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by: Madelynn Herman

CalculusIII MATH234

Madelynn Herman
Loyola Marymount University
GPA 3.95


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This 32 page Class Notes was uploaded by Madelynn Herman on Thursday October 15, 2015. The Class Notes belongs to MATH234 at Loyola Marymount University taught by Staff in Fall. Since its upload, it has received 86 views. For similar materials see /class/223481/math234-loyola-marymount-university in Mathematics (M) at Loyola Marymount University.

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Date Created: 10/15/15
H F 9 4 Classroom Voting Questions Multivariable Calculus 91 3D coordinates You awaken one morning to nd that you have been transferred onto a grid which is set up like a standard right hand coordinate system You are at the point 17 3 37 standing upright7 and facing the XZ plane You walk 2 units forward7 turn left7 and walk for another 2 units What is your nal position a 1717391 3717393 C 3757393 1717 3 The point 27 17 2 lies on the graph of the sphere x 712 y l2 22 9 a True b False Which of the following points lies closest to the point 123 a b c d 303 042 241 234 9 2 Vectors Match the vectors with their diagrams 1uv A lt C39 2 uV 3i2vu 3 D 4v12u U 03 5 00 3 True or False The vector 7 is a unit vector a True b False 93 The Dot Product Theonlywaythati7u70isif 00rw0 a True b False Two vectors have a dot product of 14 To guarantee the dot product is equal to 287 you could double the angle between the vectors double the length of both vectors double the length of one vector a b C d none of the above The angle between the vectors 7x 7 1 and 23 7 31 a b C d can be any of the above depending on the value of x is 0 degrees is less than 90 degrees is greater than 90 degrees A 100 rneter dash is run on a track in the direction of the vector 17 2i 63 The wind velocity is 13 5 3 krnhr The rules say that a legal wind speed measured in the direction of the dash must not exceed 5 krnhr Will the race results be disquali ed due to an illegal wind a Yes7 the race results will be disquali ed because the wind exceeds 5 krnhr in the direction of the race b No7 the race results will not be disquali ed because the wind exceeds 5 krnhr in the direction of the race c There is not enough information to answer this question 94 The Cross Product V 10 For any vectors 1 and 17 u gtlt 17 gtlt 17 gtlt 12 17 gtlt 12 gtlt 12 gtlt 17 a True b False 11 For the vectors 6 4 7 2 and g 7 55 31 the cross product 6 gtlt 3 is a i13i14319k b 13i143719i c i13i714519k d 1371437191 12 Eid x a a H5H2H5H b 0 C Md 5 d a N7 96 Functions and Surfaces 13 The domain of ay lnz y 71 is a 9679 y gt 1 796 b 9679 y gt z 7 1 C 9679 9H1 gt 0 d 9679 9H1 lt1 14 Match the graphs with their function i 216727y2 a 2 t 3 aeg t o ii 2 fey s quot34 4 iii 2 sin z r 4 7 iv 2 I y21 EUQQOD F rm dmv 8 A EFL do 5 va Smomm 6m do rm 3 7 aa gum w 9V 1 uu w I 55 3333th wmhhhheu gath a ham m g 00090000 V uuu h M gmquotthan w 35 o lt M hzh hhhh hh t thhth h hh hahdw xP I llI 1111111111 111111111111 111 ii 0 11111111111n1111 0 1111111111111111111 0 111111111111 quot31313 I ll i 97 Cylindrical and Spherical Coordinates 16 Match the equations in polar coordinates with the curves they de ne i 9 g a ii r1200s0 0 iii r sin 859 0 iv 7 sin 6 d V r 6 e 17 Match the equations in cylindrical coordinates with the surfaces they de ne note that only parts of the surfaces may be shown i z 6 a ii 2 7quot 0 iii 7 12sin0 iv 2 sint9 IJ A 10 4 4 l V O O S w s m iii p ii Mb I 18 Match the equations in spherical