Learning Objectives of Module 2 (Algebra and Calculus) Notes:

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1 67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under the sme lerning unit.. The notes in the Remrks olumn of the tle my e onsidered s supplementry informtion out the lerning ojetives. 4. To id tehers in judging how fr to tke given topi, suggested lesson time in hours is given ginst eh lerning unit. However, the lesson time ssigned is for their referene only. Tehers my djust the lesson time to meet their individul needs. Lerning Unit Lerning Ojetive Time Remrks Foundtion Knowledge Are Surds. rtionlise the denomintors of expressions of the form k.5 This topi n e introdued when tehing limits nd differentition.

2 68 Lerning Unit Lerning Ojetive Time Remrks. Mthemtil indution. understnd the priniple of mthemtil indution Only the First Priniple of Mthemtil Indution is required. Applitions to proving propositions relted to the summtion of finite sequene re inluded. Proving propositions involving inequlities is not required.. Binomil. expnd inomils with positive integrl indies using the Proving the Binomil Theorem is Theorem Binomil Theorem required. The use of the summtion nottion ( ) should e introdued. The following re not required: expnsion of trinomils the gretest oeffiient, the gretest term nd the properties of inomil oeffiients pplitions to numeril pproximtion

3 69 Lerning Unit Lerning Ojetive Time Remrks 4. More out trigonometri funtions 4. understnd the onept of rdin mesure 4. find r lengths nd res of setors through rdin mesure 4. understnd the funtions osent, sent nd otngent nd their grphs 4.4 understnd the identities + tn = se nd + ot = ose Simplifying trigonometri expressions y identities is required. 4.5 understnd ompound ngle formule nd doule ngle formule for the funtions sine, osine nd tngent, nd produt-to-sum nd sum-to-produt formule for the funtions sine nd osine The following formule re required: sin(a B) = sin A os B os A sin B os(a B) = os A os B sin A sin B tn A tn B tn(a B) = tn Atn B sin A = sin A os A os A = os A sin A = sin A = os A tn A = tn A tn A sin A = ( os A)

4 70 Lerning Unit Lerning Ojetive Time Remrks os A = ( + os A) sin A os B = sin(a + B) + sin(a B) os A os B = os(a + B) + os(a B) sin A sin B = os(a B) os(a + B) A B A B sin A + sin B = sin os A B A B sin A sin B = os sin A B A B os A + os B = os os A B A B os A os B = sin sin Susidiry ngle form is not required. sin os nd A A os os A A n e onsidered s formule derived from the doule ngle formule.

5 7 Lerning Unit Lerning Ojetive Time Remrks 5. Introdution to the numer e 5. reognise the definitions nd nottions of the numer e nd the nturl logrithm.5 Two pprohes for the introdution to e n e onsidered: e lim( ) n n n (proving the existene of this limit is not required) e x x x x!! This setion n e introdued when tehing Lerning Ojetive 6.. Sutotl in hours 0

6 7 Lerning Unit Lerning Ojetive Time Remrks Clulus Are Limits nd Differentition 6. Limits 6. understnd the intuitive onept of the limit of funtion Students re not required to distinguish ontinuous funtions nd disontinuous funtions from their grphs. The theorem on the limits of sum, differene, produt, quotient, slr multiple nd omposite funtions should e introdued ut the proofs re not required.

7 7 Lerning Unit Lerning Ojetive Time Remrks 6. find the limit of funtion The following formule re required: lim 0 sin = lim x0 e x x = Finding the limit of rtionl funtion t infinity is required. 7. Differentition 7. understnd the onept of the derivtive of funtion 4 Students should e le to find the derivtives of elementry funtions, inluding C, x n ( n is positive integer), x, sin x, os x, e x, ln x from first priniples. Nottions inluding y', f '(x) nd dy should e introdued. Testing differentiility of funtions is not required. 7. understnd the ddition rule, produt rule, quotient rule nd hin rule of differentition The following rules re required:

