Summarize for Principles of Mathematics (Math 500) it's Lectures for students of Computer Science and especially students of graduate studies of the Institute of Statistical Studies and Research - Cairo University

1. Math500 Sections
Functions
‫الدالة‬ ‫تعريف‬‫وليكن‬ ‫متغيرين‬ ‫بين‬ ‫العالقة‬ ‫هى‬ :𝑋‫و‬𝑌‫متغير‬ ‫والثانى‬ ‫مستقل‬ ‫أحدهما‬،،‫ال‬ ‫إن‬ ‫هنا‬ ‫ونجد‬𝑌‫قيمته‬ ‫إلن‬ ‫تابع‬ ‫متغير‬
‫المتغير‬ ‫قيمة‬ ‫بتغير‬ ‫تتغير‬𝑋..‫البسيطة‬ ‫الدالة‬ ‫هذه‬ ‫فى‬ ً‫ال‬‫فمث‬
‫لكل‬‫الدوال‬‫لها‬(𝐷𝑜𝑚𝑎𝑖𝑛( ‫و‬ )𝑅𝑎𝑛𝑔𝑒)
Algebraic functions
coefficients.lution of a polynomial equation with integerare functions that can be expressed as the so
A. Polynomials: Can be generated by addition, multiplication, and exponentiation (^) alone.
1. Constant function: polynomial of degree zero - graph is a horizontal straight line
𝑓( 𝑥) = a - 𝑓( 𝑥) = 3
2. Linear function: First degree polynomial - graph is a straight line
𝑓( 𝑥) = ax + b - 𝑓( 𝑥) = 2𝑥 + 3
3. Quadratic function: Second degree polynomial - graph is a parabola
𝑓 ( 𝑥) = a x2 + bx + c - 𝑓 ( 𝑥 ) = 2 𝑥2 + 3𝑥 + 4
4. Cubic function: Third degree polynomial
𝑓 ( 𝑥) = 𝑎 𝑥3 + 𝑏 𝑥2 + 𝑐𝑥 + 𝑑 - 𝑓 ( 𝑥 ) = 2 𝑥3 + 3 𝑥2 + 4𝑥 + 5
5. Quartic function: Fourth Quintic function: Fifth Sextic function: Sixth
B. Root function 𝑓 ( x ) = √x = 𝑓 ( 𝑥 ) = 𝑥
1
2⁄
C. Logarithmic function 𝑓 ( 𝑥 ) = 𝑙𝑒𝑛 (𝑥) = 𝑓 ( 𝑥 ) = 𝑙𝑜𝑔 (𝑥)
D. Exponential function 𝑓 ( x ) = ex e ‫قيمتها‬‫رقم‬‫ثابت‬‫وهو‬=2.7
E. Rational / Fractional functions A ratio of two polynomials 𝑓 ( 𝑥 ) =
g ( 𝑥 )
h ( 𝑦 )
‫الـ‬𝑋‫المستقل‬ ‫المتغير‬ ‫هو‬‫الـ‬𝑌‫التابع‬ ‫المتغير‬ ‫هو‬
‫الثابتة‬ ‫الدالة‬
𝐹 ( 𝑥) = 𝑎
‫الخطية‬ ‫الدالة‬
𝐹 (𝑥) = 𝑎 𝑥 + 𝑏
