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# Multivariable Calc MAC 2313

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This 14 page Class Notes was uploaded by Connor Abernathy II on Monday October 12, 2015. The Class Notes belongs to MAC 2313 at Florida International University taught by Abdelhamid Meziani in Fall. Since its upload, it has received 9 views. For similar materials see /class/221707/mac-2313-florida-international-university in Calculus and Pre Calculus at Florida International University.

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Date Created: 10/12/15

MAC2311 1of2 httpwww2 ueduNmezianiMAC2313htrnl Multivariable Calculus MAC 2313 Of ce DM 432 Phone 305 348 2957 Email meziani at uedu Of ce Hours 100 to 150 pm Mondays Wednesdays and Fridays Prerequisite MAC 2312 passed with grade C or better Textbook Calculus Early Transcendentals 9th Edition by H Anton I Biven and S Davis Publ by John Wiley amp Sons Inc The complete solution manual is available at httn39 w n solution 9htrn WileyPLUS is an online supplement to the text that will be optional for homework You will need a code to access the class site You have the following options to obtain the access code gption 1 Purchase a new copy of the textbook that comes with an access code for WileyPLUS You need to specify at the time of purchase that you need a copy that comes packaged with a WileyPLUS access code for an additional of the cost of the textbook gption 2 Buy the textbook and buy the access code separately at www wileypluscom for about gption3 Buy an Ebook from wiley that comes with an access code for a twoisemester version and for a three semester version Once you have the access code you need to register into my class at httn39 edn en wilevnln Help sessions the learning assistant Alejandro Ginory will be conducting help session for multivariable calculus the schedule is Mondays and Wednesdays from 1 to 3 pm in VH18O Thursdays and Fridays from 1 to 3pm in DM409A Math Study Hall The MatlrStats department with support from the College of Arts and Science is opening a study hall As well as other students there will be Learning Assistants available to help you if you need The learning assistants will be knowledgeable in the follong subjects Intermediate Algebra College Algebra Trig and Calculus 123 The location is VH 180 The times Mondays 170072000 Wednesdays 180072100 Fridays 160072000 Lectures and homework Answers to Tests and Quizzes Examinations There will be 3 hourly tests a lot of quizzes and a nal Each test will be worth 18 of the nal grade all the quizzes together will be worth 16 and the nal will be worth 30 of the nal grade The Wiley Plus will contribute 3 BONUS POINTS The nal grade will be assigned according to the following scale gt m 50 D Z 45 D7 240 Flt 40 Tests Schedule Test 1 Test 2 Test 3 Final Monday September 26 Monday October 24 Monday November 28 Monday December 5 945 to 1145 am Makeup policy If a student misses a test the grade on that test will be zero There will be no make ups for the quizzes For each student the lowest quiz score will be dropped Amake up for a test can be considered only in special and documented circumstances such as an illness with a doctor note to verify 111811 1115 AM MAC2311 20f2 httpWWW2 ueduNmezianiMAC2313html