25) (ln x) 2 dx 26) (ln(x + 1)) 2 dx 27) x sec 2 x dx 28) x csc 2 x dx 29) x x 2 dx 30) x x 2 2 dx 31) sec x tan x dx 32) x sec x tan x dx 33)

Transcription

1 364 Chapter Topics of Itegratio Eercises Terms ad Cocepts ) T/F: Itegratio b Parts is useful i evaluatig itegrads that cotai products of fuctios ) T/F: Itegratio b Parts ca be thought of as the opposite of the Chai Rule 3) For what is LIATE useful? I Eercises 4 33, evaluate the give idefiite itegral 4) si d ) e d 6) si d 7) 3 si d 8) e d 9) 3 e d ) e d ) e si d ) e cos d 3) e si(3) d 4) e cos() d ) si cos d 6) si d 7) ta () d 8) ta d 9) si d ) l d ) ( ) l d ) l( ) d 3) l( ) d 4) l d ) 6) 7) 8) 9) 3) 3) 3) 33) (l ) d (l( + )) d sec d csc d d d sec ta d sec ta d csc cot d I Eercises 34 38, evaluate the idefiite itegral after first makig a substitutio 34) si(l ) d 3) 36) 37) 38) si( ) d l( ) d e d e l d I Eercises 39 47, evaluate the defiite itegral Note: the correspodig idefiite itegrals appear i Eercises 4) ) π 39) si d 4) e d π/4 4) si d π/4 π/ 4) 3 si d π/ l 43) e d 44) 3 e d 4) e d π 46) e si d π/ 47) e cos d π/

2 374 Chapter Topics of Itegratio Eercises Terms ad Cocepts ) T/F: si cos d caot be evaluated usig the techiques described i this sectio sice both powers of si ad cos are eve ) T/F: si 3 cos 3 d caot be evaluated usig the techiques described i this sectio sice both powers of si ad cos are odd 3) T/F: This sectio addresses how to evaluate idefiite itegrals such as si ta 3 d I Eercises 4 6, evaluate the idefiite itegral 4) si cos 4 d ) 6) 7) 8) 9) ) ) ) 3) 4) ) si 3 cos d si 3 cos d si 3 cos 3 d si 6 cos d si cos 7 d si cos d si() cos(3) d si() cos() d si(3) si(7) d si(π) si(π) d cos() cos() d 6) 7) 8) 9) ) ) ) 3) 4) ) 6) ( π ) cos cos(π) d ta 4 sec d ta sec 4 d ta 3 sec 4 d ta 3 sec d ta 3 sec 3 d ta sec d ta 4 d sec d ta sec d ta sec 3 d I Eercises 7 33, evaluate the defiite itegral Note: the correspodig idefiite itegrals appear i the previous set π 7) si cos 4 d π 8) si 3 cos d π π/ 9) si cos 7 d π/ π/ 3) si() cos(3) d π/ 3) cos() cos() d π/ π/4 3) ta 4 sec d π/4 33) ta sec 4 d π/4

3 38 Chapter Topics of Itegratio Eercises 3 Terms ad Cocepts ) Trigoometric Substitutio works o the same priciples as Itegratio b Substitutio, though it ca feel ) If oe uses Trigoometric Substitutio o a itegrad cotaiig, the oe should set = 3) Cosider the Pthagorea Idetit si θ + cos θ = (a) What idetit is obtaied whe both sides are divided b cos θ? (b) Use the ew idetit to simplif 9 ta θ + 9 4) Wh does our Cocept state that a = a cos θ, ad ot a cos θ? I Eercises 6, appl Trigoometric Substitutio to evaluate the idefiite itegrals ) + d 6) 7) 8) 9) ) ) ) 3) 4) ) 6) + 4 d d 9 d d 6 d 4 + d 9 d 6 d 8 + d 3 7 d 8 d I Eercises 7 6, evaluate the idefiite itegrals Some ma be evaluated without Trigoometric Substitutio 7) d 8) ( + ) d 9) 3 d ) ) ) 3) 4) d ( d + 9) 3/ d ( ) d ( ) 3/ d ) 7 d 6) + 3 d I Eercises 7 3, evaluate the defiite itegrals b makig the proper trigoometric substitutio ad chagig the bouds of itegratio (Note: each of the correspodig idefiite itegrals has appeared previousl i this Eercise set) 7) 8) 9) 3) 3) 3) 8 4 d 6 d + 4 d ( + ) d 9 d d

4 4 Partial Fractios 39 Eercises 4 Terms ad Cocepts ) Fill i the blak: Partial Fractio Decompositio is a method of rewritig fuctios ) T/F: It is sometimes ecessar to use polomial divisio before usig Partial Fractio Decompositio 3) Decompose without solvig for the coefficiets 3 4) Decompose 7 without solvig for the coefficiets 9 ) Decompose 3 without solvig for the coefficiets 7 6) Decompose + 3 without solvig for the coefficiets + 7 I Eercises 7, evaluate the idefiite itegral 7) d 8) 7 + d 9) 4 3 d ) + 7 ( + ) d ) 3 ( + 8) d ) 3) ( + ) d + 33 ( )( + 3)(3 ) d 4) 94 (7 + 3)( )(3 ) d ) d 6) 3 d 7) d 8) d 9) d ) ( + )(3 + ) d ) ) 3) 4) ) ( 3)( ) d + + ( + )( + 9) d 69 ( 7)( + + 7) d ( 9)( + ) d ( + )( + + 7) d I Eercises 6 9, evaluate the defiite itegral 6) 8 + ( + )( + 3) d 7) (3 + )( + 4) d 8) + ( )( ) d 9) ( + )( + + ) d

5 Improper Itegrals 4 Eercises Terms ad Cocepts ) The defiite itegral was defied with what two stipulatios? b ) If lim f () d eists, the the itegral b f () d is said to 3) If f () d =, ad g() f () for all, the we kow that g() d 4) For what values of p will p d coverge? ) For what values of p will p d coverge? 6) For what values of p will p d coverge? I Eercises 7 33, evaluate the give improper itegral 7) e d 8) 3 d 9) 4 d ) + 9 d ) d ) ( ) d 3) + d 4) + 4 d ) ( ) d 6) ( ) d 7) d 8) d 9) d ) ) ) 3) 4) ) 6) 7) 8) 9) 3) 3) 3) 33) 3 d π sec d d e d e d e d e d + e l d l d l d l d l d e si d e cos d

