Scheme of Work 2010 – 2011
C3 - Methods for Advanced Mathematics
1-3
1
Week/
Date
Learning Outcomes
[Can be differentiated]
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Proof 1: Types of proof
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 Understand, 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.
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TEACHER A
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Natural logarithms and exponentials 1: Introduction
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 Understand and be able to use the simple properties
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of logarithmic and exponential functions including the function 
ex and the natural logarithmic function ln x.
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 Know the relationship between ln x and ex
 Know the graphs of y = ln x and y = ex
 Be able to solve problems involving exponential
growth and exponential decay.
TEACHER B
Teaching & Learning Activities
(All resources here are
hyperlinked to the MEI website)
HW and/or
Assessments
Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Links to external websites
Finding fallacies in false proofs
Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Interactive resources
Modelling population growth
Multiple choice section test
Questions
Section test solutions
4-6
TEACHER B
Functions 1: The language of functions
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Understand the definition of a function, and the associated
language (domain, co-domain,range, object, image, one-toone, 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
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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).
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Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Multiple choice section test
Questions
Section test solutions
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Understand what is meant by the terms odd function, even
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function and periodic function, and the symmetries associated 
with them
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Understand the modulus function
Be able to solve simple inequalities containing a modulus sign.
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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 solutions
Functions 3: Types of function
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Multiple choice section test
Questions
Section test solutions
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Know how to find a composite function, gf(x)
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Know the conditions needed for the inverse of a function to exist
and how to find it (algebraically and graphically)
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Understand the inverse trigonometric functions arcsin, arccos
and arctan, their graphs and appropriate restricted domains.
Functions 2: Composite and inverse functions
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Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Interactive resources
Transformations of graphs
TEACHER B NOW BEGINS TO TEACH EITHER M1 OR S1
2-5
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Techniques for differentiation 1: The chain rule
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 Understand how to differentiate composite functions using
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the chain rule (pages 63-65)
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 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 exercise
Interactive resources
The chain rule
Multiple choice section test
Questions
Section test solutions
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Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Interactive resources
The product rulefile
The quotient rule
Multiple choice section test
Questions
Section test solutions
Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Interactive resources
The gradient graph of y=a^x
Multiple choice section test
Questions
Section test solutions
TEACHER A
Techniques for differentiation 2: The product and quotient rules
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Know 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)
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exponentials
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 Know that the derivative of the exponential
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function ekx is kekx (page 82)
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 Know that the derivative of the natural logarithm function ln x is 
1/x (page 82)
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 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)
Techniques for differentiation 3: Differentiating logarithms and
6-9
Techniques for differentiation 4: Differentiating trigonometric
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functions
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 Know the derivatives of trigonometric functions sin x, cos x and 
tan x (page 93)
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 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 solutions
Techniques for differentiation 5: Differentiating implicit functions 
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 Be able to differentiate implicit functions (pages 96-97)
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 Solve the resulting equation to find dy/dx in terms
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of x and y (page 98).
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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 solutions
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Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Multiple choice section test
Questions
Section test solutions
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Study plan
Notes and Examples
Crucial points
Additional exercise
Solutions to exercise
Multiple choice section test
Questions
Section test solutions
Techniques for integration 1: Integration by substitution
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Be 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).
Techniques for integration 2: Integration of other functions
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Be 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)
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Be able to integrate sin x and cos x, and other integrals involving
trigonometric functions using substitution or inspection (pages 
123-124
Links to external websites
nrich: Integration matcher
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Be able to use the method of integration by parts for both
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definite and indefinite integration
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Know how to use the method of integration by parts to integrate
ln x.
Techniques for integration 3: Integration by parts
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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