 24 H _ D = 3;?- E=o 3 c) (5 pts) Given P | :1 = (31;, 1,2), we know that there exist certain constants a, b, c giving us an approximation PL % + aAt,1 + bAt, 2 —|— CAt). Find a, b, and c. tzl+At Yu‘r i)”: Pltg.‘ R; P\l;=\+b’t Hm $31=DFD e: lye-qt =r>e( l, 9—, 7. )=(b£,m.%‘c) =\ ’n’w‘ .‘C' 139: (>3: i‘ '1') 19"“ SP :Pi‘: (ELISE: ‘*20£.2L’2’D{7) Lfik_ kid-1S Can Lg Jon; 'm uva Mia K(+)= 3‘ xihmt)‘: “0* A“ I at ' : 134- ’Qk”. id LO I 3. a) (5 pts) Make a table of values, and use it to sketch the curve C given in polar coogdinaies by "r = 1 + cos 26. f: [-l-Gsw O, if—vrgrr:<—7T/2 93, if —7T/2S:IZSTF/2 0, if 7r/2 < :5 5 7T with period EN]. 111 the Fourier series f = an / 2 + 220:1 an cos n3: + 22:1 3),; sin mu, find ONLY the coefficient. 133. b) (7 pts) Define = [we extend to be periodic “l 5:11- [ Xmslx _— ‘S‘CDJZX 43‘ -> [XGJZX _sn3¥ '3'? _\ rm —\/' 1:953! ._ sin 37[ {ME “'—---~-—-—. 0 _/I' l 3 D 4- -o4\ “I: ’3 a C0 4. ‘We are given a. function Hm, y) which satisfies: ‘ —v 1.2:10, v =3,4. €71 =5 T = 2‘. .f(, ) f M ( ), UM ( ,6), elm (1, ) a.) (4 pts) Find the directional derivative of f at the point 130(5), 6) in the direction of the vector 17 = (3, 4). -% T? 35 MA- q-Uni\- WCJ‘Or-J O = ’w 15541 : {flue-U: 0,9125%): [3,42% ° Csie) = A 5 b) (4 pts) Find an approximate value for f(1.02, 1.99). i; P‘: (hi) A4): (A 171: (“-be "DU-D ( y’b‘j) m (1 (Par: {1(a) + VHPI ' AF). ~ = .m in chr- caag) (50913.9 : (0'07,a —-c3.o\) S\f\c,q_ Y1 (LOZI‘I ) “A {7’12 [P_ 189*), sh 16(1) 7. £0.23; (3,q)-(o.oz,-o.oi) pm.) : l 0 4r (3)(o.oz) +84%— om) okra-o) k‘i’w Cm ml:— 19 his '3 41». fivmlu‘r- haiku-1 : [0'01 (5/ =3.” CnT‘B AC1 'béfp'h * 3%[P’A1 a - C) (4 pts) Find the partial derivative '5"; [f92 + t2, 3St)](6,t):(1£)- (In other words, we have substituted 3: z 52 + t2 and y = 3815.) Us; ’Vu. Chain mic: '3‘; _ "DC ‘By L 25;, '3 _ H) "3X :3;- a; a; 1% “ W 53%) 6‘33 ,1“ , #9 wt” mafia) HM (m): (3:8, 3+2) 49,6). «A 6H =34 (34m)! 11%.») (- ‘bxln __ z ( lg (fig [(53%]; (23,39 Lfi)‘(m‘) '2 j _ . 3* ~1 . f “ m 9,9 (E'BIWEW = 0,1) {2,9 :E I. (519 (PIE) 6".” :G'y‘ ‘5 1‘) (‘1’). En. Consider an elliptit disk R in the plane, given by R 2 {(Ly) | 3:2 -$- 2;;2 S 12}. Find 3 .‘ the maximum and minimum of the function f(:1:, y) = my + 10y on the region R. Indicate ' * both the maximum / minimum values and the points where they are attained. bke \ gm! Crl‘h‘cnl FmA—(s) “C 9— & Sac. qC— Dug-1 \vaakos 'L. R I 1140 “sunk 5n; CREME ’h-u ‘Mon DC- 1 . r—‘D —o VL=(J;"£UI);()“.9 If. JM Qua-c4 Tb 5‘st 1~ x¢lo=0 ' 1.4:: n m1 H; mm W» Rhona) 13w 19° 441 CM er $.35 59 m 4:; was? mmslAQr ?.. or?