Math32B_finalexamsolutions

# Math32B_finalexamsolutions - Monday 3/i7 MATH 31B FINAL...

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Unformatted text preview: Monday 3/i7 MATH 31B FINAL EXAMINATION Lest name : First name : Section : UCLA ID number : Signaiure : Instructions : fuli credii will be given only if your solution is given with e11 the details. Notes and calculators are not allowed. CREDIT : SCORE : Problem 1 :10 points Problem 2 2E0 points Problem 3 :EO points Problem 4 :10 points Problem 5 :ED points Problem 6 :10 points WW Problem 7 :10 points 3 Problem 8 :10 points W 4.. Problem 9 :i0 points f 2 V 5/ . . g.» 3/; f f 53/ 3” TOTAL . 90 pomts Mi. 5 Problem 1 Consider ﬁx}: cosm a) Prove, without using the formuia, that f’(;v} : b) Prcwe than f is invertible. Let g 2 1°”. 1 Povethat :2 . C) T 9“) mm a} {W} W M seem f0:~ a: E {0, %) sins: cos x 3 see a: tan :y. 93 Problem 2. 5 . :13 d , ow—f—‘MMW— . Fmdf{i~s~m mm) I X g.§.§ g a: Ekég "W W L a: WLM 4» u; m xii {335;} {ﬁﬂ is, £1 3; 2 Probiem 3. ‘We want to ﬁnd f 2:3 x/E — 33%;" using two methods : a) Methoé 1. Do the substitution m :: sine? and then u : cos 9. b) Method 2. Do an integration 33y parts by setting u z: 2:2 and v’ = all — x2. ,. 5% @ﬁ ﬁ {Qaiwxhéﬂg 3-; >iﬁ§iaé§ fig? i 5 ,x Probiem 6%. . . , . \$0 1 . . (7 . . . . 1 (if; Conmder the unproper mtegrai f0 m1 2 + m2 de. To siudy this 1ntenral, we spilt 1t mto two mtegrais ft; m} 2 + \$2 gem and fog 1 dw 2 31/2+\$2?3 ' a} Prove the convergence of fol EE/zdfxz/gl 2/3 1/2 1 i E b) Prove that for a: 3 1, we have m 2 a: , Then Show ihat W 2 2 W c) Conchide about the ciévergence of [10° WM. 3? &) Does "[00“ de converge or diverge? Problem 5. 00 Let an : e'nln(”} and consider the inﬁnite series 2: an. YEW—'1 3.) Use the {001; test to prove that the series converges. ‘9) Use the compaﬁsm test to prove that the series converges. 31th for b) : use the fact. that in(n) 2 E if n, B 3. i , Problem 6. Find ihe limit of (z + I) 1n(.r + E) m wlnﬁn) when :r: m» +00. {254%} éiff‘i'?)“ MN m, 5155i?!) ¢ 1 égfgwng§m .3 \$514?) 4; 35 ﬁ {if} 2% an?) 3; 4&5" ”2% E3 Problem 7. a} Prove the foilowing reduction formuia : f ln”{m)da: = min”(m) -— n f in”_1(\$}dm. b} Deduce f Mum. Note : 1n”{x) : (In(\$))”. § § 5 f 2;? 3? 5?: is? Saga/34 “5%? NW . if: f g X f: g i 2., ' Proﬁlem 8. Mt a) Prove that the improper integrai fife Egg-Cit converges for a < IL hint : consiéer two cases. a < 0 and 0 g a, < 1. —t b) Frove that fem ert diverges. 5 if 3” °‘ 3% r- 5' V 1,: J g 3‘ 2* f =sz i; :57 gwg; ”52’ (\$2; m n: 5:“ a: * M i: i {if } J g g L '2; j 5;; m g f 3;; (£4: : y; x; g f p Probiem 9. 00 TL 3.) Find the radius of convergence, say R, of 2: ”i3”. 71:} 00 b) Does the power series 2 77%;: converge when 2: = R? when m w “R? Expiain. nwl :5 g“ {W g//‘ . ’3 f s ”5/ i ‘ f” 5 “if “15% f M W ; a»? . M ” “Mg ” 2 i s _ * g 3i i: g / a?) m WW ~~~~~~~ i M 5i 2 {“555 r x Er y: 3 ’ g E W / RM } ﬂ (5/ M”? C 3/] f? ,. g? 3 r :5; {:W s “(w/MW w w WWW WM MW e W M i m} \J / We _ z _W 5 {11 i 31W 543% f f, g «u, 5 a . f ’ g ‘ A? ﬁg? W i: :5 3; {E W; 2;; L? 1 "x A ...
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