8 74 Lerning Unit Lerning Ojetive Time Remrks d d ( u v) ( uv) du dv u v dv du d ( u v ) du v u v dv dy dy du du 7. find the derivtives of funtions involving lgeri funtions, trigonometri funtions, exponentil funtions nd logrithmi funtions The following formule re required: (C)' = 0 (x n )' = n x n (sin x)' = os x (os x)' = sin x (tn x)' = se x (ot x)' = ose x (se x)' = se x tn x (ose x)' = ose x ot x (e x )' = e x (ln x)' = x

9 75 Lerning Unit Lerning Ojetive Time Remrks The following types of lgeri funtions re required: polynomil funtions rtionl funtions power funtions x funtions formed from the ove funtions through ddition, sutrtion, multiplition, division nd omposition, for exmple x 7.4 find derivtives y impliit differentition Logrithmi differentition is required. 7.5 find the seond derivtive of n expliit funtion Nottions inluding y", f "(x) nd d y should e introdued. Third nd higher order derivtives re not required. 8. Applitions of differentition 8. find the equtions of tngents nd normls to urve 4 8. find mxim nd minim Lol nd glol extrem re required.

10 76 Lerning Unit Lerning Ojetive Time Remrks 8. sketh urves of polynomil funtions nd rtionl funtions The following points re noteworthy in urve skething: symmetry of the urve limittions on the vlues of x nd y interepts with the xes mximum nd minimum points points of inflexion vertil, horizontl nd olique symptotes to the urve Students my dedue the eqution of the olique symptote to the urve of rtionl funtion y division. 8.4 solve the prolems relting to rte of hnge, mximum nd minimum Sutotl in hours

11 77 Lerning Unit Lerning Ojetive Time Remrks Integrtion 9. Indefinite integrtion 9. reognise the onept of indefinite integrtion 6 Indefinite integrtion s the reverse proess of differentition should e introdued. 9. understnd the properties of indefinite integrls nd use the integrtion formule of lgeri funtions, trigonometri funtions nd exponentil funtions to find indefinite integrls The following formule re required: k kx C n n x x C, where n n ln x C x x x e e C sin x os x C os x sin x C se x tn x C ose ot x x C se x tn x se x C ose xot x ose x C For more omplited lultions, see Lerning Ojetives 9.4 to 9.6.

12 78 Lerning Unit Lerning Ojetive Time Remrks 9. understnd the pplitions of indefinite integrls in rel-life or mthemtil ontexts Applitions of indefinite integrls in some fields suh s geometry nd physis re required. 9.4 use integrtion y sustitution to find indefinite integrls 9.5 use trigonometri sustitutions to find the indefinite integrls involving x, x or x Nottions inluding sin x, os x nd tn x nd their relted prinipl vlues should e introdued. 9.6 use integrtion y prts to find indefinite integrls ln x n e used s n exmple to illustrte the method of integrtion y prts. The use of integrtion y prts is limited to t most two times in finding n integrl.

13 79 Lerning Unit Lerning Ojetive Time Remrks 0. Definite integrtion 0. reognise the onept of definite integrtion The definition of the definite integrl s the limit of sum nd finding definite integrl from the definition should e introdued. The use of dummy vriles, inluding f ( x) f ( t) dt, is required. Using definite integrtion to find the sum to infinity of sequene is not required. 0. understnd the properties of definite integrls The following properties re required: f ( x) f ( x) f ( x) 0 f ( x) f ( x) f ( x) k f ( x) k f ( x) [ f ( x) g( x)] = f ( x) g( x)

14 80 Lerning Unit Lerning Ojetive Time Remrks 0. find definite integrls of lgeri funtions, trigonometri funtions nd exponentil funtions Fundmentl Theorem of Clulus: f ( x) F( ) F( ), where introdued. d F(x) = f (x), should e 0.4 use integrtion y sustitution to find definite integrls 0.5 use integrtion y prts to find definite integrls The use of integrtion y prts is limited to t most two times in finding n integrl. 0.6 understnd the properties of the definite integrls of even, odd nd periodi funtions The following properties re required: f ( x) 0 if f is odd is even nt f ( x) f ( x) if f T f ( x) n f ( x) if f (x + T ) = f (x), i.e. f is periodi

