MATHEMATICS
by Julia Vasiliu
Mathematics
Summary
TOPIC 1 ...................................................................................................................................................................... 8
NUMBERS ...................................................................................................................................................................... 8
proof ......................................................................................................................................................................... 9
COMPLETENESS property of R ................................................................................................................................... 9
TOPIC 2 .................................................................................................................................................................... 10
LOGIC ........................................................................................................................................................................... 10
defini9on ................................................................................................................................................................. 10
defini9on ................................................................................................................................................................. 10
defini9on ................................................................................................................................................................. 10
defini9on ................................................................................................................................................................. 10
defini9on ................................................................................................................................................................. 11
the contradic9on strategy ....................................................................................................................................... 11
De Morgan laws ...................................................................................................................................................... 12
TOPIC 3 .................................................................................................................................................................... 12
SET THEORY.................................................................................................................................................................. 12
defini9on ................................................................................................................................................................. 12
defini9on ................................................................................................................................................................. 12
proprie9es: .............................................................................................................................................................. 13
defini9on ................................................................................................................................................................. 13
defini9on ................................................................................................................................................................. 13
proprie9es ............................................................................................................................................................... 13
power set ................................................................................................................................................................ 14
defini9on ................................................................................................................................................................. 14
defini9on ................................................................................................................................................................. 14
logic and the set theory in comparison ................................................................................................................... 14
TOPIC 4 .................................................................................................................................................................... 15
RELATIONS ................................................................................................................................................................... 15
defini9on ................................................................................................................................................................. 15
defini9on ................................................................................................................................................................. 15
proper9es ................................................................................................................................................................ 15
defini9on ................................................................................................................................................................. 16
TOPIC 5 .................................................................................................................................................................... 16
AXIOMATIZATION OF R ................................................................................................................................................. 16
Dedekind axiom ...................................................................................................................................................... 16
proposi9on .............................................................................................................................................................. 17
exercise ................................................................................................................................................................... 18
INTERVALS IN THE REAL LINE R ..................................................................................................................................... 18
defini9on ................................................................................................................................................................. 18
defini9on ................................................................................................................................................................. 19
defini9on ................................................................................................................................................................. 19
defini9on ................................................................................................................................................................. 19
defini9on ................................................................................................................................................................. 20
defini9on ................................................................................................................................................................. 20
defini9on ................................................................................................................................................................. 20
defini9on ................................................................................................................................................................. 20
defini9on ................................................................................................................................................................. 20
defini9on ................................................................................................................................................................. 20
1
defini9on ................................................................................................................................................................. 20
THEOREM: EXISTENCE OF THE SUPREMUM ............................................................................................................................. 21
proof ....................................................................................................................................................................... 21
THEOREM: CHARACTERIZATION OF THE SUPREMUM.................................................................................................................. 21
proof ....................................................................................................................................................................... 21
defini9on ................................................................................................................................................................. 22
THEOREM: CANTOR’S THEOREM ......................................................................................................................................... 22
proof ....................................................................................................................................................................... 23
TOPIC 6 .................................................................................................................................................................... 24
TOPOLOGICAL CONCEPTS IN R ..................................................................................................................................... 24
defini9on ................................................................................................................................................................. 24
defini9on ................................................................................................................................................................. 24
NEIGHBORHOOD OF +¥ ............................................................................................................................................... 25
defini9on ................................................................................................................................................................. 25
defini9on ................................................................................................................................................................. 25
THEOREM: BOLZANO – WEIRSTRASS .................................................................................................................................... 25
defini9on ................................................................................................................................................................. 25
TOPIC 7 .................................................................................................................................................................... 26
RELATIONS BETWEEN SETS: FUNCTIONS...................................................................................................................... 26
defini9on ................................................................................................................................................................. 26
defini9on ................................................................................................................................................................. 28
examples ................................................................................................................................................................. 28
defini9on ................................................................................................................................................................. 29
defini9on ................................................................................................................................................................. 30
examples: reciprocal func9ons ................................................................................................................................ 31
INJECTIVE AND SURJECTIVE FUNCTIONS ...................................................................................................................... 31
defini9on ................................................................................................................................................................. 31
defini9on ................................................................................................................................................................. 32
examples ................................................................................................................................................................. 32
defini9on ................................................................................................................................................................. 33
defini9on ................................................................................................................................................................. 33
example ................................................................................................................................................................... 33
................................................................................................................................................................................ 34
examples ................................................................................................................................................................. 34
example ................................................................................................................................................................... 35
COMPOSITION OF MAPPING........................................................................................................................................ 35
