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## Multivariate Calculus

by: Dorothea Bode

11

0

30

# Multivariate Calculus MA 26100

Marketplace > Purdue University > Mathematics (M) > MA 26100 > Multivariate Calculus
Dorothea Bode
Purdue
GPA 3.97

Staff

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COURSE
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PAGES
30
WORDS
KARMA
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## Popular in Mathematics (M)

This 30 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 26100 at Purdue University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/208129/ma-26100-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15
WE 3 WE 4 WE 5 WE 6 WE 7 WE 8 WE 9 WE 10 WE 11 WE 12 WE 1 WE 2 WARMUP EXERCISES FOR TEST 1 TURKAY YOLCU Show that an equation for the tangent plane for the surface z 4x2 y2 21 at the point 1 24 is z 8x 2y This means that the tangent plane at the point 1 2 4 passes through the origin Is this true for any tangent plane on the given surface Why Show that an equation for the tangent plane for the surface z2 4x2 y2 29 at the point 12 2 is 8x 2y 42 4 0 At what point does the tangent plane intersect with the x axis 0 At what point does the tangent plane intersect with the y axis 0 At What point does the tangent plane intersect with the z axis Show that an equation for the tangent plane for the surface 22 4x2 y2 2y at the point 122 is 8x 2y 4z 2 4 0 At what point does the tangent plane intersect with the x axis 0 At what point does the tangent plane intersect with the y axis 0 At what point does the tangent plane intersect with the z axis Show that an equation for the tangent plane for the surface x22 yzz sin7rx2y2 2 at the point 111 is 1 27rx 227ry 32 647r What is the arclength of the curve rt th 4t Zln t for 1 S t S 4 What is the curvature of the curve rt t3 t2 t at t 0 What is the arclength of the curve rt t3 3t2 6t for O lt t S 2 At what point does the tangent line of Tt 239 th 4t Zln t at 240 intersect the yz plane A particle moves in space with acceleration at at t2 t and initial velocity and position given by v0 2 101 r0 0 10 At what point is the particle at time t 2 3 39 Suppose that z is de ned implicitly as a function of x and y by the equation y e quotz xcos7tyz 2xz y What is the value of 663 at e 1 1 Suppose that z is de ned implicitly as a function of x and y by the equation 2 y e JZ xcos7ryz ey xz39 What is the value of g at e 11 Let fxy z xcos y sin z 2 x03 q 213261 1 91 q1T V q2 132 leqr qr Show that the following partial derivatives hold true 3 r2 39 39 2 2 39 p e smqr sm xq 19 8111qu a 31 q2192 9f2r2 x 2 2 3211931 r z 2 b aq p e srnlt q p smqrlt qzpz q 1 TCosCI1 l c 9 213211T2 cos Wsinm Tl 192 q V 12 132 erz COS qr Sin V 12 132 Sin qu 6r Date September 25 2011 WE13 Let maE xstst yst1sze Show that the following partial derivatives are true E t sxE 38 2m t2sz1 t2521 39af s txE at ZxS txt2sz1 t2521 62f 1 r st v t2 35 Zx 39t39xt2sz1 4stt2sz1 2mt2szi 2 3stxH ZxEtt2sz1 t2sz1 WE 14 8139 h m2 739crx T2 112 represents the surface area of a cone with base radius r and height 11 Suppose that T 80 cm 11 60 cm are the measurements Also it is known that h is correct to within 06 cm and r is correct to within 08 cm What could be the largest possible error when the surface area is calculated from these measurements Hint use the differential d5 of S What is the percentage of the largest error in question WE 15 Let u 1 2 3 and v 211 Find a such that the vectorw a 7 3 is perpendicular to u x v WE 16 Let u 1 2 3 and v 1 1 Find an equation in 1 b and c such that the vector w CL b C is perpendicular to u x v WE 17 Let u 3 b 6 v 1 Z 3 and w 4 7 3 Find b such that the vector u is perpendicular to v x w WE 18 What is the directional derivative of the function f xy xcosy y sinx yequot at the point 0 7c and in the direction from 1 1 to 2 1 WE 19 Suppose that z is de ned implicitly as a function of x and y by the equation xyz Z3 2 1 Find the unit vector in which 2 fx y increases most rapidly at 1 2 1 WE 20 Find a vector tangent to the curve of intersection of the two surfaces zx2 y2 and xyz30 at the point 32 5 WE 21 Find parametric equations of the tangent line to the curve of intersection of the paraboloid z x2 y2 and the ellipsoid 4x2 y2 z2 9 at the point 1 12 WE 22 What is the intersection of Zyz 2x2 z2 22 5 O with the xy plane WE 23 What is the intersection of ZyZ 2x2 z2 22 5 O with the xz plane WE 24 What is the intersection of 2y2 2x2 z2 Zz 5 2 O with the yz plane WE 25 What are the critical points off x y 6x5 6y5 3x5y5 6 WE 26 What are the critical points of fx y 4x4 4y4 2x4y4 5 WE 27 What are the critical points of fx y y x2 y 3x2 WE 28 What are the critical points of fx y 1 xy x y l 1 WE 29 What are the critical points off xy x y X g WE 30 Show that 02 is the only critical point of fx y 6497 2 quot92 Also f0 2 is the local maximum value of fx y Is fO 2 absolute maximum WE 31 Find and classify the critical points of fx y x4 y4 4xy 1 WE 32 Show that 00 10 10 are the only critical points of fxy x2 y2eyzquotz Also 00 is a local minimum and 110 are saddle points WE 33 Show that 00 3 1 are the only critical points of fxy x3 27xy 27y3 