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1.1 Lines
Increments
If a particle moves from the point (x1,y1) to the point
(x2,y2), the increments in its coordinates are
Δx  x2  x1 and Δy  y2  y1
1.1 Lines
Slope
Let P1= (x1,y1) and P2= (x2,y2) be points on a nonvertical
line L. The slope of L is
P2(x2,y2)
y2  y1
rise Δy
m


run
Δx x2  x1
Δy
P1(x1,y1)
Δx
Q(x2,y1)
1.1 Lines
Theorem: If two lines are parallel, then they have the
same slope and if they have the same slope, then
they are parallel.
Proof: If L1 || L2, then θ1= θ2
and m1= m2. Conversely, if
L1
L2
m1 = m2, then θ1= θ2 and
slope m1
slope m2
m1
m2
L1 || L2.
θ
θ
1
1
2
1
1.1 Lines
Theorem: If two non vertical lines L1and L2 are perpendicular,
then their slopes satisfy m1m2 = -1 and conversely.
L
Proof: Δ ADC ~ ΔCDB
L1
C
m1 = tan θ1 = a/h
m2 = tan θ2 = -h/a
2
Slope m1
θ1
A
θ1
Slope m2
θ2
h
D
a
B
so m1m2 =(a/h)(-h/a) = -1
1.1 Lines
Equations of lines
• Point-Slope Formula y = m(x – x1) + y1
• Slope-Intercept form y = mx + b
• Standard form
Ax + By = C
• y = a Horizontal line slope of zero
• x =a Vertical line
no slope
1.1 Lines
Regression Analysis
1. Plot the data
2. Find the regression equation y = mx + b
3. Superimpose the graph on the data points.
4. Use the regression equation to predict y-values.
1.1 Lines
Coordinate Proofs
1. State given and prove.
2. Draw a picture.
3. Label coordinates, use (0,0) if possible.
4. Fill in missing coordinates.
5. Use algebra to prove
• parallel/perpendicular-slope
• equidistant-distance formula
• bisect-midpoint
1.1 Lines
Prove the midpoint of the hypotenuse
of a right triangle is equidistant
from the three vertices.
Given: ΔBAC is a right triangle
Prove: AM = BM = CM
2
2
2
2
2
2
b2 a2
b
 a

AM    0     0  

2
2
4
4

 

b2 a2
b
 a

BM    0     a  

4 4
2
 2

b 
a
b2 a2

CM   b     0   

2 
2
4 4

B(0,a)
M(b/2,a/2)
A(0,0)
C(b,0)
Since AM = BM = CM, the
midpoint of the hypotenuse
of a right triangle is
equidistant from the three
vertices
1.2 Functions and Graphs
Function
A function from a set D to a set R is a rule that
assigns a unique element R to each element D.
y = f(x) y is a function of x
1.2 Functions and Graphs
Domain All possible x values
Range
All possible y values
1.2 Functions and Graphs
  x  
(, )

0
xa
( a,  )
a xb
( a, b)
open
a xb
[ a, b]
closed
a xb
( a, b]
half opened
a xb
[a, b) half opened
a
a
a
a
a
b
b
b
b
1.2 Functions and Graphs
•y = mx
•Domain (-∞ , ∞)
•Range (-∞ , ∞)
1.2 Functions and Graphs
•y = x2
•Domain (-∞ , ∞)
•Range [0, ∞)
1.2 Functions and Graphs
•y = x3
•Domain (-∞ , ∞)
•Range (-∞ , ∞)
1.2 Functions and Graphs
•y = 1/x
•Domain x ≠ 0
•Range y ≠ 0
1.2 Functions and Graphs
y x
•Domain [0, ∞)
•Range [0, ∞)
1.2 Functions and Graphs
Function
Domain
Range
y=x
( ,  )
( ,  )
y = x2
( ,  )
[0, )
y = |x|
( ,  )
[0, )
y  9  x2
[-3,3]
[0,3]
y  x2
[-2,  )
[0, )
1.2 Functions and Graphs
Definitions Even Function, Odd Function
A function y = f(x) is an
even function of x if f(-x) = f(x)
odd function of x if f(-x) = -f(x)
for every x in the function’s domain.
Even Function – symmetrical about the y-axis.
Odd Function - symmetrical about the origin.
1.2 Functions and Graphs
Odd Function
symmetrical about the origin.
(x,y)
(-x,-y)
Even Function
symmetrical about the y-axis.
(-x,y)
(x,y)
1.2 Functions and Graphs
Transformations
h(x) = af(x)
vertical stretch or shrink
h(x) = f(ax)
horizontal stretch or shrink
h(x) = f(x) + k
vertical shift
h(x) = f(x + h)
horizontal shift
h(x) = -f(x)
reflection in the x-axis
h(x) = f(-x)
reflection in the y-axis
1.2 Functions and Graphs
Piece Functions
x2
x  1
f ( x)  

