Transcript Slide 1

Distance between
2 points
m<0
m = undefined
x  x y  y 
Mid   1 2 , 1 2 
2 
 2
Terminology
D  (x2  x1 )2  (y2  y1 )2
Median – midpoint
Bisector – midpoint
Perpendicular – Right Angled
Altitude – right angled
m=0
m>0
m1.m2 = -1
Possible values
for gradient
Straight Line
y = mx + c
Form for finding
line equation
y – b = m(x - a)
(a,b) = point on line
Parallel lines
have same
gradient
For Perpendicular lines
the following is true.
m1.m2 = -1
m = gradient
m
y2  y1
x2  x1
c = y intercept (0,c)
m = tan θ
θ
f(x)
flip in
y-axis
+
Move vertically
up or downs
depending on k
f(x)
f(x)
-
Stretch or
compress
vertically
depending on k
y = f(x) ± k
y = f(-x)
Remember we
can combine
function !!
y = kf(x)
Graphs &
Functions
y = -f(x)
f(x)
flip in
x-axis
y = f(kx)
y = f(x ± k)
f(x)
f(x)
-
+
Move horizontally
left or right
depending on k
Stretch or
compress
horizontally
depending on k
f(x) =
x2
1
g(x) =
x
-4
x
g(f(x))
1
y
y = f(x)
Domain
But y = f(x) is x2 - 4
Restriction
Range
A complex function made
up of 2 or more simpler
functions
Similar to
composite
Area
=
1
g(x) =
x
x
Domain
f(x) =
y = g(x)
x2
x ≠2
- 4 f(g(x))
y2 - 4
Range
x2 - 4 ≠ 0
(x – 2)(x + 2) ≠ 0
Composite
Functions
+
1
x2 - 4
g(f(x)) =
x ≠ -2
1
x
But y = g(x) is
1
x
f(g(x)) =
Rearranging
Restriction x2 ≠ 0
1
x2
2
-4
-4
6
x5  x

5
6

1
x
7
5
6
Format for
Differentiation
/ Integration
5
x
2
 x

2
5
3
xx
x4  x
4
7
1
2
1
4
x x  x
3
4
x
1
3
Surds
n
x x
m
x
2
5
2

3
 x
Indices
m
n
x x  x
m
n
( m n )
1

x
1
3
xm
( mn )

x
xn
Basics before
Integration
Division
Working with
fractions
Adding
1 1 5
 
2 3 6
Subtracting
1 1 1
 
2 3 6
Multiplication
1 3 3
 
2 5 10
1 4

2 5
1 5 5
 
2 4 8
Nature Table
x -1 2 5
f’(x) + 0 -
Leibniz
Notation
dy
 f '( x)
dx
Max
f’(x)=0
Stationary Pts
Max. / Mini Pts
Inflection Pt
Equation of
tangent line
Gradient
at a point
Graphs
f’(x)=0
Differentiation
of Polynomials
1
2
f ( x)  2 x  x
1
2
1
2
1
f '( x)  3x  x
2
1
1
f '( x)  3x 2 
2 x
1
2
Derivative
= gradient
= rate of change
f ( x) 
f ( x)  x  2 x 1
3
2
Straight Line
Theory
f(x) = axn
then f’x) = anxn-1
2
3 4 x5
f ( x) 
9
5
4
2x
3
5
 x4
5
f '( x)  2

3
6 4 x9
Easy to graph
functions & graphs
Completing the square
f(x) = a(x + b)2 + c
Factor Theorem
x = a is a factor of f(x)
if f(a) = 0
f(x) =2x2 + 4x + 3
f(x) =2(x + 1)2 - 2 + 3
f(x) =2(x + 1)2 + 1
-2 1 4 5 2
-2 -4 -2
1 2 1 0
(x+2) is a factor
since no remainder
If finding coefficients
Sim. Equations
Discriminant of a
quadratic is
b2 -4ac
Polynomials
Functions of the type
f(x) = 3x4 + 2x3 + 2x +x + 5
Tangency
b2 -4ac > 0
Real and
distinct roots
b2 -4ac = 0
Equal roots
Degree of a polynomial
= highest power
b2 -4ac < 0
No real
roots
Limit L is equal to
L=
b
(1 - a)
Given three value in a
sequence e.g. U10 , U11 , U12
we can work out
recurrence relation
U11 = aU10 + b
U12 = aU11 + b
Use
Sim. Equations
a = sets limit
b = moves limit
Un = no effect
on limit
Recurrence Relations
next number depends on the
previous number
Un+1 = aUn + b
|a| > 1
Limit exists
when |a| < 1
|a| < 1
a > 1 then growth
a < 1 then decay
+ b = increase
- b = decrease
f(x)
g(x)
Remember to
change sign to + if
area is below axis.
Remember to work
out separately the
area above and below
the x-axis .
b
A= ∫a f(x) - g(x) dx
Finding where curve
and line intersect
f(x)=g(x) gives the
limits a and b
Integration is the
process of finding
the AREA under a
curve and the x-axis
Area between
2 curves
Integration
of Polynomials
2
I 
I   x  2x 1dx
I
5
2
3
2
4
2
x  x C
5
3
x
1
1
2
1
 32