example ................................................................................................................................................................... 36
defini9on ................................................................................................................................................................. 36
example ................................................................................................................................................................... 36
BOUNDED, EVEN, ODD FUNCTIONS ............................................................................................................................. 37
defini9on ................................................................................................................................................................. 37
defini9on ................................................................................................................................................................. 37
................................................................................................................................................................................ 37
examples of even func9ons ..................................................................................................................................... 37
examples of odd func9ons ...................................................................................................................................... 37
MONOTONIC FUNCTIONS ............................................................................................................................................ 37
defini9on ................................................................................................................................................................. 37
RELATION BETWEEN MONOTONIC AND INJECTIVE FUNCTIONS .................................................................................. 38
proposi9on .............................................................................................................................................................. 38
proof ....................................................................................................................................................................... 38
................................................................................................................................................................................ 39
contra-example ....................................................................................................................................................... 39
defini9on ................................................................................................................................................................. 39
TOPIC 8 .................................................................................................................................................................... 40
TRIGONOMETRIC FUNCTIONS ..................................................................................................................................... 40
2
defini9on ................................................................................................................................................................. 40
defini9on ................................................................................................................................................................. 41
RADIANS AND DEGREES ............................................................................................................................................... 42
SYMMETRY................................................................................................................................................................... 42
BOUNDENESS OF SINE AND COSINE ............................................................................................................................. 43
DOMAINS AND ZEROS .................................................................................................................................................. 43
GRAPH OF SINE ............................................................................................................................................................ 45
GRAPH OF COSINE........................................................................................................................................................ 45
GRAPH OF TANGENT .................................................................................................................................................... 46
TRIGONOMETRIC FORMULA ........................................................................................................................................ 46
TRIGONOMETRIC INVERSE FUNCTIONS ....................................................................................................................... 46
exercise ................................................................................................................................................................... 47
TOPIC 9 .................................................................................................................................................................... 48
CONTINUOUS FUNCTIONS ........................................................................................................................................... 48
DEFINITION OF CONTINUITY ........................................................................................................................................ 49
defini9on ................................................................................................................................................................. 49
example ................................................................................................................................................................... 50
defini9on ................................................................................................................................................................. 50
THEOREM OF THE SIGN PERMANENCE FOR CONTINUOUS FUNCTION ........................................................................ 50
proof ....................................................................................................................................................................... 51
THEOREMS AND CONTINUITY ...................................................................................................................................... 51
theorem (con9nuity of the sum) ............................................................................................................................. 51
proof ....................................................................................................................................................................... 51
theorem (con9nuity of the product)........................................................................................................................ 52
proof ....................................................................................................................................................................... 52
theorem (con9nuity of the reciprocal) .................................................................................................................... 53
proof ....................................................................................................................................................................... 53
corollary .................................................................................................................................................................. 53
proof ....................................................................................................................................................................... 53
theorem (con9nuity of the composite func9on)...................................................................................................... 54
proof ....................................................................................................................................................................... 54
theorem (con9nuity of monotone func9ons) .......................................................................................................... 54
theorem (con9nuity of the inverse func9on) ........................................................................................................... 54
theorem (Bolzano’s theorem) .................................................................................................................................. 54
proof ....................................................................................................................................................................... 55
theorem (intermediate value theorem) .................................................................................................................. 56
proof ....................................................................................................................................................................... 56
CONTINUITY ................................................................................................................................................................. 56
theorem (of compactness) ...................................................................................................................................... 56
theorem (Weierstrass theorem) .............................................................................................................................. 56
theorem (Darboux theorem) ................................................................................................................................... 56
TOPIC 10 .................................................................................................................................................................. 57
LIMITS .......................................................................................................................................................................... 57
defini9on ................................................................................................................................................................. 58
CHARACTERIZATION OF CONTINUITY BY USING THE NOTION OF LIMIT ....................................................................... 58
extension of a func9on by con9nuity ...................................................................................................................... 58
defini9on ................................................................................................................................................................. 59
defini9on ................................................................................................................................................................. 60
defini9on ................................................................................................................................................................. 60
defini9on ................................................................................................................................................................. 61
THEOREM (UNIQUENESS OF THE LIMIT) ...................................................................................................................... 61
proof ....................................................................................................................................................................... 61
THEOREM (OF THE SIGN PERMANENCE FOR LIMITS) ............................................................................................................... 62
proof ....................................................................................................................................................................... 62
THEOREM (OF COMPARISON) ........................................................................................................................................... 62
proof ....................................................................................................................................................................... 62
3
THEOREM (SANDWICH THEOREM) ..................................................................................................................................... 63
proof ....................................................................................................................................................................... 63
example ................................................................................................................................................................... 63
THEOREM (LIMIT OF THE COMPOSITION OF FUNCTIONS) ........................................................................................................ 64
OPERATIONS WITH LIMITS ........................................................................................................................................... 64
proposi9on .............................................................................................................................................................. 64
proposi9on .............................................................................................................................................................. 64
proof ....................................................................................................................................................................... 64
proposi9on .............................................................................................................................................................. 65
proof ....................................................................................................................................................................... 65
