MATHEMATICS REVISION OF FORMULAE AND RESULTS

MATHEMATICS REVISION OF FORMULAE AND RESULTS
Surds
a × b = ab
a
b =
a
b
( a) 2 = a
Absolute Value
Co-ordinate Geometry
Distance formula: d = x 2 − x 1 2 + (y 2
− y 1
) 2
Gradient formula: m = y 2 − y 1
x 2 − x 1
or m = tanθ
Midpoint Formula: midpoint = x 1+ x 2
, y 1 + y 2
2 2
Geometrically:
a = a if a ≥ 0
a = − a if a < 0
x is the distance of x from the origin on the number line
x − y is the distance between x and y on the number line
Factorisation
Real Functions
ab = a . b
a + b ≤ a + b
x 3 − y 3 = x − y (x 2 + xy + y 2 )
x 3 + y 3 = x + y (x 2 − xy + y 2 )
A function is even if f −x = f(x). The graph is
symmetrical about the y-axis.
Perpendicular distance from a point to a line:
ax 1 + by 1
+ c
a 2 + b 2
Acute angle between two lines (or tangents)
Equations of a Line
tanθ = m 1 − m 2
1 + m 1 m 2
gradient-intercept form: y = mx + b
point-gradient form: y − y 1
= m(x − x 1 )
two point formula:
intercept formula:
y − y 1
x − x 1
= y 2 − y 1
x 2 − x 1
x
a + y b = 1
general equation: ax + by + c = 0
A function is odd if f −x = − f(x). The graph has
point symmetry about the origin.
Parallel lines: m 1 = m 2
The Circle
Perpendicular lines: m 1 .m 2 = − 1
The equation of a circle with:
Centre the origin (0, 0) and radius r units is:
x 2 + y 2 = r 2
Centre (a, b) and radius r units is:
(x − a) 2 + (y − b) 2 = r 2

Trigonometric Results
The Quadratic Polynomial
sinθ =
opposite
hypotenuse
(SOH)
The general quadratics is: y = ax 2 + bx + c
cosθ =
adjacent
hypotenuse
(CAH)
The quadratic formula is: x = −b ± b2 − 4ac
2a
The discriminant is: Δ = b 2 − 4ac
tanθ = opposite
adjacent
Complementary ratios:
Pythagorean Identities
(TOA)
sin 90° − θ = cosθ
cos 90° − θ = sinθ
tan 90° − θ = cotθ
sec 90° − θ = cosecθ
cosec(90° − θ) = secθ
sin 2 θ + cos 2 θ = 1
If Δ ≥ 0 the roots are real
If Δ < 0 the roots are not real
If Δ = 0 the roots are equal
If Δ is a perfect square, the roots are rational
If α and β are the roots of the quadratic equation
ax 2 + bx + c = 0
then: α + β = − b a and αβ = c a
The axis of symmetry is: x = − b
If a quadratic function is positive for all values of x, it is
positive definite i.e. Δ < 0 and a > 0
2a
1 + cot 2 θ = cosec 2 θ
tanθ = sinθ
cosθ
tan 2 θ + 1 = sec 2 θ
and cotθ = cosθ
sinθ
If a quadratic function is negative for all values of x, it
is negative definite i.e. Δ < 0 and a < 0
If a function is sometimes positive and sometimes
negative, it is indefinite i.e. Δ > 0
The Sine Rule
a
= b
= c
sinA sinB sinC
The Cosine Rule
a 2 = b 2 + c 2 − 2bcCosA
CosA = b2 + c 2 − a 2
2bc
The Area of a Triangle
The Parabola
The parabola x 2 = 4ay has vertex (0,0), focus (0,a),
focal length ‘a’ units and directrix y = − a
The parabola (x − h) 2 = 4a(y − k) has vertex (h, k)
Area = 1 2 abSinC