coordinates with the surfaces they de ne 101 Vector Functions and Space Curves 19 Match the vector functions with the corresponding space curves 1 770 lt6 cos10tet sin101t7 6tgt a ii Ft lttt2tsin3tgt 0 iii 770 ltcostsintsin5tgt 0 iv 770 ltcostsint7 lntgt 10 d D 0 gtA 104 Motion in Space Velocity and Acceleration Which of the following describes the motion of a particle that is moving along a straight line and slowing down a nd 17 are parallel and point in the same direction b c d AA a and 17 are parallel and point in opposite directions a nd 17 are perpendicular A c7 c7 c7 None of the above particle with constant speed must have zero acceleration a True b False A particle with zero acceleration must have constant speed a True b False Which of the vectors shown could possibly be the velocity vector for a particle moving along the curve in the direction shown 24 Which of the vectors shown could possibly be the acceleration vector for a particle moving along the curve in the direction shown 105 Parametric Surfaces 25 Match the vector functions with the corresponding surfaces u l ltu 0711 7010 iv Fuv lt027u117u2gt um lt1 cosuvsinuvgt iii Fuv lt1 cosuvsinuugt V Tum ltsin2v cosusin2v sinu7 2cos1gt A wx on VV ii I lll 26 Which of the following is a parametric description of the surface created by rotating the curve z cosz in the xz plane about the z axis ltcos cos 1 7 cos sinv7 ugt a WM b WM ltu7 cos cos 17 cos sinvgt ltcos cos 1 7 u7 cos sinvgt 0 WM d WM ltsinu cos17 sinu sinv7 ugt ltcos cos 1 7 cos sinv7 vgt 8 WM 111 Functions of Several Variables 27 Distinct level curves of a function ay can never intersect a True b False 28 Using the contour plot picturedl7 which path will result in the greatest change in alti tude D A c a From A to B b From C to B c From D to B d All changes in altitude are approximately equal 29 Using the contour plot picturedl7 which path is the steepest D a From A to B b From C to B c From D to B d All paths are approximately the same steepness 13 we Z303 5 0085 23m 233 5 535 a A 1 n CE N to o t 4 oiuiouo uo hitcihocoo V 30 iii x h 31 Match the contour plots with the corresponding functions 32 Any surface that is a graph of a 2 Variable function z ay can be thought of as a level surface of a function of 3 variables a True b False 33 Match the level surfaces with the corresponding functions only parts of the level surfaces are shown 1 fz7y7zz27y272 iii fyzWi4222 ii flt7y72 zy 7962 iv fzyz z2y2 a c r r tram Ear 93 M w oea 13 2 34 Suppose the temperature at time t at the point x7 y is given by the function Tz7 y t 5t 7 2 7 yz Which of the following will not cause temperature to decrease a moving away from the origin in the positive m direction c d standing still and letting time pass b moving away from the origin in the positive y direction moving away from the origin in the direction of the line y z 113 Partial Derivan39ves as Usmglhe cmmur PM a My Wm n he Mowing m at me mm 7 as vsmg mm mm cf My whahn he knowing dam mike perms dmve uve cf 1 mm mm m at 7 3 5 03 30 40 a40 2 C7 O c d 1 4 O Suppose that the price P in dollars to purchase a used car is a function of 0 its original cost also in dollars and its age A in years So P fC A What are the signs of the partial derivatives a gt 0 b E 3P gt0 3P lt0 3P lt0 gt0 lt0 gt0 lt0 c d 38 Which of the following functions satisfy Euler7s Equation zfm yfy f zys xy1 2 yz 06 a f b f c f d f 39 There exists a function fy with f1 2y and fy 2x a True b False 4 4 4 4 4 0 H 3 9 7 114 Tangent Planes and Linear Approximation Let f23 7 fm23 71 and fy23 4 Then the tangent plane to the surface 