1. Gebze Technical University Department of Architecture
MAT120
Asst. Prof. Dr. Ferhat PAKDAMAR
(Civil Engineer)
M Blok - M106
Spring – 2014/2015
Week 3-4

2. Subjects Week Subjects Methods 1 11.02.2015 Introduction 2 18.02.2015
Set Theory and Fuzzy Logic.
Term Paper  3
Real Numbers, Complex numbers, Coordinate Systems. 4
Functions, Linear equations 5
Matrices 6
Matrice operations 7
MIDTERM EXAM MT 8
Limit. Derivatives, Basic derivative rules 9
Term Paper presentations
Dead line for TP 10
Integration by parts, 11
Area and volume Integrals 12
Introduction to Numeric Analysis 13
Introduction to Statistics. 14
Review 15 16
FINAL EXAM
FINAL

3. Numbers

4. A number is a mathematical object used to count, measure and label.

5. Numbers

6. Numbers

7. Numbers

8. Numbers

9. Numbers

10. Numbers

11. Numbers

12. Numbers Name Power SI symbol SI prefix One (none) Ten da(D) deca1
(none)
Ten 10
da(D)
deca
Hundred 2
100
h(H)
hecto
Thousand 3
1,000
k(K)
kilo
Ten Thousand (Myriad) 4
10,000
Hundred Thousand 5
100,000
Million 6
1,000,000 M
mega
Billion (Milliard) 9
1,000,000,000 G
giga
Trillion (Billion) 12
1,000,000,000,000 T
tera
Quadrillion (Billiard) 15
1,000,000,000,000,000 P
peta
Quintillion (Trillion) 18
1,000,000,000,000,000,000 E
exa
Sextillion (Trilliard) 21
1,000,000,000,000,000,000,000 Z
zetta
Septillion (Quadrillion) 24
1,000,000,000,000,000,000,000,000 Y
yotta
Octillion (Quadrilliard) 27
1,000,000,000,000,000,000,000,000,000
(X)
(xona)
Nonillion (Quintillion) 30
1,000,000,000,000,000,000,000,000,000,000
(W)
(wecta)
Decillion (Quintilliard) 33
1,000,000,000,000,000,000,000,000,000,000,000
(V)
(vinka)
Undecillion (Sextillion) 36
1,000,000,000,000,000,000,000,000,000,000,000,000
(U)
(untra)
Duodecillion (Sextilliard) 39
1,000,000,000,000,000,000,000,000,000,000,000,000,000
(S)
(sampa)
Tredecillion (Septillion) 42
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000
(R)
(rosa)
Quattordecillion (Septilliard) 45
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
(Q)
(quoda)
Quindecillion (Octillion) 48
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
(O)
(oba)
Sexdecillion (Octilliard) 51
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
Sepdecillion (Nonillion) 54
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
...
Googol
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

13. Binary and Decimal Systam Convertions

14. Romen Numbers
MDCCXCVIII
MMMDCCCLXXXVIII
MCMLXXVIII
MMXV

15. Romen Numbers

16. Numbers
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.

17. Mathematical Operations
ADDITION =4
SUBTRACTION 4-2=2
MULTIPLICATION 2×2=4
DIVISION 4÷2=2
Order of operations (PEMDAS)
Parentheses
Exponents and roots
Multiplication and Division
Addition and Subtraction

18. Mathematical Operations
48/1.2=?
23.828/0.28=??

19. LET’S HAVE A BREAK!

20. Equations

21. 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.

22. Examples of linear equations
2x + 4y =8
6y = 3 – x
x = 1
-2a + b = 5
Equation is in Ax + By =C form
Rewrite with both variables on left side … x + 6y =3
B =0 … x + 0 y =1
Multiply both sides of the equation by -1 … 2a – b = -5
Multiply both sides of the equation by 3 … 4x –y =-21

23. Examples of Nonlinear Equations !!!
The following equations are NOT in the standard form of Ax + By =C:
4x2 + y = 5
xy + x = 5
s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided

24. x and y -intercepts
The x-intercept is the point where a line crosses the x- axis.
The general form of the x-intercept is (x, 0). The y- coordinate will always be zero.
The y-intercept is the point where a line crosses the y- axis.
The general form of the y-intercept is (0, y). The x- coordinate will always be zero.

25. Finding the x-intercept
For the equation 2x + y = 6, we know that y must equal 0. What must x equal?
Plug in 0 for y and simplify.
2x + 0 = 6
2x = 6
x = 3
So (3, 0) is the x-intercept of the line.

26. 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.
2(0) + y = 6
0 + y = 6
y = 6
So (0, 6) is the y-intercept of the line.

27. To find the x-intercept, plug in 0 for y.
To summarize….
To find the x-intercept, plug in 0 for y.
To find the y-intercept, plug in 0 for x.

