September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
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Idices September 0 Laws of idices for all ratioal epoets. The equivalece of We should alrea kow from GCSE, the three Laws of idices : m m 5 ( I) a a a (e.g. a a a ) m m 7 4 ( II) a a a (e.g. a a a ) m m m a ad m a should be kow. 5 7 5 ( III) a a (e.g. a a ) I additio to these we eed to remember the followig: REMEMBER,,, a a a a a a etc. a a a a, a a So, for eample: 64 64 8 5 5 5 0 04 04 a, a a a So, for eample: 5 4 8 5 5 4 m m m a a a So, for eample, 4 4 5 5 4 5 7 7 8 4 4 4 4 65 65 5 5 8 4 8 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 7
September 0 REMEMBER 0 a ad a a for all values of a. Eam Questio Give that = ad y = 4, (a) fid the eact value of ad the eact value of y, so. Hece 5 4. Hece 5 y (b) calculate the eact value of y. 5 y 8 Edecel GCE Pure Mathematics P Jauary 00 Eam Questio (a) Give that 8 = k, write dow the value of k. k (b) Give that 4 = 8, fid the value of. 4 8 6 6 5 6 6 5 Edecel GCE Pure Mathematics P Jue 00 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 8
Surds September 0 Use ad maipulatio of surds. Studets should be able to ratioalise deomiators. The square roots of certai umbers are itegers (e.g. 9 ) but whe this is ot the case it is ofte easier to leave the square roots sig i the epressio (e.g. it is simpler to write tha it is to write the value our calculator gives, i.e..70508 ). Numbers of the form,, 5 etc. are called surds. We eed to be able to simplify epressios ivolvig surds. It is importat to realise the followig : ab a b a b e. g. 6. Now we see from the above that we ca sometimes simplify surds. For eample 75 5 5 5 5 or 50 5 5 ad 7 We ca oly simplify a epressio of the form the umber 5 was i the above eample). a if a has a factor which is a perfect square ( as Multiplyig surds: Multiply out brackets i usual way (usig FOIL or a similar method). The collect similar terms. e.g. 5 7 55 7 5 5 9 5 5 5 4 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 9
4 5 Dividig surds: For eample simplify 5 The deomiator of this fractio is 5.. September 0 This is irratioal ad we eed to be able to epress this i a form i which the deomiator is ratioal. To do this we must multiply top ad bottom of the fractio by a epressio that will ratioalise the deomiator. We saw from above that 5 5 4 so we multiply top ad bottom by 5 So we have the followig 4 5 4 5 5 5 5 5 5 0 5 4 4 5 Ratioalisig the deomiator: If the deomiator is a b the multiply top ad bottom by a b. If the deomiator is a b the multiply top ad bottom by a b. If the deomiator is a b the multiply top ad bottom by a b. If the deomiator is a b the multiply top ad bottom by a b. Eam Questio Give that ( + 7)(4 7) = a + b7, where a ad b are itegers, (a) fid the value of a ad the value of b. 74 7 8 4 7 7 7 7 so a, b.. Give that 4 7 7 = c + d7 where c ad d are ratioal umbers, (b) fid the value of c ad the value of d. 74 7 7 7 7 4 7 4 7 4 7 6 7 9 9 Edecel GCE Pure Mathematics P Jauary 00 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 0
Quadratics September 0 Quadratic fuctios ad their graphs. The graph of y a b c. (i) a 0 (ii) a 0 The turig poit ca be determied by completig the square as we will see later. The -coordiates of the poit(s) where the curve crosses the -ais are determied by solvig a b c 0. The y-coordiates of the poit where the curve crosses the y-ais is c. This is called the y-itercept. The discrimiat of a quadratic fuctio. b b 4ac I the quadratic formula, the Discrimiat is the ame give to b 4ac. a The value of the discrimiat determies how may real solutios (or roots) there are. If b 4ac 0 the a b c 0 has o real solutios. If b 4ac 0 the a b c 0 has oe real solutio. If b 4ac 0 the a b c 0 has two real solutios. Completig the square Eample Complete the square o 6. We ca write 6 9 ad so we see that 6 ( ) 9 ( ). is obtaied by halvig 6 So ad 8 ( 4) 6 ( 4) ( ) ( ). 4 4 4 4 We ca use completig the square to fid the turig poits of quadratics. Complete the square to fid the turig poit Eample : Fid the turig poit of y 6. Completig the square o y 6 gives y 6 ( ). Sice ( ) 0 we have that ( ) ad so y. It takes that miimum value of whe ( ) 0, that is whe. Hece the miimum poit at,. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
