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# Calculus II MATH 1432

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This 56 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1432 at University of Houston taught by Jeffrey Morgan in Fall. Since its upload, it has received 42 views. For similar materials see /class/208396/math-1432-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Test 3 Online Review Spring 2012 Sections 84 103 10 questions 3 multiple choice and 7 written o Trigonometric Substitution o Paititial Fraction Decomposition 0 Numerical Integration 0 Polar Coordinates o Parametric Curves 0 Sets Sequences LUB GLB Monotonicity and Limits i Part I Trigonometric Substitution For integgals involving I Use the substitution laz x2 X a 5119 ta2x2 X3 39 1Ulg 2 2 xx a x at 2 x 056 I 75 an m s new 49 5 quot 7S mm x 5 an19 e J 4 3 S I mam 7g c mote d1 Slc19d9 SuziLs E Tums Nam z 1 Lan19Xuczg 2 041 9 506 3 Gig gec 1 sum Eed 25 030 3 g z QnZ9 cume a klo Aort l Know W Kann zg swim a stdexalt 2g SCAJZEA VSecMMLQJ quoto m MALde 311 1quotst V MM 1 x I 4 x2dx2 S Li a H Su ax 2 c0309ng ej4 T x1SInB on zusM39lg z 2 CogQWM39 W W n s39m M cagLN amp us fm c049 Ll BQSZ9amp9 H i Coszoampamp gg sn19 C 6 MM 5quot j qM 20mm anwm 7 am SEA5 i g lH LLA zs n9 A 9 z X Zgg2 iaxqajf37 C 9 g t Zaras q3Z EHyl AIC 13gt Ms9 4 11 zarosin 2 EHquot 39 C 4 I Part II Pg al Fl tiOn Decomposition Setting I FEAT in where px and qx are polynomials q x 1 and the degree of q3 gt degree ofp Process 139 DE Pf 305 J 2x 1 2 2x4 x 3x 10 pgy zx alngru 2 by a WWW m 2x xzsxz 2 a B As 6 f6 x J 4 mom Xlt YX2 X Na f 9 WH 1029 Musz WM 33 2x PMM39A Jr 84 Hm x 0 Subdquot 76 61 7 9 7 R 9 EMA X 739 vS quot3973 7 B x gx1 x g x 5 2 239 imyde In M ill Jr i x 1 SV 1 H E n Na C gldx39sl 7 XA 7 my I 3 gt 392 IL 5gt 10 divmm V1W 4 MM W m 2X14 D a I1 0Y1 X10 lira0 7 1 r b l w CT aw l mu r LS ave 3 49 4 ELL A Q s WW WNW w X M WV 03 FM muHermu m b x7 1803 A ixi l 1 Agtlt1Lfr C3CXquot3 PM x7 Swim nan P LEE 9 K Iovlwaj 7 0 guLS V 7o I 2 egt 352 C gtL f I S Li 03 x1 SmbM39 Xivl 9 595 B I 3 7 3 3 g 44 W q in 2quot I Ex 5 xmx wx w XL L 2325er 7 i 39I 1x P Jr 5 s 309 22 A x Ixquotq lt xhwt 1 L oky 7 J In ll g amp 1L 7 I V Went A I LJo X 739 ALPX gyl 3H1 Xz Ak iafc gt c nX l lnx11 L gL39d V lnxz 1amw23 Q 2x2 1 x 1x21 70M thkaE 1 P 7 2x14 1 i E 4 C w 7 47 Q J LY 5ka Al B Q 1 Wt rdb YOU 1 A rmil 05313 Z N 51 W Part III Numerical Integration b 1 Tools for approximating I ak m 5441th 3 i i A l n70 whyl3 Set up M 3quot 1 13 ms 1 I H l Hng aa l I l I T a 13quot TH MMH lC M Qfxnihx tyit