MATHEMATICS REVISION OF FORMULAE AND RESULTS

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