MEI Conference 2009 Stretching students: A2 Core

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Rudolph Booker
5 years ago
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1 MEI Coferece 009 Stretchig studets: A Core Preseter: Berard Murph berard.murph@mei.org.uk Workshop G

2 How ca ou prove that these si right-agled triagles fit together eactl to make a triagle? What does it tell ou about the iscribed circle? What is the lik with the double agle formulae?. The triagle. Slidig ladders Ladder A topples awa from a wall. Ladder B slides dow a wall. Compare the paths followed b the mid-poit of each ladder. 3. Circle theorem? The spiral starts at the poit (0,-) ad the perpedicular edges are draw i a aticlockwise spiral with a commo ratio r as show. Due to similarit, after a eve umber of steps the leadig poit will be o the diagoal lie show. If this diagoal makes a agle θ with the first edge as show, fid, i terms of θ, the coordiates of the poit o which the spiral is covergig.

4. Fidig parametric equatios of a Cartesia curve Imagie a poit P o the curve as show. The = rcosθ ad = rsiθ. If we ca write r i terms of θ the we have our parametric equatios.

3 4. Fidig parametric equatios of a Cartesia curve Imagie a poit P o the curve as show. The = rcosθ ad = rsiθ. If we ca write r i terms of θ the we have our parametric equatios Fid parametric equatios for the curve ( ) =. (Substitute = rcosθ ad = rsiθ ad use this to epress r i terms of θ. Fiall substitute for r i = rcosθ ad = rsiθ ) 5. Costructig a regular petago. Draw a circle, cetre O ad bisect the radius OP. M is the midpoit of OP.. Draw a arc of a circle, cetre M, radius MA. This arc crosses OQ at N. 3. Draw a arc of a circle, cetre A, radius AN. This gives the first side, AB of the regular petago. A A A B P M P M N Q N Q Prove this would produce a regular petago. What is the lik with si 5θ or si8? 6. Equilateral triagle o grid poits Prove that a equilateral triagle i the - plae caot have all three vertices o grid poits (i.e. poits where both coordiates are itegers.)

4 7. Estimatig the harmoic series The area of the shaded regio is d The area of the shaded regio is 5 d I each case, imagie slidig the 4 shaded regios left so that each oe touches the ais. You ca see that these 4 regios fit ito the b rectagle without overlappig ad so the shaded areas are both less tha. Eplai how together these lead to l N + 3 N 8. Composite piecewise fuctios For the fuctios f( ) ad g( ) give below, fid the composite fuctio fg( ) 0 < 0 = 6 > 4 f( ) < 0 g( ) = > Primes of the form 4+3 Prove b cotradictio that there is a ifiite umber of primes of the form 4+3

5 0. Which is bigger, e p or p e? B cosiderig the turig B cosiderig the turig e l poit o the graph of = poit o the graph of = e B cosiderig the gradiets of a taget ad chords of the graph = l.. A surprisig propert? Look at the graphs of = ta ad = cos. It appears that the cross at right agles to each other. Is this true? π/ π/ π 3π/ π. All itegers? Fid the missig edge legth, a. a A

6 3. A series for l ( ) = + = Itegratig betwee = 0 ad = ou should be able to fid a ifiite series which coverges to d = l. 0 + Usig this idea ad startig with other fuctios geerate other ifiite series. The biomial epasio: ( ) 4. Series of biomial coefficiets + = B usig calculus ad/or substitutio, prove the followig: = = = = = 0 Ca ou fid a more? 5. Biomial theorem ad differetiatio 3 4 d 3 = ( ) = = = d ( ) d d = = ( ) So the epasio of ( ) 3 has triagular umbers as coefficiets. Fid the ratioal fuctio whose epasio has square umbers as coefficiets. 3 4 Evaluate

7 Lesso idea : Newto s approimatio to π The diagram shows a semi-circle with cetre (,0 ) ad radius.. Show that the area of the shaded regio is π Show that the semicircle has equatio = ( ) ad use the biomial theorem to fid the first five terms i the epasio. 3. Usig these terms, ad itegratio, fid a approimate value for the shaded area. 4. Compare this with the eact aswer foud i above*. To what level of accurac does this give the value of π? *How would Newto have evaluated 3? He might have used the biomial theorem o = 3 = = sice this would coverge quickl

8 Lesso idea : Bouds o! To calculate, sa, 00! ou eed to perform 99 multiplicatios. Is there a quicker wa to fid the approimate value of! where is a large umber? Here is oe method. Thik about the area uder the graph = l betwee = ad =. l d l d l d l l This is = = [ ] = [ ] = + We ca get lower ad upper bouds for this b approimatig the area of the regio uder the graph = l as show below ( ) l + l l < l d < l + l l (( ) ) ( ) (( ) ) ( ) l! < l d < l! l! < l + < l! Takig the two iequalities separatel: l + l + < l (! )! > e = = e e e l( ( )!) < l + ( )! < e! < e e e Combiig these gives e <! < e e e For eample, e < 00! < 00e.0 0 < 00! <.0 0 e e I fact, 00!

9 Defie si m I d m = 0 π. Evaluate I 0 ad I. Lesso idea 3: Wallis formula for π m m. Writig si = si si ad usig itegratio b parts, show that mim = ( m ) Im. 3. Usig our two aswers above, evaluate I, I4, I 6,... ad I3, I5, I 7, Usig the fact that 0< si < for 0 π π π m + m m < < si d si d si d 5. Hece show that π = π < <, eplai wh Lesso idea 4: Biet s formula i three steps The Fiboacci sequece: f =, f =, f+ = f + f for Cosider = f = Verif that ( ) f = = the divide throughout b : f = ( ). Derive, usig partial fractios: ( ) + 5 α β where 5 α + 5 = ad β =. 3. Usig the biomial epasio of the terms o the RHS ad cosiderig coefficiets of show that f = 5

10 . 3= = = Prove that ever positive iteger ca be writte i the form a + b c. Take ever iteger power (greater tha the first power) of ever positive iteger greater tha ad add the reciprocals together. What do ou get? Which of the followig umbers is bigger? 4 7 d or d 4. Prove that ever positive ratioal umber ca be writte as the sum of distict uit fractios (i.e. fractios with umerator ). 3 = Let f ( ) ( )( ) = Show that, after multiplig out, ol eve powers of remai. 6. If ou use a graph plotter to plot straight lie. Is it? = ou will fid it seems be a 7. Prove that ta 50 + ta 60 + ta 70 = ta Some positive umbers add up to 9. What is the maimum product? 9. Usig the sie rule ad the compoud agle formulae, prove that, i a triagle, A+ B a+ b ta = ( ) A B a b ta ( ) 0. Varigo's theorem: Prove that joiig the midpoits of the sides of a quadrilateral i order produces a parallelogram.. u, u, u3,..., u + is a sequece of + positive itegers. v, v, v3,..., v + is a rearragemet of u, u, u3,..., u +. Prove that the sequece { } k least oe eve umber. t where t = u v k =,,3,...,+ cotais at k k k. The particular fuctio f: It has the followig two properties: The fuctio is icreasig; i.e. f ( + ) > f ( ) for all f f = 3 for all. The composite fuctio ( ( )) Fid f( 00 )

Mathematics Extension 1

Mathematics Extension 1 016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved