Calculus 1
Lecture 1
By:
Dr. Doaa Ezzat
Dr. Doaa Mahmoud
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Important Notes
Text Book: Calculus (third edition).
Grades: 60 final exam,
15 midterm,
20 Assignments,
5 quizzes
Sections: exercises on lectures.
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Introduction
What is Calculus?
Why Calculus?
Calculus in Computer Science
Calculus in Archaeology
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Introduction
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Introduction
R
Q Z
C N
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Introduction
Mathematical Symbols:-
1. ⊂ : subset
2. ∈ : element of
3. ⋃ : Union
4. ⋂ : intersection
5. ∀ : for each
6. ∃ : there exist
7. Ʃ : Summation
8. S.T. : such that
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[
Introduction
Mathematical Symbols:-
9. ∵ : since
10. ∴ : therefore
11. → : implies
12. ↔ : iff
13. ( ) : open interval
14. [ ] : closed interval
15. LHS : Left Hand Side
16. RHS : Right Hand Side
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Notes
1. X ∈ (a , b) : {x : a < x < b}
2. X ∈ [a , b] : {x : a ≤ x ≤ b}
3. X ∈ [a , ∞) : {x : x ≥ a}
4. X ∈ (-∞ , ∞) : {x : x ∈ R}
Inequalities عدم تساوي/تباين
• If a , b and c ∈ R, then:
1. If a > b and b > c → a > c
2. If a > b → a+c > b+c
3. If a > b → a-c > b-c
4. If a > b and c is +ve → ac > bc
5. If a > b and c is -ve → ac < bc
Inequalities عدم تساوي/تباين
6. If |a| = b → a = b or a = -b
7. If |a| < b → -b < a < b
8. If |a| > b → a > b or a < -b
Notes
1. If 𝑥 2 > b → | 𝑥 | > 𝑏
i.e. 𝑥 > 𝑏 or 𝑥 < - 𝑏
2
2. If 𝑥 < b → | 𝑥 | < 𝑏
i.e. - 𝑏 < 𝑥 < 𝑏
Example: Solve and represent on line of
numbers
4−3𝑥
Ex.1:- -5 ≤ <1
2
Solution:-
-10 ≤ 4-3x < 2 (By multiplying by 2)
-14 ≤ -3x < -2 (By subtracting 4)
14 ≥ 3x > 2 (By multiplying by -1)
14 2
≥x> (By dividing by 3)
3 3
2 14 2 14
<x≤ x∈ ( , ]
3 3 3 3
(
Example: Solve and represent on line of
numbers
Ex.2:- |2x-7| > 3
Solution:-
2x-7 > 3 or 2x-7 < -3
2x > 10 or 2x < 4
x>5 or x < 2
x ∈ (-∞ , 2) ⋃ (5 , ∞)
Or x ∈ R – [2 , 5]
) (
Example: Solve and represent on line of numbers
Ex.3:- 𝑥 2 + 2𝑥 − 8 ≥ 0 (Quadratic Inequality)
Solution:-
1. Turn it to equation: 𝑥 2 + 2𝑥 − 8 = 0
2. Find the solution: (x+4) (x-2) = 0
x = -4 or x = 2
3. Draw the line of numbers:
] [
4. Try values in all intervals: (x+4) (x-2) ≥ 0
If x = 0 false, If x = -10 true
If x = 4 true, At x = -4 and 2 true
x ∈ (-∞ , -4] ⋃ [2 , ∞)
(1.1) Review of
Functions
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Figure 1.1
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Functions
The independent variable is the variable
associated with the domain.
The dependent variable belongs to the range.
The graph of a function ƒ is the set of all points
(x, y) in the xy-plane that satisfy the equation
y = ƒ(x).
The argument of a function is the expression
on which the function works. For example, x is
the argument when we write ƒ(x).
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Figure 1.2
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Functions
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Figure 1.3 (a)
It is a function
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Figure 1.3 (b)
It is not a function
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Figure 1.3 (c)
It is a function
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Figure 1.3 (d)
It is not a function
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Example
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Figure 1.4
(a)
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Figure 1.5
(b)
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Figure 1.6
(c)
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Thank you
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