September 0 Solutio of quadratic equatios. Solutio of quadratic equatios by factorisatio, use of the formula ad completig the square There are three ways of solvig quadratic equatios. Factorisig e.g. Solve 7 0. Factorisig gives 7 ( )( 4) So we have ( )( 4) 0 ad so or 4 e.g. Factorise 7 6 : (i) Fid two umbers which add to give 7 ad which multiply to give 6, i.e. 4 ad (ii) Write 7 6 4 6 (iii) Factorise first two terms ad secod two terms to give (iv) Hece write as. Completig the square. e.g. Solve 0 9 0 Completig the square gives 09 ( 5) 5 9 ( 5) 6 So we have ( 5) 6 0. Hece we see that ( 5) 6 ad so 5 6. Thus we have solutios of 5 6. Quadratic Formula e.g. Solve 0 9 0 by the formula We use the fact that the solutios to the equatio a b c 0 are b b 4ac. a I this case a, b 0 ad c 9 ad so, usig the formula, we have 0 Solvig simple cubics Eample: Solve 5 6 0. 0 4 9 0 4 Each term has i it so factorisig leads to Hece we have 0. Thus the solutios are 0, or. 5 6 0. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
Simultaeous Equatios September 0 Simultaeous equatios: aalytical solutio by substitutio. For eample, where oe equatio is liear ad oe equatio is quadratic There are two methods for solvig two liear simultaeous equatios. Method Cacellatio We cacel either the or the y by multiplyig the two equatios by suitable umbers ad the either addig or subtractig. So, for eample, solve 5 7y 9 () ad y 7 () If we multiply () by ad () by 5 we get: 5 y 57 () ad 5 0y 5 () 5 If we ow subtract bottom from top we get y ad so y. Substitute this value of y ito () gives 5 4 9 ad so. Method Substitutio We make either the or the y the subject of oe of the equatios ad the plug it back i to the other equatio to fid the other variable. So, for eample, solve y () ad 7y 44 () From () we see that y. If we ow replace i () with y That leads to 9 6y7y 44 ad so y 5. we get y 7y 44. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
September 0 We also eed to be able to solve simultaeous equatios i which oe of the equatios is liear ad oe is quadratic. So, for eample, solve y 0 () ad y y 7 () We always use the method of substitutio i these cases. So, i our eample, we use () to write y 0 We substitute this ito () to give (0 ) (0 ) 7. Now epad brackets to give a quadratic equatio 06 (00 609 ) 7 00 00 7 00 8 0 Factorise this to give 6 0 ad so or 6. We ow use () to fid y whe ad we see that y. 6 5 We also use () to fid y whe ad we see that y. It is importat to give our solutio as pairs. So we say that the solutios are ad y OR 6 5 ad y. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 4
Iequalities Solutio of liear ad quadratic iequalities (For eample p qr a b) September 0 a b c d, p q r 0, Liear Iequality Remember that whe you multiply or divide both sides by a egative umber you must flip aroud the sig of the iequality. For eample: 4 7 7 8 6 Quadratic iequality Eample Solve 40 0. We factorise to give 8( 5) 0. The critical values are 8 ad 5. Thik of the graph of y 40. We are lookig below -ais so we wat oe regio, ad hece oe iequality. So the solutio is 5 8. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 5
September 0 Eample Solve 0 0. Factorise to give ( 7) 0. Critical values are ad 7. Thik of the graph of y 0. We are lookig above -ais so we wat two regios, ad hece two iequalities. So the solutio is or 7. How do you kow if there are two regios or oe? Provided the term is positive.. If the iequality sig ivolves less tha ( or <) the there is oe regio. (Oe is less tha two). If the iequality sig ivolves more tha ( or >) the there are two regios. (Two is more tha oe). Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 6
Algebraic Maipulatio September 0 Algebraic maipulatio of polyomials, icludig epadig brackets ad collectig like terms, factorisatio. Studets should be able to use brackets. Factorisatio of polyomials of degree, eg 4. The otatio f() may be used. (Use of the factor theorem is ot required.) Epadig Whe addig (or subtractig) two polyomials we simply add together (or subtract) the coefficiets of the correspodig powers of, for eample 7 5 6 4 Whe multiplyig two polyomials we simply multiply every term i oe polyomial by every term i the other polyomial, for eample 4 5 5 4 5 4 5 4 8 0. 4 So for eample we may be asked to fid the 6 term i 5. We do ot eed to multiply out the brackets completely. We see that there are two ways of gettig a. I total, the the term is. term. Either or Factorisig Whe factorisig a quadratic epressio, such as a b c we are lookig to write it i the form m p q. We eed to cosider two cases, whe a ad whe a. Case : a For eample, factorise. Step : Fid two umbers whose product is - ad whose sum is, i.e. 4 ad. Step : Epress 4. as Case : a For eample, factorise 4 8. Step : Fid two umbers whose product is 4 (i.e. 8) ad whose sum is 4, i.e. - ad. Step : Rewrite quadratic 4 8 8. Step : Factorise this as two separate liear epressios, so 8 ( 4) ( 4). Step 4: Note that there is a commo liear factor ad use this to give 4 8 ( )( 4). Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 7
