MA261-A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM

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1 MA6-A Calculus III 006 Fall Homework 4 Soluions Due 9/9/006 00AM 97 #4 Describe in words he surface 3 A half-lane in he osiive x and y erriory (See Figure in Page 67) 97 # Idenify he surface cos We see and I suggess ha we are using sherical coordinaes Firs noice ha since mus be bigger han or equal o 0, cos 0 This imlies ha 0 Since and cos, we have x sin cos y sin sin z cos, x cos sin cos sin cos y cos sin sin sin sin z cos cos cos, where 0 and 0 Since z cos, we know ha 0 z and when 0, z and when, z 0 If we rea as a scanning line, i scans from z o z 0 By xing an angle, since x cos sin cos sin cos y cos sin sin sin sin, we know ha sin lays he role of he radius of a circle a he heigh z cos When 0, sin 0, when, sin, and when, sin 0 Thus, he 4 radius of he circles sars from 0 and hen ges bigger unil he half-way o, and hen comes back o 0 I makes our grah looked like a shere Here is he grah

2 0 5 z x05 05 y #0 Idenify he surface r z 4 We see r and z I suggess ha we are using cylindrical coordinaes In cylindrical coordinaes, we have r x + y Thus, our equaion becomes x + y z 4 By dividing by 4, we ge x 4 + y z 4 By looking a Table in age 6, we idenify his surface as a hyerboloid of one shee 97 # Skech he solid describled by 3, We see and I suggess ha we are using sherical coordinaes Firs, le us see when ells us ha we have only boom half of he shere wih radius (looks like a bowl) Similarly, when 3 ells us ha we have only boom half of he shere wih radius 3 (looks like a bigger bowl) Thus, he solid is everyhing in beween I looks like a hard-cooked egg wihou york 97 #3 (a) Find inequaliies ha describe a hollow ball wih diameer 30 cm and hickness 05 cm Exlain how you have osiioned he coordinae sysem ha you have chosen (b) Suose he ball is cu in half Wrie inequaliies ha describe one of he halves (a) If we osiion he cener of he ball in he origin of a x-y-z coordinae sysem A solid ball wih diameer 30 cm can be described as x + y + z 5 To ge he hickness 05 cm, we need o ake o a solid ball wih diameer cm which can be described as x + y + z 45 So, a hollow ball wih diameer 30 cm and hickness 05 cm can be described as 45 x + y + z 5

3 (b) If we cu his ball hrough he xy lane, we know ha half ball has eiher z 0 or z 0 So, he uer half can be described as 45 x + y + z 5 and z 0 Also, he lower half can be described as 45 x + y + z 5 and z 0 97 #36 The laiude and longiude of a oin P in he Norhern Hemishere are relaed o sherical coordinaes ; ; as follows We ake he origin o be he cener of he Earh and he osiive z-axis o ass hrough he Norh Pole The osiive x-axis asses hrough he oin where he rime meridian (he meridian hrough Greenwich, England) inersecs he equaor Then he laiude of P is 90 and he longiude is 360 Find he grea circle disance from Los Angeles (la 3406 N, long 5 W) o Monréal (la 4550 N, long 7360 W) Take he radius of he earh o be 3960 mi (A grea circle is he circle of inersecion of a shere and a lane hrough he cener of he shere) The ; ; -coordinae of Los Angeles is (; ; ) (3960; 360 5; ) (3960; 475; 5594) Thus, he x; y; z-coordinae of Los Angeles is (x; y; z) ( sin cos ; sin sin ; cos ) (3960 sin (5594) cos (475) ; 3960 sin (5594) sin (475) ; 3960 cos (5594)) (379; 3446; 34) Similarly, he ; ; -coordinae of Monréal is (; ; ) (3960; ; ) (3960; 64; 445) Thus, he x; y; z-coordinae of Monréal is (x; y; z) ( sin cos ; sin sin ; cos ) (3960 sin (445) cos (64) ; 3960 sin (445) sin (64) ; 3960 cos (445)) ( 7055; 96536; 344) Le ~v be he vecor from he origin (cener of he earh) o Los Angeles and ~w be he vecor from he origin (cener of he earh) o Monréal By using he do roduc, we have ~v ~w j~vj j~wj cos, where is he angle beween hese wo vecors which lie on a grea circle So, (379; 3446; 34) ( 7055; 96536; 344) cos j(379; 3446; 34)j j( 7055; 96536; 344)j cos

4 4 Since is , we know ha is also 496 Therefore, he grea circle disance D D (3960), ha is, D 496 (3960) # Find he domain of he vecor funcion r () + i + sin j + ln 9 k For he x-comonen funcion, 6 For he y-comonen funcion sin, here is + no resricion of For he z-comonen funcion ln (9 ), we need o have 9 > 0, or (3 ) (3 + ) > 0 This imlies ha > 3 or 3 Thus, he domain is 0 #4 Find he limi > 3 or 3 lim arcan ; e ; ln! lim arcan ; e ; ln lim arcan ; lim! e ln D E ; lim!!! ; 0; 0 0 #0 Skech he curve wih he given vecor equaion r () i + j + k Indicae wih an arrow he direcion in which increases Noice ha z In xy-lane, we have x and y This imlies ha x y So, he curve is a arabolic curve which concave u o he osiive y-axis a he heigh z When increase, i goes o he osiive y-axis direcion 0 # Skech he curve wih he given vecor equaion r () cos i cos j + sin k Indicae wih an arrow he direcion in which increases Noice ha we have x cos and z sin This imlies ha we have a circle when rojecs ino he xz-lane So, i looks like a helix ye of curve Also, since cos, his helix-like curve is wihin y Consider a cylinder x + z wih y When 0, r (0) i j + 0k, i is he boom As increases, cos increases So, he curve ravels u hrough he helix-like curve Afer i reaches he o as, he curve becomes ravelling down hrough he helix-like curve And, reea when reaches he boom ( )