Differentiation
Geometrical Applications of Differentiation
First Principles:
Stationary points:
dy
dx = 0
f ' (x) =
lim
h → ∞
f (x + h) – f (x)
h
or
Increasing function:
dy
dx > 0
f ' (c) = lim
x → c
f (x) – f (c)
h
Decreasing function:
dy
dx < 0
If y = x n then dy
dx = nxn−1
Concave up:
d 2 y
dx 2 < 0
Chain Rule: d dx
f (u) = f ' (u)
du
dx
Concave down:
d 2 y
dx 2 > 0
Product Rule: If y = uv then dy
= u dv
+ v du
dx dx dx
Quotient Rule: If y = u v
then
dy
dx = v du
dx + u dv
dx
v 2
Minimum turning point:
Maximum turning point:
dy
= 0 dx and d2 y
> 0 dx 2
dy
= 0 dx and d2 y
< 0 dx 2
Trigonometric Functions:
d
dx
sinx = cosx
Points of inflexion: d2 y
dx2 = 0 and concavity changes
about the point.
d
dx
d
dx
cosx = − sinx
tanx = sec2 x
Horizontal points of inflexion: dy
= 0 and d2 y
= 0 and
dx dx 2
concavity changes about the point.
Exponential Functions:
d
dx
ef (x) = f ' (x)e f (x)
d
dx
ax = a x .lna
Logarithmic Functions:
d
dx
log e
f (x) =
f ' (x)
f (x)

Approximation Methods
Sequences and Series
The Trapezoidal Rule:
Arithmetic Progression
a
b
f x dx = h 2
y 0 + y n + 2 y 1 + y 2 + y 3 + …+ y n−1
d = U 2 − U 1
U n = a + n − 1 d
Simpson’s Rule:
S n = n [2a + n − 1 d]
2
b
f x dx = h 3
y 0 + y n + 4 y 1 + y 3 + … + 2 y 2 + y 4 + …
S n = n [a + l] where l is the last term
2
a
Geometric Progression
In both rules, h = b − a
where n is the number of strips.
n
Integration
If f (x) ≥ 0 for a ≤ x ≤ b, the area bounded by the
curve y = f (x), the x-axis and x = a and x = b is given
by
b
a
f x dx.
r = U 2
U 1
U n = ar n−1
S n = a rn − 1
r − 1
S ∞ =
a
1 − r
= a 1 − rn
1 − r
The volume obtained by rotating the curve y = f (x)
about the x-axis between x = a and x = b is given by
b
a
π f x 2
If dx
dx = xn then y = xn+1
n + 1
If dx
= ax + b n then y = ax + b n
dx a(n + 1)
Trigonometric Functions:
The Trigonometric Functions
radians = 180⁰
Length of an arc: l = rθ
Area of a sector: A = 1 2 r2 θ
Area of a segment: A = 1 2 r2 (θ − sinθ)
[In these formulae, is measured in radians.]
sin x dx = − cosx + C
Small angle results:
cos x dx = sinx + C
sec 2 x dx = tanx + C
sinx → 0
cosx → 1
tanx → 0
lim
x → 0
sinx
x
= lim
x → 0
tanx
x
= 1
Exponential Functions:
e ax dx = eax
a + C and ax dx =
1
ln a .ax
For y = sin nx and y = cos nx the period is 2π
For y = sin nx the period is π n
n
Logarithmic Functions:
f ' (x)
f (x) dx = log e x + C

Logarithmic and Exponential Functions
The Index Laws:
a x × a y = a x+y
a x ÷ a y = a x−y
a x y = a xy
a −x = 1 a x
x
y
ay = a x
a 0 = 1
The logarithmic Laws:
If log a
b = c then a c = b
log a
x + log a
y = log a
xy
log a
x − log a
y = log a x y
log a
x n + nlog a
x
log a
a = 1 and log a
1 = 0
The Change of Base Result:
log a
b = log e b
= log 10 b
log e a log 10 a