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# Mathematical skills

## Arithmetic

### Numbers

natural

integers

rational

irrational

transcendental

algebraic numbers

in other bases, binary, octal, hexa decimal

### Use of calculator

Entering negative numbers

Entering exponents - 1.5E-4

Inverse trigonometric

Powers of negative numbers - (-2)^3

## Algebra

### indices

apply rules of indices, positive integral exponents, negative exponents, fractional powers

square root notation

meaning of a^0

### solution of equations

linear equations, a x + b = c, a/bx = c/d, cross multiply, a/(b+x) = c/(d + x)

simultaneous equations, algebraic method, graphical method, elimination method

quadratic equation, by factorization, graphical method, using the formula

polynomial equations, simple cubic; x^3 = a, simple quartic; x^3 = a, equations leading to simple cubic or quartic

polynomial

trigonometric

exponential

logarithmic

### Transformation of formulae

multiplying both sides

reciprocating both sides

taking square roots both sides

raising both sides to a fractional power

exponentiate both sides

taking logs on both sides

apply a function to both sides

### Inequalities

sketch intervals, abs(x+a) = b, Untitled, Untitled

### Evaluation of formulae

simple formulae

formulae with mixed units and mixed scales

### Algebraic Manipulation

expand brackets - mixed signs

factorization

polynomial division, long division

algebraic fractions, Combine into a single fraction, Re-arrange and simplify, Distinguish A/(x-2) vs A/x-2, Mixed algebraic - number algebraic fraction, Mixed algebraic - fraction + algebraic, Monomial plus algebraic fraction, Divide the fraction into individual terms, Monomials plus fractions with additional coefficients

Remainder theorem

### Partial Fractions

Linear factors

Repeated factors

Improper fraction - requiring long division

## Trigonometry

degrees

mills

### basic trigonometric identities

Pythogorous identity

tangent

### Worded problems

sketching appropriate figures

application of sin, cos or tan

angle of elevation and depression

### Changing Cartesian to Polar co-ordinates

Use of R->P and P->R on calculator

sin(A+B) = ...

cos(A+B) = ...

tan(A+B) = ...

## Functions and Graphs

### exponentials

y=exp(ax); a>0

y=exp(-ax); a>0

y=exp(-a(x-c)^2)

distinguish exponentials

Re-write exponentials

population models

Sketch and interpret

### logarithms

Definition - ln vs log

solve for x: y = n^x

effect of doubling the arguments of log

### trigonometric

trigonometric ratios of acute angles

trigonometric ratios of large angles

Basic trig identities

Graphs of ..., y = sin(x), y=cos(x), y=cos(x+a), y=cos(x/2+a), y=sin(2x), y=cos(2x), y=sin(x+a), y=sin(x/2+a), y=tan(x), y=tan(2x), y=tan(x/2)

Periodicity of trig functions

Sinusoidal form - A sin( omega t +/- alpha)

Solution of trig equations, Using inverse functions in the calculator, in the interval [0, 360], sin(x) = a, sin(2x+c) = a, sin(x/2+c) = a, cos(x) = a, cos(2x+c) = a, cos(x/2+c) = a, tan(x) = a

## Calculus

### Differentiation

concept of instantaneous rate of change

derivatives of polynomials

use of table look up for sin(ax), cos(ax), exp(ax), log(ax)

stationary values

inflexion points

second derivatives

determine the shape of the curve given the derivatives

determine maxima, minima

simple applications (limited to functions not involving chain rule, product rule or quotient rule)

application of product, quotient and chain rules

### Integration

opposite of differentiation

constant of integration

Integrals of polynomials - y = ax^n + ...

table look up - x^n, sin(ax), cos(ax), exp(ax), ln(ax)

algebraic substitutions

trigonometric substitutions, t = tan(theta/2) substitution

change of limits

using partial fractions

Integration by parts

Applications, area under the curve y = f(x), between x=a and x = b, area bounded between the two curves y = f(x), and y = g(x), volumes of solids of revolution, centroids of simple shapes, mean and root mean square values, second moments of area

## Vectors

### Products

scalar or dot product

cross product

## Differential equations

### First order equations

Analytical techniques, Integrating factor, seperation of variables, Laplace transform techniques

Initail value problem

Numerical methods, Euler's method, Modified Euler's method, Adam-Moulton method, Runge-Kutta methods, Adam-Bashforth method

### Second order equations

Analytical techniques, Characteristic equation, Power series method

Initial Value Problem

Boundary Value Problem

Numerical methods