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Chapter One
PREREQUISITES FOR
CALCULUS
Sets and Intervals:
Set: is a collection of things under certain conditions.
Interval: is a set of all real numbers between two points on the real number
line. (it is a subset of real numbers)
1. Open interval: is a set of all real numbers between A&B excluded (A&B are
not elements in the set). {x: A < x < B} or (A, B)
2. Closed interval: is a set of all real numbers between A&B included (A&B are
elements in the set). {x: A ≤ x ≤ B} or [A, B]
3.Half-Open interval (Half-Close): is a set of all real numbers between A & B
with one of the end-points as an element in the set.
a) (A, B]= {x: A < x ≤ B}
b) [A, B)= {x: A ≤ x < B}
Union and Intersections of intervals: if A and B are two sets then:
The union is the set whose numbers belong to A or B (or both) and denoted by
(A  B), and intersection is the set whose numbers belong to both A and B and
denoted by (A∩B).
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Inequalities:
Rules for Inequalities
Let a, b, and c are real numbers, then:
1. if a < b and b < c then a < c
2. a < b  a + c < b + c
3. a < b  a - c < b - c
4. a < b and c > o  ac < bc (c is positive)
5. a < b and c < o  ac > bc (c is negative)
special case a < b  -a > -b
6. if a < b and c < d then a+c < b+d
7. a > 0 
1
0
a
8. If a and b are both positive or both negative, and a < b 
1 1

a b
Analytical Geometry (Coordinate in the Plane)
Each point in the plane can be represented
with a pair of real numbers (a,b), the number a is
the horizontal distance from the origin to point P,
while b is the vertical distance from the origin to
point P.
The origin divides the x-axis into positive xaxis to the right and the negative x-axis to the left,
also, the origin divides the y-axis into positive
y-axis upward and the negative y-axis downward.
The axes divide the plane into four regions called quadrants, numbered I,
II, III and IV.
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Distance between Points and (Mid-Point Formula):
Distance between points in the plane is
calculated with a formula that comes from
Pythagorean Theorem:
Distance Formula for Points in the Plane
yo
The distance between P(x1,y1) and Q(x2,y2)
d  (x) 2  (y ) 2  ( x2  x1 ) 2  ( y2  y1 ) 2
and the mid-point formula:
xo 
x1  x2
;
2
yo 
y1  y2
2
Slope and Equation of Line
DEFENITION: Slope
The constant
m
rise y y2  y1


run x x2  x1
is the slope of nonvertical line P1P2
m1  
1
,
m2
m2  
1
m1
To see this, notice by inspecting similar triangles that
m1=a/h and m2=-h/a.
Hence m1. m2= (a/h). (-h/a)= -1
xo
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Point-Slope Equation:
We can write an equation for a non-vertical
straight line L if we know its slope m and the coordinate
P(x,y)
of one point P1(x1,y1) on it. If P(x,y) is any other point on
L, then we can use two points P1 and P to compute the
P1(x1,y1)
slope,
m
y  y1
x  x1
So that
y - y1 =m (x - x1)
or
y = y1 + m (x - x1)
The equation
y = y1 + m (x - x1)
is the point-slope equation of the line that passes through the point P1(x1,y1) and
has slope m.
The Distance from a Point to a Line:
d
To calculate the distance from a certain
point P(x1,y1) to a line L:
P(x1,y1)
Q(x2,y2)
1. Find an equation for the line L` that pass
L`
through a point P(x1,y1) and perpendicular to
the line L.
2. Find the point Q1(x2,y2) where L` meet with L.
3. Calculate the distance between P and Q.
4. to find d use the distance between two points formula.
Or you can use the following formula:
The distance (d) between the line L is Ax + By + C= 0 and the point P(x1,y1)
d
Ax1  By1  C
A2  B 2
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Angles between Two Lines:
L1,m1
y
To find the angle between the lines L1 and L2 which
have slopes m1 and m2 respectively, draw horizontal line