6 46 Chapter Topics of Itegratio I Eercises 34 43, use the Direct Compariso Test or the Limit Compariso Test to determie whether the give defiite itegral coverges or diverges Clearl state what test is beig used ad what fuctio the itegrad is beig compared to 3 34) 3 + d 4 3) 7 3 d ) d 37) e l d 38) 39) 4) 4) 4) 43) e +3+ d e d + si d + cos d + e d e d

7 438 Chapter 6 Applicatios of Itegratio Eercises 6 Terms ad Cocepts ) T/F: A solid of revolutio is formed b revolvig a shape aroud a ais 7) = = ) I our ow words, eplai how the Disk ad Washer Methods are related 3) Eplai the how the uits of volume are foud: if A() has uits of i, how does A() d have uits of i 3? I Eercises 4 7, a regio of the Cartesia plae is shaded Use the Disk/Washer Method to fid the volume of the solid of revolutio formed b revolvig the regio about the -ais 4) 3 = 3 I Eercises 8, a regio of the Cartesia plae is shaded Use the Disk/Washer Method to fid the volume of the solid of revolutio formed b revolvig the regio about the -ais 8) 3 = 3 9) = = ) ) = cos 6) = cos (Hit: Itegratio B Parts will be ecessar, twice First let u = arccos, the let u = arccos )

8 6 Usig Defiite Itegrals to Fid Volume 439 = I Eercises 8, a solid is described Oriet the solid alog the -ais such that a cross-sectioal area fuctio A() ca be obtaied, the appl Theorem?? to fid the volume of the solid ) = 8) A right circular coe with height of ad base radius of I Eercises 7, a regio of the Cartesia plae is described Use the Disk/Washer Method to fid the volume of the solid of revolutio formed b rotatig the regio about each of the give aes ) Regio bouded b: =, = ad = Rotate about: (a) the -ais (b) = (c) the -ais (d) = 9) A skew right circular coe with height of ad base radius of (Hit: all cross-sectios are circles) 3) Regio bouded b: = 4 ad = Rotate about: (a) the -ais (b) = 4 (c) = (d) = 4) The triagle with vertices (, ), (, ) ad (, ) Rotate about: (a) the -ais (b) = (c) the -ais (d) = ) Regio bouded b = + ad = Rotate about: (a) the -ais (b) = (c) = 6) Regio bouded b = / +, =, = ad the -ais Rotate about: ) A right triagular coe with height of ad whose base is a right, isosceles triagle with side legth ) A solid with legth with a rectagular base ad triagular top, wherei oe ed is a square with side legth ad the other ed is a triagle with base ad height of (a) the -ais (b) = (c) = 7) Regio bouded b =, = ad = Rotate about: (a) the -ais (c) the -ais (b) = 4 (d) =

9 6 Volume b The Shell Method 447 Eercises 6 Terms ad Cocepts ) T/F: A solid of revolutio is formed b revolvig a shape aroud a ais = ) T/F: The Shell Method ca ol be used whe the Washer Method fails 8) = 3) T/F: The Shell Method works b itegratig cross sectioal areas of a solid 4) T/F: Whe fidig the volume of a solid of revolutio that was revolved aroud a vertical ais, the Shell Method itegrates with respect to I Eercises 8, a regio of the Cartesia plae is shaded Use the Shell Method to fid the volume of the solid of revolutio formed b revolvig the regio about the -ais I Eercises 9, a regio of the Cartesia plae is shaded Use the Shell Method to fid the volume of the solid of revolutio formed b revolvig the regio about the -ais 3 = 3 3 = 3 ) 9) = 6) = ) 7) = cos ) = cos

10 448 Chapter 6 Applicatios of Itegratio = ) The triagle with vertices (, ), (, ) ad (, ) Rotate about: ) = (a) the -ais (c) the -ais (b) = (d) = I Eercises 3 8, a regio of the Cartesia plae is described Use the Shell Method to fid the volume of the solid of revolutio formed b rotatig the regio about each of the give aes 3) Regio bouded b: =, = ad = Rotate about: (a) the -ais (b) = (c) the -ais (d) = 4) Regio bouded b: = 4 ad = Rotate about: (a) = (b) = (c) the -ais (d) = 4 6) Regio bouded b = + ad = Rotate about: (a) the -ais (b) = (c) = 7) Regio bouded b = / +, = ad the ad -aes Rotate about: (a) the -ais (b) = 8) Regio bouded b =, = ad = Rotate about: (a) the -ais (b) = (c) the -ais (d) = 4

11 48 Chapter 6 Applicatios of Itegratio Eercises 63 Terms ad Cocepts ) T/F: The itegral formula for computig Arc Legth was foud b first approimatig arc legth with straight lie segmets ) T/F: The itegral formula for computig Arc Legth icludes a square root, meaig the itegratio is probabl eas I Eercises 3 3, fid the arc legth of the fuctio o the give iterval 3) f () = o [, ] 4) f () = 8 o [, ] ) f () = 3 3/ / o [, ] 6) f () = 3 + o [, 4] 7) f () = 3/ 6 o [, 9] 8) f () = ( e + e ) o [, l ] 9) f () = + o [, ] 3 ) f () = l ( si ) o [π/6, π/] ) f () = l ( cos ) o [, π/4] I Eercises, set up the itegral to compute the arc legth of the fuctio o the give iterval Do ot evaluate the itegral ) f () = o [, ] 3) f () = o [, ] 4) f () = o [, ] ) f () = l o [, e] 6) f () = o [, ] (Note: this describes the top half of a circle with radius ) 7) f () = /9 o [ 3, 3] (Note: this describes the top half of a ellipse with a major ais of legth 6 ad a mior ais of legth ) 8) f () = o [, ] 9) f () = sec o [ π/4, π/4] I Eercises 8, use Simpso s Rule, with = 4, to approimate the arc legth of the fuctio o the give iterval Note: these are the same problems as i Eercises ) 9) ) f () = o [, ] ) f () = o [, ] ) f () = o [, ] (Note: f () is ot defied at = ) 3) f () = l o [, e] 4) f () = o [, ] (Note: f () is ot defied at the edpoits) ) f () = /9 o [ 3, 3] (Note: f () is ot defied at the edpoits) 6) f () = o [, ] 7) f () = sec o [ π/4, π/4] I Eercises 8 3, fid the Surface Area of the described solid of revolutio 8) The solid formed b revolvig = o [, ] about the -ais 9) The solid formed b revolvig = o [, ] about the -ais 3) The solid formed b revolvig = 3 o [, ] about the -ais 3) The solid formed b revolvig = o [, ] about the -ais 3) The sphere formed b revolvig = o [, ] about the -ais