proposi9on .............................................................................................................................................................. 65
proof ....................................................................................................................................................................... 65
proposi9on .............................................................................................................................................................. 66
proof ....................................................................................................................................................................... 66
THEOREM (LIMIT OF RATIONAL FUNCTIONS) ........................................................................................................................ 67
THEOREM (LIMIT FOR MONOTONE FUNCTIONS) .................................................................................................................... 67
proof ....................................................................................................................................................................... 68
APPENDIX ................................................................................................................................................................ 68
THE NOTION OF LIMIT .................................................................................................................................................. 68
RELEVANT LIMITS FOR TRIGONOMETRIC FUNCTIONS ................................................................................................. 70
1. ............................................................................................................................................................................. 70
proof (1.) ................................................................................................................................................................. 70
2. ............................................................................................................................................................................. 70
proof (2.) ................................................................................................................................................................. 71
3. ............................................................................................................................................................................. 71
proof (3.) ................................................................................................................................................................. 71
4. ............................................................................................................................................................................. 71
proof (4.) ................................................................................................................................................................. 71
5. ............................................................................................................................................................................. 72
proof (5.) ................................................................................................................................................................. 72
TOPIC 11 .................................................................................................................................................................. 72
EXPONENTIAL AND LOGARITHMIC FUNCTIONS ........................................................................................................... 72
proof ....................................................................................................................................................................... 74
THE EXPONENTIAL FUNCTION ..................................................................................................................................... 74
proper9es ................................................................................................................................................................ 75
THE LOGARITHMIC FUNCTION ..................................................................................................................................... 76
proper9es ................................................................................................................................................................ 76
ALGEBRIC PROPPERTIES FOR THE LOGARITHMIC FUNCTIONS ..................................................................................... 78
1. ............................................................................................................................................................................. 78
proof (1.) ................................................................................................................................................................. 78
2. ............................................................................................................................................................................. 78
proof (2.) ................................................................................................................................................................. 78
3. ............................................................................................................................................................................. 78
proof (3.) – change of the basis .............................................................................................................................. 78
LIMITS WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS ...................................................................................... 79
1. ............................................................................................................................................................................. 79
proof (1.) ................................................................................................................................................................. 79
2. ............................................................................................................................................................................. 81
proof (2.) ................................................................................................................................................................. 81
3. ............................................................................................................................................................................. 81
proof (3.) ................................................................................................................................................................. 81
4. ............................................................................................................................................................................. 82
proof (4.) ................................................................................................................................................................. 82
5. ............................................................................................................................................................................. 82
proof (5.) ................................................................................................................................................................. 82
6. ............................................................................................................................................................................. 83
4
proof (6.) ................................................................................................................................................................. 83
examples of limits of type 𝒇(𝒙)𝒈(𝒙) ...................................................................................................................... 84
GRAPHICAL INTERPRETATION ...................................................................................................................................... 85
TOPIC 12 .................................................................................................................................................................. 88
DIFFERENTIAL CALCULUS ............................................................................................................................................. 88
defini9on ................................................................................................................................................................. 88
defini9on ................................................................................................................................................................. 88
defini9on ................................................................................................................................................................. 88
example 1 ................................................................................................................................................................ 89
example 2 ................................................................................................................................................................ 89
GEOMETRICAL INTERPETATION OF THE DERIVATIVE .................................................................................................... 90
example in economic ............................................................................................................................................... 90
THEOREM (“DIFFERENTIABILITY IMPLIES CONTINUITY”) .......................................................................................................... 91
proof ....................................................................................................................................................................... 91
contra-examples...................................................................................................................................................... 91
defini9on ................................................................................................................................................................. 91
BASIC DERIVATIVES ...................................................................................................................................................... 92
1. ............................................................................................................................................................................. 92
proof (1.) ................................................................................................................................................................. 92
2. ............................................................................................................................................................................. 92
proof (2.) ................................................................................................................................................................. 92
3. ............................................................................................................................................................................. 92
proof (3.) ................................................................................................................................................................. 92
4. ............................................................................................................................................................................. 93
proof (4.) ................................................................................................................................................................. 93
5. ............................................................................................................................................................................. 93
proof (4.) ................................................................................................................................................................. 93
DERIVATION RULES ...................................................................................................................................................... 94
THEOREM (deriva9ve of the sum) ........................................................................................................................... 94
proof ....................................................................................................................................................................... 94
THEOREM (deriva9ve of the product) ..................................................................................................................... 94
proof ....................................................................................................................................................................... 94
THEOREM (deriva9ve of the reciprocal) .................................................................................................................. 95
proof ....................................................................................................................................................................... 95
THEOREM (deriva9ve of the ra9o) .......................................................................................................................... 95
proof ....................................................................................................................................................................... 95
THEOREM (deriva9ve of the composi9on of func9ons) .......................................................................................... 96
THEOREM (deriva9ve of the inverse func9on) ........................................................................................................ 96