Incomplete grades It is extremely dif th to qualify for an incomplete grade An incomplete grade is not a substitute for a failing grade In order to be considered for an incomplete the student must have completed at least three fourth 34 of the course and must be passing the course with a grade of C or better Dropping the course The last day to drop the course with a DR grade is October 31 Syllabus The following sections Will be covered Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Sections 111 to 118 Sections 121 to 126 Sections 131 to 139 Sections 141 to 148 Sections 151 to 158 time permitting 111811 1115 AM Fall 2011 7 Multivariable Calculus Answers QUIZ 1 1 v 72 lt126712 gt 2 The center of the sphere is C 27 37 71 and the radius is 7 Where P 17779 We have T W 9 The standard equation of the sphere is z 7 22 y 7 32 2 12 81 QUIZ 2 1 The vectors are orthogonal since W 12 710 7 2 0 2 The work done is W E where A and B are the initial and nal position of the moved object We have T i lt 1171 gt and 144g 37 Hence W 4xg ftlb QUIZ 3 1 Let B and C be the points on the line L corresponding to the parameters t 0 and t 1respectively Thus Bl473 and C 47 27 72 A normal to the plane P is zgt mmxm 74 4 75 lt7671174gt 71 2 74 The plane P contains the point A 502 and is normal to Thus its equation is 76175711y70742720 42gt 6z11y427380 QUIZ 4 1 The surface is a hyperboloid of two sheets intersecting the yaxis at y i1 2 a We have zt2 yt2 t2 sin2 t t2 cos2t t2 Hence the curve C is contained in the paraboloid 2 12 y2 b The parametric equations of the tangent line is given by 3977r t 7r We have 3977r lt 077r77r2 gt397 39t lt sint tcost7 cost 7 tsint7 2t gt and 777r lt 77r7 712 gt The tangent line is given by 177rt7 y77r7t7 27r227rt Test 1 1 A normal vector to the plane is NEXR 74 71 73 lt718024gt 0 76 0 An equation of the plane is 718173 24271 0 or equivalently 31742 5 2 xy trace Hyperbola xztrace Hyperbola yztrace Ellipse The surface is a hyperboloid of one sheet 3 We have 22 4e 2 and 12 y2 4e 2 cos2 t 4e 2 sin2t 4e 2 cos2 t sin2 t 4e 2 22 Hence the curve C lies on the cone The curve is a spiral on the cone Winding down toward 0 We have Ft lt 72equotcos tsin t 2equotcos tisin t 72equot gt and HFtm Z equot The length of the arc forogtg 1 is 1 1 L H tHdt2 equot2 17e71 0 0 4 a WWW b 0471 c mm3 3W4 d MiMM 4272 5 The curvature is W OWWW Ni 1 WOW W3 cosh2t32 HO Quiz 4 1 The position of the ball is given by 0 7 ot where We is the initial velocity and g m 32ftsec2 In terms of z and y we have 7r 2 7r zt v0cos Zt yt 7167 v0s1n 1 The ball hits the ground 250 ft when yt 0 with t gt 0 Thus t sin Thus 2 250 v0cos sin 1 U70 and v0 403fts m 894 fts m 6095 mih 4 16 4 32 2 We have 7 7t gtlt 17 cost sint 70 lt 00cost7 1 gt sint cost 0 and the curvature is H7tx7tll 7 llicostl 7 1 Ht WOW m3 WWW In particular when t 7r2 we get H QUIZ 5 1 Since the square root and the logarithm functions are continuous in their domains then the function ln 1 7 12 y2 is continuous in the region of R2 de ned by z 2 0 and 12 y2 lt 1 This region is the following half disk 2 We have as as 7w A A8 N d8 5 5 12 1 As N i i001 QUIZ 61 The limit of f along the zaxis is ligi fz0 0397 The limit of f along the yaxis is Min f0y 13397 y Since these two limits are distincty then lim does not exist Egty0gt0 QUIZ 62 