12 6 Phsics Applicatios: Work, Force, ad Pressure 48 Eercises 6 Terms ad Cocepts ) What are the tpical uits of work? ) If a ma has a mass of 8 kg o Earth, will his mass o the moo be bigger, smaller, or the same? 3) If a woma weighs 3 lb o Earth, will her weight o the moo be bigger, smaller, or the same? 4) A ft rope, weighig lb/ft, hags over the edge of a tall buildig (a) How much work is doe pullig the etire rope to the top of the buildig? (b) How much rope is pulled i whe half of the total work is doe? ) A m rope, with a mass desit of kg/m, hags over the edge of a tall buildig (a) How much work is doe pullig the etire rope to the top of the buildig? (b) How much work is doe pullig i the first m? 6) A rope of legth l ft hags over the edge of tall cliff (Assume the cliff is taller tha the legth of the rope) The rope has a weight desit of d lb/ft (a) How much work is doe pullig the etire rope to the top of the cliff? (b) What percetage of the total work is doe pullig i the first half of the rope? (c) How much rope is pulled i whe half of the total work is doe? 7) A m rope with mass desit of kg/m hags over the edge of a m buildig How much work is doe pullig the rope to the top? 8) A crae lifts a, lb load verticall 3 ft with a cable weighig 68 lb/ft (a) How much work is doe liftig the cable aloe? (b) How much work is doe liftig the load aloe? (c) Could oe coclude that the work doe liftig the cable is egligible compared to the work doe liftig the load? 9) A lb bag of sad is lifted uiforml ft i oe miute Sad leaks from the bag at a rate of /4 lb/s What is the total work doe i liftig the bag? ) A bo weighig lb lifts lb of sad verticall ft A crack i the bo allows the sad to leak out such that 9 lb of sad is i the bo at the ed of the trip Assume the sad leaked out at a uiform rate What is the total work doe i liftig the bo ad sad? ) A force of lb compresses a sprig 3 i How much work is performed i compressig the sprig? ) A force of N stretches a sprig cm How much work is performed i stretchig the sprig? 3) A force of lb compresses a sprig from 8 i to i How much work is performed i compressig the sprig? 4) A force of lb stretches a sprig from 6 i to 8 i How much work is performed i stretchig the sprig? ) A force of 7 N stretches a sprig from cm to cm How much work is performed i stretchig the sprig? 6) A force of f N stretches a sprig d m How much work is performed i stretchig the sprig? 7) A lb weight is attached to a sprig The weight rests o the sprig, compressig the sprig from a atural legth of ft to 6 i How much work is doe i liftig the bo ft (ie, the sprig will be stretched ft beod its atural legth)? 8) A lb weight is attached to a sprig The weight rests o the sprig, compressig the sprig from a atural legth of ft to 6 i How much work is doe i liftig the bo 6 i (ie, brigig the sprig back to its atural legth)? 9) A m tall clidrical tak with radius of m is filled with 3 m of gasolie, with a mass desit of 737 kg/m 3 Compute the total work performed i pumpig all the gasolie to the top of the tak ) A 6 ft clidrical tak with a radius of 3 ft is filled with water, which has a weight desit of 64 lb/ft 3 The water is to be pumped to a poit ft above the top of the tak (a) How much work is performed i pumpig all the water from the tak? (b) How much work is performed i pumpig 3 ft of water from the tak? (c) At what poit is / of the total work doe?

13 486 Chapter 6 Applicatios of Itegratio ) A gasolie taker is filled with gasolie with a weight desit of 493 lb/ft 3 The dispesig valve at the base is jammed shut, forcig the operator to empt the tak via pumpig the gas to a poit ft above the top of the tak Assume the tak is a perfect clider, ft log with a diameter of 7 ft How much work is performed i pumpig all the gasolie from the tak? ) A fuel oil storage tak is ft deep with trapezoidal sides, ft at the top ad ft at the bottom, ad is ft wide (see diagram below) Give that fuel oil weighs 46 lb/ft 3, fid the work performed i pumpig all the oil from the tak to a poit 3 ft above the top of the tak 4) A water tak has the shape of a trucated coe, with dimesios give below, ad is filled with water with a weight desit of 64 lb/ft 3 Fid the work performed i pumpig all water to a poit ft above the top of the tak ft ft ft ) A water tak has the shape of a iverted pramid, with dimesios give below, ad is filled with water with a mass desit of kg/m 3 Fid the work performed i pumpig all water to a poit m above the top of the tak m m 7 m 3) A coical water tak is m deep with a top radius of 3 m (This is similar to Eample??) The tak is filled with pure water, with a mass desit of kg/m 3 (a) Fid the work performed i pumpig all the water to the top of the tak (b) Fid the work performed i pumpig the top m of water to the top of the tak (c) Fid the work performed i pumpig the top half of the water, b volume, to the top of the tak 6) A water tak has the shape of a trucated, iverted pramid, with dimesios give below, ad is filled with water with a mass desit of kg/m 3 Fid the work performed i pumpig all water to a poit m above the top of the tak m m m m 9 m

14 4 Chapter 6 Applicatios of Itegratio Eercises 68 Terms ad Cocepts ) I oe of our Cocepts, the equatio tah d = l(cosh ) + C is give Wh is l cosh ot used ie, wh are absolute values ot ecessar? ) The hperbolic fuctios are used to defie poits o the right had portio of the hperbola =, as show i Figure 66 How ca we use the hperbolic fuctios to defie poits o the left had portio of the hperbola? I Eercises 3, verif the give idetit 3) coth csch = 4) cosh = cosh + sih ) cosh = 6) sih = 7) cosh + cosh d [sech ] = sech tah d 8) d d [coth ] = csch 9) tah d = l(cosh ) + C ) coth d = l sih + C I Eercises, differetiate the give fuctio ) f () = cosh ) f () = tah( ) 3) f () = l(sih ) 4) f () = sih cosh ) f () = sih cosh 6) f () = sech ( ) 7) f () = sih (3) 8) f () = cosh ( ) 9) f () = tah ( + ) ) f () = tah (cos ) ) f () = cosh (sec ) I Eercises 6, produce the equatio of the lie taget to the fuctio at the give -value ) f () = sih at = 3) f () = cosh at = l 4) f () = sech at = l 3 ) f () = sih at = 6) f () = cosh at = I Eercises 7 4, evaluate the give idefiite itegral 7) tah() d 8) 9) 3) 3) 3) 33) 34) 3) 36) 37) 38) 39) cosh(3 7) d sih cosh d cosh d sih d 9 d 4 4 d + 3 d 4 6 d + d e e + d sih d tah d 4) sech d (Hit: mutipl b cosh cosh ; set u = sih ) I Eercises 4 43, evaluate the give defiite itegral 4) sih d l 4) cosh d l 43) tah d