ng Mn Mam1r mg 3arw 4i mq gt FL in 39 n Estimate 16156 using the trapezoid method the midpoint method x and Simpson39s method with n 4 RSOJL ni394 05 3 g l X 8 K 3 quot 1 O H 2 L l Lm T Gem 1CMmHaMED Lax H Z L J L L 2 g i LCE3 4 iH gt MHL 3 mle o 2 u M A X FAJ if w W T4 Ms ch Hg 4m E E 3 g XA I 111J h rt 4 P a r HAW Add f nun Use the midpoint trapezoid and Simpson39s methods to estimate f Xdx with 114 amp I C xd q 1ng I 4 3 1 L53 5 I 03H 1lt ing 2I1 1 7 391 k l l 1 l gt 2 3 T 39 M WGOEVW 2SXamp3gtBL if H V pl 0 AL 0 1 Lzzbl310 H 4 m MA 2 O39Sg 10 0 5 ML13 l15PITISXAL3927SAV 1th J lt 3 14 332 L f I 2 5 QISS Jf 390 glb LI A c A 3 quot g g 3 1 3 H 3 Part IV Polar Coordinates J 31 de a arc n Q Fpr Quadranh was 940466 m hi I M4 L1 WAwa 65 2 quot1va 9 Mg W 8quot Q0 may x A K S W a ca4WJWlt W M Mw 0 9 3 m n FL 0 6 I 277 NWA mm W m39 8i W3 0 M X V 0939 2 3 z 34 M6 1 2 C SIM8X a 12 7239 G39 t 1 d39 t f th 1 39 t 2 1V6 recLEE al cgga es or epgrPgIP 3 3 W f K gt a a r b 1V XPwsl9gt 1c gt3 35 I Vgt m E A m Points are speci ed in rectangular coordinates Give ulJpoELble polarcoordinates for each point quot P L1 at r 01 10 91c 2 2 4f34 I 1 V 1111 39BJ1 X coxm Wn39te the equation in polar coordinates a 39 s1n IN y 6 a g1n9 a at V 4a Haj x 2y3 J a Comm 2r3 93 2 7 1 x y 1 3 r2amp18 TSAnQ39 39 3 TEE93 r g m v gk zrsin9r 2 zrsmw 2 W Sketch the polar curve A J g R r cos 36 r sinZH HEX w M b r2s39u16 r I2c9 H n 0 UQ Wquot r1251u9 1 4 n etcH b eh 7 C m 1 A J W 19quot My N C we ML rdms 1 quotl 4 6 ln m Give the area inside one petal of r 300s 46 301A OUWL 3 CLIKS WW5 Give the area in the 1st quadrant that is inside the polar curve r 3 600s 69 xi W W 6TV 1V1 2 I 3 fifemm Part V Parametric Equations Wk1 Wu H x m g mj Express the parametric curve by an equation in x and y 4 2 FL 254 H xl 3sint J10 cosl L 3 3 s mlew cos a r b 1 2 3 Find a pummetrizution s l 39 m 39 m I e01 g 091 3 a J for the given cuth X L7 7 C The Ha sew ent from 3 7 to 8 5 2 The line segment from 2 6 to 6 3 The parabolic arc 39 7 39r1 from U 7 to 0 l CCJ 0U ale 3 FNMA 1 0 t Q a X 0 9 et 5 I C de lat k 20 Give tangent line and normal line to the curve at the point associated with the given value of t using both xy equations and parametric equations shah 1 1 Ema 3162 2 3 sin 521 005303 r E 193 4 I 431 E X E fib wi39 Wham f 2 239 1 a if e r situate 3W7 J 3 I l E gtlt TL 3393 Ha 43 mm t 2 3539mzzrx t 21 Find the points v r at which the curve has a a hon39zonxal tangent b a vertical tungan Then sketch lhc curve I3r73 I rll w r3 21 l39l r 12 393 4sint gt3943cosl l sian rr 2 sim o r172r