Download MAT120

30.03.2016
Subjects
Gebze Technical University
Department of Architecture
Week
MAT120
Asst. Prof. Ferhat PAKDAMAR
(Civil Engineer)
M Blok - M106
[email protected]
Spring – 2014/2015
Week 3-4
Numbers
Subjects
1
2
3
10.02.2016
17.02.2016
24.02.2016
4
03.03.2016
5
6
7
8
9
10
11
12
13
14
15
16
10.03.2016
17.03.2016
24.03.2016
30.03.2016
06.04.2016
13.04.2016
20.04.2016
27.04.2016
04.05.2016
11.05.2016
Introduction
Set Theory and Fuzzy Logic.
Technical Tour
Real Numbers, Complex numbers,
Coordinate Systems.
Functions, Linear equations
Matrices
Matrice operations
MIDTERM EXAM
Introduction to Statistics.
Term Paper presentations
Limit. Derivatives, Basic derivative rules
Integration by parts,
Area and volume Integrals
Introduction to Numeric Analysis
Review
FINAL EXAM
Methods
Term Paper
MT
Dead line for TP
FINAL
A number
is a mathematical object used
to count, measure and label.
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Numbers
Numbers
http://www.mathsisfun.com/sets/number-types.html
Numbers
Numbers
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Numbers
Numbers
Numbers
Numbers
Name
One
Ten
Hundred
Thousand
Ten Thousand (Myriad)
Hundred Thousand
Million
Billion (Milliard)
Trillion (Billion)
Quadrillion (Billiard)
Quintillion (Trillion)
Sextillion (Trilliard)
Septillion (Quadrillion)
Octillion (Quadrilliard)
Nonillion (Quintillion)
Decillion (Quintilliard)
Undecillion (Sextillion)
Duodecillion (Sextilliard)
Tredecillion (Septillion)
Quattordecillion (Septilliard)
Quindecillion (Octillion)
Sexdecillion (Octilliard)
Sepdecillion (Nonillion)
...
Googol
Power
0
1
2
3
4
5
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
...
100
Number
1
10
100
1,000
10,000
100,000
1,000,000
1,000,000,000
1,000,000,000,000
1,000,000,000,000,000
1,000,000,000,000,000,000
1,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
SI symbol
(none)
da(D)
h(H)
k(K)
SI prefix
(none)
deca
hecto
kilo
M
G
T
P
E
Z
Y
(X)
(W)
(V)
(U)
(S)
(R)
(Q)
(O)
mega
giga
tera
peta
exa
zetta
yotta
(xona)
(wecta)
(vinka)
(untra)
(sampa)
(rosa)
(quoda)
(oba)
...
...
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
...
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
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Binary and Decimal System Conventions
Romen Numbers
MDCCXCVIII
MMMDCCCLXXXVIII
MCMLXXVIII
MMXV
Romen Numbers
Some important numbers and series
Archimed’s Constant

3.141592653589793238462643383279502884197
16939937510582097494459230781640628620899
86280348253421170679821480865132823066470
93844609550582231725359408128481117450284
10270193852110555964462294895493038196442
88109756659334461284756482337867831652712
01909145648566923460348610454326648213393
607260249141273724587…
March 14
03.14
Pi day 
30.03.2016
Some important numbers and series
Archimed’s
Constant