15 38 Chapter 7 Sequeces & Series Eercises 7 Terms ad Cocepts ) Use our ow words to defie a sequece ) The domai of a sequece is the umbers 3) Use our ow words to describe the rage of a sequece 4) Describe what it meas for a sequece to be bouded I Eercises 8, give the first five terms of the give sequece { 4 } ) {a } = ( + )! {( 6) {b } = 3 ) } 7) {c } = 8) {d } = { { } + + (( + ) ( ) )} I Eercises 9, determie the th term of the give sequece 9) 4, 7,, 3, 6, ) 3, 3, 3 4, 3 8, ),, 4, 8, 6, ),,, 6, 4,, I Eercises 3 6, use the followig iformatio to determie the limit of the give sequeces { {a } = } ; lim a = {( {b } = + ) } ; lim b = e {c } = {si(3/)}; 3) {a } = { } 7 lim c = 4) {a } = {3b a } { ( ) {a } = si(3/) + ) } { ( 6) {a } = + ) } I Eercises 7 8, determie whether the sequece coverges or diverges If coverget, give the limit of the sequece { } 7) {a } = ( ) + { 4 8) {a } = } { 4 } 9) {a } = ) {a } = { ) {a } = {l()} ) {a } = 3) {a } = 4) {a } = { 3 + }, } {( + ) } { } { } ( ) + ) {a } = { } 6) {a } = { } 7) {a } = + 8) {a } = {( ) } I Eercises 9 34, determie whether the sequece is bouded, bouded above, bouded below, or oe of the above 9) {a } = {si } 3) {a } = {ta } { 3) {a } = ( ) 3 } { 3 3) {a } = } 33) {a } = { cos } 34) {a } = {!} I Eercises 3 38, determie whether the sequece is mootoicall icreasig or decreasig If it is ot, determie if there is a m such that it is mootoic for all m { } 3) {a } = + { 36) {a } = } ) {a } = {( ) } 3

16 7 Sequeces 39 { } 38) {a } = 39) Prove the defiitio of the limit of a sequece to show that if lim a =, the lim a = 4) Let {a } ad {b } be sequeces such that lim a = L ad lim b = K (a) Show that if a < b for all, the L K (b) Give a eample where L = K 4) Prove the Squeeze Theorem for sequeces: Let {a } ad {b } be such that lim a = L ad lim b = L, ad let {c } be such that a c b for all The lim c = L

17 6 Chapter 7 Sequeces & Series Eercises 7 Terms ad Cocepts ) Use our ow words to describe how sequeces ad series are related ) Use our ow words to defie a partial sum 3) Give a series a, describe the two sequeces related to the series that are = importat 4) Use our ow words to eplai what a geometric series is ) T/F: If {a } is coverget, the a is also coverget = I Eercises 6 3, a series a is give = (a) Give the first partial sums of the series (b) Give a graph of the first terms of a ad S o the same aes 6) 7) 8) 9) ) ) ) 3) ( ) = = cos(π) = =! = = = = 3 ( 9 ) ( ) I Eercises 4 9, show the give series diverges 4) ) 6) = = 3 ( + )! = 7) 8) 9) = + = = + + ( + ) I Eercises 9, state whether the give series coverges or diverges ) ) ) 3) 4) ) 6) 7) 8) 9) = = 6 = 4 = = = = =! (! + = ) ( + 8) = I Eercises 3 44, a series is give (a) Fid a formula for S, the th partial sum of the series (b) Determie whether the series coverges or diverges If it coverges, state what it coverges to 3) 4 = 3) ) = 33) 34) ( ) = e =

18 7 Ifiite Series 7 3) ) 37) 38) 39) 4) 4) = = = l = = 4) + 43) 44) ( + ) 3 ( + ) ( )( + ) ( ) + + ( + ) ( + ) ( ) + 9 = ( ) si = ( ) + 4) Break the Harmoic Series ito the sum of the odd ad eve terms: = = = + = The goal is to show that each of the series o the right diverge = (a) Show wh > = (Compare each th partial sum) (b) Show wh = < + = (c) Eplai wh (a) ad (b) demostrate that the series of odd terms is coverget, if, ad ol if, the series of eve terms is also coverget (That is, show both coverge or both diverge) (d) Eplai wh kowig the Harmoic Series is diverget determies that the eve ad odd series are also diverget 46) Show the series = ( )( + ) diverges

19 68 Chapter 7 Sequeces & Series Eercises 73 Terms ad Cocepts ) I order to appl the Itegral Test to a sequece {a }, the fuctio a() = a must be, ad ) T/F: The Itegral Test ca be used to determie the sum of a coverget series 3) What test(s) i this sectio do ot work well with factorials? 4) Suppose a is coverget, ad there are sequeces {b } ad {c } such that b a c for = all What ca be said about the series b ad = c? = I Eercises, use the Itegral Test to determie the covergece of the give series ) 6) 7) 8) 9) ) ) ) = = 4 = + l = = + = (l ) = l = 3 I Eercises 3, use the Direct Compariso Test to determie the covergece of the give series; state what series is used for compariso 3) 4) = = ) 6) 7) 8) 9) ) ) ) l =! + = = = + + = 3 = + = = l I Eercises 3 3, use the Limit Compariso Test to determie the covergece of the give series; state what series is used for compariso 3) 4) ) 6) 7) 8) 9) 3) 3) 3) = = l 3 =4 = = + + = + + si ( / ) = = = =

20 73 Itegral ad Compariso Tests 69 I Eercises 33 4, determie the covergece of the give series State the test used; more tha oe test ma be appropriate 33) 34) 3) 36) 37) 38) 39) = = ( + ) 3! = l! = = = 3 = ) cos(/) = 4) Give that a coverges, state which of the followig series coverges, ma coverge, or does ot = coverge (a) (b) a = a a + = (c) (a ) = (d) (e) a = a =

21 74 Ratio ad Root Tests 77 Eercises 74 Terms ad Cocepts ) The Ratio Test is ot effective whe the terms of a sequece ol cotai fuctios ) The Ratio Test is most effective whe the terms of a sequece cotais ad/or fuctios 3) What three covergece tests do ot work well with terms cotaiig factorials? 4) The Root Test works particularl well o series where each term is to a I Eercises 4, determie the covergece of the give series usig the Ratio Test If the Ratio Test is icoclusive, state so ad determie covergece with aother test )! = 6) 7) 8) 9) ) ) ) 3) 4) 3 4 =! ()! = + 4 = 7 + = = = 3 = = = ( 3 ) ! () I Eercises 4, determie the covergece of the give series usig the Root Test If the Root Test is icoclusive, state so ad determie covergece with aother test ) = ( ) 6) 7) 8) 9) ) ) ) 3) 4) ( 9 ) 3 = = = 3 = = = = ( + ( ( ) = l ( ) = l ) ) I Eercises 34, determie the covergece of the give series State the test used; more tha oe test ma be appropriate ) 6) 7) 8) 9) 3) 3) 3) 33) 34) = ! = = 3 = = !!! (3)! = = = = = l ( ( l ) ) ( ) +