2 x  1 x  1
Domain (-,)
Range [-3, )
1.2 Functions and Graphs
Piece Functions
| x |
 2
f ( x)   x
x 1



 2  x  1

x 1

x  2
Domain (-,)
Range [0, )
1.2 Functions and Graphs
Composite Functions
f(g(x))
f(x) = x2, g(x) = 3x - 1
Find:
1. f(g(2))
2. g(f(-1))
3. g(f(x))
4. f(g(x))
25
2
3x2 – 1
(3x – 1)2 = 9x2 – 6x + 1
•
•
•
•
1.3 Exponential Functions
Definition Exponential Function
Let a be a positive real number other than 1,
the function f(x) = ax is the exponential
function with base a.
1.3 Exponential Functions
Rules For Exponents
If a > 0 and b > 0, the following hold true for all real
numbers x and y.
1. a a  a
x
y
x y
x
a
2. y  a x  y
a
3. a

x y
a
xy
4. a b  (ab)
x
x
x
a
a
5.    x
b
b
6. a 0  1
x
x
1
7. a  x
a
-x
p
q
8. a  a p
q
1.3 Exponential Functions
Use the rules for exponents to
solve for x.
4x = 128
(2)2x = 27
2x = 7
x = 7/2
•
•
•
•
2x = 1/32
2x = 2-5
x = -5
•
•
•
1.3 Exponential Functions
x
27
•
x3/2y1/3
•
•
•
-x+1
9
=
3
x
2
-x+1
(3 ) = (3 )
33x = 3-2x+2
3x = -2x+ 2
5x = 2
x = 2/5
(x3y2/3)1/2
•
•
•
•
49
9
1/9
5
4
5
81
32
2
1
1/8
2
80/9
8
9/4
36
1/25
1/49
4
5
1.3 Exponential Functions
Properties of f (x) = ax
Domain: (-∞, ∞)
Range: (0, ∞)
Increasing for: a > 1
Decreasing for: 0 < a < 1
Point Shared On All Graphs: (0, 1)
Asymptote: y = 0
1.3 Exponential Functions
Natural Exponential Function where e
is the natural base and e  2.718…
f ( x)  e
x
x
e 
 1
1  
lim
x
x  
x
1.3 Exponential Functions
Function
f(x) = 2x
h(x) = (0.5)x
g(x) = ex
Domain
(-∞, ∞)
(0, ∞)
(-∞, ∞)
(0, ∞)
(-∞, ∞)
(0, ∞)
Inc.
Dec.
Inc.
Range
Increasing or
Decreasing
Point Shared
On All Graphs
(0, 1)
1.3 Exponential Functions
Use translation of functions to graph the following.
Determine the domain and range of each.
1. f(x) = -5(x + 2) – 3
2. g(x) = (1/3)(x – 1) + 2
1.3 Exponential Functions
Definitions Exponential Growth, Exponential Decay
The function y = k ax, k > 0 is a model for exponential
growth if a > 1, and a model for exponential decay
if 0 < a < 1.
y  yOb
t
h
y
yo
b
t
h
new amount
original amount
base
time
half life
1.3 Exponential Functions
An isotope of sodium, 24Na, has a half-life of 15
hours. A sample of this isotope has mass 2 g.
(a) Find the amount remaining after t hours.
(b) Find the amount remaining after 60 hours.
(c) Estimate the amount remaining after 4 days.
(d) Use a graph to estimate the time required for the
mass to be reduced to 0.1 g.
1.3 Exponential Functions
An isotope of sodium, Na, has a half-life of 15
hours. A sample of this isotope has mass 2 g.
(a) Find the amount remaining after t hours.
(b) Find the amount remaining after 60 hours.
• a. y = yobt/h
•
y = 2 (1/2)(t/15)
• b. y = yobt/h
•
y = 2 (1/2)(60/15)
•
y = 2(1/2)4
•
y = .125 g
1.3 Exponential Functions
An isotope of sodium, 24Na, has a half-life of 15
hours. A sample of this isotope has mass 2 g.
(c.) Estimate the amount remaining after 4 days.
(d.) Use a graph to estimate the time required for the
mass to be reduced to 0.1 g.
d.
• c. y = yobt/h
•
y = 2 (1/2)(96/15)
•
y = 2(1/2)6.4
•
y = .023 g
1.3 Exponential Functions
A bacteria double every three days. There are
50 bacteria initially present
(a) Find the amount after 2 weeks.
(b) When will there be 3000 bacteria?
• a. y = yobt/h
•
y = 50 (2)(14/3)
•
y = 1269 bacteria
1.3 Exponential Functions
A bacteria double every three days. There are
50 bacteria initially present
When will there be 3000 bacteria?
• b. y = yobt/h
•
3000 = 50 (2)(t/3)
•
60 = 2t/3
•
1.4 Parametric Equations
Equations where x and y are functions of a
third variable, such as t. That is,
x = f(t) and y = g(t).
The graph of parametric equations are called
parametric curves and are defined by (x, y)
= (f(t), g(t)).
1.4 Parametric Equations
Equations defined in terms of x and y. These
may or may not be functions. Some
examples include:
x2 + y2 = 4
y = x2 + 3x + 2
1.4 Parametric Equations
Sketch the graph of the parametric equation for t
in the interval [0,3]
x  1  2t
y  3t
t
0
1
2
3
x
1
-1
-3
-5
y
0
3
6
9
1.4 Parametric Equations
Eliminate the parameter t from the curve
x  1  2t
y  3t
2t 1  x
1 x
t
2
1 x 
y  3

