Download Maths 2 Unit Notes by Tim

Tests for special quadrilaterals:
Co-ordinate Geometry
Straight line
 Gradient form: y = m + b
 General form: A + By + C = 0

Distance: d = (1-2)2 + (y1-y2)2

Midpoint: mp = (1 + 2 , y1 + y2 )
2

Gradient: m = y2-y1
2-1

Perpendicular distance
P=
2
A1 + By1 + C
A2 + B2

m = tan
Geometrical Properties
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
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
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Complementary angles add to 90
Supplementary angle add to 180
Vertically opposite angles are equal
Angles at a point add to 360
Angle sum of a triangle is 180
The exterior angle of a triangle is
equal to the sum of the opposite
interior angles
An isosceles triangle has equal base
angles
Equilateral triangles have all angles
60
Alternate angles on parallel lines are
equal
Corresponding angles on parallel
lines are equal
Co-interior angles between parallel
lines are supplementary
The angle sum of a polygon is
(n-2)x180
The sum of the exterior angles of
any polygon is equal to 360
The angle sum of a quadrilateral is
360
Parallelograms:
 Two opposite sides equal and
parallel or
 Opposite sides are equal or
 Opposite angles are equal or
 Diagonals bisect each other
Rhombus:
 All sides equal or
 Diagonals bisect each other at right
angles
Rectangle:
 All angles are right angles or
 Parallelogram with equal diagonals
Square:
 All sides equal and one angle right
or
 All angles right and two adjacent
sides equal.
Tests for congruent triangles




SSS
SAS
AAS
RHS
Tests for similar triangles



AA
Corresponding sides proportional
(SSS)
Two sides are proportional and
included angles are equal (SAS)
Applications of Differentiation



First derivative dy/d
- Stationary point when equals 0
- Curve increasing>0
- Curve decreasing<0
-Max turning point if second
derivative negative
-Minimum turning point if second
derivative positive
Second derivative d2y/d2
- Point of inflexion when equals 0
-Concave up when >0
-Concave down when <0
Horizontal point of inflexion if both
first and second derivative equals
zero.
1
Integration
n d =
1
n+1
n+1 + c
Logarithmic Functions
loge =
 Area between curve and axis
A = ba f() d
logef() =
 Volume of revolution
V =  ba [ f() ]2 d

d = loge = c

 Area between two curves
A =  top curve -  bottom curve
 Volume between two curves
A =   (top curve)2 – (bottom curve)2
Approximating integrals
d = logef() + c
Log laws
 logee2 = 2logee = 2
 logek = logek + loge
 loge = loge ½ = ½ loge
 loge/k = loge - logek
Simpson’s Rule
A = h [ f(a) + 4 x f((a+b)/2) + f(b) ]
3
2.5
2
Trapezoidal Rule
A = ½ h [ f(a) + f(b) ]
1.5
1
Logarithmic and Exponential
Functions
0.5
Exponential functions
0
e = e
0
4
6
8
10
Trigonometric Functions
ec = cec

Arc length
l = rΘ
area of sector
A = ½ r2Θ
y = sin

ef() = f’() ef(x)

 e d = e + k
 ec d =
2
1.5
1
ec + k
0.5
0
0
1
2
3
4
5
6
7
-0.5
-1
8
-1.5
7
Period = 2
Amplitude = 1
 y = cos
6
5
4
3
1.5
2
1
1
0.5
0
-4
-3
-2
-1
0
1
2
3
0
0
1
2
3
4
5
6
7
-0.5
-1
-1.5
2

Period = 2
Amplitude = 1
 y = tan
Decay y = Ae-k
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5

cos = -sin
Exponential Growth
If the rate of change is proportional
to P, ie dP/dt = kP
Then P = Poekt
 Exponential Decay
If dP/dt = -kP
Then P = Poe-kt
Where Po is the initial value of P
k is the constant of proportionality
P is the amount of quantity present at
time t
tan = sec2
Series and Applications
Period = 

=1

Derivatives
sin = cos
 Integrals
 sina d = -1/a cos + c
 cosa d = 1/a sin + c
 sec2a d = 1/a tan + c
Rates of Change
The rate of change of some physical
quantity Q is defined as dQ/dt
 Given Q = f(t) then rate of change,
dQ/dt = f ‘(t)
 Given the rate of change, R = dQ/dt,
then Q =  R dt
Kinematics
Arithmetic Series


Tn = a + (n-1)d
Sn = n/2 (a+l) or
Sn = n/2 [2a + (n-1)d]
Geometric Series


Tn = arn-1
Sn =
if r > 1

Sn =
if r < 1

S∞ =
Superannuation
Exponential Growth and Decay
 If e = a, then  = logea
A1 = P
Growth y = aek
<1
A=P
Displacement = 
Velocity = v = d/dt
Acceleration = a = dv/dt = d2/dt2
 =  v dt
v =  a dt

where
Compound Interest
If $P is invested at the beginning of
each year in a superannuation fund
earning interest at r% pa, the investment
after n years will amount to T
A2 = P
3
And so on, so that investment = A1 +
A2…
=P
+P
…
forms a geometric series with
a=P
n = number of years
and r =
Time payments
A person borrows $P at r% per term,
where the interest is compounded per
term on the amount owing. If they pay
off the loan in equal term instalments
over n terms, their equal term instalment
is M, where
M=
Deriving the equation:
An = P (rate)n – M (1 + rate + rate2…)
After fully paid An = 0
Rearrange to find M, using (1 + rate +
rate2…) as a geometric series.
Probability
Probability of an event occurring =
The probability of two events A and B
occurring is given by:
P(AB) = P(A) x P(B)
Sum and Difference of Two Cubes
X 3+ Y 3 = (x + y)(X 2 - XY + Y 2 )
X 3 -Y 3 = (X - Y)(X 2 + XY + Y 2 )
Parabolas
(-b)2 = 4a(y-c)
where (b,c) is the vertex
a is the focal length
4