22 7 Alteratig Series ad Absolute Covergece 93 Eercises 7 Terms ad Cocepts ) Wh is ) A series si ot a alteratig series? = ( ) a coverges whe {a } = is, ad lim a = 3) Give a eample of a series where but a does ot = a coverges = 4) The sum of a coverget series ca be chaged b rearragig the order of its terms I Eercises, a alteratig series a is give =i (a) Determie if the series coverges or diverges (b) Determie if a coverges or diverges = (c) If a coverges, determie if the covergece is = coditioal or absolute ) 6) ( ) + = ( ) + =! 7) = 8) = 9) = ) = ) = ) = ( ) + 3 ( ) ( ) ( ) l + ( ) l ( ) ( ) 3) 4) ) cos ( π ) = = = 6) = 7) si ( ( + /)π ) ( ) 3 ( e) l ( )! = 8) = 9) ) ( ) ( ) = ( )! = Let S be the th partial sum of a series I Eercises 4, a coverget alteratig series is give ad a value of Compute S ad S + ad use these values to fid bouds o the sum of the series ) ) 3) 4) = ( ) l( + ), = ( ) + = 4, = 4 = = ( ), = 6! ( ), = 9 I Eercises 8, a coverget alteratig series is give alog with its sum ad a value of ɛ Fid such that the th partial sum of the series is withi ɛ of the sum of the series ) 6) 7) 8) ( ) + = 4 = 7π4 7, ɛ = ( ) =! e, ɛ = = ( ) + = π 4, ɛ = = ( ) = cos, ɛ = 8 ()! =

23 76 Power Series 67 Eercises 76 Terms ad Cocepts ) We adopt the covectio that =, regardless of the value of ) What is the differece betwee the radius of covergece ad the iterval of covergece? 3) If the radius of covergece of a is, what is = the radius of covergece of a? = 4) If the radius of covergece of a is, what is = the radius of covergece of ( ) a? = I Eercises 8, write out the sum of the first terms of the give power series ) = 6) = 7)! = 8) ( ) ()! = I Eercises 9 4, a power series is give (a) Fid the radius of covergece (b) Fid the iterval of covergece 9) ) ) ) 3) 4) ) ( ) +! = = ( ) ( 3) = ( + 4)! = = ( ) ( ) = ( ) = 6) = ( ) 7) 8) 9) = 3 = 3 ( )! = ) ( )!( ) = ) ) 3) 4) = ( + ) = 3! = ( ) ( = ) I Eercises 3, a fuctio f () = give a is = (a) Give a power series for f () ad its iterval of covergece (b) Give a power series for f () d ad its iterval of covergece ) 6) 7) = = = ( ) 8) ( 3) = 9) ( ) ()! = ( ) 3)! = I Eercises 3 36, give the first terms of the series that is a solutio to the give differetial equatio 3) = 3, () = 3) =, () = 33) =, () = 34) = +, () = 3) =, () =, () = 36) =, () =, () =

24 64 Chapter 7 Sequeces & Series Eercises 77 Terms ad Cocepts ) What is the differece betwee a Talor polomial ad a Maclauri polomial? ) T/F: I geeral, p () approimates f () better ad better as gets larger 3) For some fuctio f (), the Maclauri polomial of degree 4 is p 4 () = What is p ()? 4) For some fuctio f (), the Maclauri polomial of degree 4 is p 4 () = What is f ()? I Eercises, fid the Maclauri polomial of degree for the give fuctio ) f () = e, = 3 6) f () = si, = 8 7) f () = e, = 8) f () = ta, = 6 9) f () = e, = 4 ) f () =, = 4 ) f () = +, = 4 ) f () = +, = 7 I Eercises 3, fid the Talor polomial of degree, at = c, for the give fuctio 3) f () =, = 4, c = 4) f () = l( + ), = 4, c = ) f () = cos, = 6, c = π/4 6) f () = si, =, c = π/6 7) f () =, =, c = 8) f () =, = 8, c = 9) f () =, = 3, c = + ) f () = cos, =, c = π I Eercises 4, approimate the fuctio value with the idicated Talor polomial ad give approimate bouds o the error ) Approimate si with the Maclauri polomial of degree 3 ) Approimate cos with the Maclauri polomial of degree 4 3) Approimate with the Talor polomial of degree cetered at = 9 4) Approimate l with the Talor polomial of degree 3 cetered at = Eercises 8 ask for a to be foud such that p () approimates f () withi a certai boud of accurac ) Fid such that the Maclauri polomial of degree of f () = e approimates e withi of the actual value 6) Fid such that the Talor polomial of degree of f () =, cetered at = 4, approimates 3 withi of the actual value 7) Fid such that the Maclauri polomial of degree of f () = cos approimates cos π/3 withi of the actual value 8) Fid such that the Maclauri polomial of degree of f () = si approimates cos π withi of the actual value I Eercises 9 33, fid the th term of the idicated Talor polomial 9) Fid a formula for the th term of the Maclauri polomial for f () = e 3) Fid a formula for the th term of the Maclauri polomial for f () = cos 3) Fid a formula for the th term of the Maclauri polomial for f () = 3) Fid a formula for the th term of the Maclauri polomial for f () = + 33) Fid a formula for the th term of the Talor polomial for f () = l I Eercises 34 36, approimate the solutio to the give differetial equatio with a degree 4 Maclauri polomial 34) =, () = 3) =, () = 3 36) =, () =