September 0 e.g. Factorise 7 0 : (i) Fid two umbers which add to give 7 ad which multiply to give 0, i.e. 5 ad 7 0 5 (ii) Write e.g. Factorise 7 6 : (i) Fid two umbers which add to give 7 ad which multiply to give 6, i.e. 4 ad (ii) Write 7 6 4 6 (iii) Factorise first two terms ad secod two terms to give (iv) Hece write as Remember that 6 4 4 Remember also that 5 5 We may be asked to factorise a cubic but, for C there will be o costat term. So the epressio may be 4. We see that there is a factor of so we ca write 4 4 factorise the quadratic as described above.. We the eed to Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 8
Graphs of Fuctios September 0 Graphs of fuctios; sketchig curves defied by simple equatios. Here are some curves you should kow. y y y y y Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 9
September 0 Geometrical iterpretatio of algebraic solutio of equatios. Use of itersectio poits of graphs of fuctios to solve equatios. Fuctios to iclude simple cubic fuctios ad the reciprocal fuctio y = k/ with 0. Graphs of cubics - y a b c d If a 0 the curve goes from If a 0 the curve goes from the bottom left to the top right: the top left to the bottom right: Three distict liear factors is a factor is a factor 4 is a factor So we see that y k 4. The curve crosses the y ais at (0, 8). So k 8 4 8k ad so k. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 0
September 0 Oe liear factor repeated is a repeated factor, as it turs at this poit. is a factor So we see that y k. The curve crosses the y ais at (0, -4). So k 8 k 4 ad so k. Kowledge of the term asymptote is epected. A asymptote is a lie to which a give curve gets closer ad closer. y (as draw o the previous page) has a vertical asymptote at = 0 ad a horizotal asymptote at y = 0. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
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Trasformatios of Graphs September 0 Kowledge of the effect of simple trasformatios o the graph of y f( ) as represeted by y af( ), y f( ) a, y f( a), y f( a). Studets should be able to apply oe of these trasformatios to ay of the above fuctios [quadratics, cubics, reciprocal] ad sketch the resultig graph. Give the graph of ay fuctio y = f() studets should be able to sketch the graph resultig from oe of these trasformatios. The followig graphs are obtaied as show below: y af( ) by stretchig y f( ) by a factor of a parallel to the y-ais. y f( ) a by shiftig y f( ) by a i the positive y directio. y f( a) by shiftig y f( ) by a i the egative directio. y f( a) by stretchig y f( ) by a factor of a parallel to the -ais. The coectio betwee y f( ) ad y f( ) is show below: y f( ) y f( ) The coectio betwee y f( ) ad y f( ) is show below: y f( ) y f( ) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
The coectio betwee y f( ) ad y f( ) is show below: September 0 y f( ) y f( ) The coectio betwee y f( ) ad y f( ) is show below: y f( ) y f( ) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 4
Co-ordiate Geometry i the (, y) plae September 0 Equatio of a straight lie i forms y m c y y m ad a by c 0. To iclude (i) the equatio of a lie through two give poits, (ii) the equatio of a lie parallel (or perpedicular) to a give lie through a give poit. For eample, the lie perpedicular to the lie 4 4y 8 through the poit (, ) has equatio y., Coditios for two straight lies to be parallel or perpedicular to each other. Equatio of a straight lie From GCSE we kow that y m c is the equatio of a straight lie where m is the gradiet ad c is the y-itercept. We could use the form y y m. I this equatio m is still the gradiet ad poit o the lie Both these equatios ca be writte as a by c 0., y is a We ca calculate the equatio of a straight lie if we kow either (i) poits through which the lie passes or (ii) oe poit ad the gradiet of the lie. I case (i) the first thig we do is to fid the gradiet ad so we have ow tured it ito a problem of the type (ii) case. Eample Fid the equatio of straight lie which passes through, ad (, 7). 7 6 First of all fid the gradiet. Gradiet 4 4 y y Use the form m to write the equatio as y 4 ad the rearrage to give y 4 Two straight lies are parallel if their gradiets are equal. Two straight lies are perpedicular if the product of their gradiets is -. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 5
