Mathematical Skills

Mathematical Skills
Arithmetic
Order of evaluation BODMAS
Fractions
Numbers
natural
integers
rational
irrational
transcendental
algebraic numbers
in other bases
binary
octal
hexa decimal
Scientific notation
Decimals
Percentages
Ratios and proportions
SI Units
Arithmetic using mixed scales
Binary arithmetic
Use of calculator
Entering negative numbers
Entering exponents 1.5E4
Inverse trigonometric
Powers of negative numbers (-2)^3
Trigonometry
angular measure
radians
degrees
mills
evaluate trigonometric functions
definitions of sin, cos and tan
basic trigonometric identities
Pythogorous identity
tangent
addition formula multiple angles
double angle formulae
Worded problems
sketching appropriate figures
application of sin, cos or tan
angle of elevation and depression
sine and cosine rules
Northings and Eastings
Approximations for small angles
Series representation
Changing Cartesian to Polar
coordinates
Use of R>P and P>R on calculator
Compound angles
sin(A+B) = ...
cos(A+B) = ...
tan(A+B) = ...
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
nonlinear equations
polynomial
trigonometric
exponential
logarithmic
Transformation of formulae
adding or subtracting the same
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
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Evaluation of formulae
simple formulae
formulae with mixed units and mixed scales
Direct and inverse proportions
Algebraic Manipulation
expand brackets mixed signs
factorization
polynomial division
long division
algebraic fractions
Combine into a single fraction
Rearrange and simplify
Distinguish A/(x2) vs A/x2
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
Linear and quadratic factors
Improper fraction requiring long division
Functions and Graphs
linear relationships
polynomials
exponentials
y=exp(ax); a>0
y=exp(ax); a>0
y=exp(-a(x-c)^2)
distinguish exponentials
Rewrite exponentials
population models
Sketch and interpret
logarithms
Definition ln vs log
solve for x: y = n^x
quadratics
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
Vectors
vector addition
resolution of vectors
vector subtraction
Products
scalar or dot product
cross product
Matrix
addition, subtraction
product of matrices
determinant of a matrix
inverse of a matrix
eigenvalues
eigenvectors
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
Numerical integration
Trapezoidal rule
Simpson's rule
Advanced techniques
Gaussian, adaptive
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
AdamMoulton method
RungeKutta methods
AdamBashforth method
Second order equations
Analytical techniques
Characteristic equation
Power series method
Initial Value Problem
Boundary Value Problem
Numerical methods
System of equations
Linear and Nonlinear equations
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