25 636 Chapter 7 Sequeces & Series Eercises 78 Terms ad Cocepts ) What is the differece betwee a Talor polomial ad a Talor series? ) What theorem must we use to show that a fuctio is equal to its Talor series? I Eercises 3 6, fid the first few terms of the Talor series of the give fuctio ad idetif a patter 3) f () = e ; c = 4) f () = si ; c = ) f () = /( ); c = 6) f () = ta ; c = I Eercises 7, fid a formula for the th term of the Talor series of f (), cetered at c, b fidig the coefficiets of the first few powers of ad lookig for a patter 7) f () = cos ; c = π/ 8) f () = /; c = 9) f () = e ; c = ) f () = l( + ); c = ) f () = /( + ); c = ) f () = si ; c = π/4 I Eercises 3 6, show that the Talor series for f () is equal to f (); that is, show lim R () = 3) f () = e 4) f () = si ) f () = l 6) f () = /( ) (show equalit ol o (, )) I Eercises 7, use the Talor series to verif the give idetit 7) cos( ) = cos 8) si( ) = si 9) d d ( si ) = cos ) d d ( cos ) = si I Eercises 4, write out the first terms of the Biomial series with the give k-value ) k = / ) k = / 3) k = /3 4) k = 4 I Eercises 3, create the Talor series of the give fuctios ) f () = cos ( ) 6) f () = e 7) f () = si ( + 3 ) 8) f () = ta ( / ) 9) f () = e si (ol fid the first 4 terms) 3) f () = ( + ) / cos (ol fid the first 4 terms) I Eercises 3 3, approimate the value of the give defiite itegral b usig the first 4 ozero terms of the itegrad s Talor series π 3) si ( ) d π /4 3) cos ( ) d

26 6 Chapter 8 Curves i the Plae Eercises 8 Terms ad Cocepts ) What is the differece betwee degeerate ad odegeerate coics? ) Use our ow words to eplai what the eccetricit of a ellipse measures 3) What has the largest eccetricit: a ellipse or a hperbola? 8) 4 4 4) Eplai wh the followig is true: If the coefficiet of the term i the equatio of a ellipse i stadard form is smaller tha the coefficiet of the term, the the ellipse has a horizotal major ais ) Eplai how oe ca quickl look at the equatio of a hperbola i stadard form ad determie whether the trasverse ais is horizotal or vertical I Eercises 6 3, fid the equatio of the parabola defied b the give iformatio Sketch the parabola 6) Focus: (3, ); directri: = 7) Focus: (, 4); directri: = 8) Focus: (, ); directri: = 3 9) Focus: (/4, ); directri: = /4 ) Focus: (, ); verte: (, ) ) Focus: ( 3, ); verte: (, ) ) Verte: (, ); directri: = /6 3) Verte: (, 3); directri: = 4 I Eercises 4, the equatio of a parabola ad a poit o its graph are give Fid the focus ad directri of the parabola, ad verif that the give poit is equidistat from the focus ad directri 4) = 4, P = (, ) ) = 8 ( ) + 3, P = (, ) I Eercises 6 7, sketch the ellipse defied b the give equatio Label the ceter, foci ad vertices 6) 7) ( ) + 3 ( ) + 9 ( + 3) = = I Eercises 8 9, fid the equatio of the ellipse show i the graph Give the locatio of the foci ad the eccetricit of the ellipse 9) I Eercises 3, fid the equatio of the ellipse defied b the give iformatio Sketch the elllipse ) Foci: (±, ); vertices: (±3, ) ) Foci: (, 3) ad (, 3); vertices: ( 3, 3) ad (7, 3) ) Foci: (, ±); vertices: (, ±7) 3) Focus: (, ); verte: (, 4); ceter: (, ) I Eercises 4 7, write the equatio of the give ellipse i stadard form 4) + 8 = 7 ) + 3 = 6) = 7) = 8) Cosider the ellipse give b ( ) 4 + ( 3) (a) Verif that the foci are located at (, 3 ± ) (b) The poits P = (, 6) ad P = ( +, 3 + 6) (44, 449) lie o the ellipse Verif that the sum of distaces from each poit to the foci is the same I Eercises 9 3, fid the equatio of the hperbola show i the graph 9) =

27 8 Coic Sectios 6 I Eercises 39 4, write the equatio of the hperbola i stadard form 3) 39) 3 4 = 4) 3 + = 3) 3) I Eercises 33 34, sketch the hperbola defied b the give equatio Label the ceter ad foci 33) ( ) 6 ( + ) 9 = 34) ( 4) ( + ) = I Eercises 3 38, fid the equatio of the hperbola defied b the give iformatio Sketch the hperbola 3) Foci: (±3, ); vertices: (±, ) 36) Foci: (, ±3); vertices: (, ±) 37) Foci: (, 3) ad (8, 3); vertices: (, 3) ad (7, 3) 38) Foci: (3, ) ad (3, 8); vertices: (3, ) ad (3, 6) 4) + 4 = 3 4) (4 )(4 + ) = 4 43) Johaes Kepler discovered that the plaets of our solar sstem have elliptical orbits with the Su at oe focus The Earth s elliptical orbit is used as a stadard uit of distace; the distace from the ceter of Earth s elliptical orbit to oe verte is Astroomical Uit, or AU The followig table gives iformatio about the orbits of three plaets Distace from ceter to verte eccetricit Mercur 387 AU 6 Earth AU 67 Mars 4 AU 934 (a) I a ellipse, kowig c = a b ad e = c/a allows us to fid b i terms of a ad e Show b = a e (b) For each plaet, fid equatios of their elliptical orbit of the form a + b = (This places the ceter at (, ), but the Su is i a differet locatio for each plaet) (c) Shift the equatios so that the Su lies at the origi Plot the three elliptical orbits 44) A loud soud is recorded at three statios that lie o a lie as show i the figure below Statio A recorded the soud secod after Statio B, ad Statio C recorded the soud 3 secods after B Usig the speed of soud as 34m/s, determie the locatio of the soud s origiatio A m B m C

