MEI Coferece 009 Stretchig studets: A Core Preseter: Berard Murph berard.murph@mei.org.uk Workshop G
How ca ou prove that these si right-agled triagles fit together eactl to make a 3-4-5 triagle? What does it tell ou about the iscribed circle? What is the lik with the double agle formulae?. The 3-4-5 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. 4 4 3 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.)
7. Estimatig the harmoic series 3 4 5 The area of the shaded regio is 5 + + + d 3 4 3 4 5 The area of the shaded regio is 5 d + + + 3 4 5 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 4 0 < 0 g( ) = 0 3 6 > 3 6 8 4 3 4 5 6 6 8 4 3 4 5 6 9. Primes of the form 4+3 Prove b cotradictio that there is a ifiite umber of primes of the form 4+3
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 4 8 7 6 A
3. A series for l ( ) 3 4 5 6 7 = + = + + + +... + 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 + = + + + + 0... B usig calculus ad/or substitutio, prove the followig: + + +... = 0 4 + +... = 0 0 + + 3 +... + = 3 + + +... + = 0 + + +... + = 0 Ca ou fid a more? 5. Biomial theorem ad differetiatio 3 4 d 3 = ( ) = + + + + +... = = + + 3 + 4 +... d ( ) d d = = + + + ( ) 3 + 3 3 6 0... So the epasio of ( ) 3 has triagular umbers as coefficiets. Fid the ratioal fuctio whose epasio has square umbers as coefficiets. 3 4 Evaluate + + + +... + +.... 3 4
Lesso idea : Newto s approimatio to π 0.5 0.5 0.5 0.75 The diagram shows a semi-circle with cetre (,0 ) ad radius.. Show that the area of the shaded regio is π 3 4 3. 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 6 49 49 48 7 3 = 3 = = sice this would coverge quickl. 6 49 6 49 4 49
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. 3 3 3 4 5 6 7 8 3 4 5 6 7 8 ( ) l + l 3 +... + l < l d < l + l 3 +... + 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, 00 00 00 00 e < 00! < 00e.0 0 < 00! <.0 0 e e 57 59 57 I fact, 00! 9.33 0.
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,... 4. Usig the fact that 0< si < for 0 π π π m + m m < < 0 0 0 si d si d si d 5. Hece show that π 4 4 6 6 =........ 3 3 5 5 7 π < <, eplai wh Lesso idea 4: Biet s formula i three steps The Fiboacci sequece: f =, f =, f+ = f + f for Cosider = 3 4 5 6 f = + + + 3 + 5 + 8 +.... 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 + 5 5 coefficiets of show that f = 5
. 3= 4 + 6 7 4= 6 + 7 9 5= 4 + 5 6 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? +... + +... + +... + +... 7 3 0907 5 9583 458893590 3. Which of the followig umbers is bigger? 4 7 d or 0 7 4 0 d 4. Prove that ever positive ratioal umber ca be writte as the sum of distict uit fractios (i.e. fractios with umerator ). 3 = + + + + + + 7 7 8 9 56 57 7 39 5. Let f ( ) (... 00 )(... 00 ) = + + + + + +. Show that, after multiplig out, ol eve powers of remai. 6. If ou use a graph plotter to plot straight lie. Is it? 3 3 3 + + = ou will fid it seems be a 7. Prove that ta 50 + ta 60 + ta 70 = ta 80. 8. 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 )