2 fy at the point 23 is a 277x4y b zi4yz30 c 7x4y27 d iz4yz30 e zl7z74y Which of the following could be the equation of the tangent plane to the surface 2 2 y2 at a point ab in the rst quadrant a 273x4y7 b 22z74y5 c 266y718 d 274x74y24 A small business has 300000 worth of equipment and 100 workers The total monthly production P in thousands of dollars is a function of the total value ofthe equipment V in thousands of dollars and the total number of workers N The differential of P is given by dP 49dN 05dV If the business decides to lay off 3 workers and buy additional equipment worth 20000 then a Monthly production increases b Monthly production decreases c Monthly production stays the same 116 Directional Derivatives and Gradient Vectors In which direction is the directional derivative of z x2 y2 at the point 23 most positive a b 7 3 4 3 245 c d 239 The surface of a hill is modeled by z 25 7 2x2 7 4y2 When a hiker reaches the point 1 l 19 it begins to rain She decides to descend the hill by the most rapid way Which of the following vectors points in the direction in which she starts her descent 19 Cf 03 At which of the points ll7 QR7 S in the gure below does the gradient have the largest magnitude 11 fIv39 1 a C d cr mWQ Suppose the temperature at a point Luz in a room is given by Tzy7 Suppose heat is being radiated out from a hot spot at the origin Which of the following could be VTabc where abc are all positive 2 23 i 41 b 73 i 3339 i 51 72 23 51 3 35 51 117 Maximum and Minimum Values 47 Which of these functions has a critical point at the origin 3 1196711 962 213 b fwy 9621 4961 4y 0 f M 96213 i 964 2y d cosy 48 How would you classify the function ay zzy xy at the origin a b c This is a saddle point d We cannot tell This is a local maximum This is a local minimum 49 Which of the following guarantees a saddle point of the function ay at 17 a fm and fyy have the same sign at a7b c d none of the above b fm and fyy have opposite signs at 17 fwy is negative at a7b 118 Lagrange Multipliers 50 Find the maximum and minimum values of f on g c 11 a max 57 min 0 b max 57 min 2 A c max 4 min 0 CL max4min2 A e maX2min 0 51 This contour plot of ay also shows the circle of radius 2 centered at 00 How many local maxs and mins subject to the circle constraint does ay have l 52 Find the maximum value of xzyz subject to z y 4 a 2 b 2V2 C 4 d 8 e 16 22 121 Double Integrals over Rectangles 53 Estimate the integral fR 2 2y dA where R is the rectangle 0 S x S 4 0 S y S 2 by dividing the region into four equal pieces and using the midpoint rule 54 The integral fLx2 7 xdA over the region where L is the rectangle 71 S x S 0 71 S y S l is a positive b negative c zero 55 Below is a contour plot for a function fzy Estimate the average value of this function over the rectangle 0 S x S 4 71 S y S 3 U l l A A QQAUE 122 Iterated Integrals 56 Find the value of the iterated integral rounded to the nearest tenth 1402 dzdy 23 123 Double Integrals over General Regions 57 fol f2 fzyddy is an integral over which region a The triangle with vertices 00 20 01 b The triangle with vertices 00 02 10 c The triangle with vertices 00 20 21 d The triangle with vertices 00 10 12 58 Which of the following integrals is equal to fog f0 fxydydx7 a 0 f5 fltxygtdxdy b foulsi wwxdy 0 f0 fay4 f 967 206196611 d f0 fey4 f 967 206196611 e f0 f3 m ygtdzdy 124 Double Integrals in Polar Coordinates 59 Which of the following integrals is equivalent to fog ff rd dr a E IBM 1 dy dz b f ffQ 1dzdy wQiz 0 2 c fig fi ldx dy d f LOW 1 d 5195 60 Which integral gives the area enclosed by the curve 2 y2 