trigonometry self-study: reading · product of irreducible polynomials, i.e. as a product of linear...

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

Lecture 4: 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 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


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)


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

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:

lim𝑥→∞

𝑥𝑛

𝑎𝑥 = 0, for 𝑛 > 0, 𝑎 > 1

lim𝑥→∞

loga 𝑥

𝑥𝑛 = 0, for 𝑛 > 0, 𝑎 > 1

lim𝑥→∞

1

𝑥𝑛 = 0, for 𝑛 > 0

Be able to use those limits to calculate the limits lim𝑥→∞

𝑓(𝑥)

𝑔(𝑥) and lim

𝑥→−∞

𝑓(𝑥)

𝑔(𝑥), where 𝑓(𝑥)

and 𝑔(𝑥) are one of the following functions: a polynomial, 𝑎𝑥, loga 𝑥 (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 end behaviour of the curves

(i.e. as 𝑥 → ±∞).

http://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx

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:

𝑓(𝑥) and 𝑓(𝑥 + 𝑎)

𝑓(𝑥) and 𝑓(𝑥) + 𝑎

𝑓(𝑥) and 𝑓(𝑎𝑥)

𝑓(𝑥) and 𝑎𝑓(𝑥)

𝑓(𝑥) and |𝑓(𝑥)|

𝑓(𝑥) and 1

𝑓(𝑥)

When given a graph of 𝑦 = 𝑓(𝑥) be able to draw the graph for any of the related curves

listed above (and simple combinations of them such as 𝑦 = 𝑎𝑓(𝑥 + 𝑏)).

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 𝑎, 𝑏 ∈ ℝ):

(𝑎𝑥 + 𝑏)𝑛, 𝑛 ≠ −1; 1

𝑎𝑥+𝑏; 𝑒𝑎𝑥+𝑏; cos(𝑎𝑥 + 𝑏) ; sin(𝑎𝑥 + 𝑏) ; sec2(𝑎𝑥 + 𝑏) ;

sec(𝑎𝑥 + 𝑏) tan(𝑎𝑥 + 𝑏) ; cosec(𝑎𝑥 + 𝑏) cot(𝑎𝑥 + 𝑏) ; cosec2(𝑎𝑥 + 𝑏) ; 1

1+𝑥2 ; 1

√1−𝑥2

know that ∫ 𝑓(𝑥) + 𝑔(𝑥) 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 + ∫ 𝑔(𝑥) 𝑑𝑥 and ∫ 𝑎𝑓(𝑥) 𝑑𝑥 = 𝑎 ∫ 𝑓(𝑥) 𝑑𝑥

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 ∫ 𝑓(𝑢)𝑢′(𝑥) 𝑑𝑥

be able to used judgement about when to use integration by-parts and choose

which function should be 𝑢(𝑥) and which should be 𝑑𝑣

𝑑𝑥 including examples such as

∫ 𝑒𝑥 sin 𝑥 𝑑𝑥 or ∫ ln 𝑥 𝑑𝑥

Lecture 3: Definite integration


Students should

understand the Fundamental Theorem of Calculus

understand that the definite integral ∫ 𝑓(𝑥)𝑏

𝑎𝑑𝑥 can be positive, negative or 0

understand how to apply integration by substitution and integration by-parts to

definite integrals

Lecture 4: Integrating quotients


Students should

be able to solve integrals of the following forms

I. ∫𝑓′(𝑥)

𝑓(𝑥)𝑑𝑥

II. ∫𝑢′(𝑥)

𝑓(𝑢(𝑥))𝑑𝑥

III. ∫𝑓(𝑥)

𝑔(𝑥)𝑑𝑥 where 𝑓(𝑥) and 𝑔(𝑥) are polynomials

Lecture 5: Integrals involving trigonometric functions


Students should

know (i.e. memorise) the integrals:

1. ∫ sin 𝑥 𝑑𝑥

2. ∫ cos 𝑥 𝑑𝑥

3. ∫ tan 𝑥 𝑑𝑥

4. ∫ sec 𝑥 𝑑𝑥

5. ∫ cosec 𝑥 𝑑𝑥

6. ∫ cot 𝑥 𝑑𝑥

know (i.e. memorise) or be able to quickly calculate the integrals:

1. ∫ sin𝑛 𝑥 cos 𝑥 𝑑𝑥

2. ∫ cos𝑛 𝑥 sin 𝑥 𝑑𝑥

3. ∫ tan𝑛 𝑥 sec2 𝑥 𝑑𝑥

4. ∫ sec𝑛 𝑥 tan 𝑥 𝑑𝑥

5. ∫ cosec𝑛 𝑥 cot 𝑥 𝑑𝑥

6. ∫ cot𝑛 𝑥 cosec2 𝑥 𝑑𝑥

be able to use trigonometric identities to solve integrals to solve integrals involving

trigonometric functions including (but not limited to)

1. ∫ cos𝑛 𝑥 𝑑𝑥 where 𝑛 ≥ 2

2. ∫ sin𝑛 𝑥 𝑑𝑥 where 𝑛 ≥ 2

3. ∫ tan𝑛 𝑥 𝑑𝑥 where 𝑛 ≥ 2

4. ∫ cos(𝑎𝑥 + 𝑏) sin(𝑐𝑥 + 𝑑) 𝑑𝑥 where 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ

5. ∫ sin(𝑎𝑥 + 𝑏) sin(𝑐𝑥 + 𝑑) 𝑑𝑥 where 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ

6. ∫ cos(𝑎𝑥 + 𝑏) cos(𝑐𝑥 + 𝑑) 𝑑𝑥 where 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ

Lecture 6: Trigonometric substitution (and other inverse substitutions)


Students should

know (and be able to) use appropriate trigonometric substitutions to solve

(indefinite or definite) integrals involving √𝑎2 − 𝑥2, √𝑎2 + 𝑥2,1

𝑎2+𝑥2 or √𝑥2 − 𝑎2

be able to solve other problems involving other “inverse substitutions” if told the

substitution to use.

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 𝑦 = 𝑓(𝑥) and the 𝑥-axis

the area between a curve 𝑦 = 𝑓(𝑥) and the 𝑥-axis and the lines 𝑥 = 𝑎, 𝑥 =

𝑏 (including cases where the curve cuts the 𝑥-axis)

the area between two curves

the area between two curves and the lines 𝑥 = 𝑎, 𝑥 = 𝑏 (including cases

where the curve touch or cut each other multiple times)

Be able to write an integral either in terms of 𝑑𝑥 or 𝑑𝑦, and use their judgement to

decide which integral is easier to evaluate

Lecture 8: Volumes of revolution


Students should

Students should be able to use definite integration to evaluate volumes generated

when

the area between 𝑦 = 𝑓(𝑥), the 𝑥-axis and the lines 𝑥 = 𝑎, 𝑥 = 𝑏 is rotated

about the 𝑥 −axis

the area between the two curves 𝑦 = 𝑓(𝑥), 𝑦 = 𝑔(𝑥) and the lines 𝑥 =

𝑎, 𝑥 = 𝑏 is rotated about the 𝑥 −axis

the area between 𝑦 = 𝑓(𝑥), the 𝑦-axis and the lines 𝑦 = 𝑎, 𝑦 = 𝑏 is rotated

about the 𝑦 −axis

the area between the two curves 𝑦 = 𝑓(𝑥), 𝑦 = 𝑔(𝑥) and the lines 𝑦 =

𝑎, 𝑦 = 𝑏 is rotated about the 𝑦 −axis

Series– 2.5 weeks, 5 lectures.