Eample Fid the equatio of straight lie which passes through, 5 ad which has gradiet 4 September 0 We write y 5. 4 Cross multiplyig gives 4 y 5 Hece 4y 0 6 So we have 4y4 0. Eample Fid the equatio of straight lie which passes through, 4 ad which is parallel to y 7. The oly lie that are parallel to y 7 are lies of the form y c. We ow simply plug i the poit, 4 to give c 8 so y. Eample 4 Fid the equatio of straight lie which passes through, ad which is perpedicular to 4y 5. Gradiet of 4y 5 is 5 4 ad so gradiet of perpedicular is 5 4. It follows that the equatio must be of the form 5 y 4 c. (NB: This ca simply be obtaied by swappig aroud the umbers i frot of ad y ad chagig the sig of oe of them). We ow plug i the poit, to give c 5 4 9 so 5y 4 9. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 6
Sequeces ad Series September 0 Sequeces, icludig those give by a formula for the th term ad those geerated by a simple relatio of the form f( ). There are sequeces that be give i terms of where represets the umber of the term. So for eample we could have a sequece such as. We ca the see that the 0th term is 00 ad so o. We might have a sequece such as, 4, 7, 0,. We ca see that the th term is. Alteratively we ca see to fid ay term i the sequece we simply add to the previous term. I this case we would write. Whe we write the sequece i this way we are said to be usig a iterative formula. To calculate terms i a sequece usig a iterative formula we must be give a formula ad a startig poit. So i the above eample we would have to write,. Eample Fid the first four terms of the followig sequece:,. Pluggig i gives Pluggig i gives Pluggig i 5 gives. 5. 4 5 6. Eample Fid the first four terms of the followig sequece:,. Pluggig i gives. 6 Pluggig i gives. 5 5 6 0 Pluggig i gives 4. 5 6 5 5 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 7
September 0 Arithmetic series, icludig the formula for the sum of the first atural umbers. The geeral term ad the sum to terms of the series are required. The proof of the sum formula should be kow. At GCSE we were asked to fid the th term of sequeces such as, 8,, 8,. We will ow be lookig at these sequeces i more detail., 8,, 8,. is called a arithmetic sequece that is a sequece i which the differece betwee cosecutive terms is costat. I a arithmetic sequece the first term is represeted by a ad the commo differece is represeted by d. The first few terms of the sequece are a, a d, a d, a d,... It follows from this that the th term of this arithmetic sequece is a ( ) d. We use u to represet the th term of a sequece ad so, i this eample, we write u a( ) d. So, for eample, i the sequece, 8,, 8,. we see that the first term is ad the commo 5555. differece is 5. The th term is, therefore, I a similar way the th term of the sequece 7, 0,, 6,... is Fidig the Sum of Terms i a Arithmetic Sequece u 7 4. For eample how would we fid the sum of the first 0 terms of the sequece, 8,, 8,? We have see that th term is 5 ad so the 0 th term is 5 0 98. So we wat to fid S 8... 88 9 98. We could write the terms i reverse order to get S 98 9 88... 8 So we see that : S 98 9 88... 8 S 8... 88 9 98 Addig these up i colums gives us S 000... 00 0. There are 0 terms o the right had side so we see that S 00 00. Hece S 00. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 8
September 0 I geeral we might wat to fid S where S is the sum of the first terms of the sequece whose th term is u a ( ) d....... S a ad a d a d S a d a d ad a Addig these two up gives us that: S a d a d... a d a d So we have a d S a d If we cosider... (that is the sum of the first atural umbers) the the above formula for S gives us :... ( ) Eample (a) Fid the th term of 7,, 5 (b) Which term of 7,, 5 is equal to 5? (c) Hece fid 7 5... 5. (a) The first term, a, is 7 ad the commo differece, d, is 4. 7 4 4. The th term is, therefore, (b) For what value of is 4 equal to 5? 4 5 gives us 4 48 ad so. S a d with a 7, d 4 ad. 7 4 6 58 48. (c) We use So we have Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 9
