Formulae

The following collection of formulae is not given in the exam and so must be learnt.

Applications (Apps)

Apps 1.1 : Applying algebraic skills to rectilinear shapes.

  y - b = m(x - a),  where (a, b) are co-ordinates of a point on the line.

Gradient = tanθ    where θ is measured anticlockwise.

Apps 1.2 : Applying algebraic skills to circles

Touching circles :

If AB = r₁ +r₂, the circles touch externally.

If AB = r₁ - r₂, the circles touch internally.

Apps 1.3 :Applying algebraic skills to sequences

Expressions and Functions (E&F)

E&F 1.1 : Applying algebraic skills to logarithms and exponentials

If   y = a^x
x = log_a y

where N is the number of radioactive atoms present at time t , λ is the transformation decay constant and N_o is the original starting value.

E&F 1.2 : Applying trigonometric skills to manipulating expressions.

Exact values

Wave Theory

E&F 1.3 : Applying algebraic and trigonometric skills to functions.

The Domain is the set of input numbers,
the Codomain is the set of possible output numbers,
the Range is the set of actual output images.

Transformations

Reflection in
x axis            y axis

Translation in
x axis                          y axis

Scaling in
x axis                          y axis

Inverse

 The inverse of a function f(x) is denoted f^-1(x)
f^-1 (f(x)) = f (f^-1 (x)) = x

To inverse a function ( which must be in a one – one correspondence)
reflect it in the line y = x.

Logarithms and exponential

An ordinary exponential function always has the points
    (0, 1) , (  1 , base) and (-1, 1/base ) 
      since     a⁰ =1 ,   a¹ = a  and a^-1 =1/a  

The  exponential function f(x) = a^x  has an inverse function
f^-1(x) = log_ax  upon reflection in the line y = x

  If   y = a^x
         x = log_a y

An ordinary log function always has the points
(1, 0)   and (base, 1)
since
log_a1=0 and loga_a=1

Log graphs

Log scale Y axis only

If  y = ab^x  then log y = log a + xlogb

Compare this to     Y = mx + c
 where Y = log y,  m = log b  and c = log a

If  y = ax^b  then log y = blog x + log a   

Compare this to     Y = mX + c
where Y = log y,  X = logx  and c = log a

E&F 1.4: Applying geometric skills to vectors.

3D Vectors

Section formula

Relationships and Calculus (R&C)

R&C 1.1 : Applying algebraic skills to solve equations.

Polynomial division

The Remainder Theorem
If a polynomial f(x) is divided by x-h, then the remainder is f(h).
( h may be a fraction)

The Factor Theorem
If f(x) is a polynomial ,  f(h) = 0<=> (x-h) is  a factor of f(x)

If     b² – 4ac > 0   , the line cuts at two distinct points.
It is not a tangent.

If     b² – 4ac < 0   , the line does not touch the curve.
It is not a tangent.

If      b² – 4ac = 0   , the line touches the curve at only one point.
It is  a tangent.

R&C 1.2 : Applying trigonometric skills to solve equations.

R&C 1.3 : Applying calculus skills of differentiation.

The derivative of a function for some particular value
is a measure of the rate at which the function is changing
at that particular value.

TThe derivative of a function for some particular value
is also the gradient of the graph of the function at that point.

The chain rule

or in Leibnitz notation

Graphing

R&C 1.4 : Applying calculus skills of integration.