March 14
03.14
Pi day 
3.1415…
Some important numbers and series
NOTLAR İNGİLİZCE VERİLECEK
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
Fibonacci Sayı Dizisinin Görüldüğü ve Kullanıldığı Yerler:
1) Ayçiçeği: Ayçiçeği’nin merkezinden dışarıya doğru sağdan sola ve soldan sağa doğru taneler sayıldığında çıkan
sayılar Fibonacci Dizisinin ardışık terimleridir.
2) Papatya Çiçeği: Papatya Çiçeğinde de ayçiçeğinde olduğu gibi bir Fibonacci Dizisi mevcuttur.
3) Fibonacci Dizisinin Fark Dizisi: Fibonacci Dizisindeki ardışık terimlerin farkıyla oluşan dizi de Fibonacci Dizisidir.
4) Ömer Hayyam veya Pascal veya Binom Üçgeni: Ömer Hayyam üçgenindeki tüm katsayılar veya terimler yazılıp
çapraz toplamları alındığında Fibonacci Dizisi ortaya çıkar.
5) Tavşanlar 
6) Çam Kozalağı: Çam kozalağındaki taneler kozalağın altındaki sabit bir noktadan kozalağın tepesindeki başka bir
sabit noktaya doğru spiraller (eğriler) oluşturarak çıkarlar. İşte bu taneler soldan sağa ve sağdan sola sayıldığında
çıkan sayılar, Fibonacci Dizisi’nin ardışık terimleridir.
7) Tütün Bitkisi: Tütün Bitkisinin yapraklarının dizilişinde bir Fibonacci Dizisi söz konusudur; yani yaprakların
diziliminde bu dizi mevcuttur. Bundan dolayı tütün bitkisi Güneş’ten en iyi şekilde güneş ışığı ve havadan en iyi
şekilde Karbondioksit alarak Fotosentez’i mükemmel bir şekilde gerçekleştirir.
8) Eğrelti Otu: Tütün Bitkisindeki aynı özellik Eğrelti Otu’nda da vardır.
9) MİMAR SİNAN: Mimar Sinan’ın da bir çok eserinde Fibonacci Dizisi görülmektedir. Mesela Süleymaniye ve Selimiye
Camileri’nin minarelerinde bu dizi mevcuttur.
Some important numbers and series
Euler’s constant
e=2,718281828459...
Some important numbers and series
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946
30.03.2016
Some important numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987, 1597,
2584, 4181, 6765, 10946
Numbers

Golden ratio
=1.618033…
Mathematical Operations
 ADDITION
2+2=4
 SUBTRACTION
4-2=2
 MULTIPLICATION
2×2=4
 DIVISION
4÷2=2
Fill the grid with the numbers 1 to 9 in such that each number appears only
once in each row, column and region (3 by 3 block). Never guess the place of
a number and only fill it in when you are sure.
Mathematical Operations
Order of operations (PEMDAS)
48/1.2=?
 Parentheses
 Exponents
and roots
 Multiplication
 Addition
and Division
and Subtraction
23.828/0.28=??
30.03.2016
LET’S HAVE A BREAK!
Identifying a Linear Equation
●
●
●
●
●
●
●
Ax + By = C
The exponent of each variable is 1.
The variables are added or subtracted.
A or B can equal zero.
A>0
Besides x and y, other commonly used variables are m and n,
a and b, and r and s.
There are no radicals in the equation.
Every linear equation graphs as a line.
Equations
Examples of linear equations
2x + 4y =8
Equation is in Ax + By =C form
6y = 3 – x
Rewrite with both variables
on left side … x + 6y =3
x=1
B =0 … x + 0  y =1
-2a + b = 5
Multiply both sides of the
equation by -1 … 2a – b = -5
4x  y
 7
3
Multiply both sides of the
equation by 3 … 4x –y =-21
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Examples of Nonlinear Equations !!!
The following equations are NOT in the
standard form of Ax + By =C:
4x2 + y = 5
●
The x-intercept is the point where a line crosses the xaxis.
The general form of the x-intercept is (x, 0). The ycoordinate will always be zero.
●
The y-intercept is the point where a line crosses the yaxis.
The general form of the y-intercept is (0, y). The xcoordinate will always be zero.
The exponent is 2
There is a radical in the equation
x4
x and y -intercepts
xy + x = 5
Variables are multiplied
s/r + r = 3
Variables are divided
Finding the x-intercept
●
For the equation 2x + y = 6, we know that
●
Plug in 0 for y and simplify.
y must equal 0. What must x equal?
Finding the y-intercept
●
For the equation 2x + y = 6, we know that x must equal 0. What must y equal?
●
Plug in 0 for x and simplify.
2x + 0 = 6
2(0) + y = 6
2x = 6
0+y=6
x=3
●
So (3, 0) is the x-intercept of the line.
y=6
●
So (0, 6) is the y-intercept of the line.
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To summarize….
Find the x and y- intercepts
of x = 4y – 5
● To
find the x-intercept, plug in 0 for y.
● To
find the y-intercept, plug in 0 for x.
●
●
●
Find the x and y-intercepts
of g(x) = -3x – 1*
●
x-intercept
● Plug in y = 0
g(x) = -3x - 1
0 = -3x - 1
1
1 = -3x