‫التربيعية‬ ‫الدالة‬
𝐹 (𝑥) = 𝑎 𝑥2 + 𝑏 𝑥 + 𝑐
‫التكعيبية‬ ‫الدالة‬
𝑏 𝑥2 + 𝑐𝑥 + 𝑑𝑎 𝑥3 +𝐹 (𝑥) =
‫صحيحة‬ ‫معامالت‬ ‫مع‬ ‫الحدود‬ ‫متعددة‬ ‫معادلة‬ ‫لحل‬ ‫عنها‬ ‫التعبير‬ ‫يمكن‬ ‫التى‬ ‫الدوال‬ ‫هى‬ ‫الجبرية‬ ‫الدوال‬
2
3
4
-2
4
9
16
25
Domain Co-domain
Range
𝑓( 𝑥) = 𝑦
‫الثابتة‬ ‫الدالة‬
‫الخطية‬ ‫الدالة‬
‫التربيعية‬ ‫الدالة‬)‫المكافئ‬ ‫(القطع‬
‫التكعيبية‬ ‫الدالة‬
‫الجذرية‬ ‫الدالة‬
‫اللوغاريتمية‬ ‫الدالة‬
‫االسية‬ ‫الدالة‬
‫المنطقية‬ / ‫الكسرية‬ ‫الدالة‬
‫ملحوظة‬‫يجب‬ :‫ان‬‫تكون‬a‫ال‬‫تساوى‬‫صفر‬‫جميع‬ ‫فى‬
‫إلنه‬ ‫الدوال‬‫فى‬‫هذه‬‫الحالة‬‫ستتحول‬‫الدالة‬‫مثل‬‫الدالة‬‫الثابته‬
‫ال‬ ‫تحديد‬ ‫فيمكن‬‫ـ‬(𝑫𝒐𝒎𝒂𝒊𝒏‫المحور‬ ‫خالل‬ ‫من‬ )𝒙
‫ال‬ ‫تحديد‬ ‫و‬‫ـ‬(𝑹𝒂𝒏𝒈𝒆‫المحور‬ ‫خالل‬ ‫من‬ )𝒚
D = R
R = {a}

2. Linear function: First degree polynomial - graph is a straight line
Quadratic function: Seconddegree polynomial - graphisa parabola
Cubic function: Third degree polynomial
Domain = R , Range = R
𝑓 ( 𝑥 ) = a 𝑥2 + 𝑏𝑥 + 𝑐
+ (X + n) 𝟐
+ y
‫ـــ‬ ‫ـــ‬ ‫ـــ‬
𝑓( 𝑥) = 𝑎𝑥 + 𝑏
Slope
‫الخط‬ ‫ميل‬
y-Intersect
‫المقطوع‬ ‫الجزأ‬‫محور‬ ‫من‬y
+ (X + n) 𝟏
+ y
‫ـــ‬ ‫ـــ‬ ‫ـــ‬
x - Domain
y / b - Range y / b - Range
x - Domain
a-, x+,y+
a-, x+,y0a-, x-,y0
a-, x-,y+
a-, x-,y- a-, x+,y-
a-, x0,y+
a-, x0,y0
a-, x0,y-
a+, x+,y+
a+, x+,y0a+, x-,y0
a+, x-,y+
a+, x-,y- a+, x+,y-
a+, x0,y+
a+, x0,y0
a+, x0,y-
a-, x0,y0
a-, x+,y+
a-, x+,y0a-, x-,y0
a-, x-,y+
a-, x-,y- a-, x+,y-
a-, x0,y+
a-, x0,y-
Domain = R , Range = R
y / c - Range
x - Domain
y / c - Range
x - Domain
y / c - Range
x - Domain
a-, x+,y+
a-, x+,y0a-, x-,y0
a-, x-,y+
a-, x-,y- a-, x+,y-
a-, x0,y+
a-, x0,y0
a-, x0,y-
a+, x0,y0 a+, x+,y0
a+, x+,y+
a+, x-,y0
a+, x-,y+