V391tt3lz2r 39I2 Scost I3sinl We ny r Wok SW 039 w I We 9 313m 2 x it 2 tZ0 S4le 913 3 I MM j e iii X 5 qbo W1 5mm a Samt zt z 7 39t l 22 1 Give a formula fog the length of the curve given by A 9 xt2 3s39mtytcos3t P f mm rt39b 1H 91quot k 5 Maw 0 I J CXl 410M Ml o 3 l olah 40PM Galbdx dx 317 W1 The equations below give the position of a particle at each time t during the time interval speci ed Find the initial speed of the particle the terminal speed of the partical and the distance traveledhy the particle E f xii rill Fritzquot EL frmnr l to r 2 xii I 1i viz1t from I J to r 1 u m r3 Mfr firmn1r0 m I L 24L Eff7 MET ggt in iil iMk SP6 M 130 um 1an L magidwu alt c is Ksaj 13 bdi 9i 3 M mlm l d tiff5 lat I 73 Hill 24 ll g E uraui Part VI Sets Sequences LUB GLB Monotonicity and Limits 25 LuB GL3 Find the least upper bound if it exists and the greatest lower i0 quot4 g the given set lt4 1X llt2 v gts qrz gmt 22 2 xle2 30y Mfg 54 1 lt21 Ole n 2 26 399 NW mn3 havan 5 Dctern line the boundedness and monotonicity of the sequence 9 with 1 as indicated mongxyana 3 90 J 5quotquot 0391 1quot 7 391 n 39 n nlt L gtl m n n I I 39 bwm i n 7 k 09quot a n bbw du l 01J 3 b l v 954 quot I T Nak I2 u II 1 AIquot o 7 f O ho zI39 39 4n 39 441139 7 J43 1 Na PJLN n 1 he I 39 V 3quot 2m0 L 0 H2 6 T 21 39 A JI Inch w 1ln H 4 lt3 n1 RWY 1117 5 n 7 Ramp 9 Z V Wot is 9 W W WQ a I 1 Se LY 44 K0 17 gmmzw 3 J 154 ZELlt 0 A L 15 GuaraniQ 3 91sz 395 Wrw vumo rw I W W an m f 27 Test 3 Online Review Spring 2012 Sections 84 103 10 questions 3 multiple choice and 7 written o Trigonometric Substitution o Paititial Fraction Decomposition 0 Numerical Integration 0 Polar Coordinates o Parametric Curves 0 Sets Sequences LUB GLB Monotonicity and Limits i Part I Trigonometric Substitution For integgals involving I Use the substitution laz x2 X a 5119 ta2x2 X3 39 1Ulg 2 2 xx a x at 2 x 056 I 75 an m s new 49 5 quot 7S mm x 5 an19 e J 4 3 S I mam 7g c mote d1 Slc19d9 SuziLs E Tums Nam z 1 Lan19Xuczg 2 041 9 506 3 Gig gec 1 sum Eed 25 030 3 g z QnZ9 cume a klo Aort l Know W Kann zg swim a stdexalt 2g SCAJZEA VSecMMLQJ quoto m MALde 311 1quotst V MM 1 x I 4 x2dx2 S Li a H Su ax 2 c0309ng ej4 T x1SInB on zusM39lg z 2 CogQWM39 W W n s39m M cagLN amp us fm c049 Ll BQSZ9amp9 H i Coszoampamp gg sn19 C 6 MM 5quot j qM 20mm anwm 7 am SEA5 i g lH LLA zs n9 A 9 z X Zgg2 iaxqajf37 C 9 g t Zaras q3Z EHyl AIC 13gt Ms9 4 11 zarosin 2 EHquot 39 C 4 I Part II Pg al Fl tiOn Decomposition Setting I FEAT in where px and qx are polynomials q x 1 and the degree of q3 gt degree ofp Process 139 DE Pf 305 