proof ....................................................................................................................................................................... 97
THEOREMS FOR DIFFERENTIABLE FUNCTIONS ............................................................................................................ 98
defini9on ................................................................................................................................................................. 98
defini9on ................................................................................................................................................................. 98
THEOREM (Fermat’s theorem) ................................................................................................................................ 99
proof ....................................................................................................................................................................... 99
THEOREM (Rolle’s theorem) .................................................................................................................................. 101
proof ..................................................................................................................................................................... 101
THEOREM (Cauchy’s theorem) .............................................................................................................................. 101
proof ..................................................................................................................................................................... 101
THEOREM (Lagrange’s theorem)........................................................................................................................... 102
proof ..................................................................................................................................................................... 102
geometrical interpreta9on .................................................................................................................................... 102
consequences of Lagrange’s theorem ................................................................................................................... 103
proof (1.) ............................................................................................................................................................... 103
proof (2.) ............................................................................................................................................................... 103
proof (3.) ............................................................................................................................................................... 104
THEOREM (DE L’HÔPITAL’S THEOREM) ............................................................................................................................. 104
proof ..................................................................................................................................................................... 104
5
THEOREM (ON THE LIMIT OF THE DERIVATE) ...................................................................................................................... 105
proof ..................................................................................................................................................................... 105
TAYLOR’S FORMULA ................................................................................................................................................... 106
degrees of approxima9on ..................................................................................................................................... 107
defini9on ............................................................................................................................................................... 108
REMARK ........................................................................................................................................................................ 109
proposi9on ............................................................................................................................................................ 109
TAYLOR EXPANSION .................................................................................................................................................... 109
lemma (Peano’s Lemma) ....................................................................................................................................... 109
proof ..................................................................................................................................................................... 110
THEOREM (Taylor’s theorem with Peano’s reminder) ........................................................................................... 110
proof ..................................................................................................................................................................... 111
defini9on ............................................................................................................................................................... 111
examples for limits ................................................................................................................................................ 113
EXERCISES CONCERNING DERIVATES.......................................................................................................................... 114
LOCAL CONVEXITY AND CONCAVITY .......................................................................................................................... 115
defini9on ............................................................................................................................................................... 115
defini9on ............................................................................................................................................................... 115
defini9on ............................................................................................................................................................... 116
THEOREM (sufficient condi9on for convexity/concavity) ...................................................................................... 116
proof ..................................................................................................................................................................... 116
corollary ................................................................................................................................................................ 117
THEOREM (sufficient condi9on for points of rela9ve minimum/maximum) ......................................................... 118
proof ..................................................................................................................................................................... 118
THEOREM (sufficient condi9on for inflec9on points) ............................................................................................ 119
proof ..................................................................................................................................................................... 119
TOPIC 13 ................................................................................................................................................................ 120
STUDY OF FUNCTION ................................................................................................................................................. 120
RECAP ON THE SUPPLEMENTARY MATERIAL: PRINICPLE OF INDUCTION ................................................................... 122
TOPIC 14 ................................................................................................................................................................ 123
INTEGRALS ................................................................................................................................................................. 123
defini9on ............................................................................................................................................................... 123
consequences of Lagrange .................................................................................................................................... 123
defini9on ............................................................................................................................................................... 123
RULES OF INTEGRATION ............................................................................................................................................. 124
theorem ................................................................................................................................................................ 124
proof ..................................................................................................................................................................... 124
theorem ................................................................................................................................................................ 125
proof ..................................................................................................................................................................... 125
defini9on ............................................................................................................................................................... 125
DEFINITE INTEGRALS .................................................................................................................................................. 127
defini9on ............................................................................................................................................................... 127
defini9on ............................................................................................................................................................... 128
contra-example ..................................................................................................................................................... 128
................................................................................................................................................................................... 128
GEOMETRICAL INTERPRETATION OF THE DEFINITE INTEGRAL ................................................................................... 128
proper9es .............................................................................................................................................................. 128
THEOREM (sufficient condi9on for the integrability) ............................................................................................ 129
THEOREM (mean value theorem) ......................................................................................................................... 129
proof ..................................................................................................................................................................... 129
GEOMETRICAL INTERPRETATION OF THE DEFINITE INTEGRAL ................................................................................... 130
THEOREM (FUNDAMENTAL THEOREM OF THE INTEGRAL CALCULUS) ........................................................................................ 130
proof ..................................................................................................................................................................... 131
THEOREM (TORRICELLI’S THEOREM) ................................................................................................................................ 132
defini9on ............................................................................................................................................................... 132
proof ..................................................................................................................................................................... 132
6
7
TOPIC 1
NUMBERS
Kronecker: very famous mathema;cian (1823-1891) used to say “natural numbers are God’s
crea;on”
N = {0, 1, 2, 3, 4…} infinite set
We can add and mul;ply natural numbers.
The sum of the product sa;sfy the following proper;es (N, +) (N; *)
1. commuta;ve:
a+b = b+a and a*b = b*a
2. associa;ve:
(a+b)+c = a+(b+c)
(a*b)*c = a*(b*c)
3. distribu;ve = the sum with respect to (w.r.t.) the product
(a+b)*c = a*c + b*c
issue:
if we consider (N, +) we have that a+0=a, that is (i.e., “id est”) 0 is the neutral element w.r.t. the
sum; if we consider (N, *) we have that a*1=a, i.e., 1 is the neutral element w.r.t. the product.
How to get a+… = 0? a+ (-a) = 0
- (-a) Ï N
- (-a) Î Z, is called opposite
- Ï = “doesn’t belong to”; Î = “belongs to”
We need the integer numbers:
Z = {…, -n, …, 0, 1, 2, …, n, …} infinite set
a real case: nega;ve temperature during the winter
Integers sa;sfy the commuta;ve, associa;ve, and distribu;ve proper;es.
As the natural numbers, they sa;sfy also the following further proper;es: (N; £) (Z; £ )
1. reflexive
m £ (£ = “less or equal”)
2. an;symmetric
if m £ n and n £ m Þ (= “implies”) m=n
3. transi;ve
if m £ n and n £ s Þ m £ s
issue:
0 is the neutral element for sum; 1 is the neutral element for product
&
'
How to get a*… = 1? a+ ( ) = 1
&
&
- (') Ï Z; (') Î Q, and is called reciprocal
8
We have to introduce the ra;onal numbers
Q={
(
: m, n, Î Z, and n ¹ 0}
)
(: = “such that”)
real examples: sharing up a pie or a pizza!
-3
-2
-1
0
1
2
3
1/3 1/2
We have s;ll “holes” (= buchi) in the lane, for example √2 (= “squares of 2”)
remarks
1. elements in Q sa;sfy commuta;ve, associa;ve and distribu;ve proper;es
&
*
&
+
&
*
2. if > , then mul;plying by -1 we have - < -
&
+
proof
Now we would like to prove that √2 Ï Q. How to do that?
We prove it by CONTRADICTION:
we suppose that there exists a ra;onal number s.t. (= “such that”) is equal to √2.