Also 00 is a saddle point and f 3 1 is a local minimum 2 Samw WW WW W H 39 Q 1 t CD W FC Hxvs 1 I W 055 w mLD Pa he VF awn Mammy mme 3 0 M W V WNw ltLgx39lgl 4gt VF 9L 39 lt8 2 r gt 3 C03 ltZgtlta 23 Q L0 2 0 quot 8 6 C43 3ij HGb th 1 4X4 E 84Jrli 20 vampqwe W w r W a EWWE Cx COquot lawl t p E famp1 jaq D i Va ZVFX 39QgtDgt 2 lt 7 M1399x W gt A aquot 9M WWQOW W v I X x397 45a r a gt3 3 W b 0 1 24 1 30 3WD We xv lt2ng 33 130 v v M L I Q 1 YTgt gtX Y 391 q m 373 JVquot 5 1 7 beer 21353 25 onhla 360 lton 30 WV XDX k 997an A8 71 lfxbL V H O o o JNM WMM WMquot quotquotquotquotquotquotquot jgfyaw mwkioww WH4M W39QAWLP QVUL WWWWM W WM www wmq W231 WQ gufmmg g1 Qtfe lt23 WWWquot valvjtdrzz E2 FCC 243 VPth LW 139 lt53quot 1 13 2 gt KWCAJWD 4 393 39l quot392 4 gt Pm VW akCquot x quotWW r 4303 qu We AVEEms f HG WTELW 811 7 quotS m 1 gee02 0 620 8 KAW9M W m W39MC SYN A OI2OblxLWS A 5 gt 2 39 m m Ore265 VOWVD 2441 37 399 CFO Pp wfangMW Br a 2 quot5630103 39 396 ML a 10 a 3mm V 396 1 I aim7 x L 1 ltj 3amp 39FWMV XE I gr 9M XQHQ vF c AFX 7 FY Wah k satgt 397quot quotL lt 1 CO x1b Q wig 33 P093 QWXQX L mix 33 Qxxt f U V 1 WAVE 2 VF m 4 Harm 9mm 3 gt SSW W WN w k MAM gr QmeW 2H7 QMme 34 Ra 0 Val x 4 QiJv x E 3 GHFG III a Li A FMS g 65 t 0U a mnywx r J 0 0 it A 7 Id c mv lt3 Ill A Jmmv 151 rd A9 euzwoq v n qu Fxmsddqu x n 5 k fax Cs 3 A 33 lv 65 am 3 a H51an 6 rm H z al wmm m ray 3 3 i m ui fl W I r rx nu Nvn Nb ngm 0 NW i w LW 3 g s 4 ti gr w u PWYST a5 N 5 oil 9 x V Adm EJEV ER A p dxu swam V Hafiz 1 9lt fquotz tgt f 1 quotW39 gt Rpm k39gt16 tf L 732063 39539 43 7 I 0 47 BE Z36 3 7 haw I V 2 jg us9x A t i an L Eu 392CampJamp 3lt3 4 UGO O gtL23lt3 0 F H ntgt 30AH 0gt CEW 1amp1 gt 2 quotPM 7 4w L gt Mm F303 633 ltQ Lh0gt 39 C ltLHL gt vCm 42LF17 y L ch 1336 gt ltgtx 7 guqkzo ggt bz a go LHLvGD arlt3o2 39 m MszYSLU WM GD aim Swat Saa mwc lte I g 7 5 0 gt lt 0 0gtCquotquotM01gt C A V AC 6 36 I VWW lt yE Kgt 5 3vc FL ZltOIOHgt 393 igt cgt l 4 N 9 s L WEN SVOAOUV BLQAC ii A gt AA39 4Q 2 5 I M J r 12lt om gt 41gt MB 0gtquotf CZ My 1 CZ 4 4 no gt 3 gt A 7 L 4 g 39 0 Q 3 3 5 m w w Erma C W J w WW 6 x 1 I 39 f 9E 2 05ltW yc gxa 7 g G ggwma QHH I hwzo g I w 3 3 1 5L e gtlt TCVT8gtCWZ W3 924 v 3 C 7 WX9WWggt M W X b SmCiTj x ig 3 39 I 9 mxfbgmf q 9QX H3 33 W 9 99 bag WEWX3 mCWa EQQ x 39 2 V W39 WW 3 9396 G WM quot w M m 9 we mwwm 321 7 I rateO QQw a2 QYL39O 2zI a quotr gt I r e 6 pcL 3977 fl XI 3 9 kAP V gi l39m Q39T CQv X W95 Ww1 9421 3 9X 3gt WC jDH gW w X 29 96 R I I a WVW CWW 9 5 Cg Cm will ngqi gt f4 2 a a 39 wi c CpSQG 51 W40 M i I O O w 39 39 quot 39 mi 3 qu 39 vTr39 E39C