15 8 Lerning Unit Lerning Ojetive Time Remrks. Applitions of definite integrtion. understnd the pplition of definite integrls in finding the re of plne figure. understnd the pplition of definite integrls in finding the volume of solid of revolution out oordinte xis or line prllel to oordinte xis 4 Only dis method is required. Finding the volume of hollow solid is required. Sutotl in hours Alger Are Mtries nd Systems of Liner Equtions. Determinnts. reognise the onept nd properties of determinnts of order nd order The following properties re required:

16 8 Lerning Unit Lerning Ojetive Time Remrks k k k k k k k 0 ' ' ' ' ' ' k k k Nottions inluding A nd det(a) should e introdued.. Mtries. understnd the onept, opertions nd properties of mtries 9 The ddition, slr multiplition nd multiplition of mtries re required. The following properties re required: A + B = B + A

17 8 Lerning Unit Lerning Ojetive Time Remrks A + (B + C) = (A + B) + C ( + )A = A + A (A + B) = A + B A(BC) = (AB)C A(B + C) = AB + AC (A + B)C = AC + BC (A)(B) = ()AB AB = A B. understnd the onept, opertions nd properties of inverses of squre mtries of order nd order The following properties re required: the inverse of A is unique (A ) = A (A) = A (A n ) = (A ) n (A t ) = (A ) t A = A (AB) = B A where A nd B re invertile mtries nd is non-zero slr.

18 84 Lerning Unit Lerning Ojetive Time Remrks 4. Systems of liner equtions 4. solve the systems of liner equtions of order nd order y Crmer s rule, inverse mtries nd Gussin elimintion 6 The following theorem is required: A system of homogeneous liner equtions in three unknowns hs nontrivil solutions if nd only if the oeffiient mtrix is singulr The wording neessry nd suffiient onditions ould e introdued to students. Sutotl in hours 8 Vetors 5. Introdution to vetors 5. understnd the onepts of vetors nd slrs 5 The onepts of mgnitudes of vetors, zero vetor nd unit vetors re required. Students should reognise some ommon nottions of vetors in printed form (inluding nd AB ) nd in written form (inluding, AB nd ) ; nd some nottions for mgnitude (inluding nd ).

19 85 Lerning Unit Lerning Ojetive Time Remrks 5. understnd the opertions nd properties of vetors The ddition, sutrtion nd slr multiplition of vetors re required. The following properties re required: + = + + ( + ) = ( + ) = 0 = 0 ( ) = ( ) ( + ) = + ( + ) = + If + = + ( nd re non-zero nd re not prllel to eh other), then = nd = 5. understnd the representtion of vetor in the retngulr oordinte system The following formule re required: OP z x y in R sin = x y y nd

20 86 Lerning Unit Lerning Ojetive Time Remrks os = x x y in R The representtion of vetors in the retngulr oordinte system n e used to disuss those properties listed in the Remrks ginst Lerning Ojetive 5.. The onept of diretion osines is not required. 6. Slr produt nd vetor produt 6. understnd the definition nd properties of the slr produt (dot produt) of vetors 6. understnd the definition nd properties of the vetor produt (ross produt) of vetors in R 5 The following properties re required: = ( ) = ( ) ( + ) = + = 0 = 0 if nd only if = 0 = + ( ) The following properties re required: = 0

21 87 Lerning Unit Lerning Ojetive Time Remrks = ( ) ( + ) = + ( + ) = + ( ) = ( ) = ( ) = ( ) The following properties of slr triple produts should e introdued: ( ) = ( ) ( ) = ( ) = ( ) 7. Applitions of vetors 7. understnd the pplitions of vetors 8 Division of line segment, prllelism nd orthogonlity re required. Finding ngles etween two vetors, the projetion of vetor onto nother vetor, the volume of prllelepiped nd the re of tringle re required. Sutotl in hours 8

22 88 Lerning Unit Lerning Ojetive Time Remrks Further Lerning Unit 8. Inquiry nd investigtion Through vrious lerning tivities, disover nd onstrut knowledge, further improve the ility to inquire, ommunite, reson nd oneptulise mthemtil onepts 7 This is not n independent nd isolted lerning unit. The time is lloted for students to engge in lerning tivities from different lerning units. Sutotl in hours 7 Grnd totl: 5 hours

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