1 At a critical points on the surface 5 given by gzy2 0y we have 1 M21 7 6y v Avg 71 A76z2y lt2gt g 0 1 7A z2i6zyy27z This system has a unique solution A 711 718 y 18 2 18 The function f has a global minimum on S and no global maximum The global minimum is f7181818 718 Test 2 o 741 113va 2 HOW 72213 o 139 139 121 va 1 o 139 f 57139 7214 1310 1 1 xlglmhta 0 lim 1 does not exist rgtyH0gt0 f y dz Bzdz Bzdy i 7iii dt azdt Bydt 1 1 72 2 2272 zy e Iy 7252 3t2 7 2t 672 is 7 t2 3 The surface is given by Fzy2 0 Where Fz7 y 2 13 7 21y 23 7y 6 0 The normal to the surface at the point 1473 is the vector v1714 73 lt 75 527 gt The tangent plane is given by 751 By 272 766 and the normal line by 11757 y45t7 27327t 4 1 sin 9 a sin 9 ab cos 9 AA s dA idaidb 7 2 2 2 700sin300 500sin300 500 x 700 cos300 mm s 2 2 AA m 0022 Acres L 2 x0i25x 180 ft 0 We have lt 2767737 gt and W lt 8172y322 gti Therefore DT472 1 Wd z 1 787 0 ln direction of gradient lt 324732 gti o HWV47271gtH74M O In the direction opposite the gradient o 7HW477271H 74m 6 If zy2 are the length Width and height of the aquarium then the cost function in cents is C 28 X zy5 gtlt 2zzyz2 gtlt zy301y102zy Since the volume is 24000 zyz then after eliminating 2 in the cost function we get a function of two variables to minimize Cz y 301y 2400001 y 9 The critical points are solutions of the system C1 30y 7 7240200 0 Cy 301 7 7240200 0 z y The only solution is z 20 y 20 and 2 60 These dimensions minimize the cost This problem can also be done by using Lgrange multipliers With Fz y 2 301y102zy and the constraint Cz y 2 0 With Cz y 2 zy2724000 QUIZ 7 12 7 3 7 2 1 The volume of the solid is V d l Where the region is R Rzy 6R2 0 z 4 0g yg 12E3Ii Thus 12722 4 7 7 V2 mdydz12 0 0 4 2 1 1 2x 1 cos12dzdy cosz2dydz y cosI2 yi dx 0 yQ 01 0 0 y 0 2zcos12dz sinz2 sinl QUIZ 8 1 1 M 7r2 1 7 sinz2 y2dzdy sin39r239rd39rdt9 W 0 0 0 0 2 The surface area of the portion of the plane is 4 714 S412 22dA 144dydz24 B y 0 0 QUIZ 10 1 The solid G is de ned in cylindrical coordinates by 0 t9 27r7 Ogr sint 7 T2 2 Tsin0 7r sin9 Tsin9 a Volume V d1 szdrdt G 0 0 7 2 39 9 7 sm9 2 3 sm9 7 sm9 T3 74 T 5 Sln4 6 dz 7 s1nt9739r 39 szd39r 7 s1nt9 7 i 7 2 0 7 2 3 4 7 0 12 cos 29 cos 49 2 8 3 Since sin4 9 g 7 then 71 7r4 771 Vi O s1n Odgi b Centroid ER With TVzdv7 Vydv7 EVzdv G G G We have 7r sin9 Tsin9 7r 5 zdv 7 2 cos Odzdrdt Wdt 0 G 0 0 7 2 0 20 and so T 0 this could have been easily noticed because the solid is symmetric With respect to the yz plane 7r sin 9 7 sin 9 7r 5 ydv T2 sinOdzdrdt Ede G 0 0 7 2 0 53 293 4917429 29 W Since sin5t9 ithen sin5t9dt9 and 5 7 7r 1 y2016 5 6 Similarlyy 7r sin9 TsinQ 7r 4 5 7 sin9 zdv TzdszdO l 7 L d6 G 0 0 7 2 2 0 4 6 T0 1 7r 57r 57r 7r 5 d7 3959d97 d 77 77 240 m 384 an 2 38432 12 Then centroid is 0127512 2 The solid G consists of the union of the non overlapping parts G1 and G2 given in spherical coordinates by 0 p 2 forGl7 and 0 g p g 22cosq for G2 Their volumes are 27r 7r4 2 7 dv p2sinq5dpdq5dt9 M 01 0 0 0 3 27r 7r2 2 cos 7r dv p2sin dpdq d97 G2 0 7r4 0 3 The volume of the solid is V1 V2 QUIZ 11 a a 2 2 1 Slnce 6721 cos y 721 slny and 673y 7 1 s1n y 721 s1n y then the y 1 vector eld lt 21 cos y 3y2 7 12 siny gt is conservative in R2 A potential for F is a function R2 7gt R such that 8 B i 7 21 cosy and 7 3y2 7 12 