passes through the point of their intersections as shown.
L2,m2

So


Where: m1=tan

tan  
tan = tan(-)
tan   tan 

1  tan  . tan 
x
0
m2=tan
m1  m2
1  m1m2
Functions
DEFINITION: Function
A function from a set D (domain) to a set R (range) is a rule that assigns to
unique (single) element f(x)  R to each element x D
f: x
f(x) it means that f sends x to f(x)
1
it means that f sends x to
x2
f: x
f ( x)  y 
1
x2
R
 The set of x is called the "Domain" of the function (Df).
 The set of y is called the "Range" of the function (Rf).
 x & y are variables.
 x is independent variable.
 y is dependent variable.
Domain (Df): is the set of all possible inputs (x-values).
Range (Rf): is the set of all possible outputs (y-values).
To find Domain (Df) and the Range (Rf) the following points must be
noticed:
1. The denominator in a function must not equal zero (never divide by zero).
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2. The values under even roots must be positive.
Absolute Value Function: it is defined as:
 x if
y  x  x2  
 x if
x0
x0
Absolute Value Properties
1. |-a | = | a|
A number and its additive inverse or negative have
the same absolute value.
2. |ab | = | a||b |
The absolute value of a product is the product of the
absolute values.
3.
a
a

b
b
The absolute value of a quotient is the quotient of the
absolute values.
4. |a +b| ≤ |a | +| b| The triangle inequality. The absolute value of the sum
of two numbers is less than or equal to the sum of
their absolute values.
Absolute Values and Intervals
If a is any positive number, then
5. | x | = a
if and only if x = ±a
6. | x | < a
if and only if -a < x < a
7. | x | > a
if and only if x > a or x < - a
8. | x | ≤ a
if and only if -a ≤ x ≤ a
9. | x | ≥ a
if and only if x ≥ a or x ≤ - a
The Greatest Integer Function (Stepped Function):
The function whose values at any number x is the greatest integer less
than or equal to x is called greatest integer
function. It is denoted x , or in some books [x] or
[[x]] or int x
The greatest integer function:
[x] ≤ x <[x] +1
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Properties of greatest integer value:
1. [[[[x]]]] = [x]
2. [x+n] =[x] + n where n is integer
3. [x-n] =[x] - n where n is integer
4. -[x] ≠ [-x]
Function Defined in Pieces:
While
some
functions
defined
by
single
formulas, others are defined by applying different
formulas to different parts of their domains. One
example is the absolute value function
 x if
y  x  x2  
 x if
x0
x0
Sums, Difference, Product and Quotients of
Functions:
Definition: If f and g are functions, then we define the functions
 (f+g)(x)= f(x)+g(x)
Sum
Difference  (f-g)(x)= f(x)-g(x) or (g-f)(x)= g(x)-f(x)
Product
 (f.g)(x)= f(x).g(x)
Quotient
 (f/g)(x)= f(x)/g(x); where g(x) ≠0
…(1)
or (g/f)(x)= g(x)/f(x); where f(x) ≠0 …(2)
are also functions of (x), defined for any value of x that lies in both Df and Dg
( x  D f  Dg ), except the points which g(x) =0 in eq.(1) or f(x) =0 in eq.(2)
Composition of Functions:
Definition: If f and g are functions, the composite f o g "f composed with g"
or g o f "g composed with f" are defined by:
(f o g)(x) =f (g(x))
and
(g o f)(x) =g (f(x)) respectively.
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Graph of Functions (Graph of Curves):
To graph the curve of a function, we can follow the following steps:
1. Find the domain and range of the function.
2. Check the symmetry of the function
3. Find (if any found) points of intersection with x-axis and y-axis.
4. Choose some another points on the curve.
5. Draw s smooth line through the above points.
Symmetry Tests for Graphs:
If f(x,y) = 0 is any function then:
1. Symmetry about x-axis: If f(x,-y)= f(x,y)
2. Symmetry about y-axis: If f(-x,y)=f(x,y) It is called an even function.
3. Symmetry about the origin: If f(-x,-y)=f(x,y) It is called an odd function
DEFINITIONS
Even Function, Odd Function
A function y = f(x) is an
even function of x if f(-x) = f(x) symmetry about y-axis
odd function of x if f(-x) = -f(x) symmetry about origin
for every x in the function's domain.
Shifting, Shrinking and Stretching:
Shift formulas: (for c > 0)
Vertical shifts
y= f(x)+c
or
y-c= f(x)
y= f(x)-c
or
y+c= f(x)
Horizontal shifts
y= f(x+c)
y= f(x-c)
shifts the graph of f up by c units.
shifts the graph of f down by c units.
shifts the graph of f left by c units.
shifts the graph of f right by c units.
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Shrinking, Stretching and Reflecting Formulas:
(for c > 1)
y=c f(x)
Stretches the graph of f c units along y-axis.
y
1
f ( x)
c
Shrinks the graph of f c units along y-axis.
y= f(cx)
Shrinks the graph of f c units along x-axis.
x
y f( )
c
(for c = -1)
y=- f(x)
y= f(-x)
Stretches the graph of f c units along x-axis.
Reflects the graph of f across the x-axis.
Reflects the graph of f across the y-axis.
Trigonometric Functions
We measure angles in degrees, but in calculus it is usually best to use
radians.
1. Degree measure: One degree (1o) is the measure of an angle generated
by 1/360 of revolution.
2. Radian measure: The radian measure of the angle ACB at the center of
the unit circle (circle with radius equals one unit)
equals to the length of the arc that the angle cuts
from the unit circle.
If angle ACB cuts an arc A`B` from a second circle
centered at C, then circular sector A`CB` will be similar to
circular sector ACB. In particular,
Length of arc A`B`
Length of arc AB