DSmG O raeg e we C HQ g i mm 1 3 Wm 0va gm 39 5 WW Mn 39W W 7k3r Ag 1 3 quot5quot 3 V 3 rm C WM 39 2 quot 2 L uf 39ji V Y RT JV Tr 2Tb t 8 m 0 x h 6 O A M5 45 0 8 AS AK 3939 dhx Akk 0 6 T12 CMlt W M Mm L WTquot BLOW iii2 m i L W zokO rwampxgt0 lt 06 00 0 CB 7 947 11 I 3 0393 ANA o 339 WRKBQLLVCU Dv r 18 040 9 Q W Uk ark lt0 quotq y3gt S ltquotV 2 3 gt quotJV t lt2m gt a S2t G 31 dkzquot gt OLZQ 4 4 V Ss m MWQ H Vow M Q 133 WWJUCMQM39 0 71X 7 P3 4 Eu U0 g UKVg ax W W have M CMWSquot qu ME Q5 1t Cc aswt Yin BS 34 9 S quotquot5quot 1 919 39L Z 5 3b C 2a JC 31944211 5 5 quotb 5amp2 2C2Jo W A L I U 309 WltLhq5gt R7 ltP 217Dgt 3 4 4 M3 JCVJFSW ampglblggt ACltV397quot3Ygt quottquot g lt Lfl 413gt quot3 c Ltg quot39EA39LKS bra 25 PS O 1477LS 6 95 ugb s m 45 0 1b 0 WMCX 6980quot M Q lt2quot 4 p 7gt 14M2gt Ca T S A39Q 0 3 gt V 44clgt 5 lth7gt M r 2 a 01590 WV W SQ 0 o 0 t z 0 To 0 gem 5L01Wgtgto a a 47587 Vow M gtx W 29 wwx r Mo rgtrltWiVW I fZX 3 adv4r WLx 3 30 I 0 0 quot4 Dug hq gilt gt a 442 gt 5 QW W W 3175 ga x w X Cf gt if Orh y A K w w WWK mmw g WW Vi WinM WW i gx gg gt ltx 10 2 3 V x 9x O33 3 W0 9 L 93 fwm yaScf 3 37 wO g A 9 2 W c quot M W CWSZQJK 55L M SAW m g I 21 Fx NNW Q rhj 1 XH3 m 97 FL M 4 x33t31 2 0 23 g3 3 quot x Ci m we 0amp0 W 93 Xy f 31 ora 1 9 jayen M i h L I 9y gtlt Wr 0431 N f x 3 I J 7043 f 2 i t I I 39 39 93 3311 quot Mme frquot lt39 1a gt 4 1quot gt if m w M Z binlt92 WarL H4449 W a c 30 Cwmi 1147 Fo lm r 30 v veg22m NFCW57 71gtlts9 r lt3 P07 Xgt VF 425 lt10 43 4 gt FZKM M Vlt urgt x ltosle gt t v6 is J Ws59 39036w vv 39 10 xs 4 43 1 Iquot A A A I 1 veg ky zok y L L 7 L 1 0 SKquot FX9ugtz39j X f rt 7066118fo 7 if 7 V 3 f quotquotquotquot quotf QMQ 9mg 9M o 4 KEV mam 9pm W m aquot VG 1 5 M N 39 v m gt6 AL 39 VFW m 41x 23 4 gt M A 5 a 2 VFCMVL lt9 glgt i f f rawt WW 39 L 3 A 1 7C 7 39734 M3Mrgt I o 4 m 391 9 b awe ltymgt 4 ltob 41 gt X z 1 v M 3 141 5 CL 91ch 4M 7 WW ltgt 39ng ZZgt 231Qx1 amp10 50 quotL 1 j x z 19 WWW 42gt 30 3gt 2 7 3quot 116 W0 20 1 t 36 373 lt12 va H D 5 rquot 4 0304 z 0 O m0 938an legsol cm V 95 0 L 9A3 643 H r 1 4 1 Z I a 7 V x 656er 6g7lt9z f w 7 Ls Hlta30 D I V 5 39 x IX30XL JX 3 O gt 5 1C 0 30 t5x gf 20 5 Qf39x wtj 2 k1 I D L 1 5 E 7X C 0gt3ltRa 20 2 3 9 g3 x 433 5 mm W CWHCOQ Mg WA QED A 3 m6 mkZo 4691 wa p K ngav yda TVA Tsugzav 93 av i3 3 E O H Ayxrdvo 3 0 unrmxjmvex nw on 4qu Image V4 m T r 2 I r i rxi H m er V A XL CW SV MAL CmrHch x Pmka S j I RVOQW 3 yri Q x g 95fqu W S 9470lt 31 K gxxdf W 9 I MN W r Cyrwco gom we W L 7 39 L 1 lt3 0911 0 637 7c Am X XO x Kg 09 93 2 14 13 W9WQ sicOLQ V A 9 a ww cs m quzo a