Higher Course Plan. Calculus and Relationships Expressions and Functions

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1 Higher Course Pla Applicatios Calculus ad Relatioships Expressios ad Fuctios Topic 1: The Straight Lie Fid the gradiet of a lie Colliearity Kow the features of gradiets of: parallel lies perpedicular lies Kow the equatio of a straight lie of the form y = mx + c Kow a equatio of a straight lie of the form ax + by + c = 0 Kow the equatio of a straight lie of the form y b = m(x a) Fid the distace betwee two poits usig the distace formula Recogise the term locus Fid the equatios of: medias altitudes perpedicular bisector Kow the cocurrecy properties (i triagles) of: medias altitudes perpedicular bisectors Duratio Jue 9 periods 1. Gradiet revisio ad m ta (Ex 1A) 2. Colliearity (Ex 1B) 3. Perpedicular lies; y mx c ; Ax By C 0 (Ex 1D, 1E, 1F) 4. y b m x a (Ex 1G) 5. ( ) 6. Perpedicular bisectors (Ex 1I) 7. Altitude of a triagle (Ex 1K) 8. Medias of a triagle (Ex 1M) 9. Mixed questios (Ex 1O)

2 2: Sets ad Fuctios Set otatio Kow the meaig of the terms: domai ad rage of a fuctio Fuctios ad mappigs Composite fuctios Iverse of a fuctio Graphs of iverses Expoetial ad log fuctios 3: Recurrece Relatios Kow the meaigs of the terms: sequece, th term, limit as teds to ifiity Usig recurrece relatios to solve problems More complex recurrece relatios Liear recurrece relatios Kow the coditio for the limit of a sequece Fid ad iterpret the limit of a sequece from a recurrece relatio Defie ad iterpret a recurrece relatio of the form U +1 = au + b Solvig recurrece relatios to fid a ad b. Solvig liked recurrece relatios Special sequeces 5: Graphs of Fuctios Give the graph of f(x), draw graphs of related fuctios for: y = f(x) + a 6 periods 1. Fuctios, domais ad rages (Ex 2B) 2. Compositio of fuctios (Ex 2C) 3. Iverse of fuctios; Graphs of iverses (Ex 2D, WS) 4. Itroductio to logarithms (Ex 2H) 5. Mixed questios (Ex 2I) 6. ( ) SUMMER HOLIDAYS 5 Periods 9 Periods 1. Itroductio (Ex 5B, 5D) 2. Liear recurrece relatios i cotext (Ex 5C) 3. Limit of sequeces defied by R. R.s (Ex 5H) 4. Formig R. R.s give 3 terms (Ex 5I) 5. Mixed questios (Ex 5L) 1. Stadard graphs ad y f x a (Ex 3A, Ex 3C)

3 y = f(x + a) y = f(x) y = f( x) y = kf(x) y = f(kx) Ad combiatios of the above fuctios Kow the geeral features of the graphs of f : x a x ad f : x log a x 6: Trigoometry Graphs ad Equatios Amplitude ad period Graphs of f : x si(ax + b) ad f : x cos (ax + b) Radias Usig exact values (i degrees ad radias) Solvig problems usig exact values Solvig Trig equatios graphically ad algebraically Solvig compoud agle equatios 6 Periods 2. y kf x icludig y f x 3. y f x a (Ex 3E) 4. y f kx icludig y f x (Ex 3G, 3K) (Ex 3M, 3I) 5. Combiig trasformatios ad Summary (Ex 3P) 6. Graphs relatig to Expoetials ad Logs (Ex 3N, 3O) 1. Trig. graphs (Ex 4A, 4B) 2. Radia measure (Ex 4C) 3. Exact Values (Ex 4D, 4E) 4. Trig equatios (Ex 4H, 4I) : Differetiatio Kow that f (x) is: rate of chage of f at x gradiet of the taget to the curve at x Kow that: If f(x) = x the f ' (x) = x 1 If f(x) = g (x) + h (x) the f '(x) = g'(x) + h'(x) 13 Periods 1. Discuss Differetiatio by first priciples 2. Differetiatig x (Ex 6D) 3. Fidig gradiet at a poit (Ex 6E) 4. Differetiatig kx ad applicatios (Ex 6F, 6G, 6H) Leibitz otatio ad equatio of taget (6J) 7.

