Quiz-2_Fall-2008-2009_Makdisi

Quiz-2_Fall-2008-2009_Makdisi - Math 201 —— Fall...

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Unformatted text preview: Math 201 —— Fall 2008—09 Calculus and Analytic Geometry III,—sections 21—23 Quiz 2, December 2 — Duration: 70 minutes GRADES (each problem is worth 12 points): YOUR NAME: YOUR AUB ID#: PLEASE CIRCLE YOUR SECTION: Section 21 Section 22 Section 23 Recitation M 11 Recitation M 9 Recitation M 1 Ms. Itani Professor Makdisi Ms. Itani INSTRUCTIONSi , p 1. Write yoiir NAME and AUB ID number, and circle your SECTION above. 2. 3. Solve the. problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. 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. . Open book and notes. NO CALCULATORS ALLOWED. 'I‘urrr 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. Let f(:r,y,z) = e” — .732. (2 pts) a) Find the gradient V f . (5 pts) b) Let P(t) be a moving point with position 7"(t) and velocity 17(t). Assume we know that at time t = 0, we have 77(0) = (1,1,2) and 17(0) = (—1,3, 1). Find t D. (5 pts) 0) Let S = {(1r,y,z) e R3 ( f(:r,y,z) = —1} be the level set of f passing through the point P0(2,0,1). Find the equation of the tangent plane to S at P0. 2. (5 pts) a) Sketch the curve r = sin2 0 in polar coordinates. (2 pts) b) Write a parametrization of your curve in the form 77(9) = (22(6), 31(6)), for 0 S 0 S 27r. (Yes, I want you to use 6 instead of t as the time variable.) (5 pts) c) When 9 = 7r/4, find the position vector F|9=fl4 and the velocity vector 17 [9: Use 17 to deduce the speed of the moving point at 6 = 7T—/4. 3. Let f(:r,y) = 1n(x2 + y). (4 pts) a) Sketch the domain of f and briefly indicate why the domain is not closed. (4 pts) b) Find the directional derivative of f at the point P0(3, 1) in the direction of the vector A = (3, 4). (Suggestion: do not simplify fractions; for example, keep 40/70 the way it is, and do not reduce it to 4/7.) (4 pts) c) Approximately how much is f(3.03, 1.04)? This can be done either using (b) or by a direct calculation. You may choose whichever method you prefer. 7r/4' 4. (3 pts) a) Use integration by parts twice to show that if n is a constant, then +0. I 2 . ——a:2 cos nx 2:2.sin nm 2 cos m: I :r smnmdmz —— + —— + n n2 n3 (7 pts) b) Let f(:z:) be given by f(a:) :1 2’2, _Extend f periodically to ave period 27L Your job is to sketch the graph ~of f and to compute ONLY the coefficien s bn , in the Fourier series f(m) = (20/2 +2211 an cos m: + 22:, bn sin nz. m. Ji‘ (2 pts) c) At what points as does the Fourier series of f not' converge to f What the ,‘ "D value of the Fourier series at those points? '1 5. We wish to find the maximum and minimum of f(:c,y) = :r(y2 — 1) in the half disk D defined '1, 11. by D = {(m,y) E R2 | m2 + y? 5 10,3; 2 0}. (Note that D is closed and bounded, so f attains a maximum and minimum on D.) (3 pts) 3.) Show that f has exactly two critical points in the plane R2, but that only one of these points belongs to D. (3 pts) b) Test the top part of the boundary of D (this means the semicircle m2 + y2 = 10 where y _>_ 0) for possible maxima and minima of f. Parametrize the boundary in the following way: P(t) = (t, v 10 — 152). Remember to specify the range'of t for your parametrization. (3 pts) c) Test the bottom part of the boundary (this means the line segment) for possible maxima and minima. (3 pts) d) Make a table of values that includes the points you found above, as well as the corners, and deduce the maximum and minimum values of f, as well as the points where they are attained. 