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7390694-767644Scheme of Work 2010 – 2011C3 - Methods for Advanced MathematicsWeek/DateLearning Outcomes[Can be differentiated]Teaching & Learning Activities(All resources here are hyperlinked to the MEI website)HW and/orAssessments1TEACHER AProof 1: Types of proofUnderstand, and be able to use,?proof?by direct argument,?proof by exhaustion?and?proof by contradiction.Be able to disprove a?conjecture?by the use of a?counter-example.?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exerciseLinks to external websites?Finding fallacies in false proofs1 - 3TEACHER BNatural logarithms and exponentials 1: IntroductionUnderstand and be able to use the simple properties of?logarithmic and?exponential functions including the?function ex?and the?natural logarithmic function ln?x.Know the relationship between ln?x?and exKnow the graphs of?y?= ln?x?and?y?= exBe able to solve problems involving?exponential growth?and?exponential decay.?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exerciseInteractive resources?Modelling population growth?Multiple choice section test Questions?Section test solutions4-6TEACHER BFunctions 1: The language of functionsUnderstand the definition of a?function, and the associated language (domain,?co-domain,range,?object,?image,?one-to-one,?many-to-one,?one-to-many,?many-to-many)Know the effect of combined transformations on a graph (translations parallel to the?x?and?yaxes,?stretches parallel to the?x?and?y?axes,?reflection?in the?x?and?y?axes, and combinations of these) and be able to form the equation of the new graph.Be able, given the graph of?y?= f(x), to?sketch?the related graphs?y?= f(x?+?a),?y?= f(ax),?y?-af(x),?y?= f(x) +?a,?y?= f(-x) and?y?= -f(x).?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exerciseInteractive resources?Transformations of graphs?Multiple choice section test Questions?Section test solutionsFunctions 2: Composite and inverse functionsKnow how to find a?composite function, gf(x)Know the conditions needed for the?inverse of a function?to exist and how to find it (algebraically and graphically)Understand the?inverse trigonometric functions?arcsin, arccos and arctan, their graphs and appropriate restricted?domains.?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exercise?Multiple choice section test Questions?Section test solutionsFunctions 3: Types of functionUnderstand what is meant by the terms?odd function,?even function?and?periodic function, and the symmetries associated with themUnderstand the?modulus functionBe able to solve simple inequalities containing a modulus sign.TEACHER B NOW BEGINS TO TEACH EITHER M1 OR S1?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exercise?Multiple choice section test Questions?Section test solutions?Functions, natural logarithms and exponentials chapter assessment?Chapter assessment solutions2-5TEACHER ATechniques for differentiation 1: The chain ruleUnderstand how to differentiate?composite functions using the?chain rule?(pages 63-65)Solve problems involving associated rates of change using the chain rule (pages 65-66)?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exerciseInteractive resources?The chain rule?Multiple choice section test Questions?Section test solutionsTechniques for differentiation 2: The product and quotient rulesKnow the?product rule?formula, and be able to apply it to appropriate?functions?(pages 68-70)Know the?quotient rule?formula, and be able to apply it to appropriate functions (pages 71-72)Be able to factorise the results of product and?quotient rule?derivatives (page 70)Know that the?derivative of an inverse function?can be found using?, and use this result in associated rates of change questions (pages 77-80)?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exerciseInteractive resources?The product rulefile?The quotient rule?Multiple choice section test Questions?Section test solutionsTechniques for differentiation 3: Differentiating logarithms and exponentialsKnow that the?derivative of the exponential function?ekx?is?kekx?(page 82)Know that the?derivative of the natural logarithm function?ln x is 1/x?(page 82)Be able to use these results, together with the?chain rule,?product rule?and?quotient rule, to differentiate functions which involve?logarithmic or?exponential functions (pages 83-86)?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exerciseInteractive resources?The gradient graph of y=a^x?Multiple choice section test Questions?Section test solutionsTechniques for differentiation 4: Differentiating trigonometric functionsKnow the?derivatives of trigonometric functions?sin x, cos x and tan x (page 93)Be able to use these results, together with the?chain rule,?product rule?and?quotient rule, to differentiate?functions?which involve trigonometric?functions.?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exercise?Multiple choice section test Questions?Section test solutionsTechniques for differentiation 5: Differentiating implicit functionsBe able to differentiate?implicit functions (pages 96-97)Solve the resulting equation to find dy/dx?in terms of?x?and?y?(page 98).?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exercise?Multiple choice section test Questions?Section test solutions?Techniques for differentiation Chapter assessment?Chapter assessment solutions6-9Techniques for integration 1: Integration by substitutionBe able to use?integration by substitution?to integrate suitable?functions?(pages 103-107)Be able to use?integration by inspection?to integrate suitable functions (see Notes and Examples).?Study plan?Notes and Examples?Crucial points?Additional exercise ?Solutions to exercise?Multiple choice section test Questions?Section test solutionsTechniques for integration 2: Integration of other functionsBe able to integrate?exponential functions (page 110)Be able to integrate a quotient?function?to obtain a?natural logarithm?function?where appropriate (page 111)Be able to use?integration by inspection?to integrate suitable?functions?(page 112)Be able to integrate sin?x?and cos?x, and other integrals involving trigonometric functions using substitution or inspection (pages 123-124?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exerciseLinks to external websites?nrich: Integration matcher?Multiple choice section test Questions?Section test solutionsTechniques for integration 3: Integration by partsBe able to use the method of?integration by parts?for both definite and indefinite integrationKnow how to use the method of integration by parts to integrate ln?x.?Study plan?Notes and Examples?Crucial points?Additional exercise?Solutions to exercise?Multiple choice section test Questions?Section test solutions?Techniques for integration Chapter assessment?Chapter assessment solutions ................
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