v3 0 79v 2223 30 3 so 9 10 0 V59 3 ng 7 Cf 9 2 9 DEA SYXXUDM Sy o t I Qm w t O P in 4951 lt0 5 ON 3 k WS NV E I a 9 Max SRXWX New ampxjs3gt gm W nyxXG DZW gt0 K G 3132 w 36 5 gt0 VQDCOKQ YMMMUlkm WE 3 WE 4 WE 5 WE 6 WE 7 WE 8 WE 9 WE 10 WE 11 WE 12 WE 1 WE 2 WARMUP EXERCISES FOR TEST 1 TURKAY YOLCU Show that an equation for the tangent plane for the surface z 4x2 y2 21 at the point 1 24 is z 8x 2y This means that the tangent plane at the point 1 2 4 passes through the origin Is this true for any tangent plane on the given surface Why Show that an equation for the tangent plane for the surface z2 4x2 y2 29 at the point 12 2 is 8x 2y 42 4 0 At what point does the tangent plane intersect with the x axis 0 At what point does the tangent plane intersect with the y axis 0 At What point does the tangent plane intersect with the z axis Show that an equation for the tangent plane for the surface 22 4x2 y2 2y at the point 122 is 8x 2y 4z 2 4 0 At what point does the tangent plane intersect with the x axis 0 At what point does the tangent plane intersect with the y axis 0 At what point does the tangent plane intersect with the z axis Show that an equation for the tangent plane for the surface x22 yzz sin7rx2y2 2 at the point 111 is 1 27rx 227ry 32 647r What is the arclength of the curve rt th 4t Zln t for 1 S t S 4 What is the curvature of the curve rt t3 t2 t at t 0 What is the arclength of the curve rt t3 3t2 6t for O lt t S 2 At what point does the tangent line of Tt 239 th 4t Zln t at 240 intersect the yz plane A particle moves in space with acceleration at at t2 t and initial velocity and position given by v0 2 101 r0 0 10 At what point is the particle at time t 2 3 39 Suppose that z is de ned implicitly as a function of x and y by the equation y e quotz xcos7tyz 2xz y What is the value of 663 at e 1 1 Suppose that z is de ned implicitly as a function of x and y by the equation 2 y e JZ xcos7ryz ey xz39 What is the value of g at e 11 Let fxy z xcos y sin z 2 x03 q 213261 1 91 q1T V q2 132 leqr qr Show that the following partial derivatives hold true 3 r2 39 39 2 2 39 p e smqr sm xq 19 8111qu a 31 q2192 9f2r2 x 2 2 3211931 r z 2 b aq p e srnlt q p smqrlt qzpz q 1 TCosCI1 l c 9 213211T2 cos Wsinm Tl 192 q V 12 132 erz COS qr Sin V 12 132 Sin qu 6r Date September 25 2011 WE13 Let maE xstst yst1sze Show that the following partial derivatives are true E t sxE 38 2m t2sz1 t2521 39af s txE at ZxS txt2sz1 t2521 62f 1 r st v t2 35 Zx 39t39xt2sz1 4stt2sz1 2mt2szi 2 3stxH ZxEtt2sz1 t2sz1 WE 14 8139 h m2 739crx T2 112 represents the surface area of a cone with base radius