) + 2. Mathematics 2 Outcome 1. Further Differentiation (8/9 pers) Cumulative total = 64 periods. Lesson, Outline, Approach etc.

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Noreen West
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1 Further Differetiatio (8/9 pers Mathematics Outcome Go over the proofs of the derivatives o si ad cos. [use y = si => = siy etc.] * Ask studets to fid derivative of cos. * Ask why d d (si = (cos see graphs. d Go over Chai Rule => (ta etc + Go over the idea of Implicit Differetiatio ad show that 3y y = => dy 4 3y = etc + more eamples 3+ y Page 3 E Page 3 E Qu a, b, e, b, c, d As may as possible 3a, d i the period/h work Page 33 E 3A Qu ad 3 Page 36 E 4A page 6 E Qu, Qu 3 Go over (i equatio of a taget to a eplicit f (ii dy d dy = or by fidig the [A/B] derivative of the origial u arraged f. 4 Itroduce Logarithmic Differetiatio via y =, y = si, y = (si etc. 6 4 ( 3+ 5 ( Possibly y =?? [A/B] 5 ( Page 36 E 4A Page 6 E Qu 5, 9, 4 Qu - 6 Page 38 E 5 Qu a, d, f, k,,6 Page 40, E 6 page 0 E 4 Qu,, (7 5 Eplai idea of Parametric Equatios of Curves Page 4 E 7A Qu a - c = rcosθ, y = rsiθ, => + y = r (circle Page 44 E 8A page 3 E 5 = acosθ, y = bsiθ, => y + = (ellipse Qu ( dy a b oly = asecθ, y = btaθ, => y = (hyperbola a b c = ct, y = => y = c t (hyperbola also Eplai :- dy dy = / ad fid equatio of taget to a curve give i parametric form. 6 Fid d derivative => dy d dy dy dy = = / ad Page 44 E 8A Page 5 E 6 Qu ( dy oly,, 3 use this to check the ature of turig poits 7 Apply parametric ad implicit diff to related problems rates of chage of volumes, surface areas 8 Use parametric differetiatio to fid st ad d derivatives ad apply this to motio i a plae. Page 53 E A page 8 E 3 Qu - 6 Page 50 E Page 7 E 7 Qu,, 5, 6, 7 = f(t, y = g(t => speed = velocity = 9 Review dy ad dy ( + i ad y directios Cumulative total = 64 periods This is page umber 8

2 Further Itegratio (0/ periods Mathematics Outcome Kow = si, = + ta = si a a, a = ta + a a Itegrate usig Partial Fractios ( + 6 ad Itegrate usig Partial Fractios ( = l( + ( ( + 4 Itegrate usig Partial Fractios ( = l( + + ( + ( + l( + + c 3l( + + c Page 6 E A Page E Qu a, b, e, f a, c, f,h Page 6 E B Qu a, c, e, h Page 64 E Page 5 E Qu 5 a, b, d, e 6 a, c Page 66 E 3A Page 5 E 3 Qu, 3 a, d, e Page 66 E 3A Page 7 E 4 Qu, 3 b, c, f 7 ad + + = l + ta - + c ( + 5 Itegratio by Parts. Show that ( u dv = uv ( v du 6 Itegratio by Parts twice (or more [A/B] 7 Itegratio by Parts usig a dummy fuctio l = l, si = si Etesio :- It by Parts returig to origial fuctio e si etc 8 First Order Differetial Equatios (Variable separable 9 Particular solutio of a First Order Differetial Equatio 0 Solvig Real Problems with First Order Differetial Equatios icludig rate of chage Page 69 E 4 Page 9 E 5 Qu, Page 7 E 5A Page 9 E 6 Qu a - d, g, m, Page 30 E 7 Page 7 E 5A Page 3 E 8 Qu e, f, q, s Page 77 E 8 Qu Page 33 E 9 Page 77 E 8 Qu, 3, 5 Page 35 E 0 Page 8 E 9A Page E Qu ad 4 9 growth of bacteria ( is give by k ( = (chemical reactios, Newto s Laws of coolig etc Review Cumulative total = 75 periods This is page umber 9

