021.10f.final.sol

021.10f.final.sol - HKUST MATH021 Concise Calculus Final...

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Unformatted text preview: HKUST MATH021 Concise Calculus Final Examination Name: 21 Dec 2010 Student I.D.: 08:30—11:30 Lecture Section: Seat Number: Directions: 0 Do NOT open the exam until instructed to do so. 0 All mobile phones and pagers should be switched OFF during the examination. - You must Show the steps in order to receive full credits. 0 Electronic calculators are NOT allowed. a This is a closed book examination. 0 Answer ALL questions. Some formula: Question No. 0 sin(m + y) = sinwcosy + cosmsiny o sin(:1: — y) 2 sins cosy — cosxsiny o cos(23 —l— y) : cosa: cosy — sinmsiny - cos(x — y) = cosztcosy + sinxs‘my __ tanx+tany . tan(a: ,ng y) _._ litanmtany . tan(:l:w )_ trance—tang; y — l+tanzzttany o sian : 2sinxcossc 0 005255 = cos2xmsin2m = 2cos2m+1= 1—25in2x o secgmztan2$+1 Part I: Multiple choice questions. Only one answer in each question is correct. Each question is worth 1 point. Put your answers (a, b, c, d, e, f or g) in the boxes below. No credit will be given if you Show your answers in other places. l. The limit iim 22111332 340+ (3)-—CX) P (b)-—6 1 (C)_1 7 (@O 7 (8)1 1 (De I (3)00 ' i ‘. M?“ ,. 1;? {if}: ’1”: 1W; T 1 km? ~74 2’- 0 “((98 X 253T} $71 $ij 2. Given two positive numbers a and b such that a + b = 10. The largest possible value of ab is (a)_1o , (b)9 , (0)16 , {(3525 , (9)30 , (f)50 , (@100 . 5' acts) 03> e be 0;}: {sac—c? {3(a) crawls icqéeirQQ W? 515 [00111de f (a) converges to 0 , (b) converges to 3/5 , (c) converges to 1 , (d) converges to 5/3 , {3 diverges to +00 , (f) diverges woo , (g) diverges but not to ice Séfiflp‘ depgr'i: is 2 gain): ~ sec 4. rl‘he improper integral 3. The improper integral E .3: flw“%dx jfisgi§33§rx§$£ 0 (a) converges to 0 , (b) converges to 3/5 , (c) converges to 1 , converges to 5/3 , (e) diverges to +00 , (f) diverges ——oo , (g) diverges but not to ice 5. The improper integral ()0 2 f 1“? dm 1 (a) converges to 07 , (b) converges to 3/5 , (c) converges to 1 , (d) converges to 5/3 , @diverges to +00 , (f) diverges —oo , (g) diverges but not to ice i L :L '3‘: - %{‘§:}: {31% :1 gm 2 6. Let; C(93) = f %_ The limit '3‘ lim C(cc + h) — G’(a:) g 11—14) h (a) emz/2 , (b) 3‘32"2 , (c) (m+l)emz/2 , @25562/2 , (e) 3-2342” , (DO , (g) does not exist . . . . . . . . {x v 7. How many local maximum should f (:c) have 1f its first derivative is given by I 11(3) 2 3—3207; —- 3X33 "JP 5) ? g “My 3 WONgfil1(0)2,(d)3,(e)4,§v % +~ (f) not enough information to determine , (g) infinitely many . 8. The area of the region below the line 3; = 2 and above the curve 3; = seczm on the interval cc 6 [Om/4] is given by (a) OHMGeo x—2)d:c , (b) fgflflseczm—Zydx (c) f0"/4(tan23;—1)dg: 7 (@f wfi/4(1—tan2$)dzc (e)f021r( “see gm 2de , (f)jg]2(se02y~2)dy , (g)j:(2esec2 y)dy . §”{1"S%$35$252:{ng’jfesfi?i)§z% 0 9. The derivative of (sin film is (a) cosx(sinx)1” , (b) cosas(sin:c)1“‘1 , (e) tanm(sinx)1” 7 (d)m1 ln{sin 2:) + tanxlna: , (e) [— 1:1(si11 x) -[— tan a: 1n :13](sin 2:)1” (f) [% ln(sin 3:) + § sin(2m) In 551(8111 at?” , (QB? ln(si’n x) + cot x lo $}(sin 3;)lmc . 7 ex 3 “it; “ g“ ggfii u.