3
1
=x

● (
, 0) is the 3
x-intercept
*g(x) is the same as y
x-intercept:
●
y-intercept:
Plug in y = 0
x = 4y - 5
x = 4(0) - 5
x=0-5
x = -5
(-5, 0) is the
x-intercept
●
Plug in x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
5
=y
4
●
(0, ) is the
4
y-intercept
5
Find the x and y-intercepts of
6x - 3y =-18
●
●
y-intercept
●
Plug in x = 0
●
g(x) = -3(0) - 1
g(x) = 0 - 1
g(x) = -1
●
(0, -1) is the
y-intercept
x-intercept
Plug in y = 0
●
●
y-intercept
Plug in x = 0
6x - 3y = -18
6x -3y = -18
6x -3(0) = -18
6x - 0 = -18
6x = -18
x = -3
● (-3, 0) is the
x-intercept
6(0) -3y = -18
0 - 3y = -18
-3y = -18
y=6
● (0, 6) is the
y-intercept
30.03.2016
Find the x and y-intercepts
of x = 3
Find the x and y-intercepts
of y = -2
●
●
x-intercept
●
●
Plug in y = 0.
A vertical line never
crosses the y-axis.
There is no y. Why?
x = 3 is a vertical line
so x always equals 3.
●
●
y-intercept
●
●
There is no y-intercept.
(3, 0) is the x-intercept.
x
x-intercept
Plug in y = 0.
y cannot = 0 because
y = -2.
● y = -2 is a horizontal
line so it never crosses
the x-axis.
●
●There
●
y-intercept
●
y = -2 is a horizontal line
so y always equals -2.
●
(0,-2) is the y-intercept.
x
is no x-intercept.
y
Graphing Equations
●
Example: Graph the equation -5x + y = 2
Graphing Equations
Graph y = 5x + 2
Solve for y first.
-5x + y = 2
Add 5x to both sides
y = 5x + 2
●
x
The equation y = 5x + 2 is in slope-intercept form, y =
mx+b. The y-intercept is 2 and the slope is 5. Graph
the line on the coordinate plane.
y
30.03.2016
Graphing Equations
Graph 4x - 3y = 12
●
Graphing Equations
Graph y = x - 4
Solve for y first
4x - 3y =12 Subtract 4x from both sides
-3y = -4x + 12
Divide by -3
12
y = -4
x + -3 Simplify
-3
4
y = 3x – 4
4
● The equation y = 3x - 4 is in slope-intercept form,
y=mx+b. The y -intercept is -4 and the slope is 4 .
3
Graph the line on the coordinate plane.
x
4
3
y
Introduction to
pair of linear equation in Two variables
Pair of linear equations in two
variable
A pair of linear equation is said to form a system of
simultaneous linear equation in the standard form
a1x+b1y+c1=0
a2x+b2y+c2=0
Where ‘a’, ‘b’ and ‘c’ are not equal to real
numbers ‘a’ and ‘b’ are not equal to zero.
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6
(-1,6)
Deriving the solution through
5
Graphical Method
4
3
Let us consider the following system of two simultaneous linear equations in
two variable.
2x – y = -1 ;3x + 2y = 9
(0,1)
2
-1
0
EQUATION 1
(2,5)
2x – y = -1
(1,3)
X
0
2
Y
1
5
EQUATION 2
1
-6
-5
-4
-3
-2
We can determine the value of the a variable by substituting any value for
the other variable, as done in the given examples
1
2
3
4
-1
-2
3x + 2y = 9
5
X
3
-1
Y
0
6
-3
-4
2x – y = -1
X=(y-1)/2
3x + 2y = 9
y=2x+1
2y=9-3x
