Mocks.ie Maths LC HL Further Calculus mocks.ie Page 1

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1 Maths Leavig Cert Higher Level Further Calculus Questio Paper By Cillia Fahy ad Darro Higgis Mocks.ie Maths LC HL Further Calculus mocks.ie Page

2 Further Calculus ad Series, Paper II Q8 Table of Cotets:. Maimum ad Miimum. Itegratio by Parts a. Simple Itegratio by Parts b. Itegratio by Parts with oe Part c. Double Itegratio. The Ratio Test a. Fidig from b. Dealig with factorials c. Dealig with limits d. The differece betwee 0 ad ifiity e. Ratio Test Questio 4. Maclauri Series a. Computig Maclauri Series b. Obtaiig Approimatios c. Fidig the Geeral Term ( d. Mocks.ie Maths LC HL Further Calculus mocks.ie Page

3 Maimum ad Miimum Problems: These problems have the same basis as the Ma/Mi problems that you fid i Questio 6 ad 7 of Paper. But they are more practical based ad slightly more difficult. Also you will geerally be give a diagram to help you visualise so make the most of that. Remember from Differetiatio: at a Maimum Poit Eample: Ma ad Mi Problems Part (i) at a Maimum Poit Most questios have a part (i) to them which offers you the key doig this questio. Which is epressig everythig i terms of oe variable. Leavig Cert 008, Q8 b (i) pqr is a equilateral triagle of side 6 cm. abcd is a rectagle iscribed i the triagle as show. ab = cm ad bc = y cm. Epress y i terms of. r d c p q pq =6 ab = As we ca see from the diagram ab is i pq pa = bq bq =6- bq =- a {This is the lie pq without ab } b rqp cqb 60 {Equilateral triagle} Ta( cqb)=ta60= opp Ta= {Table Book, p. 6} adj bc y Ta( cqb)= bq y ( ) Mocks.ie Maths LC HL Further Calculus mocks.ie Page

4 Leavig Cert 008, Q8 b (ii) Fid the maimum possible area of abcd. (Cotiued o from the previous questio) Area of abcd = y= ( y ( ) ( ) A da da 0 {To fid ma, 0} d d d A d <0==>Ma 9 9 A= 9 cm WARNING! You ll otice that they asked for Area here, this is somethig which you might ot have dealt with sice Juior Cert. They usually ask for formulae which are somewhat basic but which you might be ot have used i some time. Check page 8- i the Table Book for these. It s importat also that you get lots of practice applyig these, eve though they may seem simple. Do t get caught out! Mocks.ie Maths LC HL Further Calculus mocks.ie Page 4

5 Itegratio by Parts: Just as Ma ad Mi problems had liks to Differetiatio questios so Itegratio by parts coects to Itegratio. The label Itegratio by Parts really says it all. You itegrate by splittig the sum up ito parts. There are three types of Itegratio by Parts Questios: udv=uv- vdu Simple Itegratio by Parts Itegratio by Parts with oe part Double Itegratio by Parts Table Book, p. 6 Eplaatio ad Order: Accordig to the above formula, you let U stad for oe part of the sum ad the you let for aother part. You will uderstad this better whe you see it shortly. But there is a order i which you must do this, which is:. Logs. Iverse Trig. Algebra 4. Trigoometry 5. Epoetial Fuctio ( You assig the letter U to the above i that order. I each sum you will have two of these ad you assig U to whichever is highest rak here. Eample: Simple Itegratio by Parts Use itegratio by parts to fid stad Mocks.ie Maths LC HL Further Calculus mocks.ie Page 5

6 U 4 e d du d dv v 4 e 4 e d 4 udv uv vdu 4 4 e 4 e d 4 e d e 4 e d e C 4 6 Select U=, as is Algebra ad it is higher tha epoetial i the order Eample: Itegratio by Parts with Oe Part This is very similar to ormal itegratio by parts, but you oly have oe part which is obvious ad the other is more subtle. It may seem obvious oce you kow it but it ca easily tripe you up. Leavig Cert 00 Q8 a Use itegratio by parts to fid log e U log du d dv v d d Use the formula: udv=uv- vdu {Differetiate the U to get du} {Itegrate to gai v} loge d log ( ) ( d) - d log d log ( ) C e WARNING! This is the most importat step here. Watch out for this or you ll miss it Do t forget your +C Eample: Double Itegratio by Parts Use itegratio by parts to fid Mocks.ie Maths LC HL Further Calculus mocks.ie Page 6

