Topic module: Further

Invariant points

An invariant point is a point that does not move. Not all points in a plane are moved by a transformation. These points are said to be invariant.

Convergent and Divergent Integrals

Mixed examples : Differentiating inverse trig functions

Angle between a line and a plane

Intersection of three planes

Equation of the directrix and coordinates of the focus

L’Hospital’s Rule – Still to be made

The mean value of a function

Rolle’s Theorem

Fundamental theorem of calculus

Lower and upper Riemann sums

The integral as a limit of a sum

Continuous functions and differentiable functions

Continuity and differentiability of a function at a point

Power series

Alternating series

Series that converge conditionally

Series that converge absolutely

The p-series

Integral test

Ratio test

Limit comparison test

Convergence of infinite series

Infinite sequences of real numbers and their convergence or divergence

The order of an element is unchanged by an isomorphism

Isomorphism of groups

Proof of homomorphism properties for identities and inverses

Proof that the kernel and range of a homomorphism are subgroups

Definition of the kernel of a homomorphism

Definition of a group homomorphism

The order of a combination of cycles

Result that every permutation can be written as a composition of disjoint cycles

Cycle notation for permutations

Permutations under composition of permutations

Proof that all cyclic groups are Abelian

Generators

Identity and inverse elements

Binary operations

Definition

Solutions of systems of linear equations

(a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution.

Exam Questions – Exponential Type ke^px (exponential types)

Exam Questions – Matrix transformations

Exam Questions – Finding roots

The conic sections of a circle, parabola, ellipse and hyperbola

Cyclic groups

Lagrange’s theorem

Subgroups

Order of a group

Commutative groups

Different types of groups

What defines a group

Shortest distance from a point to a plane

Range of values taken by a function

Geometrical effects of conjugating a complex number

Modulus-argument form of a complex number

Area of surface of revolution about the x-axis

Step by step methods

Graphs of rational functions

Approximating by areas of rectangles and setting bounds using inequalities

Transposed and symmetric matrices

Shortest distance between two skew lines

Vector product form of a line

Applications : Area of a triangle and parallelogram

Vector product (cross product)

Applications: volume

Triple scalar product

Standard integrals relating to hyperbolic functions

Deriving and using reduction formulae

Integrating inverse trigonometric and hyperbolic functions using integration by parts

Using trigonometric and hyperbolic substitutions

Integrating expressions involving hyperbolic functions

Length of an arc

Reducing a symmetrical matrix to diagonal form

Eigenvalues and eigenvectors

Using inverse matrices to reverse the effects of a linear transformation

Linear transformations in 3 dimensions

Finding the inverse of a 3×3 matrix where it exists

Finding the determinant of a 3×3 matrix

Finding equations of simple loci

Cartesian equation, focus and directrices

Finding equations of tangents and normals to a hyperbola

Cartesian and parametric equations and asymptotes for a hyperbola

Tangents and normals to an ellipse

The Ellipse – Cartesian and parametric forms

Derivatives of sin^-1(x), cos^-1(x) and tan^-1(x)

Relationship between the graph y=f(x) and y²=f(x)

Improper types / oblique asymptotes

Relationships between the roots and coefficients of a quartic equation

Relationships between the roots and coefficients of a cubic equation

Relationships between the roots and coefficients of a quadratic equation

Exam Questions – Maclaurin’s series

Exam Questions – General solutions where f(x) = kx (linear types)

Exam Questions – Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

Exam Questions – Matrix proofs

Integrals of the form 1/(a²+x²) and 1/√(a²-x²)

Solving quartic equations

Solving cubic equations

Solving quadratic equations with complex roots

Square roots of a complex number

Exam Questions – Tangents

Exam Questions – Area bounded by a polar curve

Exam Questions – Taylor’s series

Exam Questions – Particular solutions using boundary conditions

Exam Questions – Substitution types

Exam Questions – Exact equations (integrating factors)

Exam Questions – Transformations of the complex plane

Exam Questions – Loci in the complex plane

Exam Questions – nth roots of a complex number

Exam Questions – Further complex numbers

Exam Questions – Method of differences

Exam Questions – Modulus inequalities fractional type

Exam Questions – Recurrence relations

Exam Questions – Divisibility and multiple tests

Exam Questions – Sum of series

Exam Questions – Series

Exam Questions – Identity and inverse of a 2×2 matrix

Exam Questions – Hyperbola (rectangular)