siny 81 By B i B i Integrating 67 with respect to 1 gives 17y 12 cosy Using 67 1 9 3y2 7 12 siny hy 7 21 siny gives h y 3y2 and we can take 17y 12 cosy yg For the integral we get 21 cos yd1 3y2 7 12 sinydy 470 7 177r 17 7 7r3 0 2 Let Ca be the circle centered at the origin with radius 0 lt a lt 12 and parametrized counterclockwise by 1 acos ty y asin ty 0 g t 27L Let D be the region in R2 between the curve C and the circle Ca It follows from the Generalized Green s Theorem that 9 I 7 3 1 3i lazltz2yzgt ayltz2y2gtldA70 7 Hence 27r y I 7y I AWdIJFWdyOmmdz12y2dyO dt27r Test 3 1a 1 3y 1 4 Izy2dA zy2dxdy Sidyl R 0 2y 0 2 2 1 2 2 yg 2 yz 4 7 I elty2gtdyd eyzdzdy g 0 21 0 0 0 2 4 2a 3 20c W 3 21 3 V dzdydz mdydz2zmdz1g 0 0 0 0 0 0 2b 1b 7r 7r 25in9 7r 3 3 4 I Tcos9Tde6 wcosgdg 70 0 0 0 3 12 0 3a 27f m 47 S 1 2 2 idd68 3b WZ 1 W M2 1 7 2 I 2szde6 Mdrd97l 0 0 0 0 o 2 16 4a 2 l 2 2 MldA idydz bimb RI 1 0 I 1 I 2 4b 27r 7r4 Q 5 Q I p4 singbdpdqsde 2 L 7 COME4 71 0 0 0 5 0 5 The Questions for AMC2313 with AbdelbamidMezzhni for F3112011 Hint Refer to the answers as you tea d this Quiz 1 1 Find the vector v that has the opposite direction of a 6i 3j 6k and with twice the magnitude of a 2 Write the standard equation of the with center 231 and contains the point 179 Quiz 2 1 Show that the vectors u 3i 2j k and v 4i 5j 2k are perpendicular 2 A constant force of magnitude 4 lb has the same direction as the vector u 1i 1 j 1k 1f the distance is measured in feet find the work done if the force moves an object along the yaxis form 020 to 0 10 Quiz 3 1 Find an equation of the plane that contains the pint A 502 and the line L with parametric equations L x13t y42t z3t Quiz 4 1 Identify and sketch the quadric surface y02 x202 z3A2 1 2 a Show that the curve C give by rt tsinti tcostj t02k is contained in the paraboloid z xA2 y02 b Find parametric equations of the tangent line to C at the point where t pi Test 1 1 Find an equation of the plane through the three points A 321 B 112 amp C 341 2 A surface S is given by the equation 4yA2 202 4x02 4 a Sketch and label the xytrace39 the xztrace and the yztrace b Sketch the trace of the surface with the plane x k where k is a constant c Sketch and label the surface S 3 A curve C is given by the parametric equations x 2eAtcost y 2eAtsint z 2eAt for t gt 0 or t 0 a Verify that C lies on the cone 202 x02 yA2 b Sketch C c Find the length of the arc of the curve C corresponding to t in 01 4 The coordinates of a point P in quotR 3quot is given in one system of coordinates ConvertP to another system of coordinates a From rectangular to cylindrical coordinates P 447 b From cylindrical to rectangular coordinates P 4pi21 c From rectangular to spherical coordinates P 130052 d From spherical to rectangular coordinates P 12pi33pi4 5 Consider the curve C given by parametric equations x eAt eAt 2 OR x sinht y eAt eAt 2 OR cosht z 0 Find the curvature Kt ofthe curve C HINT CONSIDER quotORquot Quiz 4 Again 1 A baseball player throws a ball up to a distance of 250 ft 1f the ball is released at an angle of 45 degrees with the horizontal find its initial speed 2 Find the curvature of the curve C given by the parametric equations x t sint y 1 cost at the point where t pi2 Quiz 5 Assume its correct for now 1 Sketch the set of all points xy in quotR 2quot