— f‘g oi? f 3 {game} c f r r W M :f “V; v 3 Kg“ Efifigfif ‘ file: i *L M “5;” j Part II: Long questions Show all your work for full credit. ”2 \ 53’ 2 KKKConsider f($)— (Jr—Fl) 21+ 2:2: . $2+1 932+1 ® (8.) Write down the domain of f: ® (1)) Determine all (vertical and horizontal) asymptote(s) of f (2:) is“; $09-21 *4? WW W116 '- 7'“ , No \fmg 6-04 01$;- ai‘ - Sofie» f N? C @102) Find f (x) and PM [I l 1%.: Féfir’ i 22% H‘fizw'ngffl X“ {fart} ~_— 20%? J 2L1~s<7£3é~9 : ““2"” (mi)? ”P (1%)“; U1“); 4/; CD (d) Write down all critical point(s) of f(;1:); ®(e) )Find the interval(s) on which f 13 increasing and also the interva1(s) on which f is dew creasing. W .., 11 p f 69 “” f(x) is increaSing on the interval(8) .1661.) is decreaSing on the 1nterval(S) w-u (D (f) Find the interval(s) on which f is concave up and also the interval(s) on which f is cOncave down. «5:. .._, W (D (h) Write down all root(s) off: $1 *3 E ‘— 2: :5 ® (1) Sketch f(2’:). "i € { h “" I \z E is ”WWW?“ 7‘; ‘fi; — 5 g} E5 E 8 11. Evaluate the following integrals. Show all your work for full credit. (may? I, 93? d3: W i ® 44% 333:" wgzgg”? hf??? Cessgéfi-rvéki @ Cl) :1 ”fig/33?: 3%:3 m%&% : ., xgaswrg’éngfi i: C (D fxsinxd$m "’XCQSX ~§~ SEA X?“ (2/ 12. Evaluate the following integrals. Show all your work for full credit. (agggéw / /2 cossr:(cos2z—~ 1)2(sin5:r:_ l)4dx 0 é (€:q§7L—E 5"" 5 (fingf .,\‘>£Fz§ gig} ‘ (33% 25 3/? ® 15“ 2 2 - 5 4 I j; COBZ:(COS w—l) (3111 3—1) (117: 2% f (b) — 6% 3::ng 3‘ 1 efi / f0 Eda: , 3, a?“ , m % LN“: $2” 6% :: Kim (2Q W E 0939 a J; @fifié" (D 5 at hm Si 3;” GW“) afi‘afi :\%m (i aw @3135 Egagmtm) %% fl; mCGm} wagggifiaflna) Q. Ca 9 Km? QSZHLE%@};Q @ $5M QC§§£EE3®>$G 0&9 % Q J g“ ,9 "i Q S0 Sifldfififi$w Z 1 / Sin(ln$) 03$: ~—*' 0 13. Let Wé’fifi In = / tan” acdcc. ’(o’CO‘ “MT (a) [Kfig the identity seczx = tongs: + 1 and also substitution to establish the following reduction formula for n 2 2 {£42. A {3‘3 i: S m x. séwwfi far} ”2"“ ¥fi(}@éf (b) De mine 11 using substitution. e03; (c) The region bounded by y = (tanm)5/2, a: m 0 and x = Z is revolved about the :c-axis. Using (a) and (b), or otherwise, determine the volume of the generated solid. The volume can be found by computing the definite integral Rf ’ o A ‘E MEX w E ~= 3 "eff Chi %~ % . 31¢. : "Eng: 123 :f @323, ?%fi iii T“ ”2’” . e l + ’ei fgfiyfi 3? , “E 4" ‘5 _.. FE M fl 7% l :ienitdaym L? \a w "fleltegfl \ S 5., ES ”‘3 “~— H‘Ju'fi l5“: Al'GMZE Tl 1% Z lonfli‘g ,3? "fig ;,:: e2 4 fig}; Z” ENE 42 {7 {ofigefl £o9®ifi *5 , Th 1 ‘ J» u evoumeis 2’ {$32 $ /j; 11 *‘(e ‘<0* [email protected]@k®k®;:gv 14 Consider the function f(512) = ln(1 m 2x). (a) Write down the first four derivatives of fix). , 91 ,, _ LL , 7v 2 (3:0 . f($l= ”f???“ ( Z) 1 f (3) {CE-’21s? Q: 1) b“ .1 3 “J“. ‘l‘ 6.1 I" :L' E ' I / (4) m ”m l a 2 Ir “— (13.) Write down the expression for the n—th derivative of fix), i.e. f (”(02), for n 2 1. . n (nevi fl )0) — (9 (0) )Find the Taylor series expansion of f (2:) at 1': 0 i. e. the center of the power series is 0 (9 its save $001» 19:5ng l?“ 9 *~ am Hi 1% ~ i I: 2.1 (“EM ) g ”2’4 @ "—- “Ni f‘ W“ W L_ 33$. fit; { (d) Does the Taylor Series in (c) converge at m --= 31-? Why 01' why not? Show all calculations. * o 7&2 ”732’ E: 147‘: w ,3 We} 72* “W 2M?“ @ g g“; 5 K3 ,1 .0 fl * .3 no; = {i003 ., QM»! _ { Z {W {3 \Q i 2,. 51/ {W *“l W” fl'lf 5"“ 3‘0 v» lm 7} "3 eon 0 he 0’0 an W3 as I’Hti {‘3 4 w | The series (BHV’Q’ng‘B-é (oonverges/diverga) at a: = 1 r- ' 12 ,3? 459% 4 15. Determine ifihe following improper integral and infinite series converge or not. Show all calculations for full Credit. (a) 0" d3: /1 :03 +1' E z i 5‘3 g ”‘ “3 k a?“ {fif‘i‘i "7 ® 343%? ‘5 Jg *3 €@% ® The integral (aflqgfig‘eé (converges/ diverges). (b) m 311 1221 H2— ffir'i ._: , ,, 3 (D km 3 ' fl rm fl ‘3’ 2 ' W": "5 § 3 W” .,,. ‘> r‘r‘s‘v ‘13 (Ex-fr?) 3 F fiw'znfifl 3 kfiéij 3 { (converges / diverges) . 13 ...
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