That means,
$ m, n, Î Z, (n ¹ 0) s.t.
(
= √2
)
($ = “exist”)
(
(!
= √2 Þ ! = 2 Þ m2 = 2*n2
)
)
-
m2 is EVEN (= pari); m is EVEN
-
n is ODD (= dispari) because
(
is reduced to its lower terms
)
Since m is even, we have
m = 2r
2
2
2
- m = (2r) = 4r
Before we had:
m2 = 2n2 Þ 4r2 = 2n2 Þ 2r2 = n2
- n is even! it’s absurd cause it’s impossible!
So, we can say that:
√2 Î Q is FALSE Þ √2 Ï Q
Examples of irra;onal numbers: √2, √3, p, e, … ÏQ, Î R
- “greek pi” = p = 3,4321…
- Nepero number = e = 2,73…
COMPLETENESS property of R
Every real numbers correspond to a unique point on the line and vice-versa. We say that there is a
one-to-one correspondence between real numbers and the line.
We have:
!
"
!
!
!
N " Z " Q " R
= “the set is strictly included in…”
9
remark
in R we have the commuta;ve, associa;ve, and distribu;ve proper;es; moreover, we have the
neutral elements (0 and 1), the opposite and the reciprocal elements
TOPIC 2
LOGIC
A proposi;on is a statement which may be true or false.
defini@on
the nega;on of a proposi;on P in indicated by ùp, and it is the proposi;on that is true id P is false
and vice versa
-
ù = not
“truth table”
P
ùp
T
F
F
T
defini@on
the union pÚq (Ú = or) is the proposi;on which is true if at least one of the proposi;on p, q is true
p
T
T
F
F
q
T
F
T
F
pÚq
T
T
T
F
defini@on
the intersec;on pÙq (Ù = and) is a proposi;on which is true if both p and q are true
p
T
T
F
F
q
T
F
T
F
pÙq
T
F
F
F
defini@on
the implica;on pÞq (Þ = implies) is false if P is true, and q is false
p
T
T
F
F
q
T
F
T
F
pÞq
T
F
T
T
10
defini@on
the bi-implica;on pÛq (Û = if and only if…) is a proposi;on which is true if p*q have the same
logic value
p
T
T
F
F
q
T
F
T
F
pÛq
T
F
F
T
remarks
1. given p1, p2, …, pn we say that:
p1Ùp2Ùp3Ù…Ùpn is true if pi is true "i = 1, …, n (" = for all, for every…)
2. given p1, p2, p3, …, pn we say that:
p1Úp2Úp3Ú…Úpn is true if pi is true $ i = 1, …, n s.t. pi is true
3. in a theorem: pÞq
p = HYPOTHESIS, that you supposed to be true
q = THESIS, to be proven
pay aAen@on!
- $ = exists
- " = for all…
they’re quan;fiers
the contradic@on strategy
by contradic;on, instead of proving that pÞq, we can prove that ùqÞùp
- (pÞq) Û (ùqÞùp)
p
q
pÞq
T
T
T
T
F
F
F
T
T
F
F
T
p
T
T
F
F
q
T
F
T
F
ùp
ùq
ùqÞùp
F
F
T
T
F
T
F
T
T
F
T
T
11
De Morgan laws
1. ù(pÚq) Û ùpÙùq
2. ù(pÙq) Û ùpÚùq
1. ù(pÚq) Û ùpÙùq
p
T
T
F
F
q
T
F
T
F
ùp
ùq
F
F
T
T
F
T
F
T
q
T
F
T
F
ùp
ùq
F
F
T
T
F
T
F
T
pÚq
T
T
T
F
ù(pÚq)
ùqÙùp
F
F
F
T
F
F
F
T
pÙq
F
T
T
T
ù(pÙq)
ùqÚùp
T
F
F
F
F
T
T
T
2. ù(pÙq) Û ùpÚùq
p
T
T
F
F
TOPIC 3
SET THEORY
a set is a collec;on of objects called elements of the set.
examples: N, Æ, {a}, …
- Æ = “empty set”, a set without element
- {a} = “singleton”
defini@on
we say that A is a subset of B if all the elements of A are also elements of B; we indicate A as
subset of B by:
AÍB if xÎA Þ xÎB
"xÎA
A
B
defini@on
Ì
We say that A is a proper subset of B, A " B, if every element in A is also an element of B. and there
exists an element in B not belonging to A. In other words, if xÎA Þ xÎB and $yÎB s.t. yÏA
!
!
!
examples: N " Z " Q " R
12
proprie@es: (A, Í)
1. A Í A (reflexive)
2. A Í B and B Í A Þ A = B (an;symmetric)
3. A Í B and B Í C Þ A Í C (transi;ve)
C
A
B
4. Æ Í A
defini@on
given A, B we define the intersec;on AÇB and the union AÈB in the following way:
1. AÇB = {x s.t. xÎA AND xÎB}
AÇB
2. AÈB = {x s.t. xÎA OR xÎB}
A
B
AÈB
defini@on
two sets A, B are said disjoint if AÇB = Æ
A
proprie@es
1. idempotency AÈA = A
AÇA = A
2. commuta;ve AÇB = BÇA AÈB = BÈA
3. associa;ve (AÈB)ÈC = AÈ(BÈC)
4. distribu;ve (AÈB)ÇC = (AÇC)È(BÇC)
5. AÇÆ = Æ
AÈÆ = A
B
(AÇB)ÇC = AÇ(BÇC)
(AÇB)ÈC = (AÈC)Ç(BÈC)
13
power set
If all the sets in considera;ons are subset of a common set U this is called universe set.