 2 
3
3
y  x
2
2
1.4 Parametric Equations
If we let t = the angle, then:
Circle:
x  cos t
y  sin t
0  t  2
t
Since:
sin 2 t  cos 2 t  1
y 2  x2  1
We could identify the
parametric equations as a
circle.
x2  y 2  1
1.4 Parametric Equations
Ellipse:
x  3cos t
y  4sin t
x
 cos t
3
y
 sin t
4
2
2
 x  y
2
2


cos
t

sin
t
   
3  4
2
2
 x  y
    1
3  4
This is the equation of
an ellipse.
1.4 Parametric Equations
The path of a particle in two-dimensional space can
be modeled by the parametric equations x = 2 + cos t
and y = 3 + sin t. Sketch a graph of the path of the
particle for 0  t  2.
1.4 Parametric Equations
How is t
represented
on this
graph?
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
1.4 Parametric Equations
t=
t=0
1.4 Parametric Equations
Graphing calculators and other mathematical
software can plot parametric equations much more
efficiently then we can. Put your graphing
calculator and plot the following equations. In
what direction is t increasing?
(a) x = t2, y = t3
(b)
x  ln t , y  t ; t  1
(c) x = sec θ, y = tan θ; -/2 < θ < /2
1.4 Parametric Equations
Parametric equations can easily be converted to Cartesian
equations by solving one of the equations for t and
substituting the result into the other equation.
(a) x =
2
t,
y=
t  x, y 
3
t
 x  x
3
3
2
1.4 Parametric Equations
(b) x  ln t , y 
t for t  1
y  t  t  y2
2
x  ln t  ln y
x  ln y  y  e  y   e  y  e ; x  0
2
2
x
x
x
1.4 Parametric Equations
(c) x = sec t, y = tan t
where -/2 < t < /2
Hint: sec2 θ – tan2 θ = 1
x  sec t, y  tan t
2
2
2
2
x  y  sec t- tan t  1
2
2
x  y 1
2
2
2
2
1.4 Parametric Equations
Find a parametrization for the line segment with endpoints
(2,1) and (-4,5).
x = 2 + at y = 1 + bt
when t = 1, a = -6
when t = 1, b = 4
x = 2 – 6t and y = 1 + 4t
Cartesian Equation
m = (5 – 1)/(-4 – 2) = -2/3
y = mx + b
1 = (-2/3)(2) + b
b = 7/3
y = (-2/3)x + 7/3
1.5 Functions and Logarithms
A function is one-to-one if two domain values do
not have the same range value.
Algebraically, a function is one-to-one if
f (x1) ≠ f (x2) for all x1 ≠ x2.
Graphically, a function is one-to-one if its graph
passes the horizontal line test. That is, if any
horizontal line drawn through the graph of a
function crosses more than once, it is not one-toone.
1.5 Functions and Logarithms
To be one-to-one, a function must pass the horizontal line test as
well as5 the vertical line test. 5
5
4
4
4
3
3
3
2
2
2
1
1
1
-5 -4 -3 -2 -1 0
-1
1
2
3
4
5
-5 -4 -3 -2 -1 0
-1
1
2
3
4
5
-5 -4 -3 -2 -1 0
-1
1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
1 3
y x
2
1 2
y x
2
x  y2
one-to-one
not one-to-one
2
3
4
5
not a function
(also not one-to-one)
1.5 Functions and Logarithms
Determine if the following functions are one-to-one.
(a) f (x) = 1 + 3x – 2x 4
(b) g(x) = cos x + 3x 2
e e
(c) h( x) 
2
x
(d) f ( x) 
x
5 x
1.5 Functions and Logarithms
The inverse of a one-to-one function is obtained
by exchanging the domain and range of the
function. The inverse of a one-to-one function f is
denoted with f -1.
Domain of f = Range of f -1 To prove functions are
Range of f = Domain of f -1 inverses show that
f −1(x) = y <=> f (y) = x
f(f-1(x)) = f-1(f(x)) = x
1.5 Functions and Logarithms
To obtain the formula for the inverse of a
function, do the following:
1. Let f (x) = y.
2. Exchange y and x.
3. Solve for y.
4. Let y = f −1(x).
1.5 Functions and Logarithms
Given an x value, we can find a y value.
Inverse functions:
1
f  x  x 1
2
5
1
y  x 1
2
Switch x and y:
1
x  y 1
2
3
2
1
-5 -4 -3 -2 -1 0
-1
-3
-5
y  2x  2
1
2
3
4
5
-2
-4
Solve for y:
1
x 1  y
2
4
Inverse functions
are reflections about
y = x.
f 1  x   2 x  2
1.5 Functions and Logarithms
1
f  x  x 1
2
f
1
 x   2x  2
Prove f(x) and f-1(x) are inverses.
1
f ( f ( x))  f (2 x  2)  (2 x  2)  1  x  1  1  x
2
1
1