Lecture 1&2: Arithmetic/Geometric progressions and series


Students should:

be able to recognise an arithmetic or geometric progression and write down a formula

for 𝑛𝑡ℎ 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


Students should:

be able to use the “method of differences”/“telescoping series” to evaluate series where

the terms can be written in the form 𝑓(𝑛) − 𝑓(𝑛 + 𝑘), where 𝑘 ∈ ℤ (including, where it

makes sense, infinite sums)

in particular students should be able to use this method to derive the formulas

1. ∑ 𝑘2

𝑛

𝑘=1

=1

6𝑛(𝑛 + 1)(2𝑛 + 1)

2. ∑ 𝑘3

𝑛

𝑘=1

=1

4𝑛2(𝑛 + 1)2

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 𝑥3 or 𝑥4 term.

should know the Maclaurin’s series for 𝑒𝑥 , sin 𝑥 and cos 𝑥.

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 𝑥 ∈ ℝ.

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 |𝑥|, or in the case of a

Taylor series using a smaller value of |𝑥 − 𝑎|.

should be able to find derivatives and integrals of power series.

Complex numbers – 4 weeks, 8 lectures.

Lecture 1: Complex arithmetic


Students should:

understand 𝑖 to be the square root of −1, and ℂ to be the set {𝑎 + 𝑖𝑏: 𝑎, 𝑏 ∈ ℝ}, know

the functions Re(𝑎 + 𝑖𝑏) = 𝑎 and Im(𝑎 + 𝑖𝑏) = 𝑏,

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 𝑓 is a polynomial with real coefficients then 𝑓(𝑧) = 0 ⇒ 𝑓(𝑧̅) = 0, but if 𝑓

is a polynomial where some of the coefficients are not real then 𝑓(𝑧) = 0 does not

imply that 𝑓(𝑧̅) = 0,

know that a polynomial of degree 𝑛 can be written uniquely as the product of 𝑛 linear

factors (if we allow complex coefficients).

Know and be able to use the following properties of complex numbers

|𝑧1𝑧2| = |𝑧1||𝑧2|

|𝑧1

𝑧2| =

|𝑧1|

|𝑧2|

(𝑧∗)∗ = 𝑧

(𝑧∗)𝑧 = |𝑧|2 and 𝑧(𝑧∗) = |𝑧|2

|𝑧∗| = |𝑧|

(𝑤 + 𝑧)∗ = 𝑤∗ + 𝑧∗

(𝑤 − 𝑧)∗ = 𝑤∗ − 𝑧∗

(𝑤𝑧)∗ = 𝑤∗𝑧∗

(𝑤

𝑧)

∗=

𝑤∗

𝑧∗

𝑧 = 𝑧∗ ⟺ 𝑧 ∈ ℝ

(𝑧∗)𝑛 = (𝑧𝑛)∗

Lecture 2: The Argand Diagram and polar form


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 arg(𝑎 + 𝑖𝑏) is not simply arctan𝑏

𝑎 if 𝑎 + 𝑖𝑏 is

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:

arg(𝑧̅) = − arg(𝑧)

arg(𝑧1𝑧2) = arg(𝑧1) + arg(𝑧2) [up to adding or taking away 2𝜋]

arg (𝑧1

𝑧2) = arg(𝑧1) − arg(𝑧2) [up to adding or taking away 2𝜋]

understand the geometric effect on the Argand diagram of:

complex conjugation

addition or subtraction by a complex number 𝑤

multiplication/division by a real number 𝑟

multiplication/division by 𝑒𝑖𝜃 where 𝜃 is real

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 𝑒𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃, and by using either of

them be able to find formulas for

cos 𝑘𝜃 , sin 𝑘𝜃 , tan 𝑘𝜃 in terms of powers of cos 𝜃 , sin 𝜃 or tan 𝜃

formulas for cosn 𝜃 and sin𝑛 𝜃 as a sum of the form ∑ 𝑎𝑘 cos 𝑘𝜃𝑘=𝑛𝑘=1 + 𝐶 or

∑ 𝑎𝑘 sin 𝑘𝜃𝑘=𝑛𝑘=1 + 𝐶.

