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# Note for MATH 1432 with Professor Morgan at UH

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Date Created: 02/06/15

Test 3 Online Review Spring 2012 Sections 84 103 10 questions 3 multiple choice and 7 written Trigonometric Substitution o Partitial Fraction Decomposition 0 Numerical Integration 0 Polar Coordinates o Parametric Curves 0 Sets Sequences LUB GLB Monotonicity and Limits Part I Trigonometric Substitution For integgals involving I Use the substitution 07 laz xz X 01 5119 4612 xza39taa9 2 2 x a 1 Qt 2 x dx 75 an 9 s u g 49 I x san19 4 5 d1 Snc mdg 7 S 739 imam 5 CtNow SH ma m S Tan 9 2S hula 5 g me 1 739 9 Sec Vonquot9ch9 39 2g 4quot um w C192 gec9 1 SACM 3 93 2 lg g mam W HdB 9 Mia ImrdM Know WUquot Kann 1 ZS SCQCZIM 3 scc dB 2g W m v sec9xng I39o m ltme 311 1quotst m MM 139 x I 4x2dx E 2ws9 o9 eje T x FLSInB 9h QUEMllg z 2 Coglmom39 W NW u p swim ngM L1 BQSZ9J9 E Qng c0493 H Cos296 939 65gt H 39 sn19 C th 5quot j 0 b Xzzsb g 7 GE Wanda1 m 7 ffj 1 sin z zs n9 Co 91 sun98 ans93 C AX 293 I 7 9 LXI1 Arc 2 34 H 1 2015qu E L 1gt BAA A by larosjn j gl f X C4 2 4 Sewing I Process 1 PM 1 are Part II Pitial tiOn Dicgnposition dx where px and q are polynomials and le degree of qx gt degree 0fpx NataAra 2 q a W1 cm 2quotquot x sxz V r W Zx I Q E 6 inHA x ML R MMH b9 SVWA WW 33 2x4 395 66quot 73 A Bbpg mm gt3 yq 5amp3 7 3 E a 7 R 7 37 gum X 739 g 73 gt E 39 7 25v 97 SH 9 n f 139 Xgplt1 x S 3 2 i X a 5 MW 8 Z H S v ll H W MSW 7 739 My 7 3 gt 397 2x2 1 2 gt no divw 7 JWI x 1x394 u 3 v M 2X1 l be 4 39PV3 an 0 ng at CT Wow I k rub v qw lmwr 1 vJWvO bel i 4 B x A C k 1 PD W4 E wwth x X andA 1R0 MMH 1I Dw5J mmth 901 H aole 1 ix l GINA lt xgt 8045 RSVquot zg gt 9j guL 10 I gumxx mlmaj 7670 q 34 x sub X1 1 3 L s B 39XJ39 M ZB gt Br g L W 2 155 Y a x39 t X xinw t 7 1 in graib 397 i 1 39I 1 x x vx 1 X seq 9 XMJJK lnlwl ES 7 7 Erub LJ39X391iIz 1quot 4 S0 Sx x qq a XEWK ampgt nx1 anXIV 4 iarcangt 0 I M ZS 39nIMI glnbmg liomwnGB Q ve 2x2 1 x 1x22 d 70M minke J x4 7 Quay Flm k AA B CJ 1 Megan U 1 239 N Part III Numerical Integration b XX Tools for appr0X1mat1n ch t J g if MA 1 n7 0 quotWig 5 Setup 7 K 143 mg L I in ain I n I n 1 ETquot M3 M3 ma a Tn 1 11339quot 1n4l 3 Mn Hmm m a 34r m gt n ii Estimate 16156 using the trapezoid method the midpoint method x and Simpson s method with n 4 ll 5 O J L 3 i P394 05 X r o Mw 5 7 E 17 1 0 b E T 391 l L m TH was uwHaJrCKED HMS A y if 73A m LW Vi W W Ta Mg ch w Hg MUCH fa Rm lt I 11 1 J quoti mHhmvtk i 11 r a 7quot 10 Use the midpoint trapezoid and Simpson39s methods to estimate f xdx with n4 x f x Hm I a a o 1 w a 39 1 N21 h1lt mg 21g 1 7 39 l a I 5 gt i H 2 2 3 Z 3 2 T lt ZGIE 2 JFWZSgrMsBBLf 1 H L L 0 AL 0 1 Lz7quotl3 130tt quotI 2 w 39 3 4 1 7 98 g A 39 LL08 7 3 10 0 7 J ML1 L15 41hr Lz 1quot2 LCJ L lt 3 391 93 L f r 2 LgS 95 4 J c g Ib 2 a f g 3 Jiffy EM 3 3 11 Part IV Polar Coordinates J S nd f i f f g z 5a e arcf n X G Hr Qumran En6 Quirk Whirw I owl 1 mmHipM d B lt51 GUAMx NWMWM OdoA r alawt W Era ve 9g m w Ml Q0 mama K W v N r fra mwg W k M PW a 3 6 W Tr Mg 