The set of all subset of U is called power set, P(U) or 2U.
Example: U = {a, b, c,}
P(U) = {{a, b, c}, Æ, {a, b}, {b, c}, {a, c}, {a}, {b}, {c}}
cardinality = 23 = 8
- cardinality is the number of elements of the set
defini@on
the difference between A – B is defined as:
A – B = {xÎA : xÏB)
A-B
defini@on
Ì
we define the complement of A " U as the set ùA = Ac = xÎU: x Ï}
U
A
Ac
logic and the set theory in comparison
LOGIC
ùp
pÙq
pÚq
pÞq
pÛq
(“if and only if”)
LOGIC
SET THOERY
Ac
AÇB
AÈB
AÍB
A=B
SET THOERY
DE MORGAN’S LAWS
ù(pÙq) Û ùpÚùq
(AÇB)c = Ac È Bc
ù(pÚq) Û ùpÙùq
(AÈB)c = Ac Ç Bc
14
homework
prove that (AÇB)c = Ac È Bc and (AÈB)c = Ac Ç Bc
hints
1. (AÇB)c Í Ac È Bc
2. (AÈB)c Í Ac Ç Bc
[1. + 2.] Þ (AÇB)c = Ac È Bc
To prove 1. you take xÎ(AÇB)c and you have to show that xÎ Ac Ç Bc
Topic 4
RELATIONS
defini@on
the cartesian product of A and B is A*B = (a, b) s.t. aÎA and bÎB}
example: R*R is a cartesian product
defini@on
a binary rela;on R on a set A is a subset of the cartesian product A*A
proper@es
the rela;on R is:
1. reflexive: if
(e.g. x £ x
xRx
"xÎA
"xÎR)
2. a. symmetric: if
xRy Þ yRx
"x, yÎA
(e.g. r//a Þ [“and] s//r)
// = parallel
b. an;symmetric: if xRx and xRy Þ x = y "x, yÎA
(e.g. x £ y and y £ x Þ x = y "x, yÎR)
3. transi;ve: if xRy and yRz Þ "x, y, zÎA
(e.g. x £ y and y £ z Þ x = z "x, y, zÎR)
remarks
1. R is an equivalence rela2on if it is reflexive, symmetric and transi;ve
2. R is an order rela2on if it is reflexive, an;symmetric and transi;ve
15
examples:
1. R = // is an equivalence rela;on
2. P(E) and R = Í is an order rela;on
3. R and £ is a total order rela;on, i.e. it is an order rela;on and "x, yÎR either x £ y or y £ c
remark
the strict inequality < is irreflec;ve, that means ù(a < a) "aÎA, and it is transi;ve; i.e. if
a < b and b < c Þ a < c
defini@on
if ~ (= is in rela;on with…) is an equivalence rela;on on A, then the equivalence class of an
element aÎA is:
[a] = {bÎA s.t. b ~ a}
example: take Z*Z s.t.
(m, n) ~ (p, q ) Û mq = pn
&
* , -
[ * ] = { ,, -, &., …} à mq = pn à 1*4 = 2*2 à 4 = 4
Topic 5
AXIOMATIZATION OF R
R has the following proprie;es:
1. in R the sum is well-defined and
a. it is commuta;ve
b. it is associa;ve
c. $0ÎR
d. $ the opposite
2. in R the product is well defined and
a. it is commuta;ve
b. it is associa;ve
c. $1ÎR
d. $ the reciprocal
3. in R we have the distribu;ve property
4. in R we have the total order rela;on £ s.t.
i) "x, y, zÎR , x £ y Þ x + z £ y + z
(compa;bility of the sum)
ii) "x, y, zÎR , x £ y and 0 £ z Þ x*z £ y*z
(compa;bility of the product)
Dedekind axiom
5. given A, B Í R with A¹Æ, B¹Æ if "aÎA, "bÎB
a £ b then $xÎR s.t.
a £ x£ b
"aÎA, "bÎB
(this is more like Sbaiz’s symbol) = x
16
proposi@on
in Q the Dedekind axiom does NOT work
proof
take:
A = {qÎQ s.t. (q £ 0) or (q > 0 Ù q2< 2)}
B = {qÎQ s.t. (q ³ 0 Ù q2> 2)}
The sets A, B sa;sfy the condi;ons of the Dedekind axiom, i.e. "aÎA, "bÎB, we have that a £ b
Þ $xÎR s
s.t.
a£x£b
(xÎQ)
We have to prove that xÏQ.
Indeed, we have to prove that:
1. it is impossible that x2 < 2
2. it is impossible that x2 > 2
[1. + 2.] à x2 = 2 Þ xÏQ Þ √2ÏQ
We know that x £ 2.
1)
if x2 < 2 we can take xÎR s.t.
(x <
*/x!
0<x<
£1
0
*/x!