Radius of sec ond circle Radius of first circle
In notations
s 
 
r 1


s
r
or
s=r
To find the relation between degree measure and radian measure, you know that
one circle equals 360o in degree and 2 in radians so:
2 radians= 360o

 radians= 180o
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 1o 

180
rad  0.01745rad
o
180
and 1rad     57 o17'44.8"
  
The six basic trigonometric functions:
sine: sin  
y
,
r
cosecant: csc 
x
r
cosine: cos  ,
tangent: tan  
secant: sec  
1
r
 ,
sin  y
1
r

cos  x
sin 
y
1
cos  x
 , cotangent: cot  


cos  x
tan  sin 
y
from Pythagoras theorem:
x2+ y2 = r2

x2  y2
1
r2

cos2sin2

x2 y 2

1
r2 r2
true for all values of  
When we divided eq.(1) by cos2yields:
cos 2 sin 2
1


2
2
cos  cos  cos 2

1tan2 sec2
And when we divided eq.(1) by sin2yields:
cos 2 sin 2
1


2
2
sin  sin  sin 2

cot2 +1 csc2
Identities:
- Periodicity: A function is periodic with period p if
f(x+p)= f(x)
for every value of x.
cos(±2) =cos
sin(±2) =sin
tan(±2) =tan
cot(±2) =cot
sec(±2) =sec
csc(±2) =csc
- Symmetry:


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Even functions
Odd functions
cos(-x)=cosx
sec(-x)=secx
sin(-x)=-sinx
tan(-x)=-tanx
cot(-x)=-cotx
csc(-x)=-cscx
- Shift formulas:
sin(x + /2) = cos(x);
cos(x + /2) = -sin(x)
sin(x - /2) = -cos(x);
cos(x - /2) = sin(x)
- Addition formulas:
cos(A+B)=cosA cosB – sinA sinB
sin(A+B)=sinA cosB + cosA sinB
- Double angle formulas:
cos2=cos2sin2
sin2 2.sin .cos
- Half angle formulas:
cos 2  
1  cos 2
2
sin 2  
1  cos 2
2
Graph of trigonometric functions:
1. y = sin x
 Domain and Range of the function
v
Df=(-∞,∞)
From Figure nearby, we conclude that:
-r ≤ v ≤ r
1 
v
1
r
(divide the inequality by r)