6. (3 pts) a) State the first three'terms in the Maclaurin series of (1 +u)1/3. Express your answer in the form (1 + u)1/3 = (2—)- bu + cu2 + 0(u3). (In other words, find specific numbers a, b, and c.) (4 pts) b) Show that :r + y = 0(As), where A5 = 3/32 + y?. Use this to show that lim (9” + W = 0. (am—«0.0) r2 + y2 (5 pts) (3) Use the above results to find the limit mm (1 +m+y)1/3 — 1 — 111/3 — y/3 +2my/9. (x,y)~(o.o> $2 + y2 ii 1. Let f(;z;,y,z) = ezy — :rz. (2 pts) 8.) Find the gradient V f . (5 pts) b) Let P(t) be a moving point with position P(t) and velocity 17(t). Assume we know that at time t = 0, we have 77(0) = (1,1,2) and «3(0) = (—1,3, 1). Find gtma» H. (5 pts) 0) Let S = {(r,y,z) E R3 I f(:z:,y, z) = —1} be the level set of f passing through the point Po(2,0,1). Find the equation of the tangent lane to S at P0. 09 $9 = (fiance): is) fitczmfl: film)- -—- (9.4;,43- (-1.3,0 2. (5 pts) a) Sketch the curve r = sin2 6 in polar coordinates. (2 pts) b) Write a parametrization of your curve in the form 77(0) = (33(6), y(0)), for 0 S 6 S 27r. (Yes, I want you to use 0 instead of t as the time variable.) (5 pts) c) When 0 = 7r/4, find the position vector Fle=fl4 and the velocity vector 17' [emf/4. Use 17 to-deduce the speed of the moving point at 6 = 7r/4. a) 5 O 77/4 1'71. 3&4 w 5 5‘4 3 ‘71... 9 r4? 74 l \ ~ 1r : 5‘91: 0 t, i ’2' 0 L l ‘1 o ( “ ‘1 f3. , a CA°+¢ 4'“th Leta-Ru 41M cum am} “An. y..aw'.s) 3. Let f(:z:,y) = ln(a:2 + (4 pts) a) Sketch the domain of f and briefly indicate why the domain is not closed. (4 pts) b) Find the directional derivative of f at the point Po(3, 1) in the direction of the vector A = (3,4). (Suggestion: do not simplify fractions; for example, keep 40/70 the way it is, and do not reduce it to 4/7.) (4 pts) c) Approximately how much is f(3.03, 1.04)? This can be done either using (b) or by a direct calculation. You may choose whichever method you prefer. f9 Dev/Min :9 1:; Rim)“ X1+17 ‘7 'chr‘ M \LEN'thM'fi mam. 9’ D= gape 1?} K £1170? ' 7' Rowe at" l P “>53? I D "5 in M £2 um. I. 1:.“- Dr LSKA‘CAbS-cd \azcwso. (op) ( Mono gag—u :5 q lam)», P);- -H) fi‘d‘ Lu 91“th Nd- bel-xa «(pl ._ t y) fl. « u 2» x 6 9 VL;(%{:1 ’ '15?) 5' file: (an "07") K “5 NIL-9 vm‘ vector-l Ll)— U: E) Ii; 5' UM)" Veal-f m “MSW Cliftckon' (A “m 5°= (3”) —, (3w) _ (2 ") erm— ? d 5’ f’ e. e. ' = 19 ,r '4 24. a £6 £3 Jcsalaawhl- 9 M K, (3,!) P, (3,031 MW), A? __ @072? = (0.03) 0.0%) ‘5 U : Opt ., 72 1:: c- ‘r’ T!‘ __ _, 'V A A ( é; 0.059 POM, gram? w M ' h g‘ M L'retllaa s? 9 we. \avv. mm) 0.05 Vn“ fl _3_ m5 5‘5 6.: '-' ‘g—O V °'°5 = Undo -o.o’LL = \n(3m+|) 4&(1 lnlo + 0911-) . M «q v». 4?:61630-0'0 x M I: 3%; A? z ‘5; i - (“gm-0‘4) = 01240-004 t: A 1— 41m ‘10» 39.? N SWNWNhn‘u-t-DaLLS O 0 2 154m :- mo +0? "‘ 4. (3 pts) a) Use integration by parts twice to show that if n is a constant, then 2 , —x2 cos m: 2m sin m: Zoos n11: msmnmda:=———————+——2—+ 3 +0 n n n (7 pts) b) Let f(:r) be given by f(z) {25, 0 period 27r. Your job is to sketch the grapm’of f and to compute ONLY the coefficients bn in the Fourier series f(x) = (10/2 + 2:11 0.72 cos m: + 22:1 bn sin ms. (2 pts) 0) At what points a: does the Fourier series of f not converge to f What is the value of the Fourier series at those points? ' {:M" (*Lr XEE3N,3I] i m x) 05 '1. _ a. —casm< — L A SXS-nnx-lx —— S‘)‘ ,K— " "‘x QXJX Extend f periodically to have ._ L 1. ‘ — —>9_{.’11‘ + 3 cha thX = ’1932‘ +— 311 SXJ(SM’\X) n h n n “5' E L ' 1 Coin n I \ _ _ x _ : - 1/69,," 4. 