“ SJfl-EZ Skch art "' cr'~‘~4¢-R‘ Evin-3f; I"‘ E, M Mi] kafw‘n W Li. qt— m 8W4 L ~- 5/ Luumé £~ K - I n t. a K J 1*,1 Lzl _v “'1 ‘ IS (nMu-u;:l km. 4-: QTM—Efi #41 d. mku\xph $ (HIP-r; hi- i :5, A g MU“: “Eh ’NNQ ffhdA—Q Mayhem 436‘“) ; xj+ NJ _ ‘ 1. glad—- Are M CoaCLme¥ 305.1) 3 75'2"} = {1-- MAQ .63: (DC/*1); $¥=QIXH9 C7Lm€§ . . _ , GD , . Lrjrwju MUKPLU' (rufihas, g 3' if) Q cK-mrd't r}: (2) 3%) z m “1‘0 ' :6‘1 12.11:- 1+me 231m ‘21:)(‘4A'o-x .L 1‘ ¥*11bIL© 7 7 k2) { 2 Q1 255:th F) '11 “H” 1-116) x® X 4- x1 rrl'L 21L \}+ x1!» “3“ = ‘25 m cfiusK-a Lr x Cl‘uMZ. u\ 2x1&\0¥-‘L:g F3 ICX—IqXX—i. L) .: t: g. :r | our X 2—“: u \l\J "— Vtkox“ u '11“ 11"” - 2c—co < o ,1 , - JD MIG. m M Fag-Jul. vdw; \\ D - .- 6— :9‘ "c- X:—5 A“; 1-1:: CPM'JQ-nl +0 m3 Vohxb on ’fln-L Lounéanl. memmmmuw. 6, ’C fl 7— ‘kSC/‘I qR-al‘ui c4- L HA1. Ml'n;MUM Valug 6 Mn): \-K~@>¥KD-Q—$;L) L -\\’:1’;_:_ smaller __U\F/ 2.) 5 «Heme; G‘- P1(ll”\ig)_ . ", $2 3 6. at) (4 pts) Show that lim 2 2 _ (1;,3;)-—-:(U,0) :r; + y 05a Jam ’2.th 4:5» Mm Lima 4 Pam, “46:8, ’1. l __ K a \x» ————L — \w t?— ’ a} W °" L _ 1H a Ewe t-w I _, m 3° UM LC‘WH) JXQMAC an ’hu. Amiga. '9‘ walk ‘Cor A‘Qeeml‘ Ll J does NOT exist. W) 1(‘E'QE) / q EVA) {3—50) 53?(Q—-ae.°) {3609 “M pow) D065 par €XI5T‘ fl______*____________, ear-+0”) 1 m4 . I 't 1' =. b) (4 pts) Show tm (may)fl010)x2+y2 O ’1. L. . 1- % 3‘ My. 0 9. 53'; Y “1 Melbll bunk M X ‘ 2' "' 1. ‘54- ..z ‘1 I 5‘ X—9 D “5(‘33-1, I; \uunéecl J lnmfl-fl— 4MVMA' ‘5 O . *1 1 L “a. \ ' "ko- x“ = ><- L 4: x 1m gNJmla 92-22 as «m. w M ‘ of «a» ( l mm. D O l h - -__ - [whim on Bohr-marl mm- (-0 k WWW ‘1 ‘T Fido P‘Hdg'. LMLL J ‘I H H9 '1. I1 Now “M “5" l L XL“ ‘%—F{. = if?” 1 r C“ e I uf- CMka—L- OE: {3&ng S f' LU“ n“- ‘ha—u‘ (' eI2/2 _ cosy C) (4 pts) Find the value of 11:11 W". Hint: use Taylor series and (am—40,0) $2 + 1:2 notation. Even if you have not done part (b), you may use the result from there as well 4 as the similar result limwamm —3"— 2 me+y2 'L 2/ x1, 7' 'L. E1: lAr %?C%—).l|L—— ‘; l*%*0(xq) I q ~ L H (DU-l: l’jfl'qu/t—H — l—jal’onl) -.1;F€m~°L‘W a ‘L. . (mk‘wirfidfifg) gift-9(1) (I? 5:]. L q C/Zflos—j ~624- L 4.. O(¥)¥O(\1) X’L‘P‘m XL*‘;‘\ L‘ >4” (Ll—— =‘€*O(§€:’$"Ofiw" Y _—-—90 H a; c,~1)__9(o,9 M ha” ’61» J fight—ac: 6 ...
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This note was uploaded on 03/01/2010 for the course MATH 201 taught by Professor Variousteachers during the Spring '10 term at American University of Beirut.

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Quiz-2_Fall-2005-2006_Makdisi_Jaber - ,5; Math 201 — Fall...

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