r and height 11 Suppose that T 80 cm 11 60 cm are the measurements Also it is known that h is correct to within 06 cm and r is correct to within 08 cm What could be the largest possible error when the surface area is calculated from these measurements Hint use the differential d5 of S What is the percentage of the largest error in question WE 15 Let u 1 2 3 and v 211 Find a such that the vectorw a 7 3 is perpendicular to u x v WE 16 Let u 1 2 3 and v 1 1 Find an equation in 1 b and c such that the vector w CL b C is perpendicular to u x v WE 17 Let u 3 b 6 v 1 Z 3 and w 4 7 3 Find b such that the vector u is perpendicular to v x w WE 18 What is the directional derivative of the function f xy xcosy y sinx yequot at the point 0 7c and in the direction from 1 1 to 2 1 WE 19 Suppose that z is de ned implicitly as a function of x and y by the equation xyz Z3 2 1 Find the unit vector in which 2 fx y increases most rapidly at 1 2 1 WE 20 Find a vector tangent to the curve of intersection of the two surfaces zx2 y2 and xyz30 at the point 32 5 WE 21 Find parametric equations of the tangent line to the curve of intersection of the paraboloid z x2 y2 and the ellipsoid 4x2 y2 z2 9 at the point 1 12 WE 22 What is the intersection of Zyz 2x2 z2 22 5 O with the xy plane WE 23 What is the intersection of ZyZ 2x2 z2 22 5 O with the xz plane WE 24 What is the intersection of 2y2 2x2 z2 Zz 5 2 O with the yz plane WE 25 What are the critical points off x y 6x5 6y5 3x5y5 6 WE 26 What are the critical points of fx y 4x4 4y4 2x4y4 5 WE 27 What are the critical points of fx y y x2 y 3x2 WE 28 What are the critical points of fx y 1 xy x y l 1 WE 29 What are the critical points off xy x y X g WE 30 Show that 02 is the only critical point of fx y 6497 2 quot92 Also f0 2 is the local maximum value of fx y Is fO 2 absolute maximum WE 31 Find and classify the critical points of fx y x4 y4 4xy 1 WE 32 Show that 00 10 10 are the only critical points of fxy x2 y2eyzquotz Also 00 is a local minimum and 110 are saddle points WE 33 Show that 00 3 1 are the only critical points of fxy x3 27xy 27y3 Also 00 is a saddle point and f 3 1 is a local minimum 2 Samw WW WW W H 39 Q 1 t CD W FC Hxvs 1 I W 055 w mLD Pa he VF awn Mammy mme 3 0 M W V WNw ltLgx39lgl 4gt VF 9L 39 lt8 2 r gt 3 C03 ltZgtlta 23 Q L0 2 0 quot 8 6 C43 3ij HGb th 1 4X4 E 84Jrli 20 vampqwe W w r W a EWWE Cx COquot lawl t p E famp1 jaq D i Va ZVFX 39QgtDgt 2 lt 7 M1399x W gt A aquot 9M WWQOW W v I X x397 45a r a gt3 3 W b 0 1 24 1 30 3WD We xv lt2ng 33 130 v v M L I Q 1 YTgt gtX Y 391 q m 373 JVquot 5 1 7 beer 21353 25 onhla 360 