Þ 5x < 2 – x2 Þ x2 + 5x < 2)
0
Now,
(x + e)2 Þ x2 + 2ex + e2 £ x2 + 4e + e2 £ x2+ 4e + e2 = x2 + 4e + e
x£2
-
x2 + 4e + e = x2 + 5e Þ x2 + 5e < 2
-
&
&
&
(= e), (= e2) £
*
,
*
0< x ≤ 1
Þ! #
x + 5x < 2
(e.g. x = 1 : 2e*1 £ 2e*2)
e£1
17
This means that (x + e)2 < 2
Þ (x + e)ÎA but this is impossible because every element of A must be £ x
Þ it is impossible that x2 < 2
2) in a similar way if you take xÎR s.t. 0 < x £
x! /*
£ 1 you can prove that it is impossible that
,
x2 < 2.
Þ x2 = 2 Þ x = √2 Þ thanks to the proposi;on proved yesterday, we have that xÏQ
exercise
4 < 5 à {a, b} < {a, b, c}
4 £ 5 à {a, b} £ {a, b, c} à A £ B (could be that A = B)
INTERVALS IN THE REAL LINE R
defini@on
take a, bÎR and a < b. We call bonded intervals the sets.
[a, b] = {x : a £ x £ b} close interval
]a, b[ = {x : a < x > b} open interval
]a, b] = {x : a > x £ b} half-open and half-closed interval
[a, b[ = {x : a £ x < b} half-closed and half-open interval
18
we call unbounded intervals the sets
[a, +¥[ = {x : x ³ a} closed and posi;ve half-line
]a, +¥[ = {x : x > a} open and posi;ve half-line
]-¥, b] = {x : x £ b} closed and nega;ve half line
]-¥, b[ = {x : x < b} open and nega;ve half line
remark
all the concepts in the sequel could be generalized to a set A
defini@on
a set a is said upper bounded if
$mÎR s.t.
"xÎA : x £ m
$mÎR s.t.
"xÎA : x ³ k
defini@on
a set A is said lower bounded if
defini@on
a set is bounded if it is upper bounded and lower bounded that means
$m, k, Î R s.t.
k£x£m
"xÎA (m > k)
19
defini@on
A is unbounded if it is not bounded
defini@on
given A £ R, µÎR is an upper bounded for A if "aÎA we have a ³ µ
defini@on
l is said to be a lower bounded for A if "aÎA we have a ³ l
defini@on
M is maximum for A if "aÎA, a £ M and MÎA
defini@on
m is minimum for A if "aÎA, a ³ M and mÎA
remark
if M, m exist they are unique
defini@on
given A £ R upper bounded, we call supremum supA, the minimum of the upper bounds for A
remark
if supAÎA, it coincides with the maximum of A, max A
defini@on
given A £ R lower bounded, we call infimum, infA, the maximum of the lower bounds for A
remark
if infAÎA, it coincides with the minimum of A, minA
20
Theorem: existence of the supremum
let A£R, A¹Æ and A is upper bounded (HYPOTHESIS)
then,
there exists the supremum of A (THESIS)
proof
we indicate by A* the set of the upper bounds for A
we no;ce that:
- A¹Æ
- A*¹Æ (because A* £ R)
- "aÎA, "bÎA* we have a£b
thanks to the Dedekind axiom we have that
$xÎR s.t.
a£x£b
Þ x is the supremum of A
"aÎA, "bÎA*
indeed,
- x is an upper bound x ³ a "aÎA
- x is the minimum of the upper bounds because x £ b
"bÎA*
remark
an analogous theorem can be stated also for the infimum
theorem: characterizaGon of the supremum
let A£R, A¹Æ, A is upper bounded and let xÎR,
then: (THESIS)
x = supA Û !
(HYPOTHESIS)
1) ∀𝑎 ∈ 𝐴, 𝑎 ≤ 𝜉
2) ∀𝜀 > 0, ∃𝑎4 ∈ 𝐴 ∶ 𝑎4 > 𝜉 − 𝜀
𝑎4
proof
let x be the supremum; then:
- x is an upper bound Þ"aÎA, a £ x
(1)
- x is the minimum of the upper bounds Þ"e > A, x £ e is NOT an upper bound
this is NOT an upper bound
So, it is NOT true ("aÎA, a £ x - e), then it means that holds
ù
Þ $𝑎4ÎA, 𝑎7 > x - e
("aÎA, a £ x - e)
(2)
A similar theorem holds for the infimum.
21
Theorem: characteriza;on of the infimum
let A £ R, A¹Æ, A is lower bounded and let hÎR
then,
1) ∀𝑎 ∈ 𝐴, 𝜂 ≤ 𝑎
h = infA Û !
2) ∀𝜀 > 0, ∃𝑎4 ∈ 𝐴 ∶ 𝑎4 < 𝜂 + 𝜀
defini@on
if A is not upper bounded, we have that supA = +¥
if A is not lower bounded, we have that infA = -¥
examples
1. for N we have:
infN = minN = 0
supN = +¥ Þ N is not upper bounded
&
)
2. A = { : nÎN – {0}}
infA = 0
(∄ minA)
maxA = supA = 1
because 1ÎA
3. B = {
&
: nÎN}
)/&
minB = infB = 0
(0ÎB)
∄maxB but supB = 1 indeed,
a) 1 ³
)
)1&
"nÎN : ok
b) We have to prove that "e > 0
How to prove that?
1Þ
&
)1&
$𝑛4ÎB :
)
&
=1)1&
)/3
>1–e Þ –
&
)1&
>–e Þ
&
)1&
)2
>1-e
)21&
< e Þ 1 < e (n+1)
&
&
<n+1 Þ n> –1
4
4
Since N is upper unbounded, we are able to find 𝑛4 s.t. 𝑛4 >
&
– 1 (à 2)
4
Theorem: Cantor’s theorem
let I1 ÉI2 ÉI3 É … É In É … on infinite decreasing sequence of closed and bounded intervals.