1

f ( f ( x))  f  x  1  2 x  1  2  x  2  2  x
2

2

1
1
1.5 Functions and Logarithms
Determine the formula for the inverse of the
following one-to-one functions.
3x  1
(a) h( x) 
x2
(b) f ( x)  2 x 3  3
(c)
g ( x)  3
x
1.5 Functions and Logarithms
You can obtain the graph of the inverse of a oneto-one function by reflecting the graph of the
original function through the line y = x.
1.5 Functions and Logarithms
1.5 Functions and Logarithms
1.5 Functions and Logarithms
Sketch a graph of f (x) = 2x and sketch a graph of
its inverse. What is the domain and range of the
inverse of f.
Domain: (0, ∞)
Range: (-∞, ∞)
1.5 Functions and Logarithms
The inverse of an exponential function is called a
logarithmic function.
Definition: x = a y if and only if y = log a x
1.5 Functions and Logarithms
The function f (x) = log a x is called a logarithmic
function.
Domain: (0, ∞)
Range: (-∞, ∞)
Asymptote: x = 0
Increasing for a > 1
Decreasing for 0 < a < 1
Common Point: (1, 0)
1.5 Functions and Logarithms
Find the inverse of g(x) = 3x.
Definition: x = a y if and only if y = log a x
1
g ( x)  log 3 x
1.5 Functions and Logarithms
1.
2.
3.
4.
5.
log a (ax) = x for all x  
alog ax = x for all x > 0
log a (xy) = log a x + log a y
log a (x/y) = log a x – log a y
log a xn = n log a x
Common Logarithm: log 10 x = log x
Natural Logarithm: log e x = ln x
All the above properties hold.
1.5 Functions and Logarithms
The natural and common logarithms can be
found on your calculator. Logarithms of other
bases are not. You need the change of base
formula.
log x
log a x 
b
log b a
where b is any other appropriate base.
1.5 Functions and Logarithms
$1000 is invested at 5.25 % interest compounded annually.
How long will it take to reach $2500?
1000 1.0525   2500
t
1.0525
t
 2.5
ln 1.0525   ln 2.5
We use logs when we have an
unknown exponent.
t
t ln 1.0525  ln 2.5
17.9 years
In real life you would have to
ln 2.5
t
 17.9 wait 18 years.
ln 1.0525
1.5 Functions and Logarithms
Example 7:
Indonesian Oil Production (million barrels per year):
1960 20.56
1970 42.10
1990 70.10
Use the natural logarithm
regression equation to estimate
oil production in 1982 and 2000.
How do we know that a logarithmic equation is appropriate?
In real life, we would need more points or past experience.
1.5 Functions and Logarithms
1.
2.
3.
4.
5.
Determine the exact value of log 8 2.
Determine the exact value of ln e 2.3.
Evaluate log 7.3 5 to four decimal places.
Write as a single logarithm: ln x + 2ln y – 3ln z.
Solve 2x + 5 = 3 for x.
1.6 Trigonometric Functions
A
The Radian measure of angle ACB
at the center of the unit circle equals
the length of the arc that ACB cuts