be able to write cos 𝜃 or sin 𝜃 in exponential form and be able to use these forms to

prove trigonometric identities

Lecture 4: Simple Loci


Students should:

be able to draw and describe in words loci such as

|𝑧 − 𝑎| = 𝑏

|𝑧 − 𝑎| = |𝑧 − 𝑏|

arg(𝑧 − 𝑎) = 𝜃

be able to find the equations (in terms of 𝑥 and 𝑦) of loci such as the equations below:

|𝑧 − 𝑎| = 𝑏

|𝑧 − 𝑎| = |𝑧 − 𝑏|

arg(𝑧 − 𝑎) = 𝜃

Re(𝑧) = 𝑐

Im(𝑧) = 𝑑

|𝑧 − 𝑐1| = 𝐾|𝑧 − 𝑐2|

be able to draw loci based on inequalities such as

|𝑧 − 𝑎| < 𝑏

|𝑧 − 𝑎| ≥ |𝑧 − 𝑏|

𝜃1 ≤ arg(𝑧 − 𝑎) ≤ 𝜃2

Re(𝑧) < 𝑐

𝑑 ≤ Im(𝑧)

Convert simple equations written in terms of modulus and argument into equations

written in terms of 𝑥 and 𝑦, and where appropriate sketch this curve,

e.g. write |𝑧 − 3 + 4𝑖| = 2 and arg(𝑧 − 3 + 4𝑖) =𝜋

3 as equations in terms of 𝑥 and 𝑦.

Note: You will not be tested on the following types of equations:

|𝑧 − 𝑐1| + |𝑧 − 𝑐2| = 𝑅

|𝑧 − 𝑐1| − |𝑧 − 𝑐2| = 𝑅

arg(𝑧 − 𝑐1) + arg(𝑧 − 𝑐2) = 𝜙

arg(𝑧 − 𝑐1) − arg(𝑧 − 𝑐2) = 𝜙

Lecture 5: Transformations of the complex plane: composite functions and fixed points

Be able to write down a function that is the composite of two of the following

transformations

Reflection in the 𝑥-axis

Translation in the plane

Expansion centred at the origin

Rotation about the origin

Be able to identify two transformations above that make a given transformation.

Be able to find the fixed points of a given transformation

Lecture 6: Transformations of the complex plane – the 𝑧-plane and 𝑤-plane


Students should:

Be able to find the equation of a curve after a transformation has been applied to it, i.e.

if 𝑤 = 𝑓(𝑧) and 𝑤 = 𝑢 + 𝑖𝑣 then find the image of a curve 𝐶(𝑥, 𝑦) in terms of 𝑢 and 𝑣.

Lecture 7: Roots of a general complex number


Students should:

be able to find all the complex roots of the equation 𝑧𝑛 − 𝑐 = 0 for any complex

number 𝑐 and any positive integer value of 𝑛, and write them in Cartesian, polar or

exponential form,

know the relationship between the roots of 𝑧𝑛 − 1 = 0 and the roots of 𝑧𝑛 − 𝑐 = 0

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 |𝑧| = |𝑐|1

𝑛,

the 𝑛-fold rotational symmetry

Lecture 8: Roots of unity


Students should:

be able to find all the complex roots of the equation 𝑧𝑛 − 1 = 0 for any positive integer

value of 𝑛, 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 𝑛 roots,

the roots all lie on the unit circle,

the 𝑛-fold rotational symmetry, and the symmetry in the 𝑥-axis

know and be able to use the following properties of the roots

the sum of the roots equals 0

if 𝛼 is a root of 𝑧𝑛 − 1 = 0 then �̅� = 𝛼−1

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 𝒂 ∙ 𝒃 = |𝒂||𝒃| cos 𝜃, and be able to use it to calculate the angle

between two vectors.

Lecture 2: Cross product (vector product) and triple product


Students should:

Be able to calculate the determinant of a 3 × 3 matrix

Know how to calculate the cross product of two vectors; know & use the properties of

the cross product.

Know the identity |𝒂 × 𝒃| = |𝒂||𝒃| sin 𝜃, 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


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


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.

trigonometry self-study: reading · product of irreducible polynomials, i.e. as a product of linear...

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