
Daniel L.
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0
Very excellent resource. This glossary has many terms that have been covered in this course. 
XSIQ
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Advanced maths  Glossary
Glossary
Fractions involving algebraic expressions. For example:
(May be simplified using partial fractions.)
ANGLE OF FRICTION
The angle between the resultant reaction and the normal reaction. Used in
problems involving friction and inclined planes.
ANTI DIFFERENTIATE
The opposite process to differentiation, where the original function can
be found from the differentiated function.
ARGAND DIAGRAM
Used to display complex numbers where the horizontal axis is real (x) and
the vertical axis is imaginary (_iy_), shows both Cartesian (_x_ + _iy_)
and polar (_r_ cisq).
ASYMPTOTES
Denoted by a dotted line or curve on a graph this shows where the function
is undefined.
BOUNDARY POINTS
For definite integration you are finding the area under the function
_f(x)_ between two values of _x_ (boundaries). These boundaries are shown
as numbers below the integral sign (the lower value) and above (the higher
value). Also called terminals of integration.
CARTESIAN EQUATION
Can be represented on a Cartesian plane (two variables only). May be found
from parametric equations using substitution.
CHAIN RULE
Used in substitution problems:
COLLINEAR
Points that are on the same line.
COMPLEX CONJUGATE
A complex number that has the imaginary part the opposite sign to the
original complex number. When a complex number and its conjugate are
multiplied together the result is a real number.
COMPLEX FACTORS
When a polynomial is factorised and not all factors are real, the complex
factors are of the form (_z_  _ai_) or (_z_ + _ai_).
COMPLEX ROOTS
The solution of a polynomial with complex factors of form _z_ = _ai_ or z
= _ai_.
COMPLIMENTARY
Complimentary angles are angles that add together to total 90 degrees. The
trigonometric relationships:
and:
can be shown using a right angled triangle.
COMPOUND ANGLE FORMULAE
These formulae are given on the formula sheet in the exam along with the
double angle formulae. They are used to rewrite trigonometric functions in
a form that can be anti differentiated easily. The compound angles are:
CONCEPT DIAGRAMS
A way of organizing, sorting and linking information by placing individual
concepts in a circle (or oval) and linking with other concepts using lines
and comments about the connection between the two concepts.
CONJUGATE PAIRS
If one root is complex then another root must be the conjugate of the
first, and the roots are said to be in conjugate pairs.
DE' MOIVRES THEOREM
Used to find powers of complex numbers given in polar form:
This is given on the formula sheet.
DECIMAL PLACES
Where an answer is given to a certain number of figures after the decimal
place. For example 16.95 is given to two decimal places.
DEFLECTION
When an object is moved (by a force) resulting in a change from its usual
position. Usually used with beams fixed at one end.
DIFFERENCE OF TWO CUBES
Used to factorize polynomials:
DIFFERENCE OF TWO SQUARES
Used in factorizing polynomials:
DILATION
Where a basic graph can be made larger or smaller to obtain the graph of
another function. For example:
is narrower than:
by a factor of 5.
DOT PRODUCT
Used to multiply two vectors together. Only multiply the same direction
together  all other combinations will equal zero (perpendicular).
DOUBLE ANGLE FORMULAE
These formulae are given on the formula sheet in the exam along with the
compound angle formulae. They are used to rewrite trigonometric functions
in a form that can be anti differentiated easily:
are double angles.
EXACT ANSWER
Where the answer is left in surd form or in terms of pi (p).
FACTOR THEOREM
This theorem states that if a polynomial is divided by (_x_  _a_) and the
remainder is zero then (_x_  _a_) is a factor.
When a function can be expressed as the product of its factors, the roots
(_x _intercepts) can be found.
FORMULA SHEET
List of formulae given in exams. Can be obtained from your school.
FUNDAMENTAL IDENTITY
Obtained from a right angled triangle in the unit circle, (hypotenuse =
1), and using Pythagoras' theorem to get the expression:
FUNDAMENTAL THEOREM OF CALCULUS
Area between a graph and the _x_ axis can be found using the integral
between two values of:
x
GENERAL SOLUTION
When integrating a function always include (+ c) to allow for the many
functions that could have been differentiated to obtain the original
function.
GRADIENT
The slope of a graph can be obtained by use of differentiation.
GRAVITY
The pull of the planet's gravitational field on a body on or near the
surface.
The constant acceleration due to gravity (_g_ = 9.8 ms2 on Earth).
INITIAL CONDITIONS
This enables a general solution to be changed into a specific solution. By
substituting for _x_ and _f(x)_ the specific constant can be evaluated for
the specific conditions. (_Initial_ usually refers to the start when _t_ =
0 or _x_ = 0)
INTEGRAL
The area under a curve in a certain domain is found using anti
differentiation and substituting for the end point of the domain, then
subtracting the lower end from the upper end:
INTERCEPTS
_x_ and _y_ intercepts are where a graph crosses the _x_ and _y_ axes (at
_y _= 0 and _x_ = 0 respectively); or where two graphs cross over each
other.
INTERVAL OF INTEGRATION
For definite integration you are finding the area under the function
_f(x)_ between two values of _x_  this is the interval of integration.
These values are shown as numbers below the integral sign (the lower value)
and above (the higher value). They are called _boundary points_ or the
_terminals of integration_.
LAMI'S THEOREM
Used to solve problems where three forces are in equilibrium:
where the force is divided by the sine of the angle between the other two
forces.