0 5 277quot ldasSgA WfIWJL K guns n 2 Eyes A LN L X V 0939 1 2 gt4quot 1 P Q sin8 9 9 12 E G1ve rec iglal cggmltes for the pill Pgt ff 3 W K gt 5 a r b 1 C08 35 I g I X ps9 3 M 42mm33 dE 3V3 3 4 m 13 Points are speci ed in rectmlgulzu coordinates Give all possible quot polar coordinates for each point A 0 at l 01 7310 91 1 2 4f34 I T z K V ws Im Wnte the equation in polar coordinates a 39 sln M y 6 a s1n9vg ar y u cswaj X 2y3 9 Cws9 Zr3n93 2 2 a x y 1 3 gt CZQDC8 mum3 3 HMS93 r g m m 2rsin9 r 2 2 V 2F v3 7 2 15 MA A d Sketch the polar curve FlAm Am 5 quot r cos 36 r sinzg 1493 w M K r2s39u16 r 1209 v h 90 UQKWA r1251u6 I394 n 377 CECEz 9 OX nut 19 OMAN A q J lute nb F541quot Us 6 LI 47quot 6 16 Give the area inside one petal of r 300s 49 014 00 1 CLMS WU5 3 Give the area in the 1st quadrant that is inside the polar curve r 3 600s Kx mm iono 69 a 3 9591quot gt r O g4le gt 30cosmto Cosbhl z e W TYZ 2 e v J Fwd B v LU 3 05059330 9 17 Part V Parametric Equations xl ayf WW H x39m g f j 18 Express the parametric curve by an equation in x and y 2 19 Find a pununetrizution 5 x m 39 2m IE 0 l g B 5 9 4 3 t J tor the glvcn cunt X 39 L7 7 C 3 The we 563191 from 3 7 to bk 5 The line segment 1quotmm2 6 to 6 3 The parabolic arc 39 1 7 39r3 from 0 7 L to 0 l CCJ A a 3 VW ll 0 t Q 3 X 39 0 g at e I C de 333 i 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 p ght a arametric equations a DE xt2 3si11rytcos3t3 7 Tug TY 2 1E eEJ 739 if x fig 39 r aim7 32 gnaw U n 3gmzzr s t 39F mt d u g x 8 i BAULQ asWbo trait X 2 3 3 61 e 41 lt 3 You C ln hmamp 05 it 21 Find the points v r at which the curve has a a horizontal tangent b 21 Vertical tangent Then skelch the curve rl3r7r I IIl w t3 21 r r 122 393 4sinl 39lb43cosl 39 sinZI rr 2 sim o I1172f ymxh3r12a 39I2 5ltosr I3sinl We WW quot NM SMFQ 039 J 0 1eg390 X39 M1 x io Nam Q 34539 at 2 34 2t 1 El 1TZU QIVL 9730 3 a M 0267 Sandi a Wild Xi g X0M a W 939 Sam WV LE 22 1 Give a formula fog the length of the curve given by xt2 3sinr yfcos3t Mo WNWNJ 39 39 DELQ FWWMNHL M 1W 2 2M quot t L 3 gmth 4 gsf memlu39 o r23cos6 0 37 3 l 0 M 40PM f H lbdx dx T ly Elm9 T41 23 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 traveled hty thg particle a 1th I2 PHI t from U to r xix I la it from I D to r L I Mr r2 Wis1ft 2 I3 t from 0 to I l g r r Ij Paula Kill Part VI Sets Sequences LUB GLB Monotonicity and Limits 25 LuB GL3 MM Find the least upper bound if it exists and the greatest lower iaOW A g the given set 39 x3 lt 4 x1 Ix 1 lt 2 v x3 3 8L r v 5 16 252525 N 3392 30 1 l l l 1 Z 3 4 In gt432 Z A 6 7 Q 0 399 t a ws bawdy Detern linc the boundedness and monotonicity of the sequence 9 with 1 as indicated mvhgxf me39 Ox E 54 3 360 J b an 71 yhd 12 lt O I 1 I n 1quot GA lew fquot moon v L 395 a band nil MCIa n 3 O k 09 quot l g 3 17 59 W 395 bbvmd on O 139 3 b 1 quotww We dwqu du 000 f M01 71I39 L w 239 4 9 WWW 41 V 4 x M I W New n r 4 13 M 31 2quot100 RWY ARInf l 2 A x H n quot1 7 1Hn i W lt1 quot117 29H WW SEW u gQQ 530 5 n 2 5 4 1 m V WOW 39 3 9 Q 5w 0 W N a N I A s 901 05 W 44 L 17 27

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