a+, x-,y- a+, x+,y-
a+, x0,y+
a+, x0,y-
𝑓 ( 𝑥 ) = 𝑎 𝑥3 + 𝑏 𝑥2 + 𝑐𝑥 + 𝑑
Domain = R , Range = Ry / c - Range
x - Domain
+ (X + n) 𝟐
+ y
‫ـــ‬ ‫ـــ‬ ‫ـــ‬
: ‫ملحوظة‬‫قيمة‬ ‫كانت‬ ‫كلما‬a‫كبيرة‬‫كلما‬‫القوس‬ ‫قل‬‫محور‬ ‫على‬ ‫وانضم‬y‫صحيح‬ ‫والعكس‬
Polynomials
a+, x0,y0
a+, x+,y0
a+, x+,y+
a+, x-,y0
a+, x-,y+
a+, x-,y- a+, x+,y-
a+, x0,y+
a+, x0,y-

3. Root function
Root function
𝑓 ( x ) =+ √x ||𝑓 ( 𝑥 ) = 𝑥
1
2⁄
Domain = R , Range = Ry / c - Range y / c - Range
x - Domain
a-, x+,y+
a-, x+,y0a-, x-,y0
a-, x-,y+
a-, x-,y- a-, x+,y-
a-, x0,y+
a-, x0,y0
a-, x0,y-
+ (X + n) 𝟐
+ y
‫ـــ‬ ‫ـــ‬ ‫ـــ‬
a-, x+,y+
a-, x+,y0a-, x-,y0
a-, x-,y+
a-, x-,y- a-, x+,y-
a-, x0,y+
a-, x0,y0
a-, x0,y-
y / c - Range
x - Domain
x - Domain
a+, x0,y0 a+, x+,y0
a+, x+,y+
a+, x-,y0
a+, x-,y+
a+, x-,y- a+, x+,y-
a+, x0,y+
a+, x0,y-
a+, x0,y0 a+, x+,y0
a+, x+,y+
a+, x-,y0
a+, x-,y+
a+, x-,y- a+, x+,y-
a+, x0,y+
a+, x0,y-
y / c - Range
x - Domain
Domain = R , Range = R
+ (X + n) 𝟐
+ y
‫ـــ‬ ‫ـــ‬ ‫ـــ‬
y / c - Range
x - Domain
y / c - Range
x - Domain
F(x) = √1 − 𝑥2
𝑥 =
4
3
F(x) = - √2𝑥 + 6
𝑥 = −3
𝑥 = ±1−𝑥2
≥ −1
−1 ≤ 𝑥 ≤ 1
x - Domain
y / c - Range
F(x) = √𝑥2 − 1 𝑥2
≥ 1 𝑥 = ±1
−1 ≥ 𝑥 ≥ 1
y / c - Range
F(x) = √𝑥3 −1
𝑥 = 1
‫و‬ ‫السالبة‬ ‫االشارة‬ ‫تحمل‬ ‫ان‬ ‫يمكن‬
‫تربيع‬ ‫الدالة‬ ‫الن‬ ‫الموجبة‬ ‫االشارة‬‫ية‬
‫واحدة‬ ‫اشارة‬ ‫ستحمل‬
‫تكعيبي‬ ‫الدالة‬ ‫الن‬‫ة‬
x - Domain

4. Rational / Fractional functions A ratio of two polynomials 𝑓 ( 𝑥 ) =
g ( 𝑥 )
h ( 𝑦 )
x - Domain
Rational / Fractional functions
y / c - Range
±
𝟏
𝒙
+ 𝒏 ± y
‫ـــ‬ ‫ـــ‬ ‫ـــ‬x - Domain
y / c - Range
Domain = R-{0} ,Range = R

5. Composition of Function
