# 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 1.5E4

Inverse trigonometric

Powers of negative numbers (-2)^3

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 andchain 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

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 Polarcoordinates

Use of R&gt;P and P&gt;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 simplecubic 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 inthe 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

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

Matrix

addition, subtraction

product of matrices

determinant of a matrix

inverse of a matrix

eigenvalues

eigenvectors