28 8 Parametric Equatios 66 Eercises 8 Terms ad Cocepts ) T/F: Whe sketchig the graph of parametric equatios, the ad values are foud separatel, the plotted together ) The directio i which a graph is movig is called the of the graph 3) A equatio writte as = f () is writte i form 4) Create parametric equatios = f (t), = g(t) ad sketch their graph Eplai a iterestig features of our graph based o the fuctios f ad g I Eercises 8, sketch the graph of the give parametric equatios b had, makig a table of poits to plot Be sure to idicate the orietatio of the graph ) = t + t, = t, 3 t 3 6) =, = si t, π/ t π/ 7) = t, =, t 8) = t 3 t + 3, = t +, t I Eercises 9 7, sketch the graph of the give parametric equatios; usig a graphig utilit is advisable Be sure to idicate the orietatio of the graph 9) = t 3 t, = t, t 3 ) = /t, = si t, < t ) = 3 cos t, = si t, t π ) = 3 cos t +, = si t + 3, t π 3) = cos t, = cos(t), t π 4) = cos t, = si(t), t π ) = sec t, = 3 ta t, π/ < t < π/ 6) = cos t + 4 cos(8t), = si t + 4 si(8t), t π 7) = cos t + 4 si(8t), = si t + 4 cos(8t), t π I Eercises 8 9, four sets of parametric equatios are give Describe how their graphs are similar ad differet Be sure to discuss orietatio ad rages 8) (a) = t = t, < t < (b) = si t = si t, < t < (c) = e t = e t, < t < (d) = t = t, < t < 9) (a) = cos t = si t, t π (b) = cos(t ) = si(t ), t π (c) = cos(/t) = si(/t), < t < (d) = cos(cos t) = si(cos t), t π I Eercises 9, elimiate the parameter i the give parametric equatios ) = t +, = 3t + ) = sec t, = ta t ) = 4 si t +, = 3 cos t 3) = t, = t 3 4) = t +, = 3t + t + ) = e t, = e 3t 3 6) = l t, = t 7) = cot t, = csc t 8) = cosh t, = sih t 9) = cos(t), = si t I Eercises 3 33, elimiate the parameter i the give parametric equatios Describe the curve defied b the parametric equatios based o its rectagular form 3) = at +, = bt + 3) = r cos t, = r si t 3) = a cos t + h, = b si t + k 33) = a sec t + h, = b ta t + k I Eercises 34 37, fid parametric equatios for the give rectagular equatio usig the parameter t = d Verif that at t =, the poit o the graph d has a taget lie with slope of 34) = 3 + 3) = e 36) = si o [, π] 37) = o [, ) I Eercises 38 4, fid the values of t where the graph of the parametric equatios crosses itself 38) = t 3 t + 3, = t 3 39) = t 3 4t + t + 7, = t t 4) = cos t, = si(t) o [, π]

29 66 Chapter 8 Curves i the Plae 4) = cos t cos(3t), = si t cos(3t) o [, π] I Eercises 4 4, fid the value(s) of t where the curve defied b the parametric equatios is ot smooth 4) = t 3 + t t, = t + t ) = t 4t, = t 3 t 4t 44) = cos t, = cos t 4) = cos t cos(t), = si t si(t) I Eercises 46 4, fid parametric equatios that describe the give situatio 46) A projectile is fired from a height of ft, ladig 6ft awa i 4s 47) A projectile is fired from a height of ft, ladig ft awa i 4s 48) A projectile is fired from a height of ft, ladig ft awa i s 49) A circle of radius, cetered at the origi, that is traced clockwise oce o [, π] ) A circle of radius 3, cetered at (, ), that is traced oce couter clockwise o [, ] ) A ellipse cetered at (, 3) with vertical major ais of legth 6 ad mior ais of legth ) A ellipse with foci at (±, ) ad vertices at (±, ) 3) A hperbola with foci at (, 3) ad (, 3), ad with vertices at (, 3) ad (3, 3) 4) A hperbola with vertices at (, ±6) ad asmptotes = ±3

30 83 Calculus ad Parametric Equatios 673 Eercises 83 Terms ad Cocepts ) T/F: Give parametric equatios = f (t) ad = g(t), d d = f (t)/g (t), as log as g (t) = ) Give parametric equatios = f (t) ad = g(t), the derivative d d is a fuctio of? 3) T/F: Give parametric equatios = f (t) ( ad ) = g(t), to fid d, oe simpl computes d d d dt d 4) T/F: If d d = at t = t, the the ormal lie to the curve at t = t is a vertical lie I Eercises, parametric equatios for a curve are give (a) Fid d d (b) Fid the equatios of the taget ad ormal lie(s) at the poit(s) give (c) Sketch the graph of the parametric fuctios alog with the foud taget ad ormal lies ) = t, = t ; t = 6) = t, = t + ; t = 4 7) = t t, = t + t; t = 8) = t, = t 3 t; t = ad t = 9) = sec t, = ta t o ( π/, π/); t = π/4 ) = cos t, = si(t) o [, π]; t = π/4 ) = cos t si(t), = si t si(t) o [, π]; t = 3π/4 ) = e t/ cos t, = e t/ si t; t = π/ I Eercises 3, fid t-values where the curve defied b the give parametric equatios has a horizotal taget lie Note: these are the same equatios as i Eercises ) ) 3) = t, = t 4) = t, = t + ) = t t, = t + t 6) = t, = t 3 t 7) = sec t, = ta t o ( π/, π/) 8) = cos t, = si(t) o [, π] 9) = cos t si(t), = si t si(t) o [, π] ) = e t/ cos t, = e t/ si t I Eercises 4, fid t = t where the graph of the give parametric equatios is ot smooth, the fid d lim t t d ) = t +, = t3 ) = t 3 + 7t 6t + 3, = t 3 t + 8t 3) = t 3 3t + 3t, = t t + 4) = cos t, = si t I Eercises 3, parametric equatios for a curve are give Fid d, the determie the itervals o d which the graph of the curve is cocave up/dow Note: these are the same equatios as i Eercises ) ) ) = t, = t 6) = t, = t + 7) = t t, = t + t 8) = t, = t 3 t 9) = sec t, = ta t o ( π/, π/) 3) = cos t, = si(t) o [, π] 3) = cos t si(t), = si t si(t) o [ π/, π/] 3) = e t/ cos t, = e t/ si t I Eercises 33 36, fid the arc legth of the graph of the parametric equatios o the give iterval(s) 33) = 3 si(t), = 3 cos(t) o [, π] 34) = e t/ cos t, = e t/ si t o [, π] ad [π, 4π] 3) = t +, = 3t o [, ] 36) = t 3/, = 3t o [, ] I Eercises 37 4, umericall approimate the give arc legth 37) Approimate the arc legth of oe petal of the rose curve = cos t cos(t), = si t cos(t) usig Simpso s Rule ad = 4 38) Approimate the arc legth of the bow tie curve = cos t, = si(t) usig Simpso s Rule ad = 6 39) Approimate the arc legth of the parabola = t t, = t + t o [, ] usig Simpso s Rule ad = 4

31 674 Chapter 8 Curves i the Plae 4) A commo approimate of the circumferece of a ellipse give b = a cos t, = b si t is C a π + b Use this formula to approimate the circumferece of = cos t, = 3 si t ad compare this to the approimatio give b Simpso s Rule ad = 6 I Eercises 4 44, a solid of revolutio is described Fid or approimate its surface area as specified 4) Fid the surface area of the sphere formed b rotatig the circle = cos t, = si t about: (a) the -ais ad (b) the -ais 4) Fid the surface area of the torus (or dout ) formed b rotatig the circle = cos t +, = si t about the -ais 43) Approimate the surface area of the solid formed b rotatig the upper right half of the bow tie curve = cos t, = si(t) o [, π/] about the -ais, usig Simpso s Rule ad = 4 44) Approimate the surface area of the solid formed b rotatig the oe petal of the rose curve = cos t cos(t), = si t cos(t) o [, π/4] about the -ais, usig Simpso s Rule ad = 4