at which the function fxy x005 ln 1 x02 yA2 is continuous 2 The specific gravity of an object more dense than water is given by s a a w where a and w are the weights in pounds of the object in air and in water respectively The measurements are a 12 lb and w 5 lb with maximum errors of or 05 oz and or 1 oz respectively Use differentials to estimate the error in the calculated specific gravity s Quiz 61 1 Show that the function fxy y02 xA2 3y02 does not have a limit as xy approaches 00 2 Find the direction in which the function fxy x02 sin4y increases most rapidly at the point P 10 and find the derivative of f in this direction Quiz 62 1 Find the maximum and minimum values of the function fxyz x y 2 subject to the constraint gxyz 0 where gxyz x02 6xy yA2 2 Test 2 1 Let fxy 2x02 y x04 yA2 a Find the lim of fxy as xy approaches 00 along y 2x b Find the lim of fxy as xy approaches 00 along y mx where m is a constant c Find the lim of fxy as xy approaches 00 along y x02 d What is the lim of fxy as xy approaches 00 2 Find dz dt if x eA2t y tA2t 1 z lnx y HINT SUBSTITUTE INTO Z WHAT X AND Y AND DIFFERERENTIATE IMMEDIATELY BUT THIS UNDERMINES THE CHAIN RULE 3 Find the equation of the tangent plane and the parametric equations of the normal line to the surface givenby x03 2xy 203 7y 6 0 at the point P l43 4 A surveyor wants to find the area in acres which 1 acre 43560 ftA2 of a certain triangular field She measures two different sides and finds that a 500 ft and b 700 ft with a possible error of as much as or 1 ft in each measurement She finds the angle between these two sides to be x 30 degrees with a possible error of as much as or 025 degrees Use differentials to estimate the resulting error in acres in the calculated area A Hint The area of a triangle is A 05ab sinx and 1 degree pi l80 radians 5 The temperature T at pointP xyz in quotR 3quot is given by Txyz 4x02 y02 16202 a Find the rate of change ofT atP 42l in the direction of the vector 2i 6j 3k b In what direction does T increase most rapidly atP c What is the maximum rate of change d In what direction does T decrease most rapidly at P e What is the minimum rate of change 6 You must find the minimum cost of building an aquarium that will have a volume of 24000 inA3 The bottom will be made of slate which will cost 28 cents per inA2 the sides will be made of glass which will cost 5 cents per in02 and the top will be made of stainless steel which will cost 2 cents per in02 What are the dimensions of the least expensive aquarium HINT YOU MAY APPLY LAGRANGE lIULTIPLIERS USING PART OF THE CONSTRAINT IN THE FIRST SENTENCE ALSO THE VOLUME OF A RECTANGULAR PRISM LIKE THIS AQUARHM IS V X Y Z Quiz 7 1 Use double integration to find the volume of the solid in the first octant bounded above by the plane 3x 2y 42 l2 2 Evaluate the integral by first reversing the order of integration for the integral From y 0 to y 2 and from x y 2 to x l integrate cosx02 dx dy Quiz 8 1 Evaluate the integral From y 0 to y l and from x 0 to x l y02005 integrate sinx02 yA2 dx dy by first converting to polar coordinates 2 Find the area of the portion of the plane 2x 2y z 8 in the first octant Quiz 10 Finally he added one 1 Use cylindrical coordinates to find a and b of the solid S bounded above by the plane 2 y and below by the