4 If f(x) = ax the f '(x) = ax 1 Applicatios of derivatives Kow how to use Leibiz otatio to differetiate d y d x Differetiatio of si x ad cos x 8. Icreasig ad decreasig fuctios (Ex 6L) 9. Statioary poits ad their ature (Ex 6M) 10. Curve Sketchig (Ex 6N) 11. Closed Itervals (Ex 6O) 12. Graph of derived fuctio (Ex 6P) 13. Differetiatio of si x ad cos x (Ex 14B) Equatio of the taget to a curve y = f(x) at x = a Icreasig ad decreasig fuctios Statioary poits ad graph sketchig Closed itervals Graphs of derived fuctios 7: Polyomials Evaluatig fuctios usig the ested form Sythetic Divisio by (x a) Remaider theorem Factor theorem Fidig a polyomial s coefficiets Solvig polyomial equatios Fuctios from graphs Curve sketchig Approximate roots / Iteratio Assessmet: A OCTOBER HOLIDAYS 10 Periods 1. Itroductio ad sythetic divisio (Ex 7A, 7B) 2. Remaider Theorem (Ex 7C) 3. Factor Theorem (Ex 7E) 4. Fidig coefficiets of polyomial (Ex 7F) 5. Solvig polyomial equatios (Ex 7G) 6. ( ) 7. Fidig equ of polyomial curve with give roots (Ex 7H) 8. Sketchig polyomial curves (Ex 7I) 9. ( ) Note: Behaviour for large pos/eg x ot required 10. Iteratio (Ex 7J)

5 8: Quadratic Fuctios Graphs of quadric fuctios Sketchig of quadric fuctios Remider completig the square (from NAT 5) Solvig quadric equatios ad i-equatios The quadratic formula Usig the discrimiat Tagecy 9: Itegratio Kow that if f(x) = F'(x) the f(x) dx = F(x) + C Uderstad the defiite itegral, limits of itegratio Idefiite itegrals Rules of itegratio Defiite itegrals Area uder a curve (approximate ad exact usig itegratio) Area betwee two curves Differetial equatios 10: Optimizatio Optimizatio : maximum ad miimum 8 Periods 9 Periods 1. Sketchig quadratic graphs (Ex 8C) 2. Completig the square remider from NAT 5 (Ex 8D) 3. Iterpretig completed square form (Ex 8D) 4. Derivig the quadratic formula May be omitted for lower sets or if time is tight 5. Solvig quadratic iequatios (Ex 8F) 6. Discrimiat ad the ature of roots (Ex 8H) Note: Ratioal/irratioal roots are ot i the textbook but are required! 7. Fidig coefficiets give the ature of roots (Ex 8I) 8. Discrimiat ad tagecy (Ex 8J) 1. Itroductio ad Itegratig x (Ex 9G) 2. Itegratig sums ad differeces of kx (Ex 9H, Ex 9I) Fudametal theorem of calculus (Ex 9L) Area betwee curve ad x-axis (Ex 9N) 7. Area betwee two curves (Ex 9P) 8. Differetial equatios (Ex 9Q) 2 Periods 9. Optimisatio (Ex 6Q, 6R) 10.