32 84 Itroductio to Polar Coordiates 687 Eercises 84 Terms ad Cocepts ) I our ow words, describe how to plot the polar poit P(r, θ) ) T/F: Whe plottig a poit with polar coordiate P(r, θ), r must be positive 3) T/F: Ever poit i the Cartesia plae ca be represeted b a polar coordiate 4) T/F: Ever poit i the Cartesia plae ca be represeted uiquel b a polar coordiate ) Plot the poits with the give polar coordiates (a) A = P(, ) (b) B = P(, π) (c) C = P(, π/) (d) D = P(, π/4) 6) Plot the poits with the give polar coordiates (a) A = P(, 3π) (b) B = P(, π) (c) C = P(, ) (d) D = P(/, π/6) 7) For each of the give poits give two sets of polar coordiates that idetif it, where θ π C D A O 3 B 8) For each of the give poits give two sets of polar coordiates that idetif it, where π θ π D C A O 3 B 9) Covert each of the followig polar coordiates to rectagular, ad each of the followig rectagular coordiates to polar (a) A = P(, π/4) (b) B = P(, π/4) (c) C = (, ) (d) D = (, ) ) Covert each of the followig polar coordiates to rectagular, ad each of the followig rectagular coordiates to polar (a) A = P(3, π) (b) B = P(, π/3) (c) C = (, 4) (d) D = (, 3) I Eercises 9, graph the polar fuctio o the give iterval ) r =, θ π/ ) θ = π/6, r 3) r = cos θ, [, π] 4) r = + si θ, [, π] ) r = si θ, [, π] 6) r = si θ, [, π] 7) r = + si θ, [, π] 8) r = cos(θ), [, π] 9) r = si(3θ), [, π] ) r = cos(θ/3), [, 3π] ) r = cos(θ/3), [, 6π] ) r = θ/, [, 4π] 3) r = 3 si(θ), [, π] 4) r = cos θ si θ, [, π] ) r = θ (π/), [ π, π] 3 6) r =, si θ cos θ [, π] 7) r =, 3 cos θ si θ [, π] 8) r = 3 sec θ, ( π/, π/) 9) r = 3 csc θ, (, π) I Eercises 3 38, covert the polar equatio to a rectagular equatio 3) r = cos θ 3) r = 4 si θ 3) r = cos θ + si θ 7 33) r = si θ cos θ 34) r = 3 cos θ 3) r = 4 si θ 36) r = ta θ 37) r = 38) θ = π/6

33 688 Chapter 8 Curves i the Plae I Eercises 39 46, covert the rectagular equatio to a polar equatio 39) = 4) = ) = 4) = 43) = 44) = 4) + = 7 46) ( + ) + = I Eercises 47 4, fid the poits of itersectio of the polar graphs 47) r = si(θ) ad r = cos θ o [, π] 48) r = cos(θ) ad r = cos θ o [, π] 49) r = cos θ ad r = si θ o [, π] ) r = si θ ad r = si θ o [, π] ) r = si(3θ) ad r = cos(3θ) o [, π] ) r = 3 cos θ ad r = + cos θ o [ π, π] 3) r = ad r = si(θ) o [, π] 4) r = cos θ ad r = + si θ o [, π] ) Pick a iteger value for, ( where =, 3, ad use m ) techolog to plot r = si θ for three differet iteger values of m Sketch these ad determie a miimal iterval o which the etire graph is show 6) Create our ow polar fuctio, r = f (θ) ad sketch it Describe wh the graph looks as it does

34 8 Calculus ad Polar Fuctios 699 Eercises 8 Terms ad Cocepts ) Give polar equatio r = f (θ), how ca oe create parametric equatios of the same curve? ) With rectagular coordiates, it is atural to approimate area with ; with polar coordiates, it is atural to approimate area with I Eercises 3, fid: (a) d d (b) the equatio of the taget ad ormal lies to the curve at the idicated θ value 3) r = ; θ = π/4 4) r = cos θ; θ = π/4 ) r = + si θ; θ = π/6 6) r = 3 cos θ; θ = 3π/4 7) r = θ; θ = π/ I Eercises 7 7, fid the area of the described regio 7) Eclosed b the circle: r = 4 si θ 8) Eclosed b the circle r = 9) Eclosed b oe petal of r = si(3θ) ) Eclosed b the cardiod r = si θ ) Eclosed b the ier loop of the limaço r = + cos t ) Eclosed b the outer loop of the limaço r = + cos t (icludig area eclosed b the ier loop) 3) Eclosed betwee the ier ad outer loop of the limaço r = + cos t 4) Eclosed b r = cos θ ad r = si θ, as show: 8) r = cos(3θ); θ = π/6 9) r = si(4θ); θ = π/3 ) r = si θ cos θ ; θ = π I Eercises 4, fid the values of θ i the give iterval where the graph of the polar fuctio has horizotal ad vertical taget lies ) Eclosed b r = cos(3θ) ad r = si(3θ), as show: ) r = 3; [, π] ) r = si θ; [, π] 3) r = cos(θ); [, π] 4) r = + cos θ; [, π] I Eercises 6, fid the equatio of the lies taget to the graph at the pole 6) Eclosed b r = cos θ ad r = si(θ), as show: ) r = si θ; [, π] 6) r = si(3θ); [, π]

35 7 Chapter 8 Curves i the Plae 7) Eclosed b r = 3 cos θ ad r = cos θ, as show: 3 I Eercises 8 3, aswer the questios ivolvig arc legth 8) Let (θ) = f (θ) cos θ ad (θ) = f (θ) si θ Show, as suggested b the tet, that (θ) + (θ) = f (θ) + f (θ) 9) Use the arc legth formula to compute the arc legth of the circle r = 3) Use the arc legth formula to compute the arc legth of the circle r = 4 si θ 3) Approimate the arc legth of oe petal of the rose curve r = si(3θ) with Simpso s Rule ad = 4 3) Approimate the arc legth of the cardiod r = + cos θ with Simpso s Rule ad = 6 I Eercises 33 37, aswer the questios ivolvig surface area 33) Fid the surface area of the sphere formed b revolvig the circle r = about the iitial ra 34) Fid the surface area of the sphere formed b revolvig the circle r = cos θ about the iitial ra 3) Fid the surface area of the solid formed b revolvig the cardiod r = + cos θ about the iitial ra 36) Fid the surface area of the solid formed b revolvig the circle r = cos θ about the lie θ = π/ 37) Fid the surface area of the solid formed b revolvig the lie r = 3 sec θ, π/4 θ π/4, about the lie θ = π/