paraboloid z xA2 y02 HINT FIND THE INTERSECTION OF THE TWO SURFACES AND IDENTIFY THE UNUSUAL UPPER AND LOWER LIMITS OF INTEGRATION FOR Z DON39T EXPECT PART B TO BE CORRECT IF PART A IS INCORRECT a The volume of S b the centroid of S 2 Use spherical coordinates to find the volume of the solid common to the two spheres rho 2 and rho 2 2005 cosphi HINT ADD UP THE TWO SEPARATE VOLUMES WHEN IDENTIFIED ERROR ON THE ANSWER SHEET FOR VOLUME 2 IT39S 2015 PI 3 NOT PI 3 uiz ll 1 Is the line integral Along the curve C integrate 2x cosy dx 3y02 x02 siny dy independent of the path of integration If yes evaluate the integral when C is a path connecting the points from Pl lpi to P2 40 HINT ASSUME YES39 OTHERWISE IT39S A WASTE OF TIME JUST SHOW YOUR WORK ALSO THIS IS A GOOD TIME TO SHOW OFF THAT YOU CAN USE THE GROUPING METHOD GRANTED THAT YOU HAVE TAKEN OR ARE ATTENDING THE INTRODUCTION TO DIFFERENTIAL EQUATIONS COURSE I COULDN39T GOOGLE THE GROUPING METHOD EITHER SO IF YOU39RE LOST YOU39LL STAY CONFUSED FOR A VERY LONG TIME 2 Evaluate the line integral Along the curve C integrate y x02 y02 dx x x02 y02 dy where C is the simpley closed curve sketched below with a counterclockwise orientation HINT THE REGION R WITHIN C IS A M39ULTIPLY CONNECTED REGION AND YES YOU DO NEED GREEN S THEOREM BASED ON THIS REASONING HOWEVER YOU MAY USE USE THE METHODS OF A PREVIOUS SECTION IN THE TEXT TO ARRIVE AT THE SAME SOLUTION I KNOW I THOUGHT THAT COULDN39T HAPPEN AT ALL THE MISSING DRAWING IS LIKE THIS C C1 C2 C3 C4 C5 YES YOU quotPIEC quot THEM TOGETHER AS THE PROBLEM SAYS YOU DO39 REMEMBER THAT YOU CAN USE A GRAPHING CALCULATOR AWAY FROM THE CLASSROOM THAT39S WHAT I DID Cl x t y l which is orientedfrom t 0 totl C2 x l l t02 005 y t which is oriented from t l to t l C3 x t y l which is oriented from t l to t 0 C4 x t y t l whichis oriented from t0 to t 1 C5 x t y l t which is oriented from tl to t 0 Test 3 1 Evaluate these integrals a I the double integral in the region R39 integrate xy02 dA where R is the triangular region with vertices 00 2 l and 31 b I From x 0 to x l and from y 2x to y 239 integrate eAQIAZ dy dx Hint Reverse the order of integration 2 Evaluate more integrals a Find the volume V of the solid in the first octant planes xyz 0 that is bounded by x02 y02 9 and y 2x b Use polar coordinates for I from y 0 to y 2 amp from x 2y y02005 to x 2y y0200539 integrate x dx dy 3 Evaluate even more integrals a Find the surface area of the portion of the sphere rho 4 between the planes 2 l and z 2 b Use cylindrical coordinates for I from y 0 to y 1 from x 0 to x ly02005 amp from z 0 to z 4 x02 y02 39 integrate 2 dz dx dy 4 Evaluate these integrals too a Find the mass of the lamina that has the shape of the region bounded by y lnx y 0 and x 2 with the mass density function deltaxy x0l b Convert I to spherical coordinates I from x 2 to x 2 y 4 x02005 to y 4 x02005 and from z x02 y02005 to z 8 x02 y0200539 integrate x02 y02 202 dz dy dx End of Class Material oi er For the final exam know the following theorems asI did 1 Extreme Value Theorem from chapter 13 2 Fundam enta1 Theorem for Line Integrals from chapter 15 3 Conservative Fields Theorem from chapter 15 4 Green39s Theorem from chapter 15 5 Divergence Theorem from chapter 15

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