3t LXS'IA nx —- gm AX chi] - Fang/“.5 i—ETLXS'MX +3, ,-——- h 0 h h V\ L I _ lug, A! n .___ X645 nx + “L A + 2c“ 2 ( +- C ) . h h-5 ( IOJQL‘AA" "Aifjntli‘ (DJ: W A1T:.(-))n Q [5 CGR'L‘AVD‘IU l 4-— ’WUL—Fow-SQF .Seriq 9440331 "F 'i: 4-) “\RU‘WU' k“ 0‘ "lei" 2‘ “a” JCVIV‘J‘W. . TIM; .1; «xi—6“ (sub Exceh‘ ai- 5611') X=Sfi, “‘5'.- ‘ Xm, xm, ““Sm air.- git-$in» (this- M‘Fww sum canwa ~1~ 90+) 443:) 0+ 1" _ 7" ice? x 5. We Wish to find the maximum and minimum of f (x, y) = 22(312 — 1) in the half disk D defined ' ‘by D = {(as,y)-E R2 l 2:2 -{-y2 3 10,3; 2 0}. (Note that D is closed and bounded, so f attains 3. ~37- 'fi‘ _ maximum and minimum on D.) (3 pts) a) Show that f has exactly two critical points in the plane R2, but that only one of these points belongs to D. (3 pts) b) Test the top part of the boundary of D (this means the semicircle m2 + y: = 10 where y 2 0) for possible maxima and minima of f. Parametrize the boundary in the following way: P(t) = (t, v10 — t2). Remember to specify the range of t for your parametrization. (3 pts) c) Test the bottom part of the boundary (this means the line segment) for possible maxima and minima. (3 pts) d) Make a table of values that includes the points you found above, as well as the corners, and deduce the maximum and minimum values of f, as well as the points where they are attained. a) z (.114) "Lucio . Crikml 8.243 occur wt.“ vl‘zazo m avg 39 1:31 J M1 in equcuw1$mflu 2x~1=o€=3x=é 5:““1'éb' 2'" ‘“ 3:2. W: (fishy-i) ¢D 'Qu'ggfl 5) Pat): (Eng—5e") be L—‘GDIJ-WD] 5° Lam) = (.1 them) : E(o\_£1) r cu. -g. ssh at which '” flaws). R‘s-l" an lemmas-s. £- :, amy- um. MSW ‘P°“\ \ I i 3 i l ‘- K, I - 1:1, J ) g Second, m M J‘pfid, (COCW ‘ ('5 tfin D lam . ' ‘ z J9me) =Ho—I) -.-& u. —-\,:r- M an. :3 “Judd-gs '91P mxu'm v- Sn UL Mu. Q 53 l", gum-H1 «l—‘lko. ado-sit}; (HA is Aim mm 4, «3%). fAlh rel-0. Mir GE ‘7 6 ? ‘t-> We Y... "l {’3 (“r (665‘: 3-6 -3 = 1»: >10 = (IT-b“) \ g; m Wm “a .9 Q- ; WWCE» ,qlml 4 Mes) - "Am-(L nu MWWM “(K4 (1 a“ D is (301)2— (‘5’ «Halal al~ Msgfi). 5 6. (3 pts) a) State the first three terms in the Maclaurin series of (1+ 101/3. Express your answer in the form (1 + 101/3 = a. + bu + cu2 + 0(u3). (In other words, find specific enumbers a, b, and c.) (4 pts) b) Show that :3 + y = 0(A3), where As = \/$2 + 3/2. Use this to show that 3 lim (2: + y) = (mm-40,0) 2:2 + y2 0’ (5 pts) 0) Use the above results to find the limit hm (1+m+y)1/3—1—m/3—y/3+21y/9 (mm—40,0) $2 + y? ' on) ’n‘fl B:nsm‘.:\ yer-Eu cmemh,‘ (or Sen“) J m w ow) \rkébw came; + wow-0% + -- 3 3 = 1+ 530 " 'z- 01— + O (03) scmkk' 1 ’33("3"9=(‘a\(~%) > (Hnfi- “’“K m“ ‘3 “ET \ 7‘ - X a — Q 4A5 g N) we" “AN (A91; XHWLz XL 15" AS? hd 4 (0‘9 x S;M:\U‘\v\ A; 2 I | 41% (xml é (xl+(1\ s ASH); .1115: (wk-fr) As (“an ‘ flfiésww m» *1») is |[ \m("*‘3h Dang—3) a»; (was _ ("*"‘\)3 (S A“ ‘3'"? W)" UM)1 fit w; he.” m~~ 5,1)q(.,.),ug )Mvm MAO! m (BOND-so W (Xe-~43 A b L) ?w 027.23 in “ark-(a) ‘ y aw-vo’L ~ E 1%? = H 5“ ex 3 3 '1 : I ~ xx+ k *— 0 (“$03) Em other! WM ‘1 CW1 ‘x UNA (l*x*‘13—|-3‘-l’r?_‘:1 be )L 3 3 q = -K/ _\_ Kai-11" % 4—.1 \0V\" ‘3 - Ia .. “an MP km .SaML (LugL)—so as (am) 1(010) ,Wl' . aVr Ash‘s! “wk (‘m (ea r O ))= _\6‘ k0 : —‘ %L*.,L— / q ...
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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-2008-2009_Makdisi - Math 201 —— Fall...

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