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# Calculus and Analytic Geometry III MATH 153

OSU

GPA 3.8

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This 13 page Class Notes was uploaded by Theresia Hyatt DDS on Monday September 21, 2015. The Class Notes belongs to MATH 153 at Ohio State University taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/209944/math-153-ohio-state-university in Mathematics (M) at Ohio State University.

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Date Created: 09/21/15

Math 15301 AU 10 7 Gangyong Lee Name BE 0120 or PS 014 Signature December 6 or 77 2010 Name mm Final Exam 7 Solution Lecturer Gangyong Lee 13 pages Class Time 830AM 930AM General Instructions 0 Do all the problems Answer each part thoroughly 0 Show all Of your work lncorrect answers with work shown may receive partial credit but unsubstantiated answers will receive NO credit You do not need to simplify numerical answers7 eg 3 25 7 76 Calculators are permitted EXCEPT those calculators that have symbolic algebra or calculus capabilities In particular 7 the following calculators and their upgrades are not permitted T1 897 Tl 927and HP 49 ln addition7 neither PDAs7 laptops nor cell phones are permitted 0 Special Instructions 0 The Exam Duration is 108 minutes 0 This exam consists of 12 problems on 12 pages excluding this cover sheet Make sure that your exam paper is not missing any pages before you start 0 Answers without supporting work will receive no credit except problem 1 o A random sample of graded exams will be xeroxed before being returned Good Luck 1 Problem Points Score Problem Points Score 1 18 7 12 2 12 8 16 3 20 9 20 4 14 10 20 5 20 11 16 6 20 12 12 Total 200 1 Mark at the correct answer 18 points a Determine whether the following in nite series converges or diverges 4 points 71 2 Converges 1 Z nl V5716 4713 8 Converges 2 Zltiln b Find the sum of the following in nite series 6 points M o m 05 m 1 6 15 e 2 lt3 2 nl c Determine whether the given vectors orthogonal7 parallel7 or neither 8 points Neither 5 u i 7j7 V ij 7 k Orthogonal 6 u lt 7131 gt7 V lt 57271gt Parallel 7 u 2i 3j k7 V 74i 6j 2k Orthogonal 8 u lt abc gt7 V lt 49410 gt with ab 7 0 2 Find the sum of the following series 12 points 00 3WL5n 3W 5 1 5 94 2536 ltagt J Emit a H 6 H 6 6 2 2 2 6 1712 1756 9 2572 1 2 6 6 3 1 1 1 1 i i13 713 ltbgt1 quot Te 0 Obtain the Cartesian equation of the following curve by eliminating the parameter 2 Z ycost7 y 2sint7 with 0 g t g 2 2 Sol Since 0 g t g g 0 g x g 3 and 0 gy 2 Also7 Cos2tsin2t 1 2 2 The answer is 17 with 0 g x g 3 and 0 g y g 2 3 Find the radius of convergence and interval of convergence of the series 20 points 00 2n1 a Z 3712 x 7 2 Sol Let an 2W1 x 7 2 7 3n2 By the Absolute Ratio Test 2 2 a 2n1 L hm a hm 3lt 1gtf n7gtoo an n7gtoo WW 7 2n 2712 7 WQ72 7 12272l lt1 Thus lx 7 2 lt12 gt 32 lt z lt 52 Therefore the radius of convergence is 12 So the series converges absolutely when 32 lt z lt 52 2 71 When s 32 2 3712 converges because of p Test p 2 oo 2 When z 52 2 W converges because of p Test p 2 n n1 Therefore the interval of convergence is 32 s g 52 b 591 1 7 55z12 57 13 i 5914 5595 12 57z 13 59z 14 52 1 i 3 7 7 7 17 Sol Since 5 x 1 4 7 10 E71 1 3n 7 2 1 7 52n1 let an 71 1 2 1 n 7 By the Absolute Ratio Test 1 n252n3 1 n1 L hm a 7 hm n7gt00 an n7gtoo 71n m1n 7253n 7 2 7 7 l25z1l lt1 Thus ls 1 lt 125 7 72625 lt x lt 72425 Therefore the radius of convergence is 125 So the series converges absolutely when 72625 lt x lt 72425 When z 72625 Z n 1 75 3 2 diverges because it is a harmonic series n 7 571 1 i i i i 1 When z 72425 Z W converges because it is an alternating harmonic series n 7 n1 Therefore the interval of convergence is 72625 lt x g 72425 4 Find a power series representation for the following functions 14 points 1 a fx 3 2 6 points Solfzlt11 gti3ni b g W Hint Use a 8 points 8 1 s f eiifemg O