5 0 #4 Find a vecor equaion and arameric equaions for he line segmen ha joins P (; 0; ) o Q (; 3; ) The direcional vecor is! P Q h; 3; i is h; 0; i + h; 3; 0i, where 0 Also, he arameric equaions are x + y 3 z h; 0; i h; 3; 0i Thus, he vecor equaion, where 0 0 #4 Show ha he curve wih arameric equaions x sin, y cos, z sin is he curve of inersecion of he surfaces z x and x + y Use his fac o hel skech he curve The surface x + y can be described as (sin ; cos ; z) The inersecion of wo surface will have oins of his form and saisfy z x Thus, he oins in he inersecion looks like sin ; cos ; sin Also, when luging x sin, y cos, z sin ino boh surfaces, hey are boh sais ed So, we can conclude ha he curve wih arameric equaions x sin, y cos, z sin is he curve of inersecion of he surfaces z x and x + y x + y is a cylinder z x is a arabolic curve in he xz lane So, our curve looks like a arabolic curve on he surface of he cylinder x + y 0 #3 Find a vecor funcion ha reresens he curve of inersecion of he cylinder x +y 4 and he surface z xy The cylinder x + y 4 can be described as ( cos ; sin ; z), where 0 The inersecion wih z xy makes z 4 sin cos Thus, we can wrie down he vecor funcion r () h cos ; sin ; 4 sin cos i 0 #3 Two aricles ravel along he sace curves r () ; ; 3 and r () h + ; + 6; + 4i Do he aricles collide? Do heir ahs inersec? Le us re-wrie r as r (s) h + s; + 6s; + 4si If hese wo aricles collide, we will have a leas a air of (; s) such ha r () r (s) This imlies ha + s + 6s 3 + 4s By uing he rs equaion ino he second one, we have ( + s) + 6s This ells us ha s 0 or 5

6 6 When s 0, This air sais es he hird equaion So, i reresens a collision oin When s, This air also sais es he hird equaion So, i reresens anoher collision oin Thus, hese wo aricles collide a (; ; ) and (; 4; ) 0 #4 r () + ; (a) Skech he lane curve (b) Find r 0 () (c) Skech he osiion vecor r () and he angen vecor r 0 () for (a) Since x + and y, we have x + y So, he curve is y x D E (b) r 0 () ; (c) When, r () h; i and r 0 () ; 0 #0 Find he derivaive of he vecor funcion r () hcos 3; ; sin 3i r 0 () h 3 sin 3; ; 3 cos 3i 0 #4 Find he derivaive of he vecor funcion r () a (b + c)

7 7 r 0 () (a) 0 (b + c) + a (b + c) 0 a (b + c) + a c a b + a c + a c a b + (a c) 0 #6 Find he uni angen vecor T () of a he oin 4 We have Thus, jr 0 ()j So, he angen vecor is r () sin i + cos j + an k r 0 () cos i sin j + sec k q ( cos ) + ( sin ) + (sec ) 4 cos +4 sin + sec sec 4 T () r0 () jr 0 ()j 4 + sec4 cos 4 + sec4 i cos i sin j + sec k sin 4 + sec4 j + sec 4 + sec4 k 0 #0 Find arameric equaions for he angen line o he curve x cos, y sin, and z 4 cos a he oin 3; ; Illusrae by grahing boh he curve and he angen line on a common screen Le he curve be reresened by a vecor funcion r () h cos ; sin ; 4 cos i When, we have he oin 6 3; ; So, he direcional vecor of he angen line a 3; ; is D r 0 h sin ; cos ; sin ij 6 6 Thus, he arameric equaions for he angen line is x 3 + ( ) 3 y z ; 3; 4 E 3 0 # A wha oin do he curves r () h; ; 3 + i and r (s) h3 s; s ; s i inersec? Find heir angle of inersecion correc o he neares angle

8 To ge he inersecion oins, we assume ha r () r (s) This gives us a air (; s) (; ) So, a he oin (; 0; 4), hey inersec The angen vecor of r () a (; 0; 4) is r 0 () h; ; ij h; ; i The angen vecor of r () a (; 0; 4) is r 0 () h ; ; sij s h ; ; 4i Thus, he angle sais es r 0 () r 0 () jr 0 ()j jr 0 ()j cos I ells us ha cos h; ; i h ; ; 4i q + ( ) + q( ) So, he angle is cos #3 Evalue he inegral Z i + j + sin k d Z i + j + sin k d Z Z d i + Z d j i j 3 k 0 #36 Find r () if r 0 () i + e j + e k and r (0) i + j + k Since r () is he aniderivaive of r 0 (), we have Z Z r () r 0 () d i + e j + e k Z d d i + + C i + e + C j + e ( ) + C 3 k Since r (0) i + j + k, we have () + C e () + C e () (() ) + C 3 sin d k These imly ha C, C e, and C 3 Thus, we have r () + i + e + e j + e e + k 0 #44 Find an exression for d [u () (v () w ())] d Z Z e d j + e d k

9 9 d [u () (v () w ())] d d d u () (v () w ()) + u () d (v () w ()) d d u 0 () (v () w ()) + u () d v () w () + v () d d w () u 0 () (v () w ()) + u () (v 0 () w ()) + u () (v () w 0 ())

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