7 e si d STEP ONE: Itegrate the first time U si du cos d dv V e e d udv uv vdu si e si d e e cos d This requires itegratio by parts agai STEP TWO: Itegrate the secod time e cos d e cos d {Take out the costat for ease} U cos du si d dv e d V e Udv uv vdu cos e e cos d e ( si ) d + e (si ) d Notice that this is the eact same as your questio. This is how you kow how to stop STEP THREE: Combie agai si e si d e e cos d cos e - + e (si ) si cos e e si d = e e si cos e si = 9 e d e 9 d 0 e si d =si e cos e {Multiply by 9} e d C (si ) d si e cos e si = {Divide by 0} 0 Maipulate here: Just brig the last term across to the other side. Mocks.ie Maths LC HL Further Calculus mocks.ie Page 7

8 The Ratio Test: =R R< ==> The series is coverget R> ==> The series is diverget R= ==> The test is icoclusive You may ot uderstad how to use limits or you may but for this test a real uderstadig of limits is ot ecessary. There are three areas we eed to touch o before dealig with the Test: Fidig from Dealig with factorials Dealig with limits The differece betwee 0 ad ifiity Fidig All you do is take your (which will be the umber give to you i terms of ) ad aywhere there is a you chage to +. Eample : Mocks.ie Maths LC HL Further Calculus mocks.ie Page 8

9 Eample : Dealig with Factorials: Factorials are give the sig: So basically a eclamatio mark. Factorials work i the followig way:! =. Note: There is a butto o your calculator for factorials as well. Become familiar with it. (!) Mocks.ie Maths LC HL Further Calculus mocks.ie Page 9

10 So whatever umber the factorial is you start there ad multiply by every umber before it all the way back to. What is!? Well! = ()(... What happes frequetly i the ratio test is that you will have to cacel. Here s how: (!)= What is the differece? Well the oly differece is that has a etra (+) So = + Secod eample; Dealig with limits: Not kowig limits is ot a problem for this questio. Your oly real use of limits is at the fial part ad that is a fairly predictable system. To fid, all you do is divide every umber by the highest power of. The highest power of here is. So we divide everythig by. You the simplify. Whatever is still divide by or to ay power becomes 0. = {Highest power is } = {Highest power is } The differece betwee zero ad ifiity: The distictio betwee 0 ad pay particular attetio to it., i the ratio test questio is quite small ad therefore you eed to As Not i the Table Book Mocks.ie Maths LC HL Further Calculus mocks.ie Page 0

11 You ca see that just the positio of the 0 ca decide which is coverget ad diverget so pay particular attetio. Leavig Cert 008 Q8 a Use the ratio test to show that is coverget. I wat it to be coverget therefore I eed < where u u! ( ) 4 ( )! ( )! u!!.. u ( )! ( )! 4 4 4! 8.. ( )! lim 0 0 As 0<, the series coverges. WARNING! Pay careful attetio to the power of here whe you icrease from to + It is (+)+=++ Not ++ The Maclauri Series: This is the mai elemet of the questio. Usually a Maclauri Series of some sort will feature i this questio. We will tackle this questio i the followig way: Computig Maclauri Series Obtaiig Approimatios Fidig the Geeral Term ( Mocks.ie Maths LC HL Further Calculus mocks.ie Page

12 Computig the Maclauri Series: The Maclauri Series is foud from the followig formula: (Table Book, page 7) Maclauri Series ( ) f ' 0 f '' 0 f ''' 0 f 0 f f!!!! Usig this formula you are required to compute the followig Maclauri Series: Eample: Leavig Cert 004 Q8 b (i) Derive the first five terms of the Maclauri Series of f ' 0 f '' 0 f ''' 0 f e f!!! f e f 0 e f e f e 0 ' ' 0 f e f e 0 '' '' 0 f e f e 0 ''' ''' f e f 0 e e e Maclauri Series (First five terms) 4!! 4! Eample: log (+) Leavig Cert 005 Q8 b (i) Derive the Maclauri series for f () lup to ad icludig the term cotaiig Mocks.ie Maths LC HL Further Calculus mocks.ie Page

13 f '' 0 f ''' 0 f l f 0 f ' 0!! f ( ) l f 0 l 0 f '( ) f ' 0 f ''( ) f '' 0 0 f ''' f ''' 0 0 Note that to is coverted for the rest of the terms ad you apply the Chai Rule. log( ) Maclauri Series (First four terms) l 0.!! l 0 Eample: Leavig Cert 008 Q8 c (i) Derive the Maclauri series for icludig the term cotaiig up to ad f ' 0 f '' 0 f ''' 0 f f!!! cos 0... f cos f (0) cos 0 f ' si f ' 0 si 0 0 f '' cos f '' 0 cos0 f ''' si f ''' 0 si f cos f 0 cos 0 Note that it begis to repeat itself cos Maclauri Series (First five terms) (0) cos (0).!! 4! 4 cos 4! Eample: Si 4 Derive the Maclauri series for cotaiig up to ad icludig the term Mocks.ie Maths LC HL Further Calculus mocks.ie Page

