Trigonometry – Self-study:
Reading: Red Bostock and Chandler p137-151, p157-234, p244-254


Trigonometric functions
Students should:
 be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant,
secant, cotangent.
 be able to draw the graphs of the six trigonometric functions
Inverse trigonometric functions
Students should:
 be familiar with the six trigonometric functions
 be able to draw the graphs of the six inverse trigonometric functions
 understand the notations for inverse functions, e.g. the inverse function of
could be written as
or
Note that
.
 Be able to find all solutions of equations of the form
, where is one
of the six trigonometric functions and is a specified range such as [
E.g. find
the values of

in the range [
for which
.
Trigonometric identities
Students should:
 be familiar with the formulas on the formula sheet and be able to use them to do
the following
 find the possible values of
given the value of
, where and are
any of the six trigonometric functions e.g. given



values of
write expressions of the form
four forms
,
hence be able to solve equations of the form
solve trigonometric equations,
prove trigonometric identities.
find the possible
in any one of the following
, or
,.and
Algebra – 4 weeks, 8 lectures.

Lecture 1: Quadratic functions.
Reading: Red Bostock and Chandler p 10-14 (ignore example 2), p48-58.
Students should:
 be familiar with the shape of a quadratic curve
(i.e a parabola), its
symmetry about its minimum/maximum point, and be able to sketch this curve.
 be familiar with the method of ‘completing the square’ and be able to use it to
determine the coordinates of the minimum/maximum point of a quadratic, determine
the range of a quadratic function, and prove the formula for the roots of a quadratic
function.


 know the formula for the roots of a quadratic equation, understand the significance of
the discriminant and know what different values of the discriminant mean.
 be able to solve quadratic inequalities.
Lecture 2: Long division and factorisation
Reading: Photocopied notes from Precalculus.
Optional reading: Red Bostock and Chandler p32-34, and yellow p342-349 Bostock and
Chandler
Students should:
 be able to use polynomial long division to find the quotient and remainder when
one polynomial is divided by another, understand that the remainder will always
have a lower degree than the divisor.
 understand that all polynomials with real coefficients can be factorised uniquely as a
product of irreducible polynomials, i.e. as a product of linear factors and quadratics
with negative discriminant.
Lecture 3 and 4: Remainder and factor theorem
Reading: Red Bostock and Chandler p32-35, yellow Bostock and Chandler p342-349 (ignore
the material on repeated roots).
Students should
 be able to both use and prove the remainder and factor theorem.
 be able to generalise this technique to find the remainder when a polynomial is
divided by a quadratic
 know the ‘rational root test’ i.e. that if (where p and q are coprime) is a root of the



polynomial with integer coefficients
then is a factor
of and is a factor of .
 be able to factorise a given polynomial using the factor theorem, long division and
the quadratic formula as necessary
 be able to use the factorisation of a polynomial
to determine the range of
values of for which f(x) is positive or negative
Lecture 5: Partial fractions
Reading: Partial fractions of Paul’s notes:
http://tutorial.math.lamar.edu/Classes/Alg/PartialFractions.aspx
Optional Reading: Red Bostock and Chandler p5-9 and p271-272.
Students should
 be able to find the partial fraction decomposition of a rational function including
examples where the numerator has a higher degree than the denominator. Note:
Questions involving repeated quadratic factors will not be asked in exams.
 The cover-up rule may be used – but if used then it must be justified somehow e.g. by
saying ‘by cover-up rule’.
Lecture 6: Proof by Induction
Reading: Red B&C p 629-631 and Yellow B&C p162-166
 Students should be able to use proof by induction to prove given statements about
integers.
Lecture 7 and 8: Binomial Theorem and Generalised Binomial theorem
Reading: Binomial Theorem: Red B&C p 37-38 and p603-610.
Generalised Binomial Theorem: Red B&C p610-616
Optional Reading: Precalculus Mathematics: A problem solving approach p 434-440.
Students should:
 be able to find the expansion of an expression of the form
 find the coefficient of a particular term of the expansion
without
calculating the whole expansion
 know when it is appropriate to use the Binomial Theorem and when it is appropriate
to use the Generalised Binomial Theorem, and know the range of validity of the
expansion
 be able to use the Generalised Binomial Theorem to find approximations,
understand how to improve approximations.
Differentiation – 4 weeks, 8 lectures


Lecture 1: Limit of a function at a point
Students should:
 Understand the concept of continuity as a curve you can “draw without taking your
pen off the page”
 Understand left-limits, right-limits, limits at a point.
Lecture 2: Definition of the derivative as a limit, derivative of
Reading: Red Bostock and Chandler p106-119.
Students should:
 Know the definition of derivative in terms of limits
 Given a specific function, for example e.g.