from the unit circle.
s
θ  but for the unit circle, r  1
r
so θ  s
s
r
θ
C
B
1.6 Trigonometric Functions
y
sine : sin θ 
r
y
tangent : tan θ 
x
r
cosecant : csc θ 
y
x
cosine : cos θ 
r
x
cotangent : cot θ 
y
r
secant : sec θ 
x
terminal ray
y
P(x,y)
r
θ
x
y
x
initial ray
1.6 Trigonometric Functions
105
90
75
120
(2,/4)
60
135
45
150
(5,5 /6)
30
165
15
180
0
195
(4, 11/6)
345
210
330
225
315
240
300
255
270
285
(-4, /2)
1.6 Trigonometric Functions
Let a point P have rectangular coordinates (x,y)
and polar coordinates (r,). Then
x  r cos 
y  r sin 
x y r
2
2
2
y
tan   x  0
x
1.6 Trigonometric Functions
 1 3 
 , 

 2 2  
2 2 



, 

 2 2 
(0,1)
 3 1 

, 


 2 2 
1 3 

 , 

2 2   2 2 

 , 

 2 2 
 3 1 

 , 

 2 2 
2
1
45°
1
30°
3
2
(-1,0)
60°
45°
(1,0)
1
 3 1 

,- 


 2 2 
 2
2 

,- 


2 
 2
 3 1 

 ,- 

 2 2 
1 3 
 1 3


,



 , 

 2 2  (0,-1) 2 2 
 2
2 

 ,- 

2 
 2
S
A
T
C
1.6 Trigonometric Functions
1.6 Trigonometric Functions
Even and Odd Trig Functions:
“Even” functions behave like polynomials with even exponents,
in that when you change the sign of x, the y value doesn’t
change.
Cosine is an even function because:
cos     cos  
Secant is also an even function, because it is the reciprocal of
cosine.
Even functions are symmetric about the y - axis.
1.6 Trigonometric Functions
Even and Odd Trig Functions:
“Odd” functions behave like polynomials with odd exponents,
in that when you change the sign of x, the sign of the y value
also changes.
Sine is an odd function because:
sin      sin  
Cosecant, tangent and cotangent are also odd, because their
formulas contain the sine function.
Odd functions have origin symmetry.
1.6 Trigonometric Functions
Definition Periodic Function, Period
A function f(x) is periodic if there is a positive
number p such that f(x + p) = f(x)
for every value of x. The smallest
such value of p is the period of p.
1.6 Trigonometric Functions
Vertical stretch or shrink;
reflection about x-axis
a  1 is a stretch.
Vertical shift
Positive d moves up.
y  a f b  x  c   d
Horizontal stretch or shrink;
reflection about y-axis
b  1 is a shrink.
Horizontal shift
Positive c moves left.
1.6 Trigonometric Functions
 2

f  x   A sin   x  C    D
B

Vertical shift
A is the
amplitude.
B is the period.
Horizontal shift
B
4
A
3
C
2
D
1
-1
0
-1
 2

y  1.5sin   x  1   2
 4

1
2
x
3
4
5
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
1.6 Trigonometric Functions
Let  be the acute angle of a right triangle with sin  = 3/5.
Find the exact values of the other five trig functions.
Show all your work.
y 3
r 5

csc   
r 5
y 3
x 4
r 5
cos   
sec   
r 5
x 4
y 3
x 4
tan   
cot   
x 4
y 3
sin  
5
3