3 Comple Numbers (9/0 pers Mathematics Outcome 3 / Itroduce Imagiary umber i = ( ad Comple Number z = a + ib Show that i =, i 3 = i, i 4 = etc. Show how to add, subtract, multiply ad itroduce the comple cojugate z = a ib to divide Show also how to fid simple powers ( + i 4 (biomial. Page 90 E Pages E Qu,, 3, 6, 7, 8 Page 9 E E (some if eeded Qu a, b, c, c, e 3 a, b, f 5 a, b 3/4 Show how to represet a comple umber p a Argad Diagram (treat additio etc as vector additio etc Defie modulus ad argumet ad show how to fid r ad θ. Show z z = r(cosθ + i si θ s(cos φ + i si φ = rs(cos( θ + φ + i si( θ + φ Same for divisio 5 Show De Moivre s Theorem z = r (cos θ + i si θ (Itroduce z = rcisθ ad/or z = re i θ as shorteed form 6 Itroduce Fudametal Theorem of Algebra to fid all the roots (factors of a polyomial equatio (fuctio 7 Show the Geometric Iterpretatio of equatios ad iequalities i the Comple Plae Page 94 E Page E 3 Qu 3 a, b, d, e, i Qu 6 a, b, f 7 a, b, c Page 98 E 5 Page 55 E 4 Qu a, b, f, g page 0 E 6 Page E 5 Qu,, 3a, 4 g, h, i, j Page 08/09 E 8 Page 6 E 6 Qu a, d 3 a, b, 4, 5, 6a, b Page 96 E 4 Page 63 E 7 Qu a, b, d, f, j 3 a, b, 4 a, c 8 Apply De Moivre s Theorem to solve Multiple Trig formulae Epress si5θ i terms of siθ oly 9 use De Moivre s Theorem to fid the Roots of Uity (etesio :- fid roots of other comple umbers 0 Review Page 0 E 6 Qu 5, 6, 7 Page 06 E 7 Qu + selectio from Ow Eamples ow Eamples Cumulative total = 85 periods This is page umber 0

4 Sequeces ad Series (7/8 pers Mathematics Outcome 4 Revise Fifth Year defiitios of Sequeces ad Series Defie fiite ad ifiite sequeces & series Page 7 E A Page 7 E Qu a, c, d, a-e, 3, 4, 6 Qu - 3 Defie otatio u = Itroduce idea of Arithmetic Sequece u+ = u + d Build up to u = a + ( d - geeral term of A.S. Show how to fid sum of terms of a Arithmetic Series => S = ( a + d ( Show two types fid S 5 = ad S = Show how to use simultaeous equatios i :- u = 9, u 5 = 7, fid a, d, ad S 0 3 Itroduce the Geometric Sequece :- u+ = ru Build up to u = ar - the geeral term of a G.S. Fid u 0 = Page 0 E 3A Page 7 E Qu, 3, 4, 5, 8 Qu 4-9 Page 3 E 4A Page 75 E A Qu a - e,, 3, 5, 7 Qu Show how to fid sum of terms of a Geometric Series ar ( Build up to S = (or a r ( r r Fid S 9 = Page 7 E 5A Page 75 E A Qu,, 3, 4 Qu 5-7 Use sim equ s to solve u 3 = 3, u 6 = 4 => fid S 8 5 Go over S of a G.S. ad defie its eistece (Graphics Calculators are useful here 6 Epad r as a G.S. with commo ratio r, first term. as 3 4 = + r + r + r + r +... r show alterative usig biomial o ( r or by dividig => ( r etc. Epad a+ b as a + b etc. a ( [A/B] 7 Show that Lim + = e (calculator Reitroduce the otatio to show that Page 3 E 6A Page 77 E B Qu - 4 ad 7 Page 34 E 7A Page 79 E 3 Qu, 4 Page 35 E 7B Qu, 5 Page 37 E 8 Page 8 E 4 Qu a - d, a, b, 4 page 8 E 5 k= k = = ( + k = 8 Review, ( 4 k etc. to fid ( k + (use A.S. ad show how Cumulative total = 93 periods This is page umber

5 Elemetary Number Theory & Proofs (0/ periods Mathematics Outcome 5 Go over idea of statemet, true/false, implicatios (=>, <=, <=> ad equivalece if a => b ad b => a. Coverse Statemets. Direct Proof ad Disproof by Couter eample Page 3 E A Page E, & 3 A selectio Proof by Cotradictio. set of primes is ifiite is irratioal etc. Page 4 E 3A Page 88 E 4 Qu,, 3, 6,, 3/4 Simple proof by Iductio (At this stage, depedet o time, you may wish to do ALL the proofs by iductio by combiig the work of this Uit with that of the Iductio Chapter i Maths 3 (a Prove that 8 3 W k= (b Prove that r = ( + ( + 6 r = Prove Biomial ad De Moivre s Theorem by Iductio Page 0 E 4 Page 89 E 5 Qu 6, 7, 8, 9,,, 3 + whole sheet of questios 5 Review For Sessio Ed of Mathematics Total = 43 periods for Mathematics Cumulative Total for M & M = 98 periods Assumig Maths started aroud st Nov, this uit should ed aroud Friday 8th Ja. =>3 periods revisio (icludig specime NAB3 + period for test = 47 periods Cumulative total for M & M = 0 periods TEST aroud Thurs 4th Jauary Actual Test Date = PRELIMS start Wed 0th Feb for weeks- Suggest doig at least two weeks work o Maths 3 before begiig revisio for Prelims (possibly Matrices could be doe or Vectors if a little more time is available This is page umber

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( ) ( ) ( ) ( ) ( + ) ( ) LSM Nov. 00 Cotet List Mathematics (AH). Algebra... kow ad use the otatio!, C r ad r.. kow the results = r r + + = r r r..3 kow Pascal's triagle. Pascal's triagle should be eteded up to = 7...4 kow ad