2y a f0 fozsme 7 dr d0 b f0quot foe 7 dr d6 c fosme fOZW 7 d0 dr d f0quot fem 7 dr d6 e fOZW fozsme 7 dr d0 127 Triple Integrals 2 W 10 61 The region for integration for the integral fxyzdzdyd is a 72 7W 0 a sphere b cylinder c cone d none of the above 1 W x 17z27y2 62 2 y2 22dzdydx describes the mass of 71 ix17z2 ixlim2iy2 a A cone that gets heavier toward the outside b c d A ball that gets lighter toward the outside A cone that gets lighter toward the outside A ball that gets heavier toward the outside 63 Which of the following integrals does not make sense a 13 y Oyfyzdzdxdy b 13 0y 71 fzyzdxdydz c gOWfx7yzdzdydz d Am my dzdxdy 128 Triple Integrals in Cylindrical and Spherical Coordinates 64 Which of the following is equivalent to 3 5 m z dy dz dz 0 75 712542 7r 3 3 a r2 cosddz dr d0 0 0 0 7r 5 3 b r2 cos 6 dz dr d0 0 0 0 27r 5 3 C rcosddzdrd 0 0 0 27r 5 3 d r2 cos 6 dz dr d0 0 0 0 65 Which of the following integrals give the volume of the unit sphere 27r 27r 1 a dp d0 dab 0 0 0 7r 27r 1 b dp d6 do 0 0 0 7r 27r 1 c pzsina dpdd d 0 0 0 7r 27r 1 d p2 sin dpd d0 0 0 0 7r 27r 1 e p dp dab d0 0 0 0 lt27yzgt lt 0gt 1V Fwy iii Fwy 4 x 4411 x4 441 y yvxx A Allrvypirv4144 xxx Alfddvvp pvvddel A144444v v4144AJ k AIAAAII zn AAAI 4 y z 1 1 t Ir xx f4Alilv y AAaArAI xxx Alf444391v444f11 Aflrvyprv444A 44xxr yvxx 4 xx4 2 4 AVAVAVAAAAAAAIAVAAAA 444444444444444 AAAAAAALAAAAAAA ltywgt lt07zgt aaaaaaaaaaaaaaa gtgtgtgtgtgtgt5gtgtgtgtgtgtgt 444444444444444 4 eeee i Fwy ii Fwy 131 Vector Fields 66 Match the vector elds with the functions 27 VVVVVVVVVVVVVVV WVVVVVV39VWVVVVV tttttttttttt VIVIVKVKVKVKVKVIVIVIVIYKVKVKV ekkkekeeekkekk 1 2 1 67 Identify the vector eld CT a 967972 lt070796gt 0 967972 lt27070gt 0 967972 hag72gt d 967972 lt967070gt 8 967972 lt27y70gt True or False The vector eld Fxyz lty27x27ygt is conservative 132 Line Integrals Which of the following gives the line integral of x7 y with respect to arc length along the line from 10 to 14 a fol f14t1 4t2dt b fo f011t2dt c f f1 4t 4dt d f f1741t4dt 7 70 C is the line segment from 10 to 14 Compute fcyds to the nearest tenth 71 Which of the following is equivalent to the line integral of on the line segment from 11 to 34 a f02F1t115t i15jdt b f02 1t115t 2 35W 0 fol F 1t115t i15jdt d fol F 1t115t 2 33W 72 Along which curve is the line integral of the vector eld greatest moving left to right 133 Fundamental Theorem of Line Integrals 73 Which vector eld is not path independent ka tr k k 4l k Al LU5714411V xs 44111 44 4 ri rE IS I S I D krll A AL xp 4117 Kli ri siV V39 V f v WU Y gt gt gt gt gt l p 4444 p gtaa gt 4gt4gt4gt p gtgt p kke4 4 4447474 4 kkee4 4 4747474 4 4 136 Surface Integrals 7A For wheh surface 5 lg ffszds the meet helmet 75 mm whleh sneee ls the ux at mm 2 2a hegtltev e A square a mdA length 2 m the 112 hlehe melted lh t A ere a mdA length 2 m the u hlehe mehtel a A square a nae length 2 lh the m1 plene ehehterl lh the lemme z dn ecnm 76 Let F zQw 115 Whlah d the sunsets hdnw hss pmtlve lm e vhlt dlsk lh the z plehe mentel upwexd h 512th 0 hell l eeltehel et the algn c vhlt dlsk lh the plsne z z mmmd malexd the engn a None 0 the above 77 Chase the teeter hell wnh the lexgest ux thhemgh the surface hdnw e a estez 51 137 Stokes Theorem he an mm mhdnwshnwsl v a in Nommulsimmevm ddi s e mmled curve C gauche pmpmwulsx m v x The F wanketh I I I t 14 m 1 r 39 x g mm a s m a m l he demnmned mm mum miameum 133 Divergence Theorem 79 cm men we xmmgm me 211 pm wnh mews at way We may and may mm m m Wm mm mm ux mm me my a F a h F 11 a F2 zk a


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