lton 30 WV XDX k 997an A8 71 lfxbL V H O o o JNM WMM WMquot quotquotquotquotquotquotquot jgfyaw mwkioww WH4M W39QAWLP QVUL WWWWM W WM www wmq W231 WQ gufmmg g1 Qtfe lt23 WWWquot valvjtdrzz E2 FCC 243 VPth LW 139 lt53quot 1 13 2 gt KWCAJWD 4 393 39l quot392 4 gt Pm VW akCquot x quotWW r 4303 qu We AVEEms f HG WTELW 811 7 quotS m 1 gee02 0 620 8 KAW9M W m W39MC SYN A OI2OblxLWS A 5 gt 2 39 m m Ore265 VOWVD 2441 37 399 CFO Pp wfangMW Br a 2 quot5630103 39 396 ML a 10 a 3mm V 396 1 I aim7 x L 1 ltj 3amp 39FWMV XE I gr 9M XQHQ vF c AFX 7 FY Wah k satgt 397quot quotL lt 1 CO x1b Q wig 33 P093 QWXQX L mix 33 Qxxt f U V 1 WAVE 2 VF m 4 Harm 9mm 3 gt SSW W WN w k MAM gr QmeW 2H7 QMme 34 Ra 0 Val x 4 QiJv x E 3 GHFG III a Li A FMS g 65 t 0U a mnywx r J 0 0 it A 7 Id c mv lt3 Ill A Jmmv 151 rd A9 euzwoq v n qu Fxmsddqu x n 5 k fax Cs 3 A 33 lv 65 am 3 a H51an 6 rm H z al wmm m ray 3 3 i m ui fl W I r rx nu Nvn Nb ngm 0 NW i w LW 3 g s 4 ti gr w u PWYST a5 N 5 oil 9 x V Adm EJEV ER A p dxu swam V Hafiz 1 9lt fquotz tgt f 1 quotW39 gt Rpm k39gt16 tf L 732063 39539 43 7 I 0 47 BE Z36 3 7 haw I V 2 jg us9x A t i an L Eu 392CampJamp 3lt3 4 UGO O gtL23lt3 0 F H ntgt 30AH 0gt CEW 1amp1 gt 2 quotPM 7 4w L gt Mm F303 633 ltQ Lh0gt 39 C ltLHL gt vCm 42LF17 y L ch 1336 gt ltgtx 7 guqkzo ggt bz a go LHLvGD arlt3o2 39 m MszYSLU WM GD aim Swat Saa mwc lte I g 7 5 0 gt lt 0 0gtCquotquotM01gt C A V AC 6 36 I VWW lt yE Kgt 5 3vc FL ZltOIOHgt 393 igt cgt l 4 N 9 s L WEN SVOAOUV BLQAC ii A gt AA39 4Q 2 5 I M J r 12lt om gt 41gt MB 0gtquotf CZ My 1 CZ 4 4 no gt 3 gt A 7 L 4 g 39 0 Q 3 3 5 m w w Erma C W J w WW 6 x 1 I 39 f 9E 2 05ltW yc gxa 7 g G ggwma QHH I hwzo g I w 3 3 1 5L e gtlt TCVT8gtCWZ W3 924 v 3 C 7 WX9WWggt M W X b SmCiTj x ig 3 39 I 9 mxfbgmf q 9QX H3 33 W 9 99 bag WEWX3 mCWa EQQ x 39 2 V W39 WW 3 9396 G WM quot w M m 9 we mwwm 321 7 I rateO QQw a2 QYL39O 2zI a quotr gt I r e 6 pcL 3977 fl XI 3 9 kAP V gi l39m Q39T CQv X W95 Ww1 9421 3 9X 3gt WC jDH gW w X 29 96 R I I a WVW CWW 9 5 Cg Cm will ngqi gt f4 2 a a 39 wi c CpSQG 51 W40 M i I O O w 39 39 quot 39 mi 3 qu 39 vTr39 E39C DSmG O raeg e we C HQ g i mm 1 3 Wm 0va gm 39 5 WW Mn 39W W 7k3r Ag 1 3 quot5quot 3 V 3 rm C WM 39 2 quot 2 L uf 39ji V Y RT JV Tr 2Tb t 8 m 0 x h 6 O A M5 45 0 8 AS AK 3939 dhx Akk 0 6 T12 CMlt W M Mm L WTquot BLOW iii2 m i L W zokO rwampxgt0 lt 06 00 0 CB 7 947 11 I 3 0393 ANA o 339 WRKBQLLVCU Dv r 18 040 9 Q W Uk ark lt0 quotq y3gt S ltquotV 2 3 gt quotJV t lt2m gt a S2t G 31 dkzquot gt OLZQ 4 4 V Ss m MWQ H Vow M Q 133 WWJUCMQM39 0 71X 7 P3 4 Eu U0 g UKVg ax W W have M CMWSquot qu ME Q5 1t Cc