ince 7 x3 22 i 1 n 2n1 7 is 73 00 996 7 xSnil i 71 3717 by 51 3 x322 3 1 2 1 271n1 n 3n n1 n1 2 5 a Find the Taylor Series for f cm at a 2 Use proper summation notation which includes the correct general nth term 10 points We 62 Sol Since f em for all n E N f 7 2 Z 7 2 n0 39 n0 39 b Find the 3 th degree Taylor polynomial of g e ma centered at a 1 10 points Sol Since f 7326 m3 f z 76xe w3 9z46 m3 and f z 766 m3 543 m3 7 27x66 w3 f1 6 17f 1 736 17f 1 766 1 964 36 1 and f 1 766 1 54671 7 276 1 216 1 le X 77 3 3 7 Thus7 fx T3 Z x 71 6 1 7 36 1 71 Eeilw 712 Eeilw 7 DB n0 6 a Evaluate the function f as an in nite series f e w3dx 10 points n Sol Since em 20 7 a 00 43 00 1 3 m d d n1 O fa 6 z nZO n z gn3n1 b Evaluate the function f correct to within an error of 001 10 points 1 3 67m dx 0 1 lt71 1 lt71 Sol Frorn problem 4a7 fx 0 67m dx Z mzh 0 Z n0 n0 Let an 7732 Since Rn g an i lt 001 by hypothesis7 n 3 because 13 13 10 160 gt 001 and a4 14 13 1312 lt 001 1 3 1 n fz 0 Nadya u go 17 14 114 7160 338420 m 080476 within an error of 001 7 Consider the cardioid r 1 cos 9 12 points a Find the area of the above cardioid SOI2 1COS 2d 1QCOS COS2 d 1200S 10S2 d 0 0 0 t 1 sin26 7r 7T 37T 231n T 77r b Find the length of the above cardioid Sol2 11COS 2S1n2 d 22200S d 2 14COSngt9 0 0 0 6 7 t9 4 cosid 831ni 8 0 2 20 8 16 points a Find its center and radius of the sphere 2 y2 22 7 62 7 16 0 4 points Sol Since 2 y2 z 7 32 16 9 25 527 its center is 07 07 3 and the radius is 5 b What is the intersection of this sphere with the yz plane 4 points Sol y2227627160 c Write the equation in cylindrical coordinates 4 points Sol r22276z7160 d Write the equation in spherical coordinates 4 points Sol p2 7 6pcos 716 0 9 20 points a Find an equation of the plane that contains the three points P01733 P172764 and 1323 715 6 points Sol Since P0131 lt 727621 gt i lt 1 733 gtlt 73731gt and P0 2 lt 3515 gt i lt1733 gtlt 222 gt i j k Po lxpo z 43 43 1 7672i77672j766k78i8j 2 2 2 Thus 78x718y3 0gt7y4 b Find an equation of the plane that contains the three points P01 711 13171 2 5 and 33 4 5 6 points Sol Since P0131 lt 7125 gt i lt 1711 gtlt 7234 gt and P0 2 lt 345 gt i lt1711 gtlt 254 gt i j k Po lxpo z 72 3 4 12720i77878j71076k78i16j716k 2 5 4 Thus 78x7116y17162710gtx72y225 c Find symmetric equations for the line of intersection L of these two planes 4 points Sol Sincexy4zi2y2275y472y22757y22710 y2zi1 74711 1 2 2 239 thus74y2zi1 d Find the angle between these two planes 4 points Sol Since two normal vectors are a lt 1 71 0 gt and b lt 1 72 2 gt ab1203 lal 12722223and lbl 112712022 b 3 cos L lallbl 3V2 7 12 Thus 19 I 4 10 a Find the limit of rt at the point t 2 10 points 6t 7 62 Jr sinm Jr 2 t 7 2k 21 2372 p2 hm rt hm H2 H2 61762 sinm 2t72 SODlErngg ig NH H 1 t7 2 t 2 1H2 1 hm ii hm j hm k hm eti hmvrcos 7th hm 7k H2 t72 H2 t72 H2 t72 H2 H2 H2 242t 1 ezi 7Tj 1k by 17H0spital7s Rule b Evaluate the integral if rt EZti 3th 1n tk 10 points 1 e rtdt 8 1 nth 1ntkdt EEZtit3jtlnt71k 1 SOD rtdt 525752i5371jk 1 11 Let I39t e ti xitj etk be the position vector Simplify your answer 16 points a Find the tangent vector7 r t 4 points Sol r t 764i j etk b Find the unit tangent vector7 Tt 4 points Sol Since r t 764 2 62 61 aquot 6t 647 Tt amp767ti jetk 71 g 62 7 7 7k r t 6te t 62t1162t1J62t1 c Find the length of the curve I39t with 0 g t g 1 8 points 1 1 Sol L rtdt ete tdt 6t767t3676717171 67671 0 0 12 Let rt lt t37tsin t7tcost gt be the position vector of a particle Simplify your answer 12 points a Find the velocity vector7 Vt 4 points Sol Vt r t lt 19 tcost sint7 7tsint cost gt b Find the speedl7 4 points Sol lvtl lr tl tZV tcos t sin t2 7tsint cos It2 t4 t4 1 1 t2cos2tsin2t sin2tcos2t Zt2 1 Evt 4t2 4 t2 2 c Find the acceleration vector7 at 4 points Sol at I39 t lt t 7tsint cos t cos t 7tcost 7 sint 7 sint gt lt t 7tsint 2cost7 7tcost 7 2sint gt

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