‫الدوال‬ )‫(تحصيل‬ ‫تركيب‬
(f ∘ g)(x) = f (g(x))
(g ∘ f)(x) = g (f (x))
Ex: Given f(x) =5x-4 and g(x) = x2
+3 find (f ∘ g) (x) and (g∘ f) (x)
(f ∘ g)(x) = f (g(x)) = f (x2
+3)
= 5(x2
+3)-4
= 5x2
+15-4
= 5x2
+11
Ex: Given f(𝑥) = √ 𝑥 + 23
and g(𝑥) = 𝑥3
− 2 find (f ∘ g) (𝑥) and (g∘ f) (𝑥)
Ex: Given f(x) = √ 𝑥 − 1 and g(x) = 2𝑥2
+ 3 find (f ∘ g) (𝑥) and (g∘ f) (𝑥)
(g ∘ f)(x) = g (f(x)) = g (5x − 4)
= (5x − 4)2
+ 3
= 25x2
- 40x + 16 +3
= 25x2
- 40x + 19
(f ∘ g)(𝑥) = f (g(𝑥)) = f (𝑥3
− 2 )
= √(𝑥3 − 2 ) + 23
= √𝑥33
= 𝑥
(g ∘ f)( 𝑥) = g (f(𝑥)) = g (√ 𝑥 + 2
3
)
= (√ 𝑥 + 2
3
)3
- 2
= 𝑥 + 2 − 2
= 𝑥
(f ∘ g)(𝑥) = f (g(𝑥)) = f (2𝑥2
+ 3)
= √(2𝑥2 + 3) − 1
= √2𝑥2 + 2
(g ∘ f)( 𝑥) = g (f(𝑥)) = g ( √ 𝑥 − 1)
= 2( √ 𝑥 − 1)2
+3
= 2(x-1)+3
= 2𝑥 − 2 + 3
= 2x+1
‫قيمة‬ ‫تعويض‬g‫دالة‬ ‫فى‬‫االصلية‬f
‫قيمة‬ ‫تعويض‬f‫دالة‬ ‫فى‬‫االصلية‬g

6. Differentiation- Derivation
‫التفاضل‬ - ‫شتقاق‬ ‫اال‬
y= f (x) f(x) =𝑥2 𝑑𝑦
𝑑𝑥
= 𝑦 𝑥
= 𝑓′
(x) =
𝑑𝑓(𝑥)
𝑑𝑥
Rules:
1- F(x) = C , 𝑦′ = 0
2- y = 𝑥 𝑛 , 𝑦′ = n𝑥 𝑛−1
3- y = ( 𝑥) 𝑛 , 𝑦′ = n(𝑥) 𝑛−1. 𝑥′
4- y = 𝑒 𝑥 , 𝑦′ = 𝑒 𝑥. 𝑥′.ln 𝑥
5- Ln (x ) , 𝑦′ =
𝑥′
𝑥
| ln(A.B) =ln A + lnB
| ln(A/B) =ln A - ln B
| lnAn = n(lnA)
6-log 𝑎(𝑥) , 𝑦′ =
𝑥′
𝑙𝑛𝑎.𝑥
7- y = f.g , 𝑦′ = (𝑓′.𝑔) + (𝑓. 𝑔′)
8- y =
𝑓
𝑔
, 𝑦′ =
(𝑓′.𝑔)−(𝑓.𝑔′)
(𝑔)2
9- √ 𝑥 𝑥 = 𝑥
1
2 , 𝑦′=
1
2
𝑥
−1
2 =
1
2√ 𝑥
=
𝑥′
2 √𝑥2−12 =
1
2√ 𝑥
10-
1
√ 𝑥 𝑥
=
1
𝑥
1
2
= 𝑥−
1
2 ,
1
2
𝑥−
3
2 =
1
2 √𝑥32
=
𝑥′
2 √𝑥2+12 =
1
2 √𝑥32
11-
‫اخري‬ ‫قيمة‬ ‫على‬ ‫قيمة‬ ‫تغيير‬ ‫معدل‬ ‫هو‬
‫صفر‬ ‫فالمشتقة‬ ‫ثابتة‬ ‫الدالة‬ ‫كانت‬ ‫اذا‬Ex: y = 3 , 𝑦′= Zero
Ex: y = 𝑥5 , 𝑦′= 5𝑥4
Ex: y =√ 𝑥 → 𝑦 = 𝑥
1
2 , 𝑦′ =
1
2
𝑥
1
2
−1
=
1
2√ 𝑥
tan(𝑥) =
sin 𝑋
cos 𝑋
,
1
𝑐𝑜𝑠2 𝑋
− sin(𝑥)
cos(𝑥)
sin(𝑥)
− cos(𝑥)