J 2x 1 2 2x4 x 3x 10 pgy zx alngru 2 by a WWW m 2x xzsxz 2 a B As 6 f6 x J 4 mom Xlt YX2 X Na f 9 WH 1029 Musz WM 33 2x PMM39A Jr 84 Hm x 0 Subdquot 76 61 7 9 7 R 9 EMA X 739 vS quot3973 7 B x gx1 x g x 5 2 239 imyde In M ill Jr i x 1 SV 1 H E n Na C gldx39sl 7 XA 7 my I 3 gt 392 IL 5gt 10 divmm V1W 4 MM W m 2X14 D a I1 0Y1 X10 lira0 7 1 r b l w CT aw l mu r LS ave 3 49 4 ELL A Q s WW WNW w X M WV 03 FM muHermu m b x7 1803 A ixi l 1 Agtlt1Lfr C3CXquot3 PM x7 Swim nan P LEE 9 K Iovlwaj 7 0 guLS V 7o I 2 egt 352 C gtL f I S Li 03 x1 SmbM39 Xivl 9 595 B I 3 7 3 3 g 44 W q in 2quot I Ex 5 xmx wx w XL L 2325er 7 i 39I 1x P Jr 5 s 309 22 A x Ixquotq lt xhwt 1 L oky 7 J In ll g amp 1L 7 I V Went A I LJo X 739 ALPX gyl 3H1 Xz Ak iafc gt c nX l lnx11 L gL39d V lnxz 1amw23 Q 2x2 1 x 1x21 70M thkaE 1 P 7 2x14 1 i E 4 C w 7 47 Q J LY 5ka Al B Q 1 Wt rdb YOU 1 A rmil 05313 Z N 51 W Part III Numerical Integration b 1 Tools for approximating I ak m 5441th 3 i i A l n70 whyl3 Set up M 3quot 1 13 ms 1 I H l Hng aa l I l I T a 13quot TH MMH lC M Qfxnihx tyit ng Mn Mam1r mg 3arw 4i mq gt FL in 39 n Estimate 16156 using the trapezoid method the midpoint method x and Simpson39s method with n 4 RSOJL ni394 05 3 g l X 8 K 3 quot 1 O H 2 L l Lm T Gem 1CMmHaMED Lax H Z L J L L 2 g i LCE3 4 iH gt MHL 3 mle o 2 u M A X FAJ if w W T4 Ms ch Hg 4m E E 3 g XA I 111J h rt 4 P a r HAW Add f nun Use the midpoint trapezoid and Simpson39s methods to estimate f Xdx with 114 amp I C xd q 1ng I 4 3 1 L53 5 I 03H 1lt ing 2I1 1 7 391 k l l 1 l gt 2 3 T 39 M WGOEVW 2SXamp3gtBL if H V pl 0 AL 0 1 Lzzbl310 H 4 m MA 2 O39Sg 10 0 5 ML13 l15PITISXAL3927SAV 1th J lt 3 14 332 L f I 2 5 QISS Jf 390 glb LI A c A 3 quot g g 3 1 3 H 3 Part IV Polar Coordinates J 31 de a arc n Q Fpr Quadranh was 940466 m hi I M4 L1 WAwa 65 2 quot1va 9 Mg W 8quot Q0 may x A K S W a ca4WJWlt W M Mw 0 9 3 m n FL 0 6 I 277 NWA mm W m39 8i W3 0 M X V 0939 2 3 z 34 M6 1 2 C SIM8X a 12 7239 G39 t 1 d39 t f th 1 39 t 2 1V6 recLEE al cgga es or epgrPgIP 3 3 W f K gt a a r b 1V XPwsl9gt 1c gt3 35 I Vgt m E A m Points are speci ed in rectangular coordinates Give ulJpoELble polarcoordinates for each point quot P L1 at r 01 10 91c 2 2 4f34 I 1 V 1111 39BJ1 X coxm Wn39te the equation in polar coordinates a 39 s1n IN y 6 a g1n9 a at V 4a Haj x 2y3 J a Comm 2r3 93 2 7 1 x y 1 3 r2amp18 TSAnQ39 39 3 TEE93 r g m v gk zrsin9r 2 zrsmw 2 W Sketch the polar curve A J g R r cos 36 r sinZH HEX w M b r2s39u16 r I2c9 H n 0 UQ Wquot r1251u9 1 4 n etcH b eh 7 C m 1 A J W 19quot My N C we ML rdms 1 quotl 4 6 ln m Give the