Eample The sum of the first terms of, 8, is 00. Fid. September 0 The first term, a, is ad the commo differece, d, is 5. Usig S a d with a ad d 5 we have that S 65 5. We eed to solve 5 00. Multiplyig both sides by gives This factorises to give 5 00 ad so 5 00 0. 0 5 0 0 ad so 0. Eample The teth term of a arithmetic sequece is 67 ad the sum of the first twety terms is 80. Fid the commo differece, d ad the first term, a. The 0 th term is a 9d (usig a( ) d) so we have a 9d 67. 0 9 S a d) so we have 0 a9d 80. This ca be rearraged to give a9d 8. The sum of the first twety terms is a d (usig So we have two simultaeous equatios. If we double both sides of a9d 67 we get a8d 4. We ow subtract this from a9d 8 to get d 6 ad so a. Uderstadig of otatio will be epected. Kow that NB: r r u r u u u... e.g. r 4 7... r... u, that r... etc.. r. Hece this is a arithmetic sequece with a 4 ad d, so r 8 5. it follows that r Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 0
Sigma Notatio September 0 Uderstadig of the otatio will be epected. Suppose we wated to fid 7... 79. It would make sese to have some sort of otatio that eables us to write dow +7++5+..79 i a simpler form. We alrea kow that the rth term of the sequece is 4r. So we wat to fid the sum of all terms of the form 4r from r to r 0. The otatio we use to represet the sum is. We the have to use a otatio which idicates that we wat the sum of terms of the form 4r, that we start with r ad that we ed with r 0. The otatio we use is 4r 0 0. r So r r 4 4 4... 4 0 7... 79 So, for eample, r 4 7.... Hece this is a arithmetic sequece with 4 r r 8 5. it follows that NB: r r... a ad d, so Aother eample is r.... r Some more eamples 4 r 4 ad r It is worth recogisig that So The ed value for r The start value for r r... 0 r 5 r r 4r 4 5 The epressio ito which the values of r are put. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
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Differetiatio September 0 The derivative of f() as the gradiet of the taget to the graph of y = f () at a poit; the gradiet of the taget as a limit; iterpretatio as a rate of chage. For eample, kowledge that d y is the rate of chage of y with respect to. Kowledge of the chai d rule is ot required. Fidig the gradiet of the curve Cosider the curve show below. Q ( h,f( h)) y f( ) P (,f( )) How do we fid the gradiet to the curve at the poit P? At GCSE we would have foud the gradiet by drawig a taget to the curve at that poit ad the measurig the gradiet of that taget. This was oly estimate we ow wat to fid the eact value of this gradiet. If P is a poit o the curve y f ( ) the the gradiet of the curve at P is the gradiet of the taget to the curve at P. If Q is aother poit o the curve the, as Q moves closer ad closer to P, the gradiet of the chord PQ get closer ad closer to the gradiet of the taget to the curve at P. The gradiet of PQ is f h f f h f h h. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets
September 0 The gradiet of the curve at P is the gradiet of the taget to the curve at P. This is obtaied by movig Q closer ad closer to P (i.e. by makig h ted to 0). The gradiet of the curve at P is deoted by f ( ) or d y d. So f( h) f( ) f( ) lim h0 h where lim meas that we make h ted to 0. h 0 For eample if f( ) the f( h) ( h) h h. So we have that f( h) f( ) f( ) lim h0 h h h lim h0 h h h lim h0 h lim h h0 Thus we have proved that if f( ). the f ( ) Alteratively we could write that if y the d y or simply say differetiates to. d This meas that the gradiet at ay poit o the curve y is. So, for eample, at the poit (, 9) the gradiet is 6. Sice d y d is the gradiet of the curve, we see that d to. represets the rate of chage of y with respect Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 4
September 0 The otatio f() may be used. Differetiatio of, ad related sums ad differeces. 5 E.g., for, the ability to differetiate epressios such as 5 ad epected. Applicatios of differetiatio to gradiets, tagets ad ormals. is Use of differetiatio to fid equatios of tagets ad ormals at specific poits o a curve. We have just proved that if y the d y d. We could have proved that if y the d. We alrea kow that the gradiet of y is 0 (it is a horizotal lie) ad that the gradiet of the lie y is. We could write this more formally by sayig: If y the d y 0 d. If y the d y d So we see the followig: y d 0 We see a patter here which cotiues. The patter is that that if y the d holds for all values of positive ad egative, fractios ad whole umbers.. This 6 So, for eample if y the d The geeral result is as follows: 6 5. If y k the d k. So, for eample if y 5 the d 0 4. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 5