28. Find the x and y- intercepts of x = 4y – 5
x-intercept:
Plug in y = 0
x = 4y - 5
x = 4(0) - 5
x = 0 - 5
x = -5
(-5, 0) is the
x-intercept
y-intercept:
Plug in x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
= y
(0, ) is the
y-intercept

29. Find the x and y-intercepts of g(x) = -3x – 1*
Plug in x = 0
g(x) = -3(0) - 1
g(x) = 0 - 1
g(x) = -1
(0, -1) is the
x-intercept
Plug in y = 0
g(x) = -3x - 1
0 = -3x - 1
1 = -3x
= x
( , 0) is the
*g(x) is the same as y

30. Find the x and y-intercepts of 6x - 3y =-18
Plug in x = 0
6x -3y = -18
6(0) -3y = -18
0 - 3y = -18
-3y = -18
y = 6
(0, 6) is the
x-intercept
Plug in y = 0
6x - 3y = -18
6x -3(0) = -18
6x - 0 = -18
6x = -18
x = -3
(-3, 0) is the

31. Find the x and y-intercepts of x = 3
x-intercept
Plug in y = 0.
There is no y. Why?
x = 3 is a vertical line so x always equals 3.
(3, 0) is the x-intercept.
y-intercept
A vertical line never crosses the y-axis.
There is no y-intercept. x y

32. Find the x and y-intercepts of y = -2
y = -2 is a horizontal line
so y always equals -2.
(0,-2) is the y-intercept.
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 is no x-intercept. x y

33. Graphing Equations
Example: Graph the equation -5x + y = 2
Solve for y first.
-5x + y = 2 Add 5x to both sides
y = 5x + 2
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.

34. Graphing Equations
Graph y = 5x + 2 x y

35. Graphing Equations Graph 4x - 3y = 12 Solve for y first
4x - 3y =12 Subtract 4x from both sides
-3y = -4x Divide by -3
y = x Simplify
y = x – 4
The equation y = x - 4 is in slope-intercept form, y=mx+b. The y -intercept is -4 and the slope is Graph the line on the coordinate plane.

36. Graphing Equations
Graph y = x - 4 x y

37. Pair of linear equations in two variable
Anushka

38. Introduction to pair of linear equation in Two variables
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.
Nikunj

39. Deriving the solution through Graphical Method
Let us consider the following system of two simultaneous linear equations in two variable. 2x – y = -1 ;3x + 2y = 9 We can determine the value of the a variable by substituting any value for the other variable, as done in the given examples
Shubham
2x – y = -1
3x + 2y = 9
X=(y-1)/2 y=2x+1
2y=9-3x x=(9-2y)/3 X 2 Y 1 5 X 3 -1 Y 6

40. X=1 Y=3 (-1,6) EQUATION 1 (2,5) 2x – y = -1 X 2 Y 1 5 (1,3) (0,1)4 3 2 1 -6 -5 -4 -3 -2 -1
(-1,6)
EQUATION 1
(2,5)
2x – y = -1 X 2 Y 1 5
(1,3)
(0,1)
EQUATION 2
3x + 2y = 9 X 3 -1 Y 6
‘X’ intercept = 1
‘Y’ intercept = 3
X=1 Y=3

41. ax1 + by1 + c1 = 0; ax2 + by2 + c2 = 0 a1 b1 c1 = i) =
Intervening Lines; Infinite Solutions a2 b2 c2 a1 b1
ii) =
Intersecting Lines; Definite Solution a2 b2 a1 b1 c1
iii)
Parallel Lines; No Solution = = a2 b2 c2

42. Deriving the solution through Substitution Method
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
x + 2y = -1 ;2x – 3y = 12
Amel

43. x + 2y = (i)
2x – 3y = (ii)
x + 2y = -1
x = -2y (iii)
2x – 3y = 12
2 ( -2y – 1) – 3y
= y – 2 – 3y
= y = 14
= = 7y
y = -2
Substituting the value of x inequation (ii), we get
Putting the value of y in eq. (iii), we get
x = -2y -1
x = -2 x (-2) – 1
= 4–1
x = 3
Hence the solution of the equation is ( 3, - 2 )

44. Deriving the solution through 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.
For example:
3x + 2y = 11 ;2x + 3y = 4
Siddhartha

45. Hence, x = 5 and y = -2 3x + 2y = 11 --------- (i)
2x + 3y = (ii)
3x + 2y = x3-
9x - 3y = (iii) x3
2x + 3y = 4
4x + 6y = (ii) x2
(iii) – (iv) =>
=>9x + 6y = (iii)
4x + 6y = (iv)
(-) (-) (-)
5x = 25
x = 5
Putting the value of x in equation (ii) we get, =>
2x + 3y = 4
2 x 5 + 3y = 4
10 + 3y = 4
3y = 4 – 10
3y = - 6
y=-2
Hence, x = 5 and y = -2