, students should be able to use the
definition to find derivatives of it at a particular point, or at a general point
Lecture 3: Derivatives of
,
,
Reading: Red Bostock and Chandler p255-264.
Students should:
 know the proofs that
results
and
and
. The proofs of the
will not be assessed and they may be
used without proof.
 know the proof that
where is defined as the number such that
evaluated at 0 is one.


Lecture 4: Rules of differentiation (chain, product and quotient)
Reading: Red Bostock and Chandler p265-274.
Students should
 know the proofs of the product rule and the quotient rule (the proof of the chain
rule will not be examined),
 be able to apply these rules appropriately to find the derivatives of a wide range of
functions.
Lecture 5 & 6: Implicit differentiation: derivatives of inverse functions, tangents and normal
Reading: Red Bostock and Chandler p274-283 & p119-121
Students should:
 Be able to find
when a curve that cannot be written in the form
 Be able to find the tangent and normal of a curve at a specified point
 Be able to find derivatives of inverse functions such as
, and inverse trig
functions by using implicit differentiation.

 Be able to differentiate functions of the form
.
Lecture 7: Finding and classifying stationary points + finding global and local minima/maxima
Reading: Red Bostock and Chandler p122-132
Students should:
 be familiar with the 1st and 2nd derivative test; they should be able to use their
judgement about which might be more appropriate/easier to use in a given context,
but they should also be able to use a specific test if told to do so.
 Understand the concept of concavity
 understand that global min/max can occur at end points, where

or where
is undefined.
 understand that a function may have multiple local min/max or none.
Lecture 8: Optimisation
Reading: Photocopied notes from Calculus.
Students should:
 Be able to find use differentiation to solve practical problems involving optimisation.
 Note: Problems will only be asked about situations where the variables are defined
on a closed interval.
Curve sketching – 2.5 weeks, 5 lectures.

Lecture 1: Basics of graph sketching
Reading: Scanned notes from Understanding Pure Mathematics pages 275-280
Students should:
 Know the main features that should be included on graphs of
:

-intercepts

-intercepts (where
can be reasonably solved, otherwise some note
should be made of what range the root is in)
 stationary points

places where
is not defined

vertical and horizontal asymptotes (these two features to be covered in more detail
later)
 Places where the function is not defined
 Understand what a point of inflection is and be able to find points of inflection if asked
(but non-stationary points of inflection do not need to be found and put on graphs,
unless that is specifically asked for).

Lecture 2: Numerical methods for finding roots
Reading: Photocopied notes from Further Pure Mathematics 1" by Geoff Mannall and
Michael Kenwood
Students should:
 Understand that if a function is continuous on [
and
and
have opposite
signs, then must have a root in the range
.
 Understand that that sometimes roots cannot be found exactly, and that sometimes
numerical methods are needed get estimates of roots
 Be able to use bisection method and the Newton-Raphson Method.

Lecture 3: Horizontal asymptotes, the power of functions/race to infinity
Reading: See page 183 of Calculus,
Possibly also: http://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx
http://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityII.aspx
Students should:
 Know the following limits:



for
for
for
 Be able to use those limits to calculate the limits


and
, where
and
are one of the following functions: a polynomial, ,
(or if the
limits do not exist then work out if the function tends to or
)
 Be able to use this information about the limits to find the horizontal asymptotes of
functions.
Lecture 4: Vertical asymptotes
Reading:
Students should:
 Know that asymptotes can occur in graphs of functions of trig functions, rational
functions and functions involving logarithms.
 Using the information from this lecture, the previous lecture, and the material covered
in differentiation students should be able to determine if a function has a global
minimum or maximum (or if the function is unbounded).
Lecture 5: Transformations of curves
Reading: Scanned notes from Understanding Pure Mathematics page 280 & pages 284-290.
Students should:
 Know the relationships between:

and

and

and

and

and

and
 When given a graph of
be able to draw the graph for any of the related curves
listed above (and simple combinations of them such as
).
Integration – 4 weeks, 8 lectures.

Lecture 1: Indefinite integration and standard integrals
Reading: Red Bostock and Chandler p 299-307
Students should:
 understand definite integration as the “reverse of differentiation”
 know (i.e. memorise) the integrals of following functions (where
):
√

 know that ∫
and ∫
∫
∫
∫
Lecture 2: Integrating products
Reading: Red B&C p308-316
Students should:
 understand the methods of integration by substitution and integration by-parts
 know to use substitution to solve integrals of the form ∫
 be able to used judgement about when to use integration by-parts and choose
which function should be

and which should be
or ∫
∫
Lecture 3: Definite integrals
Reading: Red Bostock and Chandler p337-344
Students should
 understand the Fundamental Theorem of Calculus
 understand that the definite integral ∫

including examples such as
can be positive, negative or 0
 understand how to apply integration by substitution and integration by-parts to
definite integrals
Lecture 4: Integrating quotients
Reading: Red Bostock and Chandler p316-322
Students should
 be able to solve integrals of the following forms
∫
∫
∫