-5
x=(9-2y)/3
X
0
2
X
3
-1
Y
1
5
Y
0
6
ax1 + by1 + c1 = 0;
ax2 + by2 + c2 = 0
i)
ii)
a1
a2
a1
a2
iii) a1
a2
=
=
=
b1
=
b2
c1
c2
b1
b1
b2
=
c1
c2
‘Y’ intercept = 3
X=1
Y=3
Deriving the solution through
Substitution Method
Intervening Lines; Infinite
Solutions
Intersecting Lines; Definite
Solution
b2
‘X’ intercept = 1
This method involves substituting the value of one
variable, say x , in terms of the other in the
equation to turn the expression into a Linear
Equation in one variable, in order to derive the
solution of the equation .
For example
Parallel Lines; No Solution
x + 2y = -1 ;2x – 3y = 12
30.03.2016
2x – 3y = 12 ----------(ii)
x + 2y = -1 -------- (i)
Deriving the solution through
x + 2y = -1
x = -2y -1 ------- (iii)
2x – 3y = 12
2 ( -2y – 1) – 3y
= 12 - 4y – 2 – 3y
= 12 - 7y = 14
= 12 - 14 = 7y
Substituting the value of x
inequation (ii), we get
elimination Method
In this method, we eliminate one of the two variables to obtain an equation in
one variable which can easily be solved. The value of the other variable can be
obtained by putting the value of this variable in any of the given equations.
y = -2
Putting the value of y
x = -2y -1
x = -2 x (-2) – 1
= 4–1
x=3
in eq. (iii), we get
Hence the solution of the equation is (
3, - 2 )
3x + 2y = 11 --------- (i)
2x + 3y = 4 ---------(ii)
3x + 2y = 11 x3 x39x - 3y = 33---------(iii)
2x + 3y = 4 x2
4x + 6y = 8---------(ii)
(iii) – (iv) =>
=>9x + 6y = 33-----------(iii)
4x + 6y = 8------------(iv)
(-) (-)
(-)
5x
For example:
3x + 2y = 11 ;2x + 3y = 4
Deriving the solution through
Cross-multiplication Method
The method of obtaining solution of simultaneous equation by using
determinants is known as Cramer’s rule. In this method we have to
follow this equation and diagram
= 25
x=5
Putting the value of x in
equation (ii) we get, =>
Hence, x = 5 and y = -2
2x + 3y = 4
2 x 5 + 3y = 4
10 + 3y = 4
3y = 4 – 10
3y = - 6
y=-2
ax1 + by1 + c1 = 0;
ax2 + by2 + c2 = 0
X=
b1c2 –b2c1
a1b2 –a2b1
Y=
c1a2 –c2a1
a1b2 –a2b1
30.03.2016
Example:
8x + 5y – 9 = 0
3x + 2y – 4 = 0
X
X
=
B1c2-b2c1
Y
=
c1a2 –c2a1
=
B1c2-b2c1
1
a1b2 –a2b1
X
X=
a1b2 –a2b1
Y=
X
c1a2 –c2a1
-2
a1b2 –a2b1
Y
=
5
1
-2
2
x
+
3
= 1
y
3
5
x
-
4
x
= p
16-15
Y
1
and
Y=5
‘p’ = 2 ;‘q’ = 3
We know that
1
p =
x
1
y
= 1
5
-2
We can turn the equations into linear equations by substituting
1
1
These equations can now be solved by any of the
aforementioned methods to derive the value of
‘p’ and ‘q’.
=
y
a1b2 –a2b1
=
-27-(-32)
X
1
The resulting equations are
2p + 3q = 13 ; 5p - 4q = -2
Equations reducible to pair of linear
equation in two variables
In case of equations which are not linear, like
Y
1
X = -2
=
c1a2 –c2a1
=
-20-(-18)
b1c2 –b2c1
Y
=
q
X =
q =
1
2
&
Y =
1
3
1
y
30.03.2016
Summary