Exam Questions – Parabola

Exam Questions – Complex numbers

Further transformations

The angle between two planes

The equation of the line of intersection between two non parallel planes

Finding the point of intersection between a line and a plane

Scalar product forms of a plane

Cartesian form of a plane

Parametric vector form of a plane

Cartesian form of a line

Differentiation of inverse hyperbolic functions

Differentiation of hyperbolic functions

Solving hyperbolic equations using hyperbolic identities

Expressing inverse hyperbolic functions as natural logarithms

Inverse hyperbolic functions and their graphs

Osborn’s rule

Hyperbolic identities

Solving equations using inverse and exponential functions

Graphs of sech(x), cosech(x) and coth(x)

Graphs of sinh(x), cosh(x) and tanh(x)

Definitions

Finding equations of tangents parallel to and perpendicular to the initial line

Area bounded by a cardioid and a loop

Area bounded by a polar curve

Sketching curves the curve r = asin 2θ

Sketching curves the cardioid r = a (1+cosθ)

Sketching curves a circle and arc

Sketching polar curves a half-line

Sketching polar curves a spiral

Converting the equation of a Cartesian curve to polar form

Converting the equation of a polar curve to Cartesian form

Converting polar coordinates to Cartesian coordinates

Converting Cartesian coordinates to polar coordinates

Defining the position of a point

Solution to differential equations using Taylor’s series

Taylor’s series

Further series

Series expansion for ln(1+x)

Series expansion for sin(x) and cos(x)

Series expansion for e^x

Maclaurin’s series expansion

Using substitution to reduce a differential equation to a known form

Particular solutions using boundary conditions to solve differential equations

Special types of particular integrals

General solutions where f(x) = λ cosωx + µ sinωx (trig types)

General solutions where f(x) = ke^px (exponential types)

General solutions where f(x) = kx² (quadratic types)

General solutions where f(x) = kx (linear types)

General solutions where f(x) = k (constant types)

How to solve second order linear differential equations that equal zero

Using substitution to reduce a differential equation to a known form

Solving equations of the form dy/dx + Py = Q using an integrating factor

Exact equations where one side is the exact derivative of a product

Transformations of the complex plane

Using complex numbers to represent regions on an Argand diagram

The locus of a point moving on the arc of a circle

The locus of a point moving along a half-line

The locus of a point moving along a perpendicular bisector

The locus of a point moving in a circle

nth roots of a complex number

Expressing sinⁿθ and cosⁿθ in terms of sin(kθ) and cos(kθ)

Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

De Moivre’s theorem

Multiplication and division rules for mod and argument of two complex numbers

Exponential Form (Euler’s relation)

Method of differences

Modulus inequalities fractional type

Fractional inequalities

Matrix proofs

Recurrence relationship proofs

Divisibility and multiple test proofs

Proof for other series

Proof of the sum of the series ∑r³

Proof of the sum of the series ∑r²

Proof of the sum of the series ∑r

Using known formulae to sum more complex series

Sum of the cubes of the first n natural numbers ∑r³

Sum of the squares of the first n natural numbers ∑r²

Determinant as the area scale factor of a transformation

Inverse matrices to reverse linear transformations

Combinations of transformations

How well do you know your transformations?

Linear transformations – enlargement

Linear transformations – reflections

Linear transformations – rotations

Solving simultaneous linear equations in 2 unknowns

Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)

Identity and Inverse of a 2×2 matrix

Matrix multiplication

Addition and subtraction and multiplying a matrix by a scalar

Introduction and dimension of a matrix

Tangents and normals: parametric type

Tangents and normals : Cartesian type

Cartesian and parametric type

Tangents and normals: parametric types

Tangents and normals: Cartesian types

Parabola parametric form

Coordinate geometry: Parabola directrix, focus, locus and equation

Solving problems with complex numbers

Modulus and argument of a complex number

Argand diagrams

Division of a complex number by a complex number

Complex conjugates

Addition, subtraction and multiplying complex numbers and simplifying powers of i

Real and imaginary numbers

Infinite integrals

In this tutorial I show you how to handle integrals where a limit is infinite as in the example below. Example:

Sigma notation