Eample Fid d whe y 5 y d We first eed to multiply out the bracket to get y 5. September 0 We ow differetiate usig the rule o the previous page to get d y 4 d. Eample Fid 5 whe y d We first eed to rewrite y i the form y 5, usig the laws of idices. We ow differetiate usig the rule o the previous page to get 5. d Eample If y the fid the gradiet of the curve at the poit (, 5). First we fid d y 4 d. We the eed to replace i this epressio with the coordiate of the poit we are lookig at. The coordiate of the poit (, 5) is so we substitute ito the equatio d y 4 d to get d. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 6
September 0 Some more eamples... y d 5 6 5 6 6 6 4 7 7 44 4 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 7
Tagets The gradiet of a taget to the curve at ay poit is the gradiet of the curve at that poit. Eample 4 Fid the equatio of the taget to the curve y at the poit (, -). September 0 6 d. We wat to kow at the poit (, -) so we substitute ito the equatio 6 d d to get d y d. So the gradiet of the taget at (, -) is. Usig the earlier work o straight lies we kow that the equatio is y. Rearragig gives y. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 8
September 0 Normals The gradiet of a ormal to the curve at ay poit is the egative reciprocal of the gradiet of the curve at that poit. So.. if d y d at a poit o the curve, the gradiet of the ormal is. Eample 5 if d y 4 d at a poit o the curve, the gradiet of the ormal is 4. if d y d at a poit o the curve, the gradiet of the ormal is 5 if d y d at a poit o the curve, the gradiet of the ormal is -. Fid the equatio of the ormal to the curve y 5. at the poit (, 0). 6 d. We wat to kow at the poit (, 0) so we substitute ito the equatio 6 d d to get d y d. So the gradiet of the ormal at (, 0) is. Usig the earlier work o straight lies we kow that the equatio is y 0. Rearragig gives y 0. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 9
Eample 6 September 0 A curve C has equatio y = 5 + 5 +. (a) Fid i terms of. d The poits P ad Q lie o C. The gradiet of C at both P ad Q is. The -coordiate of P is. (b) Fid the -coordiate of Q. (c) Fid a equatio for the taget to C at P, givig your aswer i the form y = m + c, where m ad c are costats. (d) Fid a equatio for the ormal to C at P, givig your aswer i the form a by c 0, where m ad c are costats. Edecel GCE Pure Mathematics P Jauary 00 (a) 0 5 d (b) At P ad Q, 0 5 d 0 0 This gives 0 ad so or. So coordiate of Q is. (c) At P,, so y 5 5. Also at P d y d Hece the equatio is y. That is y 7. (d) At Q,, ad y. Also at P d y d. So gradiet of ormal is. y Hece the equatio is. That is y. Hece y 0. Secod order derivatives. Cosider the curve y 9. We see that 6 9 d. The gradiet of the gradiet is deoted by d y d. We see that i this eample d d y 6 6 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 40
Idefiite Itegratio September 0 Idefiite itegratio as the reverse of differetiatio. Studets should be aware that a costat of itegratio is required., Itegratio of. For eample, the ability to itegrate epressios such as is epected. Give f( ) ad a poit o the curve, studets should be able to fid a equatio of the curve i the form y f( ). We kow that if y 5 the d 6 5. However we also that if y 5 the d 6 5. or if y 5 the d 6 5. So if we were give d ad asked to fid y we caot get a defiite aswer. 6 5 As we have see y could be 5 or it could be 5 or it could be 5. All we ca say is that if 6 5 d the y 5 c where c ca be ay costat. To fid a particular solutio of d we eed to have more iformatio. 6 5 Eample If 6 5 ad the curve passes through the poit (, ) the fid y. d We see that y 5 c Sice the curve passes through the poit (, ), we have that 5 c ad so c 6. Hece see that the solutio is y 5 6. Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 4
Notatio The idefiite itegral of f ( ) is deoted by f ( ) d. September 0 The geeral rule we use is p p k d k c ( p ) p Here are some eamples dc 5 d 5 c 8 4 d 4 d c 6 d 6 d 6 c NB We ca oly itegrate a epressio made up of terms of the form Eample Fid d. p k. We first of all eed to epress as 4 8 We ow use the above rule to see that d 4 8 d 4 4 c Eample 8 Fid d. We first of all eed to epress 8 as 8 8 We ow use the above rule to see that 8 d 8 d 5 6 5 c Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 4
September 0 Some more eamples... y y d 5 6 5 6 c c 6 6 6 4 c 4 7 7 5 4 4 c 5 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets 4
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