46. 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
ax1 + by1 + c1 = 0;
ax2 + by2 + c2 = 0
Karthik
b1c2 –b2c1
c1a2 –c2a1 X= Y=
a1b2 –a2b1
a1b2 –a2b1

47. X Y 1 = =
B1c2-b2c1
c1a2 –c2a1
a1b2 –a2b1
b1c2 –b2c1
c1a2 –c2a1 X= Y=
a1b2 –a2b1
a1b2 –a2b1

48. X Y 1 = = B1c2-b2c1 X Y 1 = = -20-(-18) -27-(-32) 16-15 = =
Example:
8x + 5y – 9 = 0
3x + 2y – 4 = 0 X Y 1 = =
B1c2-b2c1
c1a2 –c2a1
a1b2 –a2b1 X Y 1 = =
-20-(-18)
-27-(-32)
16-15 Y X Y 1 X = = 1 1 -2 5 5 1 -2
X = -2 and Y = 5

49. Equations reducible to pair of linear equation in two variables
In case of equations which are not linear, like We can turn the equations into linear equations by substituting 2 3 5 4 + = 13 - = -2 x y x y
Anushka 1 1 p = q = x y

50. The resulting equations are
2p + 3q = 13 ; 5p - 4q = -2
These equations can now be solved by any of the aforementioned methods to derive the value of ‘p’ and ‘q’.
‘p’ = 2 ;‘q’ = 3
We know that 1 1 p = q = x y 1 1 X Y = =
& 2 3

51. 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
Reducing Complex Situation to a Pair of Linear Equations to derive their solution
Anushka

52. Exponential Functions and Their Graphs

53. The exponential function f with base a is defined by
f(x) = ax
where a > 0, a  1, and x is any real number.
For instance,
f(x) = 3x and g(x) = 0.5x
are exponential functions.

54. The value of f(x) = 3x when x = 2 is9
The value of f(x) = 3x when x = –2 is
f(–2) = 3–2 =
The value of g(x) = 0.5x when x = 4 is
g(4) = 0.54 =
0.0625

55. The Graph of f(x) = ax, a > 1y
Range: (0, )
(0, 1) x
Horizontal Asymptote y = 0
Domain: (–, )

56. The Graph of f(x) = ax, 0 < a <1y
Range: (0, )
Horizontal Asymptote y = 0
(0, 1) x
Domain: (–, )

57. Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) -2 ¼ (-2, ¼) y x
f(x)
(x, f(x)) -2
(-2, ¼) -1
(-1, ½) 1
(0, 1) 2
(1, 2) 4
(2, 4) 4 2 x –2 2

58. Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.y
f(x) = 2x
The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2
Domain: (–, ) x
y = –1
Range: (–1, )

59. Example: Sketch the graph of g(x) = 2-x. State the domain and range.y
f(x) = 2x
The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4
Domain: (–, ) x –2 2
Range: (0, )

60. The irrational number e, where
is used in applications involving growth and decay.
Using techniques of calculus, it can be shown that

61. The Graph of f(x) = ex x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 6 4 2 –2 2y x
f(x) -2
0.14 -1
0.38 1
2.72 2
7.39 6 4 2 x –2 2

62. Euler’s Formula
𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃+𝑖𝑠𝑖𝑛𝜃

63. Graph of Some Common Functions
f(x) = c
y = x
y = x2
y = x 3

64. Cube root f (x) x -8 -10 -6 -4 -2 2 4 6 8 10 Intercepts? Domain?
Range?
Even, odd, neither?
Increasing?
Decreasing?
Constant?
Maxima?
Minima?

65. Reciprocal f (x) x 5 4 3 2 1 Intercepts? Domain? Range?-4 -5 -3 -2 -1 1 2 3 4 5
Reciprocal
Intercepts?
Domain?
Range?
Even, odd, neither?
Increasing?
Decreasing?
Constant?
Maxima?
Minima?

66. Greatest integer f (x) x 5 4 3 2 1 Intercepts? Domain? Range?-4 -5 -3 -2 -1 1 2 3 4 5
Greatest integer
Intercepts?
Domain?
Range?
Even, odd, neither?
Increasing?
Decreasing?
Constant?
Maxima?
Minima?

67. Circle Not a function! y x 5 4 3 2 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2

68. Semicircle f (x) x 5 4 3 2 1 Intercepts? Domain? Range?-4 -5 -3 -2 -1 1 2 3 4 5
Semicircle
Intercepts?
Domain?
Range?
Even, odd, neither?
Increasing?
Decreasing?
Constant?
Maxima?
Minima?

69. Graph of Some Important Functions
Sine
Cosine
Functions

70. Graph in Excell!

71. Have a nice week!