M1101 E
1. Real Numbers System
1.1 Real Numbers System R
1.1.1 Rational Numbers
Set of natural numbers: N = {0, 1, 2, ...}
Non-zero natural numbers: N∗ = N - {0}
The sum of two natural numbers n and m is defined as n + m ∈ N.
No solutions for n + m = 0 in N, necessitating the set of integers Z:
Z = {..., -2, -1, 0, 1, 2, ...}
Non-zero integers: Z∗ = Z - {0}.
The product of two integers n and m is defined as n × m.
No solutions for n × m = 1 in Z, leading to the definition of rational numbers Q:
Q = { p/q | (p, q) ∈ Z × N∗ }
Any rational number r ∈ Q is a ratio of integers p (numerator) and non-zero integer q (denominator).
For two rational numbers r = p/q and r₀ = p₀/q₀, r = r₀ iff pq₀ = qp₀.
1.2 Properties of Real Numbers
Trivial inclusions: N ⊂ Z ⊂ Q.
Not all physical quantities can be expressed with rational numbers; for example, the length of the diagonal of a unit square is √2, which is irrational.
A number is called a real number if it's rational (Q) or irrational (R - Q).
0 ∈ Q, yet 0 ∉ (R - Q), whereas √2 ∈ (R - Q) and √2 ∉ Q.
The inclusion is structured as N ⊂ Z ⊂ Q ⊂ R and Q ∩ (R - Q) = ∅.
1.3 Order of the Set of Real Numbers R
The field of real numbers R is equipped with the relation "less than or equal to".
The relation ≤ satisfies:
Reflexive: a ≤ a ∀a ∈ R
Antisymmetric: if (a ≤ b and b ≤ a), then a = b.
Transitive: if (a ≤ b and b ≤ c), then a ≤ c.
Total: either a ≤ b or b ≤ a ∀a, b ∈ R.
Notations:
a < b iff a ≤ b and a ≠ b
b ≥ a iff a ≤ b
b > a iff b ≥ a and b ≠ a.
1.4 Extended Real Number Line R
Extended R includes two unique elements not in R, denoted -∞ and +∞: R = R ∪ {-∞, +∞}.
Extended operations yield:
x + (+∞) = +∞; x + (-∞) = -∞.
x > 0: x(+∞) = +∞; x(-∞) = -∞.
x < 0: x(+∞) = -∞; x(-∞) = +∞.
(+∞) + (+∞) = +∞; (-∞) + (-∞) = -∞.
(+∞)(+∞) = +∞; (-∞)(-∞) = +∞; (+∞)(-∞) = -∞.
∀x ∈ R, -∞ < x < +∞.
(-∞) ≤ -∞ and (+∞) ≤ (+∞).
Indeterminate forms include (+∞) + (-∞), (-∞) + (+∞), 0(+∞), (+∞)0, 0(-∞), (-∞)0.
1.5 Intervals of R
Interval definitions:
Segment: [a, b] = {x ∈ R | a ≤ x ≤ b}.
Interval I is a subset of R that satisfies, ∀a, b ∈ I, a ≤ b: the segment [a, b] ⊂ I.
Types of intervals:
[a, b], ]a, b[, [a, b[, ]a, b], [a, +∞[, ]a, +∞[, ]-∞, a], ]-∞, a[, ]-∞, +∞[ = R.
Subsets defined:
R+ = [0, +∞[, R- = ]-∞, 0], R∗+ = ]0, +∞[, R∗- = ]-∞, 0[.
1.6 Absolute Value of a Real Number
Definition: |x| =
x if x ≥ 0
-x if x ≤ 0.
Properties:
|x| ≥ 0.
|x| = 0 iff x = 0.
x ≤ |x|.
|xy| = |x| |y|.
Triangle inequality: |x + y| ≤ |x| + |y|.
|x| − |y| ≤ |x - y|.
1.7 Distance on R
Definition: d(x, y) = |x − y|.
Properties:
Positivity: d(x, y) ≥ 0.
Symmetry: d(x, y) = d(y, x).
Separation: d(x, y) = 0 iff x = y.
Triangle inequality: d(x,z) ≤ d(x, y) + d(y, z).
1.8 Upper Bound and Lower Bound of a Subset of R
1.8.1 Upper Bound, Lower Bound of a Subset
Definitions:
M is an upper bound of A if ∀x ∈ A, x ≤ M.
m is a lower bound of A if ∀x ∈ A, m ≤ x.
Remarks:
An upper bound exists if any real number ≥ M is also an upper bound.
A lower bound exists if any real number ≤ m is also a lower bound.
Definitions of bounded subsets:
Upper bounded = has upper bound.
Lower bounded = has lower bound.
Bounded = has both.
1.8.2 Greatest Element, Least Element
Definitions:
M is the greatest element of A if M ∈ A.
m is the least element of A if m ∈ A.
Notation:
Max(A) = greatest element.
Min(A) = least element.
1.8.3 Supremum, Infimum
Definitions:
Supremum = least upper bound.
Infimum = greatest lower bound.
