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Mathematics
No.
Lecture title
1. Mathematics: general concepts: coordinate and
graph in plane; inequality; absolute value or
magnitude; function and their graphs
displacement function; slope and equation of
lines.
2. Limits and continuity; theorem of limits; limit
involving infinity; continuity; continuity
conditions.
3. Derivatives: line tangent
and derivatives;
differentiation rules; derivative of trigonometric
function; practice exercises.
4. Integration: indefinite integrals; rules for
indefinite integrals; integration formulas for
basic trigonometric function; definite integrals;
properties of definite integrals;
Practice exercises
hours
6
4
6
6
Lecture No.1
Reviews the basic ideas you need to start calculating. The
topics include Cartesian coordinates in the plane, straight
lines.
General concepts:
Real number: are numbers that can be expressed as
decimals, such as:
3
  0.750000 , 2  1.4
4
The real number can be represented geometrically as points
on a number line called the real line.
-3
+3
-2
-1
0
+1
+2
The real number properties:
1. Algebraic: can be added, subtracted, multiplied and
divided.
2. Order properties :
n
n
n
k 1
k 1
k 1
n
n
n
k 1
k 1
k 1
 (ak  bk )   ak   bk
 (ak  bk )   ak   bk
n
n
k 1
k 1
 cak  c ak
Where c= constant value
Inequalities:
Rule of inequalities, if a, b and c are real numbers then :
1. a < b, a+c < b+c
2. a < b, a-c < b-c
3. a < b and c > 0 , ac < bc
4. a < b and c < 0 , ac > bc
Special case a <b , -b <-a
5. a >0 , 1/a > 0
6. if a and b are both positive or negative, then a < b , 1/b
<1/a
Notice: Multiplying by a negative number reverses the
inequality: 2<5 but -2>-5.
Intervals:
A subset of the real line is called an interval if it contains at
least an interval if it contains at least two numbers and
contains all the real numbers lying between any two of its
elements.
Example : x> b , -2< x <5.
Types of intervals:
Finite : (a,b){x: a< x< b} open
[a, b]{x / a  x  b}
Closed
[a, b){x / a  x  b}
Infinite:
Half
( a, ){ x / x  a}
[ a, ){ x / x  a}
( , b){ x / x  b}
open
open
Half open
open
( , b]{ x / x  b} half Closed
[ a, b]
closed
Example: solve the following inequalities and show their
solution sets on the real line.
a. 2x-1<x+3
b. –x/3<2x+1
c. 6/(x-1)>5
Solution:
a. 2x-1<x+3
2x-x<4
X<4 interval (-  ,4) open
b. –x<6x+3
7x>-3
x>-3/7
(-3/7,  )
c. 6>5x-5
11>5x
X<11/5
(-  ,11/5) open.
Absolute value:
The absolute value of a number X denoted by x ; is
defined by the formula:
X , X  0 
X 

 X , X  0 
3  3 , 0  0 ,  5  (5)  5 ,  a  a
0  3
3
5
 0
Geometrically the absolute value of x is the distance
from x to zero on the real number line.
The distance between x and y because the
x y 
distance are always positive or zero.
Absolute value properties:
1.
2.
3.
4.
a  a  a
ab  a b
a
a

b
b
ab  a  b
a  a
 3  3 while  3  3
Example: solve
2x-3=  7
2x= - 7+3
2x= - 4
X= -2
2x  3  7
or 2x-3=7
2x=10
x=5
Solution set [-2,5]
Note : if D is any positive number then
a D , DaD
a D , DaD
Example: Solve x  5  8
-8< x  5 <8
-8 < x – 5
-3 < x
Homework: solve
or 8> x-5
x < 13
7
3
1
2x
Note: Reciprocal and multiply by negative number reverse
the inequality
Example: solve the inequality and show the solution on real
line :
2x  3  1
Solution:
2x  3  1
1  2x  3  1
 3 for both side
2  2x  4
1 x  2
[1,2],
1
2
Home works: Solve the following inequality:
1.
4
1
( x  2)  ( x  6)
5
3
2.
 x  5 12  3 x