aswt Yin BS 34 9 S quotquot5quot 1 919 39L Z 5 3b C 2a JC 31944211 5 5 quotb 5amp2 2C2Jo W A L I U 309 WltLhq5gt R7 ltP 217Dgt 3 4 4 M3 JCVJFSW ampglblggt ACltV397quot3Ygt quottquot g lt Lfl 413gt quot3 c Ltg quot39EA39LKS bra 25 PS O 1477LS 6 95 ugb s m 45 0 1b 0 WMCX 6980quot M Q lt2quot 4 p 7gt 14M2gt Ca T S A39Q 0 3 gt V 44clgt 5 lth7gt M r 2 a 01590 WV W SQ 0 o 0 t z 0 To 0 gem 5L01Wgtgto a a 47587 Vow M gtx W 29 wwx r Mo rgtrltWiVW I fZX 3 adv4r WLx 3 30 I 0 0 quot4 Dug hq gilt gt a 442 gt 5 QW W W 3175 ga x w X Cf gt if Orh y A K w w WWK mmw g WW Vi WinM WW i gx gg gt ltx 10 2 3 V x 9x O33 3 W0 9 L 93 fwm yaScf 3 37 wO g A 9 2 W c quot M W CWSZQJK 55L M SAW m g I 21 Fx NNW Q rhj 1 XH3 m 97 FL M 4 x33t31 2 0 23 g3 3 quot x Ci m we 0amp0 W 93 Xy f 31 ora 1 9 jayen M i h L I 9y gtlt Wr 0431 N f x 3 I J 7043 f 2 i t I I 39 39 93 3311 quot Mme frquot lt39 1a gt 4 1quot gt if m w M Z binlt92 WarL H4449 W a c 30 Cwmi 1147 Fo lm r 30 v veg22m NFCW57 71gtlts9 r lt3 P07 Xgt VF 425 lt10 43 4 gt FZKM M Vlt urgt x ltosle gt t v6 is J Ws59 39036w vv 39 10 xs 4 43 1 Iquot A A A I 1 veg ky zok y L L 7 L 1 0 SKquot FX9ugtz39j X f rt 7066118fo 7 if 7 V 3 f quotquotquotquot quotf QMQ 9mg 9M o 4 KEV mam 9pm W m aquot VG 1 5 M N 39 v m gt6 AL 39 VFW m 41x 23 4 gt M A 5 a 2 VFCMVL lt9 glgt i f f rawt WW 39 L 3 A 1 7C 7 39734 M3Mrgt I o 4 m 391 9 b awe ltymgt 4 ltob 41 gt X z 1 v M 3 141 5 CL 91ch 4M 7 WW ltgt 39ng ZZgt 231Qx1 amp10 50 quotL 1 j x z 19 WWW 42gt 30 3gt 2 7 3quot 116 W0 20 1 t 36 373 lt12 va H D 5 rquot 4 0304 z 0 O m0 938an legsol cm V 95 0 L 9A3 643 H r 1 4 1 Z I a 7 V x 656er 6g7lt9z f w 7 Ls Hlta30 D I V 5 39 x IX30XL JX 3 O gt 5 1C 0 30 t5x gf 20 5 Qf39x wtj 2 k1 I D L 1 5 E 7X C 0gt3ltRa 20 2 3 9 g3 x 433 5 mm W CWHCOQ Mg WA QED A 3 m6 mkZo 4691 wa p K ngav yda TVA Tsugzav 93 av i3 3 E O H Ayxrdvo 3 0 unrmxjmvex nw on 4qu Image V4 m T r 2 I r i rxi H m er V A XL CW SV MAL CmrHch x Pmka S j I RVOQW 3 yri Q x g 95fqu W S 9470lt 31 K gxxdf W 9 I MN W r Cyrwco gom we W L 7 39 L 1 lt3 0911 0 637 7c Am X XO x Kg 09 93 2 14 13 W9WQ sicOLQ V A 9 a ww cs m quzo a v3 0 79v 2223 30 3 so 9 10 0 V59 3 ng 7 Cf 9 2 9 DEA SYXXUDM Sy o t I Qm w t O P in 4951 lt0 5 ON 3 k WS NV E I a 9 Max SRXWX New ampxjs3gt gm W nyxXG DZW gt0 K G 3132 w 36 5 gt0 VQDCOKQ YMMMUlkm

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## Why people love StudySoup

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#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

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Forbes

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