I    2 x  x 2 dx


2
1
2
I 
IF f’(x) = axn
Then I = f(x) = ax

1
n 1
n 1

1
2
dx
x
dx
2
2
 
I  x 
 1
 2 1
1
2
Quadratic Theory
Discriminant
Graph sketching
r  g2  f 2 c
Move the circle from the origin
a units to the right
b units upwards
b2 - 4ac  0
Used for
intersection problems
between circles and lines
NO intersection
b2 - 4ac  0
Centre
(a,b)
x2 y2 2g x2f  yc0
Centre
(-g,-f)
Factorisation
The
Circle
(x a)2 (y b)2  r2
Special case
x2  y2  r2
Centre
(0,0)
2 pts of intersection
b2 - 4ac  0
line is a tangent
Pythagoras Theorem
Rotated
through 360 deg.
Two circles touch
externally if
the distance
Straight line Theory
Two circles touch
internally if
the distance
C1C2  (r2  r1)
C1C2  (r1  r2 )
Distance formula
2
2
CC
1 2  (y2  y1) (x2 x1)
Perpendicular
equation
m1  m2  1
Double Angle Formulae
sin2A = 2sinAcosA
cos2A = 2cos2A - 1
= 1 - 2sin2A
= cos2A – sin2A
Addition Formulae
sin(A ± B) = sinAcosB
cos(A ± B) = cosAcosB
 cosAsinB
sinAsinB
90o
The exact
value of sinx
4
xo
4
2
o
180
Trig Formulae
and Trig equations
S
A
T
C
270o
3cos2x – 5cosx – 2 = 0
Let p = cosx 3p2 – 5p - 2 = 0
sinx = 2 (¼ + √(42 - 12) )
(3p + 1)(p -2) = 0
cosx = 2
p = cosx = 1/3
x = no soln
x = cos-1( 1/3)
sinx = ½ + 2√15)
x = 109.5o and 250.5o
sinx = 2sin(x/2)cos(x/2)
0o
1.
2.
3.

Rearrange into sin =
Find solution in Basic Quads
Remember Multiple solutions
290
S
180o
o
A
C
T

0
30
o
0
Period
360o
-1
sin x
1 Amplitude
-1
cos x

÷180 then X π
30o
6
45o
radians
then x 180 ÷ π
360o
90o
sin
cos
tan
0
1
0
1
2
1
2
3
2
1
2
1
3
3
2
1
1
2
0
1
3
undefined
Trigonometry
sin, cos , tan
Amplitude
Period
0
o
60o
Basic Strategy
for Solving
Trig Equations
0
o
degrees
3
o
270
2
1
Exact Value Table
Complex Graph
3
Basic Graphs
-1
Amplitude
1
Period
0
1
0
2
-1
o 180o
90
Period
tan x
90o
180o
270o
360o
y = 2sin(4x + 45o) + 1
Max. Value =2+1= 3 Period = 360 ÷4 = 90o
Mini. Value = -2+1 -1 Amplitude = 2
 u1   a1   u1  a1 
    

PQ  a   u2    a2    u2  a2 
u   a  u  a 
3
 3  3  3
Addition
Scalar product
 ku1 


k a   ku2 
 ku 
 3
Basic
properties
Q
B
P
A
a  b  a1b1  a2b2  a3b3  0
2 vectors
perpendicular if
a  b  a1b1  a2b2  a3b3
Component form
a  a1  a2  a3
Magnitude
PQ
same for
subtraction
a
Vectors are equal if
they have the same
magnitude &
direction
a  b  a b cos
scalar product
Vector Theory
Magnitude &
Direction
Notation
 u1 
 
Component form PQ   u2 
u 
 3
Unit vector form
a  a1i  a2 j  a3k
b
Tail to tail
a  (b  c )  a  b  a  c
a b
cos  
a b
a b  b a
Vector Theory
Magnitude &
Direction
Section formula
B
B
A
m
Points A, B and C are said to
be Collinear if
AB  k BC
B is a point in common.
a
Angle between
two vectors
properties
C
θ
A
b
n
m n
C
n
c
b
a
a
m
m n
O
c
d
d
d
(inside) n   (outside the bracket)  (inside the bracket)
dx
dx
dx

d 2
4
x  2x
dx
 
d n
x
dx
3
2
2
4 
3

3 x  2x  2x  8x 
Harder functions
Use Chain Rule
Trig.
d
sin( a x)
dx
d
cos ( a x)
dx
Rules of
Indices
n 1
n x
Polynomial
s
Differentiations
Real life
d
 (distance)
dx
= velocity
d
= (velocity)
dx
= acceleration
a sinx ( a x)
Factorisation
Graphs
dy
0
dx
Meaning
Stationary Pts
Mini / Max Pts
Inflection Pts
Rate of change
of a function.
Gradient at
a point.
a cos( a x)
Tangent
equation
Straight
line Theory
y
y = logax
(a,1)
To undo log take exponential
loga1 = 0
logaa = 1
(1,0) x
log A + log B = log AB
To undo exponential take log
Basic
log graph
y = axb
Can be transformed into
a graph of the form
Can be transformed into
a graph of the form
log y = x log b + log a
x
a0 = 1
a1 = a
Basic
exponential
graph
abx
(0,C)
(1,a)
Logs &
Exponentials
Basic log rules
log y
(0,1)
x
log A - log B = log A
B
n
log (A) = n log A
y=
y
y = ax
Y = mX + C
Y = (log b) X + C
C = log a m = log b
log y = b log x + log a
Y = mX + C
Y = bX + C
C = log a
m=b
log y
(0,C)
log x
f(x) = a sinx + b cosx
Compare coefficients
compare to required
trigonometric identities
a = k cos β
f(x) = k sin(x + β)
= k sinx cos β + k cosx sin β
b = k sin β
Square and add then
square root gives
k  a2  b2
Process
example
Divide and inverse tan gives
  tan 1
Wave Function
b
a
a and b values
decide which
quadrant
transforms
f(x)= a sinx + b cosx
Write out required form
into the form
f ( x)  k sin( x   )
f ( x)  k sin( x   )
OR
f ( x)  k cos( x   )
Related topic
Solving trig equations