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Mathematical skills by Mind Map: Mathematical skills
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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 - 1.5E-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

non-linear 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, Untitled, Untitled

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, 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

Linear and quadratic factors

Improper fraction - requiring long division

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 co-ordinates

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

Compound angles

sin(A+B) = ...

cos(A+B) = ...

tan(A+B) = ...

Functions and Graphs

Can add custom made graphs and upload them as .png files

linear relationships

polynomials

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

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

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

Vectors

vector addition

resolution of vectors

vector subtraction

Products

scalar or dot product

cross product

Matrices

addition, subtraction

product of matrices

determinant of a matrix

inverse of a matrix

eigenvalues

eigenvectors

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

System of equations

Linear and Non-linear equations