Lecture 5: Integrals involving trigonometric functions
Reading: Red Bostock and Chandler p322-325
Students should
 know (i.e. memorise ) the integrals:
1. ∫
2. ∫
3. ∫
4. ∫
5. ∫
6. ∫
 be able to use trigonometric identities to solve integrals to solve integrals involving
trigonometric functions including (but not limited to)
1. ∫
where
2. ∫
where
3. ∫
where
4. ∫
where
5. ∫
where
6. ∫
where
Lecture 6: Trigonometric substitution
Reading: Red Bostock and Chandler p326-328
Students should
 know (and be able to) use appropriate trigonometric substitutions to solve
(indefinite or definite) integrals involving √


√
or √
Lecture 7: Finding areas
Reading: Red Bostock and Chandler p344-348 and p682-686
Students should
 be able to use definite integration to calculate areas including areas such as:
 the area between a curve
and the -axis
 the area between a curve
and the -axis and the lines
(including cases where the curve cuts the -axis)
 the area between two curves
 the area between two curves and the lines
(including cases
where the curve touch or cut each other multiple times)
 Be able to write an integral either in terms of
or , and use their judgement to
decide which integral is easier to evaluate
Lecture 8: Volumes of revolution
Reading: Red Bostock and Chandler p687-694
Students should
 Students should be able to use definite integration to evaluate volumes generated
when
 the area between
, the -axis and the lines
is rotated
about the
axis
 the area between the two curves
,
and the lines
is rotated about the
axis


the area between
, the -axis and the lines
is rotated
about the
axis
the area between the two curves
,
and the lines
is rotated about the
axis
Series– 2.5 weeks, 5 lectures.

Lecture 1&2: Arithmetic/Geometric progressions and series
Reading: Red Bostock and Chandler p 586-603
Students should:
 be able to recognise an arithmetic or geometric progression and write down a formula
for
term
 be able to derive the formulas for an arithmetic series or geometric series
 be able decide whether a given geometric series will converge or diverge, and if it
converges calculate the infinite sum
 be able to use the formulas for arithmetic progressions, geometric progressions,
arithmetic series and geometric series

Lecture 3: Method of differences and sum of squares and cubes
Reading: Red Bostock and Chandler p 616-625
Students should:
 be able to use the “method of differences”/“telescoping series” to evaluate series where
the terms can be written in the form
, where
(including, where it
makes sense, infinite sums)
 in particular students should be able to use this method to derive the formulas
∑
∑
 be able to use the formulas for sum of square and cubes to evaluate series involving
squares or cubes

Lecture 4&5: Maclaurin’s series and Taylor series
Reading: Yellow Bostock and Chandler p250-263
Students:
 should be able to find the Maclaurin’s series or Taylor series expansion of a given
function up to a specified term, e.g. up to the
or term.
 should know the Maclaurin’s series for
and
 are not expected to know or be able to find the valid range for a Maclaurin’s series or
Taylor series expansion, but they should understand that some expansions are not valid
for all
 should be able to use Maclaurin’s series or Taylor series to make approximations to
numbers, and understand that (roughly speaking) the accuracy of the approximations
are improved by
1. increasing the number of terms used in the expansion
2. in the case of a Maclaurin’s series using a smaller value of , or in the case of a
Taylor series using a smaller value of
.
 should be able to find derivatives and integrals of power series.
Complex numbers – 3.5 weeks, 7 lectures.

Lecture 1: Complex arithmetic
Reading: Red Bostock and Chandler p 532-542
Students should:
} know
 understand to be the square root of
and to be the set {
the functions
and
 be able to do basic arithmetic with complex numbers i.e. be able to familiar with and
perform the following operations: conjugation, taking the modulus of a complex
number, addition, subtraction, multiplication and division,
 know that if is a polynomial with real coefficients then
̅
but if
is a polynomial where some of the coefficients are not real then
does not
imply that
̅
 know that a polynomial of degree can be written uniquely as the product of linear
factors (if we allow complex coefficients).