Remarks:
Supremum and infimum are unique even if they do not belong to A.
Example: A = [0,1[; supA = 1, infA = 0.
1.9 Integer Part of a Real Number
Definition of Integer Part E(x): unique integer p such that p ≤ x < p + 1.
Examples:
E(3.4) = 3, E(-2.1) = -2.
1.10 Neighborhood of a Real Number
Definition: Neighborhood N of x if it contains an open interval centered at x.
Properties:
If N is a neighborhood of x, then x ∈ N.
If N ⊂ M, then M is a neighborhood of x.
Intersection of neighborhoods of x is also a neighborhood.
Any open interval is a neighborhood of any of its points.
Neighborhoods of ±∞:
Neighborhood of +∞: ]A, +∞[, for A ∈ R.
Neighborhood of -∞: ]-∞, A[, for A ∈ R.
1.11 Adherent (Closure) Point
Definition: a is an adherent point of A if any neighborhood of a intersects A.
Example: A = ]0, 1[; adherent points include [0,1].
Theorem: Any real number is an adherent point to Q; Q is dense in R.
2. Sequences
2.1 Definition of a Sequence
Definition: An infinite numerical sequence is defined as a mapping from N (or an infinite subset of N) to R: u: N → R.
General term denoted as u(n).
2.1.1 Bounded Sequences
A sequence (un) is bounded above if ∃M ∈ R; ∀n ∈ N, un ≤ M.
A sequence (un) is bounded below if ∃m ∈ R; ∀n ∈ N, un ≥ m.
A sequence is bounded if it is both bounded above and below.
2.1.2 Arithmetic and Geometric Sequences
Arithmetic Sequence: Difference between any two consecutive terms is constant (common difference d).
Example: u0 = 2, u1 = u0 + 3 = 5, u2 = u1 + 3 = 8, ...
Geometric Sequence: Ratio of any two consecutive terms is constant (common ratio r).
Example: u0 = 2, u1 = u0 × 3 = 6, u2 = u1 × 3 = 18, ...
2.2 Limit of a Sequence
Definition: A sequence (un) tends to l as n -> +∞ if and only if ∀ε > 0, ∃N ∈ N such that ∀n ∈ N, (n ≥ N ⇒ |un - l| ≤ ε).
Convergent Sequence: un tends to a finite limit l as n tends to +∞.
Divergent Sequence: Not convergent.
2.2.1 Eventually Constant Sequence
A sequence is eventually constant if it is constant beyond some index n.
2.2.2 Operations and Limits with Sequences
Theorem: Limits and arithmetic operations on sequences.
2.2.3 Sign of the Limit
Theorem: If all terms of a convergent sequence (un) are positive, then the limit l ≥ 0.
2.3 Monotonic Sequence
A sequence is increasing if un+1 - un ≥ 0.
If it is increasing and bounded above, then it is convergent.
2.4 Adjacent Sequences
Two real sequences (un) and (vn) are adjacent if:
(un) is increasing.
(vn) is decreasing.
vn - un tends to 0 as n tends to ∞.
2.5 Subsequences
A subsequence is defined by a strictly increasing mapping.
3. Functions
3.1 Functions and Mappings
Definition: A function f from set E to set F associates each element of E with a unique element in F.
3.2 The Set RI
Set of mappings from I to R.
3.3 Order in RI
The relation ≤ defines an order on RI.
3.4 Monotonic Mappings
Monotonic function is either only increasing or decreasing.
3.5 Parity and Periodicity
Even functions: f(-x) = f(x).
Odd functions: f(-x) = -f(x).
3.6 Bounded Functions
A function is bounded if its image set is bounded.
3.7 Examples of Functions
Study specific functions such as f(x) = x², explaining properties of each as defined.
4. Limits and Continuity
4.1 Limit of a Function
Definition: A function has a limit if the values approach that limit as the input approaches some point.
4.2 Right and Left Side Limits
Definitions for limits approaching from different directions.
4.3 Examples
Provide specific limits to illustrate the concept.
5. Differentiation
5.1 Derivatives
Definition: A function is differentiable at point a if the limit of the difference quotient exists.
5.2 Algebraic Properties
Properties relating to derivatives in calculus.
5.3 Higher Order Derivatives
Definition of nth derivatives and differentiability.
6. Transcendental Functions
6.1 Inverse Functions
Define relationships between a function and its inverse.
6.2 Natural Logarithm
Definitions and properties of the natural logarithm function.
6.3 Exponential Function
Properties and definitions of the exponential function.
6.4 Power Functions
Definition and properties of power functions.
6.5 Hyperbolic Functions
Definition and properties of hyperbolic sine and cosine functions.
6.6 Inverse Circular Functions
Definitions for arcsine, arccosine, and arctangent functions and their properties.
7. Taylor Expansions and Finite Expansions
7.1 Introduction
Provide foundational concepts leading to finite expansions.
7.2 Taylor Expansions
Definitions and theorems related to Taylor expansions.
7.3 Finite Expansions
Define and explore polynomial approximations of functions.