area inside one petal of r 300s 46 301A OUWL 3 CLIKS WW5 Give the area in the 1st quadrant that is inside the polar curve r 3 600s 69 xi W W 6TV 1V1 2 I 3 fifemm Part V Parametric Equations Wk1 Wu H x m g mj Express the parametric curve by an equation in x and y 4 2 FL 254 H xl 3sint J10 cosl L 3 3 s mlew cos a r b 1 2 3 Find a pummetrizution s l 39 m 39 m I e01 g 091 3 a J for the given cuth X L7 7 C The Ha sew ent from 3 7 to 8 5 2 The line segment from 2 6 to 6 3 The parabolic arc 39 7 39r1 from U 7 to 0 l CCJ 0U ale 3 FNMA 1 0 t Q a X 0 9 et 5 I C de lat k 20 Give tangent line and normal line to the curve at the point associated with the given value of t using both xy equations and parametric equations shah 1 1 Ema 3162 2 3 sin 521 005303 r E 193 4 I 431 E X E fib wi39 Wham f 2 239 1 a if e r situate 3W7 J 3 I l E gtlt TL 3393 Ha 43 mm t 2 3539mzzrx t 21 Find the points v r at which the curve has a a hon39zonxal tangent b a vertical tungan Then sketch lhc curve I3r73 I rll w r3 21 l39l r 12 393 4sint gt3943cosl l sian rr 2 sim o r172r V391tt3lz2r 39I2 Scost I3sinl We ny r Wok SW 039 w I We 9 313m 2 x it 2 tZ0 S4le 913 3 I MM j e iii X 5 qbo W1 5mm a Samt zt z 7 39t l 22 1 Give a formula fog the length of the curve given by A 9 xt2 3s39mtytcos3t P f mm rt39b 1H 91quot k 5 Maw 0 I J CXl 410M Ml o 3 l olah 40PM Galbdx dx 317 W1 The equations below give the position of a particle at each time t during the time interval speci ed Find the initial speed of the particle the terminal speed of the partical and the distance traveledhy the particle E f xii rill Fritzquot EL frmnr l to r 2 xii I 1i viz1t from I J to r 1 u m r3 Mfr firmn1r0 m I L 24L Eff7 MET ggt in iil iMk SP6 M 130 um 1an L magidwu alt c is Ksaj 13 bdi 9i 3 M mlm l d tiff5 lat I 73 Hill 24 ll g E uraui Part VI Sets Sequences LUB GLB Monotonicity and Limits 25 LuB GL3 Find the least upper bound if it exists and the greatest lower i0 quot4 g the given set lt4 1X llt2 v gts qrz gmt 22 2 xle2 30y Mfg 54 1 lt21 Ole n 2 26 399 NW mn3 havan 5 Dctern line the boundedness and monotonicity of the sequence 9 with 1 as indicated mongxyana 3 90 J 5quotquot 0391 1quot 7 391 n 39 n nlt L gtl m n n I I 39 bwm i n 7 k 09quot a n bbw du l 01J 3 b l v 954 quot I T Nak I2 u II 1 AIquot o 7 f O ho zI39 39 4n 39 441139 7 J43 1 Na PJLN n 1 he I 39 V 3quot 2m0 L 0 H2 6 T 21 39 A JI Inch w 1ln H 4 lt3 n1 RWY 1117 5 n 7 Ramp 9 Z V Wot is 9 W W WQ a I 1 Se LY 44 K0 17 gmmzw 3 J 154 ZELlt 0 A L 15 GuaraniQ 3 91sz 395 Wrw vumo rw I W W an m f 27

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