Download ADEDEX424 Formula Sheet Arithmetic and Algebra Rules of Indices

ADEDEX424 Formula Sheet
Arithmetic and Algebra
Rules of Indices:
xm × xn = xm+n .
(xm )n = xmn .
xm ÷ xn = xm−n .
x0 = 1.
x1 = x.
1
x−n = n .
x
(x × y)n = xn × y n .
√
1
n
x = xn .
Rules of Logarithms:
Definition: loga x, is the number y such that x = ay .
loga (xy) = loga x + loga y.
m
log
) = m loga x.
a (x
x
= loga x − loga y.
loga
y
loga 1 = 0.
loga a = 1.
logb x
loga x =
.
logb a
aloga x = x.
loga (ax ) = x.
The Binomial Theorem:
n X
n n−i i
n
x y.
(x + y) =
i
i=0 n n−2 2
n n−1
n
n
x y + · · · + y n.
x y+
(x + y) = x +
2
1
Binomial Coefficients:
n!
n
n(n − 1)(n − 2) · · · (n − i + 1)
=
=
.
i
i!(n − i)! i!
n
n
.
=
n
−
i
i
n
n
n+1
.
+
=
i
i−1
i
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Lines and their Equations
y2 − y1
.
x2 − x1
The equation of a line is y = mx + c, where m is the slope and c is the y-intercept.
p
The length of a line segment between two points (x1 , y1 ) and (x2 , y2 ) is (x2− x1 )2 + (y2 − y1 )2 .
x1 + x2 y1 + y2
The midpoint of the line seqment between two points (x1 , y1) and (x2 , y2) is
.
,
2
2
Given two points (x1 , y1 ) and (x2 , y2 ) on a line, then the slope of the line is m =
Quadratic Equations
√
b2 − 4ac
.
2a
b2 − 4ac
b
2
.
The turning point of the graph of y = ax + bx + c lies at − , −
2a
4a
The equation ax2 + bx + c = 0 has solutions x =
−b ±
Trigonometry
Figure 3: Definition of Trigonometric Functions.
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θ
sin(θ)
0
π
6
π
4
0
1
2
1
√
2
cos(θ)
1
tan(θ)
0
√
π
3
√
π
2
3
2
1
0
3
2
1
√
2
1
2
1
√
3
1
√
3
∗
Table 1: Values of sin(θ), cos(θ) and tan(θ) for important values of θ.
sin(−θ) = − sin(θ)
cos(−θ) = cos(θ)
tan(−θ) = − tan(θ)
cot(−θ) = − cot(θ)
cosec(−θ) = − cosec(θ)
sec(−θ) = sec(θ)
Table 2: Parity Identities.
sin
π
2
− θ = cos(θ)
tan
π
− θ = cot(θ)
cosec
2
π
2
− θ = sec(θ)
cos
π
2
− θ = sin(θ)
cot
π
− θ = tan(θ)
sec
2
π
2
− θ = cosec(θ)
Table 3: Co-function Identities.
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sin(A ± B) = sin(A) cos(B) ± cos(A) sin(B)
cos(A ± B) = cos(A) cos(B) ∓ sin(A) sin(B)
tan(A ± B) =
tan(A) ± tan(B)
1 ∓ tan(A) tan(B)
Table 4: Sum and Difference Formulae.
sin2 (θ) =
1 − cos(2θ)
2
cos2 (θ) =
1 + cos(2θ)
2
tan2 (θ) =
1 − cos(2θ)
1 + cos(2θ)
Table 5: Half Angle Formulae.
b
c
a
=
=
.
sin(A)
sin(B)
sin(C)
The Cosine Rule: a2 = b2 + c2 − 2bc cos(A).
The Sine Rule:
Differential Calculus
f (x)
f ′ (x)
c
0
xn
nxn−1
eax
aeax
1
x
a cos(ax)
ln(ax)
sin(ax)
cos(ax)
−a sin(ax)
Comments
Here c is any real number
Here we must have ax > 0
Note the change of sign
Table 6: Some Common Derivatives
f (x + h) − f (x)
.
h→0
h
′
′
′
The sum rule for differentiation: (f + g) (x) = f (x) + g (x).
The multiple rule for differentiation: (cf )′ (x) = cf ′ (x).
Differentiation using first principles: f ′ (x) = lim
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Integral Calculus
Z
f (x)
k
xn
1
x
eax
sin(ax)
cos(ax)
f (x) dx
kx + c
1
xn+1 + c
n+1
ln(x) + c
1 ax
e +c
a
1
− cos(ax) + c
a
1
sin(ax) + c
a
Comments
Here k is any real number
Here we must have n 6= −1
Here we must have x > 0
Note the change of sign
Table 7: Some Common Integrals
Evaluating definite integrals:
Z
b
If F is an antiderivative of f then
f (x) dx = [F (x)]ba = F (b) − F (a).
Z b a
Z b
Z b
The sum rule for integration:
(f + g)(x) dx =
f (x) dx +
g(x) dx.
a Z
a Z
a
b
b
The multiple rule for integration:
(kf )(x) dx = k
f (x) dx.
a
a
Statistics
n
1
1P
xi .
(x1 + x2 + · · · + xn ) =
n
n i=1
Given a list numbers x1 , x2 , . . . , xn in ascending order, then their median is defined to be
x n2 + x n2 +1
if n is even and x n+1 if n is odd.
2
2
n
P
(xi − x)2
.
The variance is given by Var(x) = i=1
n
p
The standard deviation is given by σ = Var(x).
The line of best fit y = mx + c is found using the formulae:
n
n n P
P
P
n
xi yi −
xi
yi
i=1
i=1
i=1
m=
and c = y − mx.
n n 2
P 2
P
n
xi −
xi
The mean is given by x =
i=1
c
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