14 f ' 0 f '' 0 f ''' 0 f f!!! si 0... f si f (0) si 0 0 f ' cos f ' 0 cos 0 f '' si f '' 0 si 0 0 f ''' cos f ''' 0 cos0 4 4 f si f 0 si f cos f 0 cos 0 Note the similarities betwee si Maclauri Series (First si terms) 0 ( ) 0 si (0) ().!! 4! 5! 5 si! 5! Eample: Biomial 4 5 WARNING! You ca be asked to fid or, oce you have si or cos. Simply do this by applyig the formula o p.4 Table Book. Ad lettig =A f ' 0 f '' 0 f ''' 0 f f!!! ( ) 0... f ( ) f (0) () f ' ( ) f ' 0 () f f '' ( )( ) '' 0 ( )() ( ) f f ''' ( )( )( ) ''' 0 ( )( )() ( )( ) ( ) Maclauri Series (First four terms) ( ) ( )( ) ( ) () ( ).!! Approimatios: Mocks.ie Maths LC HL Further Calculus mocks.ie Page 4

15 You ca usually be asked to fid a approimatio of the Maclauri Series. There are two types of approimatios: Geeral Approimatio Geeral Approimatio: The geeral approimatio is the simplest of approimatios ad is the same for all of the Maclauri Series above ecept for. Eample: Approimatio for l(+) Leavig Cert 005 Q8 b (ii) Use those terms to fid a approimatio for l These terms are those foud earlier i the questio: ( ) ( ) l l( ) {Use Calculator} WARNING! It is very easy to sub i = if you do t look closely. Be careful of this with every approimatio. l 0 Approimatio for These are a little differet. First of all you will probably have to sub i two values oe for the value of ad the other the value for. This series oly coverges for. You eed it to coverge ad therefore whatever you sub i for must fit this. If ot maipulate it so it will. Eample: Mocks.ie Maths LC HL Further Calculus mocks.ie Page 5

16 Leavig Cert 009 Q8 b (i)derive the first four terms of the Maclauri Series WARNING! This is somethig of a implicit approimatio. You eed to otice that here you. You will usually be asked to compute your Biomial Maclauri like this. You would solve it as we did above subbig i =. Which meas you would get: ( ) ( )( ) ( ) () ( ).!! ( ) ( ) () ( ). 4 8! ( ) () ( ). ( ) ( ) 8 6 Leavig Cert 009 Q8 b (ii)derive the first four terms of the Maclauri Series. Give that this series coverges for < <, use these four terms to fid a approimatio for, as a fractio. 7 6 But remember < so it caot be 6 6 6( ) Therefore = ( ) ( ) () ( ). ( ) ( ) {sub = } With that we have + but we wat 4 +. Multiply by A good check step is to check if o your calculator is close to your fial aswer. If they are the you should be right! Mocks.ie Maths LC HL Further Calculus mocks.ie Page 6

17 Fidig the Geeral Term: We fid the geeral term usually so we ca apply the ratio test as eplaied earlier i this chapter. There are two ways you ca fid the geeral term: How to Work it Out: It does t take much work to figure out the Geeral term o your ow so log as you kow what to look for: Begi by umberig each term accordig to order. You should see how their factorial value ad the power of relate to this umber If sigs alterate every term the you must put either a or with the geeral term. E.g. For that whe is eve the the term should be positive ad whe is odd the term should be egative. The opposite holds for List of the Geeral Terms: Workig the term out yourself ca be much easier oce you ve see the actual Geeral term for each. But if you ca t lear how to work the Geeral term out yourself the you should cosider learig these off. Not i the table book! e u! l( ) u ( ) si u cos u ( ) ( )! ( ) ( )!! r ( )( )...( r) ( ) ur Use r here as is part of the sum ( r )! Mocks.ie Maths LC HL Further Calculus mocks.ie Page 7

18 : The Maclauri Series for is differet tha the oes we have see previously as it is formed i a differet way. As a matter of fact, you do t eve use the Maclauri Series formula at all. ta a a a Table book, page 6 This tells us that to get. All we eed do is use log divisio ad divide ito etc So d (...) d 5 7 ta {Itegrate} Mocks.ie Maths LC HL Further Calculus mocks.ie Page 8

19 Approimatig for : y ta ta yta y Importat Formula: Not i the Table Book. Although, if you look at p. 9. The formula for the agle betwee two lies is somewhat similar ad you ca lear that as a basis for this oe. Eample: Usig the first four terms of, get a approimatio for ad ta ta ( ) ta ( ) ta ( ) ta ( ) Mocks.ie Maths LC HL Further Calculus mocks.ie Page 9

20 Eample : Show. From this get a approimatio for correct to 4 decimal places. y ta ta yta y ta ta ta 5 6 = ta ta 5 6 ta ta ta 4 4 ta ta 4 4( )= 4(ta ta ) {Sub i previous appro.}.408 Mocks.ie Maths LC HL Further Calculus mocks.ie Page 0