mathematics

Mathematics

No. Lecture title hours
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. 6
2. Limits and continuity; theorem of limits; limit involving infinity; continuity; continuity conditions. 4
3. Derivatives: line tangent and derivatives; differentiation rules; derivative of trigonometric function; practice exercises. 6
4. Integration: indefinite integrals; rules for indefinite integrals; integration formulas for basic trigonometric function; definite integrals; properties of definite integrals;
Practice exercises 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:
,
The real number can be represented geometrically as points on a number line called the real line.

-3 -2 -1 0 +1 +2 +3

The real number properties:

1. Algebraic: can be added, subtracted, multiplied and divided.
2. Order properties :







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 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

Closed

Half open

Infinite:


Example: solve the following inequalities and show their solution sets on the real line.
a. 2x-15

Solution:

a. 2x-12x-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 ; is defined by the formula:





3

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 distance are always positive or zero.

Absolute value properties:

1.
2.
3.
4.



Example: solve
2x-3= 7
2x= - 7+3 or 2x-3=7
2x= - 4 2x=10
X= -2 x=5

Solution set [-2,5]


Note : if D is any positive number then



Example: Solve

-8< <8 or 8> x-5
-8 < x – 5 x < 13
-3 < x

Homework: solve

Note: Reciprocal and multiply by negative number reverse the inequality

Example: solve the inequality and show the solution on real line :


Solution:



1 2



Home works: Solve the following inequality:








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
P(1,1)
1 .
-3 -2 -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=

Has the same value for every choice of the two points on the line , note that
The angle of inclination of a line horizontal is zero, for vertical line is 90, if we expressed the angle of inclination by then .
The relation between the slop m and inclination line is :

M=

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 or y=4+1(x-3)
Y=-1+1(x-(-2)) =4+x-3
Y=x+1 y= x+1

Distance in the plane :


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 in the particles coordinate also find the distance from A to B .
A (-3,2), B (-1,-2)
A , 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. (- )

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.

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


Y=
[0, )
[0, )

Y=
( ]
[0, )


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)= , g(x)=x+1 then

(g )(x)=g( )
(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)=

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))).

**********…………**********………….***************************************************////////////////////////////////////////////////////////////////////////////
Limits of the functions

Provide the limit exist if f exist we say f is differtiable function

Example : find if f(x)=x/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