Insight to Pair of Linear Equations in Two Variable

Deriving the value of the variable through


Graphical Method

Substitution Method

Elimination Method

Cross-Multiplication Method
Exponential Functions
and Their Graphs
Reducing Complex Situation to a Pair of Linear Equations to derive their
solution
The exponential function f with base a is
defined by
The value of f(x) = 3x when x = 2 is
f(2) = 32 9
=
The value of f(x) = 3x when x = –2 is
f(x) = ax
where a > 0, a  1, and x is any real
number.
For instance,
f(–2) = 3–2 =
The value of g(x) = 0.5x when x = 4 is
f(x) = 3x and g(x) = 0.5x
g(4) = 0.54 0.062
=
5
are exponential functions.
59
1
9
60
30.03.2016
The Graph of f(x) = ax, a > 1
The Graph of f(x) = ax, 0 < a <1
y
y
Range: (0, )
(0,
1)
Range: (0, )
Horizontal
Asymptote
y=
0
x
Horizontal
Asymptote
y=
0
Domain: (–, )
61
x
Domain: (–, )
62
Example: Sketch the graph of g(x) = 2x – 1. State the
domain and range.
Example: Sketch the graph of f(x) = 2x.
x
-2
-1
0
1
2
63
(0,
1)
y
f(x) (x, f(x))
¼
½
1
2
4
(-2, ¼)
(-1, ½)
(0, 1)
(1, 2)
(2, 4)
The graph of this
function is a vertical
translation of the graph
of f(x) = 2x down one
unit .
4
f(x) = 2x
4
2
2
x
–2
y
2
64
Domain: (–, )
x
Range: (–1, )
y = –1
30.03.2016
Example: Sketch the graph of g(x) = 2-x. State the
domain and range.
y
The graph of this
function is a reflection
the graph of f(x) = 2x in
the y-axis.
The irrational number e, where
e  2.718281828…
f(x) = 2x
is used in applications involving growth and decay.
4
Using techniques of calculus, it can be shown that
Domain: (–, )
x
–2
Range: (0, )
n
2
 1
1    e as n  
 n
65
66
The Graph of f(x) = ex
Euler’s Formula
y
x
-2
-1
0
1
2
6
4
2
x
–2
67
2
f(x)
0.14
0.38
1
2.72
7.39
݁ ௜ఏ ൌ ܿ‫ ߠݏ݋‬൅ ݅‫ߠ݊݅ݏ‬
30.03.2016
Graph of Some Common Functions
f (x)
Cube root
10
8
f(x) = c
f ( x)  3 x
6
4
y x
y=x
2
Intercepts?
Domain?
-10
-8
-6
-4
-2
0
Range?
2
4
6
8
10
x
1
2
3
4
5
x
-2
Even, odd, neither?
Increasing?
-4
Decreasing?
-6
Constant?
Maxima?
-8
Minima?
y x
y=x3
-10
y = x2
f (x)
Reciprocal
4
1
f ( x) 
x
f (x)
Greatest
integer
5
3
5
4
3
f ( x )  int( x)
2
2
1
1
Intercepts?
Domain?
Range?
-5
-4
-3
-2
-1
0
1
-1
Even, odd, neither?
Increasing?
Maxima?
3
4
5
x
Domain?
Range?
-5
-4
-3
-2
-1
0
-1
Even, odd, neither?
-2
Decreasing?
Constant?
2
Intercepts?
Increasing?
-2
Decreasing?
-3
-4
Minima?
Constant?
Maxima?
-3
-4
Minima?
-5
-5
30.03.2016
y
Circle
x2  y2  r 2
f (x)
Semicircle
5
Not a function!
4
5
4
f ( x)  r 2  x 2
f ( x)  9  x 2
3
r  radius
3
r  radius
2
2
1
-5
-4
-3
-2
-1
0
1
1
2
3
4
5
x
Intercepts?
Domain?
-5
-4
-3
-2
-1
Range?
-1
0
1
-1
Even, odd, neither?
-2
x2  y2  9
-3
-4
Increasing?
-2
Decreasing?
Constant?
Maxima?
-3
-4
Minima?
-5
-5
Graph of Some Important Functions
Sine
Cosine
Functions
Graph in
Excell!
2
3
4
5
x
30.03.2016
Have a nice
week!