2
4
3.
2x 
1
7
 7x 
2
6
Rectangular coordinate system or Cartesian coordinate in
the plane:
Point in the plane can be identified with ordered pairs of
real numbers, to begin we draw two perpendicular
coordinate lines that intersect at the o-point of each line ,
these line are called coordinate axes in the plane .
y
3
2
1
-3
-2
-1
P(1,1)
.
1
2
3
x
-1
Q(-2,-2)
-2
-3
Slope : any non-vertical line in the plane has the property
that the ratio :
M= rise  y 
run
x
y 2  y1
x 2  x1
Has the same value for every choice of the two points on the
line , note that x  0 for vertical line , y  0 for horizontal line
The angle of inclination of a line horizontal is zero, for
vertical line is 90, if we expressed the angle of inclination by
 then 0    180 .
The relation between the slop m and inclination line is :
M= tan 
The equation of non-vertical straight line with slope m
M= y-y1/x-x1
Y=y1+m(x-x1)
Example : write an equation of the line that passes (2,3) with
slope -3/2
Y=3-3/2(x-2)= 3- 3/2 x+6/2
When x=0 then y=6
Y=0 then x=4
Example : write the equation of the line passing through (2,-1), (3,4)
Slope= (-1-4)/(-2-3)=1
Y=-1+1(x-(-2))
Y=x+1
or y=4+1(x-3)
=4+x-3
y= x+1
Distance in the plane :
1
2 2
d  [( x2  x1 )2  ( y2  y1 ) ]
Q(x2-y2)
(y2-y1)
(x2-x1)
P(x1-y1)
Examples : calculate the distance between :
1. (-1,2), (3,4)
2. the origin and (x, y)
3. the radius of a circle has center (h,k) and the point
passing through p(x,y).
4. a particle moves from A to B in the coordinate plane
find the increments x and y in the particles
coordinate also find the distance from A to B .
A (-3,2), B (-1,-2)
A ( 2 ,4) , B( 0, 1.5)
5. describe the graph of the equations
x2+y2=1
x2+y2<3
x2+y2=0
6. find the equation for a. vertical line and b. the
horizontal line through the given point:
a. (-1,4/3)
b. (-  ,0 )
Functions and their graphs:
Functions are the key to describe the real world in
mathematical terms.
The area of a circle depends on the radius, the distance an
object travels at constant speed from an initial position a
long straight line path depends on the elapsed time, so the
value of one variable y depend on the value of another
variable x so we say y is a function of x.
A   r2 a rule to calculate the area of a circle from its
radius r
So y= f(x) in general, x independent variable, y dependent
variable.
Definition
Domain: all possible input value
Range: the set of all values of f(x) as x varies throughout d is
called the range of the function.
Function
Domain
Range
Y=x2
(- ,  )
[0,  )
Y=1/x
(,0)  (0, )
(,0)  (0, )
Y=
[0,  )
[0,  )
(  , 4 ]
[0,  )
x
Y= 4  x
Example: Graph the function y=x2 over the interval [-2,2]
Solution:
X=-2, y=4
X=-1, y=1
X=3/2, y=9/4
y=x2
Example: Graph y=1/x+1
x
y
(x, y)
2
1/3
(2,1/3)
1
1/2
(1,1/2)
0
1
(0,1)
-1
?
?
-2
-1
(-2, -1)
Homework: graph the function
y=1/x , y=(4-x)1/2
……------………..----------…………----------……….---Sums, differences, products, and quotients of functions
Like numbers, functions can be added, subtract, multiplied
and divided to produce new functions.
If f and g are functions then:
(f+g) (x) = f(x)+g(x)
(f-g)(x)= f(x) –g(x)
(f.g) (x)=f(x).g (x)
(f/g) (x) =f (x) /g (x) where g (x) doesn't equal zero.
(c f) (x) = c f (x), c= constant
Composite functions
If f(x)= x , g(x)=x+1 then
( f  g )( x)  f ( g ( x))  x  1
(g  f )(x)=g(
x  1)
(f  f )(x)= f(f(x))=x1/4
H.W: find f+g and f.g and their domain and ranges:
1. f(x)=x, g(x)= x  1
2. F (x) =x+5, g (x) =x2-3 find: F (g (0)), f (f (-5)), g (f (2)).
3. Find f/g and g/f and D, R,
f (X) =2, g (x) =x2+1.
4. u(x)= 4x-5, v(x)=x2, f(x)=1/x find u(v(f(x))), v(u(f(x))),
f(v(u(x))).
**********…………**********………….*************
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Limits of the functions
Provide the limit exist if
f exist we say f is differtiable
function
Example : find
f  lim
h 0
f
if f(x)=x/x-1
f ( x  h)  f ( x )
h
xh
x

lim x  h 1 x 1
h
lim
( x  h)( x  1)  ( x )( x  h  1)
/h
( x  h  1)( x  1)
lim
h 0
x 2  xh  x  h  x 2  xh  x
1
( x  h  1)( x  1)
 lim
h
h 0 ( x  h  1)( x  1)
Differentiation rules :
1. dc/dx=0, c=const.
2. dxn/dx=n xn-1
f=1.x.x2.x3.x-2 .x-3.x-4
=0.1.2x.3x2.-2x-3.-3x-4.-4x-5
3. if u and v two differentaible functions of x then u+v
and u.v is different.
d/dx (u+v)=du/dx+dv/dx
d/dx(uv)=du/dx.dv/dx
d/dx(u/v)=(du/dx.v-u.dv/dx)/u2
Example :
1. Y=(X2+1)(X3+3)
dx/dy= 3x2(x2+1)+(x3+3).2x
= 5x4+3x2+6x.
2. Y=t2-1/t2+1
Dx/dy=4t/(t2+1)2
3.y=(x-1)(x2-2x)/x4
Dx/dy=-1/x2
4. y=4/x3