1  sin   1
Rf=[-1,1]
 Symmetry:
f(-x) = sin(-x) =-sinx = - f(x)
≠ f(x)
u
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So it is an odd function (it is symmetric about the origin).
 Additional points:

1
0
0
x
y

0
3
-1
2
0
2. y = cos x
 Domain and Range of the function
Df=(-∞,∞)
From Figure, we conclude that:
-r ≤ u ≤ r
1 
(divide the inequality by r)
u
1
r
1  cos  1

Rf=[-1,1]
 Symmetry:
f(-x) = cos(-x) =cosx = f(x)
≠- f(x)
So it is an even function (it is symmetric
about the y-axis).
 Additional points:
x
y
3. y = tan x =
sin x
cos x
 Domain and Range of the function
cosx ≠ 0 
Df= R\{ x  n

2
xn

2
; n=±1, ±3, ±5
; n=±1, ±3, ±5}
Rf=(-∞,∞)
 Symmetry:
f(-x) = tan(-x) =
sin(  x)  sin x
=-tanx ≠ f(x)

cos(  x)
cos x
0
1

0

-1
3
0
2
1
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=- f(x)
So it is an odd function (it is symmetric about the origin).
 Asymptotes:
To find vertical asymptote put the denominator equal to zero.
xn

cosx = 0

2
; n=±1, ±3, ±5
 Additional points:
x
y

±∞
0
0
4. y = cot x =

0
3
±∞
2
0
cos x
sin x
 Domain and Range of the function
x=n ; n=0, ±1, ±2, ±3

sinx ≠ 0
Df= R\{x=n; n=0, ±1, ±2, ±3}
Rf=(-∞,∞)
 Symmetry:
f(-x) = cot(-x) =
cos(  x)
cos x
=-cotx ≠ f(x)

sin(  x)  sin x
=- f(x)
So it is an odd function (it is symmetric about the origin).
 Asymptotes:
To find vertical asymptote put the denominator equal to zero.
sinx = 0

x≠n ; n=0, ±1, ±2, ±3
 Additional points:
x
y
0
±∞
5. y = sec x =

0

±∞
3
0
1
cos x
 Domain and Range of the function
2
±∞
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cosx ≠ 0
xn

Df= R\{ x  n


2
; n=±1, ±3, ±5
; n=±1, ±3, ±5}
2
From Figure, we conclude that:
-r ≤ u ≤ r
1 
(divide the inequality by r)
u
1
r
1  cos  1

 sec  1 
sec   1
or
cos  1


1
1
cos 
sec   1
Rf= R\(-1,1)
 Symmetry:
f(-x) = sec(-x) =
1
1
=sec x = f(x)

cos( x) cos x
≠- f(x)
So it is an even function (it is symmetric about the y-axis).
 Asymptotes:
To find vertical asymptote put the denominator equal to
zero.
cosx = 0
xn


2
; n=±1, ±3, ±5
 Additional points:
x
y
6. y = csc x =
0
1

±∞

-1
3
±∞
2
1
sin   1

1
sin x
 Domain and Range of the function
sinx ≠ 0

x≠n; n=0, ±1, ±2, ±3
Df= R\{x=n; n=0, ±1, ±2, ±3}
From Figure, we conclude that:
-r ≤ v ≤ r
1 
(divide the inequality by r)
v
1
r
 csc  1 

1  sin   1
csc  1
or

csc  1
1
1
sin 
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Rf= R\(-1,1)
 Symmetry:
f(-x) = csc(-x) =
1
1
=-cscx ≠ f(x)

sin(  x)  sin x
=- f(x)
So it is an odd function (it is symmetric about the
origin).
 Asymptotes:
To find vertical asymptote put the denominator
equal to zero.
sinx = 0

x=n ; n=0, ±1, ±2, ±3
 Additional points:
x
y
x3  x
y
x
0
±∞

1

±∞
3
-1
2
±∞