LINEAR SUBSTITUTION
Used to solve equations of the type:
or:
LOG LAWS
These are used in differential equations to simplify solutions:
MAGNITUDE
Refers to the positive numerical value.
MAGNITUDE OF A COMPLEX NUMBER
The numerical value of the complex number found using:
from the formula sheet; where _r_ is the length of the hypotenuse (polar
form).
MAGNITUDE OF A VECTOR
The numerical value of the vector found using:
from the formula sheet.
MAGNITUDE OF AREA UNDER GRAPH
When integrating a function that crosses the axis, you will need to
consider each section separately. Find the magnitude of the negative side,
and then add this to the area from the positive side. This is denoted by
straight lines on either side of the integral. For example _f(x)_ crosses
over the _x_ axis at _x _= _b _and the section from _x_ = _b_ to _x_ = _c_
is under the _x_ axis:
MIDPOINT RULE
Used to find the area under a function by approximating with a series of
rectangles with height obtained from the midpoint of each interval.
NON REAL ROOTS
Also called complex roots, found by solving polynomials with complex
factors.
PARAMETRIC EQUATION
An equation where each direction is in terms of a variable (usually time)
for example:
PERFECT SQUARES
Used when finding complex roots:
(Complete the square, then use difference of two squares letting )
PERIODIC
If a function is periodic it will repeat after a certain period of time.
This is used in trigonometry to sketch functions or find equations.
is a sine function with amplitude _a_, and period .
PERPENDICULAR VECTOR
A vector that is at right angles to another vector.
POLAR COORDINATES
Where a complex number is defined using an angle and the length of the
hypotenuse of the triangle made from the Cartesian form in the Argand
diagram. [_r_,q]
POSITION VECTOR
The location of an object relative to a reference point (origin):
PRODUCT RULE
This is on the formula sheet and is used to differentiate the product of
two functions:
QUADRATIC FORMULA
Used to find the roots of a quadratic polynomial when factorization is not
possible:
where:
(Not on the formula sheet.)
QUOTIENT RULE
This is on the formula sheet and used to differentiate the quotient of two
functions:
RATE OF CHANGE
Differential equation problems using the chain rule. For example, finding
the rate of change of volume:
REALISE DENOMINATOR
To make the denominator real by multiplying by the complex conjugate and
eliminating the unreal part of the number. For example:
as:
RECIPROCAL FUNCTIONS
If a function can be represented as the reciprocal of a function that can
be sketched, then this function can also be sketched:
RECTILINEAR MOTION
Motion in a straight line. Used in kinematics examples
RESOLUTION OF A FORCE
Replacing a force with two other forces at right angles to each other.
RESTRICTED DOMAIN
This is used when only part of the function is needed. For example, when
finding the area between two curves you will need to find the points of
intersection and integrate between these two values. This is also used for
inverse trigonometric functions.
RESULTANT REACTION
Found by adding the friction force and the normal reaction.
SCALAR RESOLUTE
When resolving a vector into two directions, the scalar resolute is the
magnitude of the vector resolute.
SIGNIFICANT FIGURES
Where an answer is given to a total number of figures including those
before and after the decimal place, for example 16.95 has 4 significant
figures.
SPECIFIC SOLUTION
Found from substituting the initial (or specific) conditions into the
general solution and finding the constant 'c'.
STATIONARY POINT
This includes all cases of the gradient being momentarily zero  maximums,
minimums and the stationary point of inflection where the gradient is the
same sign before and after it, is zero. For example, (positive, zero,
positive) or (negative, zero, negative).
STUDY DESIGN
A summary of the skills that should be learnt in this study in the form of
a series of dot points. Available in schools.
SUBSTITUTION
To replace part of an expression with a variable that simplifies the
expression and enables the solution to be found.
SUM OF TWO CUBES
SURDS
A square root that cannot be given as an exact answer by evaluating. For
example, the square root of 6 is 2.44949 to 6 decimal places
(approximately).
SYMMETRIC
Finding alternative expressions for trigonometric functions using:
p + q
p  q
2p + q
2p  q.
TANGENT
The straight line that touches a curve (or circle) at one point only and
has the same gradient as the curve at that point.
TENSION
The force exerted on an object through a rope or string (T).
TERMINALS OF INTEGRATION
For definite integration you are finding the area under the function
_f(x)_ between two values of _x_. These are shown as numbers below the
integral sign (the lower value) and above (the higher value). Also called_
boundary points_.
TILDE
The symbol (~) under a letter that tells you it is a vector.
TRANSFORMATIONS
Used where a basic function can be identified and moved across and up to
obtain the graph of another function. For example:
is the basic _ graph with b_ translated to the right and _c_ translated
up.
TRAPEZOIDAL RULE
Used to find the area under a function by approximating with a series of
trapeziums using the heights at the ends of each interval.
TURNING POINT
Where the function changes its gradient and is momentarily zero. For a
change of positive to negative, the turning point is a local maximum; for a
negative to positive change, the turning point is a local minimum.
UNIT CIRCLE
A circle of radius magnitude one. Used to define trigonometric functions.
VARIABLE ACCELERATION
Acceleration varies with time (usually), and to find position or velocity
vectors you will need to use calculus techniques.
VECTOR RESOLUTE
When resolving a vector into two directions, the vector resolute has both
magnitude and direction.
VELOCITY VECTOR
A vector that represents the velocity of an object. It is found by
differentiating the position vector, or anti differentiating the
acceleration vector.
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