6 Assessmet: B 1 11: The Circle Revise distace betwee two poits (straight lie topic) Equatio of a circle cetred o the origi ad cetre ( a, b ) Expaded form of the equatio of a circle (the geeral equatio of a circle) Itersectio of a lie ad a circle Tagets to circles Fidig taget equatios 12: Additio Formulae Compoud agles si( A ± B) = si Acos B ± cos Asi B cos( A ± B) = cos Acos B si Asi B Trig idetities si 2A = 2siAcos A cos 2A = cos 2 A si 2 A cos 2A = 2cos 2 A 1 cos 2A = 1 2si 2 A Applicatios of additio formulae Trig equatios Formulae for cos^2 x ad si^2 x 8 Periods 6 Periods 1. Distace formula ad equ of circle, cetre O (Ex 12B, 12D) 2. Equatio of circle, cetre (a, b), radius r (Ex 12F) 3. Geeral equatio of a circle (Ex 12H) Itersectio of lie ad circle (Ex 12J) Taget to a circle (Ex 12K, Ex 12L) Additio formulae (Ex 11B, 11C, 11D) Trig idetities (Ex 11E, 11F) 5. Double agle formulae (Ex 11G) 6. Trig equatios ivolvig double agles (Ex 11H) CHRISTMAS HOLIDAYS Assessmet: B 2

7 13: Vectors Revise scalars ad vectors, compoets, magitude, equal vectors, additio ad subtractios, multiplicatio by a scalar, positio vectors ad 3D coordiates (from NAT 5) Uit vectors Colliearity Sectio Formula 3D vectors Scalar Product (formula ad compoet form) Agle betwee vectors Perpedicular vectors Applicatios Properties of the scalar product 14: Further Calculus Itegratio of si x ad cos x Derivative of (x + a) Derivative of (ax + b) Chai rule Applicatios Itegratig (ax + b) Itegratig si (ax + b) ad cos(ax + b) 11 Periods 1. Revisio of Nat 5 Vectors (Ex 13A, 13B, 13C, 13D, 13E) Uit vectors & positio vectors (Ex 13F, 13G) 4. Colliearity (Ex 13I) 5. Vectors ad ratios (NB. Sectio formula may ot be the best approach!) (Ex 13K) 6. 3D ad basis vectors (Ex 13L, 13M) 7. Extedig rules to 3D (Ex 13N) 8. Scalar product (both forms) (Ex 13O, 13P) 9. Agle betwee vectors (Ex 13Q) 10. Perpedicular vectors (Ex 13R) 11. Commutative & distributive laws for scalar product (Ex 13U) (NB Ex 13S is well worth doig but may have to be left util after the prelims) 5 Periods 1. Itegratig si x ad cos x (Ex 14C) 2. Chai rule (Ex 14H) 3. Applicatios of chai rule (Ex 14I) 4. Itegratig ax b (Ex 14J) 5. Itegratig si ax b ad cos ax b (Ex 14K) 15: Expoetial ad Log Idices Expoetial graphs 9 Periods 1. Revisio of expoetial & log graphs (Teacher)

8 Expoetial growth ad decay The expoetial fuctio Laws of logs Logarithmic equatios Natural logs Formulae from experimetal data Related graphs 16: The wave fuctio Revisio - waves ad graphs (earlier i the course) Addig two waves The differece of two waves Express acos + bsi i the form rcos( ± ) or rsi( ± ) Multiple agles Maximum ad miimum values Solvig equatios Applicatios 6 Periods 2. Expoetial growth ad decay (Ex 15C) 3. Expoetial/log coversio (Ex 15E) 4. Laws of logarithms (Ex 15F) 5. Log equatios (Ex 15G) 6. Natural logs (Ex 15H) 7. Models of the form y kx (Ex 15I) 8. Models of the form y x ab (Ex 15J) 9. Related graphs (Ex 15K) 1. Writig i form kcos x 2. Writig i forms k cos x ad k si x (Ex 16D) (Ex 16E) 3. Wave fuctio with multiple agles (Ex 16F) (Ca be omitted for lower sectios) 4. Fidig max ad mi of wave fuctios (Ex 16G) 5. Solvig equatios ivolvig wave fuctios (Ex 16H) 6. Applicatios (Ex 16I) Assessmet: C FEBRUARY HOLIDAYS EXAM PREPARATION

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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