Lecture 2: The Argand Diagram and polar form
Reading: Red Bostock and Chandler p 542-563
Students should:
 be able to plot complex numbers on the Argand Diagram, whether they are in Cartesian
form or polar form,
 be able to change a complex number from Cartesian form to polar form, or from polar
form to Cartesian form (note that finding
is not simply
if
in the second or third quadrant),
 be able to multiply and divide numbers that are in polar form,
 be familiar with and able to use the following properties of complex numbers:

̅

̅

̅
is


| |=



( )
[up to adding or taking away
]
[up to adding or taking away
]
Lecture 3: Exponential form and De Moivre’s Theorem
Reading: Yellow Bostock and Chandler p290-299, and p310-312
Students should:

 be able to change a complex number from Cartesian or polar form into exponential
form, or from exponential form into either polar or Cartesian form,
 know De Moivre’s Theorem and be able to prove it using either Maclaurin’s Series, or
(for positive integer values) by proof by induction,
 know De Moivre’s Theorem and the identity
and by using either of
them be able to find formulas for

in terms of powers of
or
 formulas for
and
as a sum of the form ∑
or
∑
.
 be able to write
or
in exponential form and be able to use these forms to
prove trigonometric identities
Lecture 4: Simple Loci
Reading: Yellow Bostock and Chandler p315-328
Students should:
 be able to draw loci such as:





 be able to draw loci based on inequalities such as





 Convert simple equations written in terms of modulus and argument into equations
written in terms of and , and where appropriate sketch this curve, e.g. write
and

as equations in terms of
Lecture 5: Simple transformations
Reading: Yellow Bostock and Chandler p328-336
Students should:
and .

 understand the geometric effect on the Argand diagram of:
 complex conjugation
 addition or subtraction by a complex number
 multiplication/division by a real number
 multiplication/division by
where is real
 and simple combinations of the above transformations
 be able to interpret what the geometric meaning of a given function is, or be able to
write down the function which represents a given transformation
 be able to find the fixed points of a given transformation
 find the equation of a curve after a transformation has been applied to it, i.e. if
and
then find the image of a curve
in terms of and .
Lecture 6: Roots of unity
Reading: Yellow Bostock and Chandler p299-305
Students should:

 be able to find all the complex roots of the equation
for any positive integer
value of , and write them in Cartesian, polar or exponential form,
 be able to plot these points on an Argand diagram without first calculating their values,
in particularly they should know
 1 is one of the roots,
 the roots all lie on the unit circle,
 the -fold rotational symmetry, and the symmetry in the -axis
 know and be able to use the following properties of the roots
 the sum of the roots equals 0
 if is a root of
then ̅
Lecture 7: Roots of a general complex number
Reading: Yellow Bostock and Chandler p305-309
Students should:
 be able to find all the complex roots of the equation
for any complex
number and any positive integer value of and write them in Cartesian, polar or
exponential form,
 know the relationship between the roots of
and the roots of
 be able to plot these points on an Argand diagram without first calculating all of their
values, in particularly they should know


the roots all lie on the circle
the -fold rotational symmetry
Vectors – 3 weeks, 6 lectures.

Lecture 1: Basic definitions & operations
,
Reading: Red Bostock and Chandler p455-485 & p496-504
Students should:

 Be able to perform the following operations on vectors and understand the geometric
meaning of these operations: addition, subtraction and scalar multiplication.
 Be able to find the modulus of a vector, and the unit vector in the same direction as a
given vector.
 Know how to calculate the dot product of two vectors; know & use the properties of the
dot product.
 Know the identity
and be able to use it to calculate the angle
between two vectors.
Lecture 2: Cross product (vector product) and triple product
Reading: Yellow Bostock and Chandler p64-71
Students should:

 Be able to calculate the determinant of a
matrix
 Know how to calculate the cross product of two vectors; know & use the properties of
the cross product.
 Know the identity
and be able to use it to calculate the area of a
triangle or a parallelogram.
 Know how to calculate the triple product of three vectors; know & use the properties of
the triple product.
 Be able to use the triple product to calculate the volume of a parallelepiped or a
tetrahedron.
Lecture 3: Equations of lines
Reading: Red Bostock and Chandler p486-93
Students should:

 Be familiar with the 3 forms of describing lines in 3D i.e. the vector equation, parametric
equations, Cartesian equations.
 Be able to convert one form of the description of a line into another.
 Be able to find the equation of a line when given enough geometric information to
define the line, e.g. find the line that goes through 2 specified points, or that goes
through a specified point and in a specific direction.
Lecture 4: Equations of planes
Reading: Red Bostock and Chandler p504-10
Students should:
 Be familiar with the 3 forms of describing planes in 3D i.e. scalar product form,
parametric form, the Cartesian equation.
 Be able to convert one form of the description of a plane into another.

 Be able to find the equation of a plane when given enough geometric information to
define the line, e.g. find the plane that goes through 3 specified points, or that goes
through a specified point and is parallel to 2 specified directions, etc.
Lecture 5&6: 3d Geometry problems
Reading: Red Bostock and Chandler p493-496, p498-500 & p512-7
Students should:
 Be able to find the point(s) of intersection between lines & planes.
 Be able to solve geometric problems involving the distance between and/or the angle
between lines, planes and points.