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 sin could be written as arcsin or sin −1 . Note that sin −1 ≠ 1 sin . 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 [0,2). E.g. find the values of in the range [0,2) for which sin(2 + 3 )= 1 3 . 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 sin = 3 4 find the possible values of tan . write expressions of the form cos + sin in any one of the following four forms cos( − ) , cos( + ), sin( + ), or sin( − ), and hence be able to solve equations of the form cos + sin = . solve trigonometric equations, prove trigonometric identities. Algebra – 4 weeks, 8 lectures. Lecture 0 (Self-study): Quadratic functions. Reading: See Moodle for lecture, Red Bostock and Chandler p 10-14 (ignore example 2), p48-58. Students should: be familiar with the shape of a quadratic curve = 2 + + (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.
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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 sin 𝑥
could be written as arcsin 𝑥 or sin−1 𝑥. Note that sin−1 𝑥 ≠1
sin 𝑥.
Be able to find all solutions of equations of the form 𝑓(𝑎𝑥 + 𝑏) = 𝑐, where 𝑓 is one
of the six trigonometric functions and 𝑥 is a specified range such as [0,2𝜋). E.g. find
the values of 𝑥 in the range [0,2𝜋) for which sin(2𝑥 +𝜋
3) =
1
3.
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 𝑓(𝑥) given the value of 𝑔(𝑥), where 𝑓 and 𝑔 are
any of the six trigonometric functions e.g. given sin 𝑥 =3
4 find the possible
values of tan 𝑥.
write expressions of the form 𝑎 cos 𝜃 + 𝑏 sin 𝜃 in any one of the following
four forms 𝑅 cos(𝜃 − 𝛼) , 𝑅 cos(𝜃 + 𝛼), 𝑅 sin(𝜃 + 𝛼), or 𝑅 sin(𝜃 − 𝛼), and
hence be able to solve equations of the form 𝑎 cos 𝜃 + 𝑏 sin 𝜃 = 𝑐.
solve trigonometric equations,
prove trigonometric identities.
Algebra – 4 weeks, 8 lectures.
Lecture 0 (Self-study): Quadratic functions.
Reading: See Moodle for lecture, Red Bostock and Chandler p 10-14 (ignore example 2),
p48-58.
Students should:
be familiar with the shape of a quadratic curve 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 (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 1: Long division and factorisation
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 2 and 3: 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 𝑝
𝑞 (where p and q are coprime) is a root of the
polynomial with integer coefficients 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎0 then 𝑝 is a factor
of 𝑎0 and 𝑞 is a factor of 𝑎𝑛.
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 𝑓(𝑥) to determine the range of
values of 𝑥 for which f(x) is positive or negative
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 5 and 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 (𝑎 + 𝑏)𝑛
find the coefficient of a particular term of the expansion (𝑎 + 𝑥)𝑛 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 𝑥𝑛
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. 1
3𝑥+2, 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 sin 𝑥, cos 𝑥, 𝑒𝑥
Reading: Red Bostock and Chandler p255-264.
Students should:
know the proofs that 𝑑
𝑑𝑥(sin 𝑥) = cos 𝑥 and
𝑑
𝑑𝑥(cos 𝑥) = sin 𝑥. The proofs of the
results lim𝑥→0
sin 𝑥
𝑥= 1 and lim
𝑥→0
cos 𝑥−1
𝑥= 0 will not be assessed and they may be used
without proof.
know the proof that 𝑑
𝑑𝑥(𝑒𝑥) = 𝑒𝑥 , where 𝑒 is defined as the number such that
𝑑
𝑑𝑥(𝑒𝑥) 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 𝑑𝑦
𝑑𝑥 when a curve that cannot be written in the form 𝑦 = 𝑓(𝑥)
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 ln 𝑥, and inverse trig
functions by using implicit differentiation.
Be able to differentiate functions of the form 𝑓(𝑥)𝑔(𝑥).
Understand what a differential equation is.
Be able to show that a given function is a solution to a given differential equation
(they are NOT expected to be able to find the solution of a given equation).
Lecture 7: Finding and classifying stationary points + finding global and local minima/maxima
Reading: Red Bostock and Chandler p122-132
Students should:
Understand the following concepts: unbounded, bounded, bounded above,
bounded below, local maximum, local minimum, global maximum, global minimum.
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 𝑑𝑦
𝑑𝑥= 0 or where
𝑑𝑦
𝑑𝑥
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 𝑦 = 𝑓(𝑥):
𝑦-intercepts
𝑥-intercepts (where 𝑓(𝑥) = 0 can be reasonably solved, otherwise some note
should be made of what range the root is in)
stationary points
places where 𝑑𝑦
𝑑𝑥 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 [𝑎, 𝑏] and 𝑓(𝑎) and 𝑓(𝑏) have opposite
signs, then 𝑓 must have a root in the range (𝑎, 𝑏).
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: End behaviour: horizontal asymptotes, the power of functions/race to infinity