4
1.6 Trigonometric Functions
If sec  2 3 / 3 and sin   1/ 2
find the exact value of cot 
3
x
3
cot   
 3
y 1

2
-1
1.6 Trigonometric Functions
Find the amplitude, period, and frequency of the
simple harmonic motion.
3 t
y
4
sin
2
Amplitude ¾
2 2
Period

4

b
2
Frequency ¼
1.6 Trigonometric Functions
Find the exact values without using a calculator:
(a) tan (11/6)
3

3
(b) sec(-3/4)
 2
(c) cot (-5/3)
3
3
1.6 Trigonometric Functions
Given that tan  = 3/5,  in quadrant III,
and cos  = -1/2,  in quadrant II, Find
(a) cos( - )
(b) sin 2
-5

-3
34
(a) cos  cos  + sin  sin 
 5 1  3
3 5  3 3 5 34  3 102





68
34 2
34 2
2 34
(b) sin 2
2
sin 2 = 2 sin  cos 
 3  5 30 15



34 34 34 17
2
3

-1
1.6 Trigonometric Functions
Verify the identities. Show all your work.
(a)
cos x  1
cot x  csc x 
sin x
1  tan 4 x
2

1

tan
x
(b) sec 2 x
cos x
1
cos x  1


sin x sin x
sin x
(1  tan 2 x)(1  tan 2 x)
2

1

tan
x
2
1  tan x
(c) sin(π + x) = -sin x
sin  cos x + sin x cos 
0 cos x + sin x (-1) = -sin x
1.6 Trigonometric Functions
Find the exact values without using a calculator.
(a)
 1
cos 1   
 2
120º 240º
(b)

3

tan  

 3 
(c) sec-1 (2)
150º 330º
60º 300º
1
1.6 Trigonometric Functions
Find the exact values without a calculator.
(a) sin  tan 1 3 
5


1 4 
cos
2
sin
(b)

5 

34
5
3

5
3
3 34

34
34
(c) tan(sec-1x)
x
4

3
cos 2 =cos2 - sin2
9 16
7


25 25
25
x2 1

1
x2 1
1.6 Trigonometric Functions
Solve each equation for exact solutions in the interval [0,2).
(a) cos2 x – 1 = 0
cos2 x = +1
cos x = 1 or cos x = -1
x = 0, x = 
(c) sin2x = 0
2sin x cos x = 0
sin x = 0 or cos x = 0
x = 0, /2, ,3/2
(b) 2 cos2 x + 1 = -3 cos x
2 cos2 x + 3 cos x+ 1 = 0
(2cos x + 1)(cos x + 1) = 0
x = 2/3, 4/3 x = 
1.6 Trigonometric Functions
Solve each equation for exact solutions in the interval [0,2).
(a) tan x sin x – sin x = 0
(b) 2cos x sin x - cos x = 0
sin x(tan x – 1) = 0
cos x(2sin x – 1) = 0
sin x= 0 or tan x – 1 = 0
cos x = 0 or 2 sin x – 1 = 0
x = 0,  x = /4, 5/4
x = /2, 3/2 or sin x = ½
x = /6, 5/6
(c) tan2 x = 3
tan x   3
x = /3, 2/3 4/3 5/3


1
0
1
0
1/2
3 /2
3 /3
2
2 3 /3
3
1
2 /2
1
2 /2
2
2
3 /2 1/2  3 

3 /3
2 3 /3  2
0
   1  
 1  0


2 3 /3 2  3 /3
 3 /2 1/2

 3 
 2 /2
1

 2 /2 
 1  2   2

1/2 
 3 /3  3
 3 /2  3 /3  2 2


 0  1  0    1
1/2  3 /2  3 /3  2 
2 3 /3
3


1
1
  2 /2    2  2
   2 /2
 3 /3 2
3 /3
1/2  3 2


 3 /2 
0


 1  0   1  

  3 /2 1/2  3 2 3 /3  2  3 /3

 2 
2
1

2
/2
2
/2
1
 




1/2  3 /2 


 3 /3 2 2 3 / 3  3
   1

 0  1  0