(HYPOYHESIS)
Then, (THESIS)
+¥
Ç In = I1 ÉI2 ÉI3 É … É In É …
n=1
¹Æ
22
proof
We define In = [an, bn] "n = 1, …
Then, we first prove that "k, nÎN – {0}
an £ bk
Indeed,
- if n £ k Þ an £ ak £ bk £ bn
- if n ³ k Þ ak ³ an ³bn ³ bk
Now, we consider A = {an : n Î N – {0}} and B = {bk : k Î N – {0}}
Then, bk is an upper bound for A Þ $supA
Since supA is the minimum of the upper bounds we have that supA £ bk "k
Þ supA is a lower bound for B
Þ $infB
Then, we have that supA £ infB
At the end, "nÎN – {0} we have that an £ supA £ infB £ bn
A
+¥
Þ Ç In Ê [supA, infB] ¹ Æ
B
because in the worst case we have that supA = infB
n=1
remark
actually, one can prove that
+¥
Ç In = [supA, infB]
n=1
contra-examples
We would like to show that if on hypothesis is missed, then the theorem does NOT hold
1. we can consider In = ]0,
&
)
[ open and bounded interval; in this case,
+¥
Ç
c In = Æ
n=1
2. We can consider In = [n, +¥[ closed and unbounded interval; in this case,
+¥
Ç In = Æ
n c= 1
23
TOPIC 6
TOPOLOGICAL CONCEPTS IN R
defini@on
let x0ÎR, e > 0
we define the open interval centered in x0 with radius e as:
Iex0 = {xÎR |x – x0| < e} = ]x0 – e, x0[
e < x – x0 < e
x0 - e < x < x0 + e
remark
|y| = !
𝑦 if 𝑦 ≥ 0
−𝑦 if 𝑦 < 0
absolute value func;on
defini@on
a set U<R is said to be a neighborhood of a point x0ÎR if Ux0 contains am open interval Iex0
example
consider I = ]0, 1]; then, I is a neighborhood of xÎI – {1}
I is NOT a neighborhood of 1 since Ie1 contains points Ï I
example
&
consider U = { , nÎN – {0}}
)
then, U is a neighborhood of no points!!!
So, there is no point xÎU s.t. U is a neighborhood of x
24
NEIGHBORHOOD OF +¥
U+¥ = {xÎR : x > x4}
fixed (x2 > 0)
U-¥ = {xÎR : x < - x4}
fixed
defini@on
1. A Í R is open if "x0ÎA, $e > 0
s.t.
Iex0 Ì A
2. A Í R is said to be closed if AÌ = R – A is open
remark
A Í R is open if A is a neighborhood of each of its points
defini@on
A Í R then,
1. x0ÎR is a boundary point of A if every Ux0 contains
points of A and points of AÌ
2. x0ÎA is an interior point of A if is a neighborhood of x0
3. x0ÎA is an isolated point of A if $ Ux0 s.t. Ux0ÇA = {x0}
4. x0ÎR is an accumula;on point of A if EVERY ($) Ux0 contains infinitely many points of A
Theorem: Bolzano – Weirstrass
let A be a bound ed subset of R which has infinitely many elements; then, there exists at least one
accumula;on point
defini@on
A Í R is compact if it is closed and bounded
25
TOPIC 7
RELATIONS BETWEEN SETS: FUNCTIONS
What do we mean by saying “the price of rice is a func;on of how much rice is supplied”?
We mean that there is a “LAW” which allows us to deduce the market price of the rice, given its
supplies quan;ty: if we have a plenty period, of course the price of rice will decrease in order to
sell all the quan;;es produced.
We have the opposite, in proud of under–produc;on.
supply à func;on à price
defini@on
Given two sets A, B a func;on from A to B is LAW that associates with each element in A one
element in B.
A is called domain of the func;on and B is called co–domain of the func;on.
This is a func;on.
This is NOT a func;on.
We refer to func;ons as:
f:AàB
f (A) = { f (x) s.t. xÎA} is called image of f and it is denoted by Imf
26
remark
the domain of f is denoted by Domf
remark
if f : N à R
f : n à an
we call it real value sequence
1. f: A à B
Then the restric;on of f to C £ A is f ½C = C à B
- f ½C = f restricts to C
2. f: A à B is the iden;ty if f (x) = x
3. f: A à B is the constant func;on if f (x) = K
"xÎA;
and f (A) = {k}
27
defini@on
the graph of func;on f : A à B is the set of pairs (x, y) with xÎA, yÎB
we consider
s.t.
f (x) = y
RxR = {(x,y)ÎRxR : xÎR, yÎR}
given a func;on f: R à R the graph of f could be wri|en as
Gf = {(x, f (x)) : x Î Domf }
- f (x) = y
examples
28
f (x)
y=x
y = x2
y = x3
y = √𝑥
y = ∛𝑥
y = √1 − 𝑥 #
DOMAIN
R
R
R
IMAGE
R
+
R = [0, +¥[
R
R+
R
[-1, 1]
R+
R
[0, 1]
defini@on
M = max f is called maximum (belongs to y-axis)
of f in A and x0ÎA is called maximum point
(belongs to x-axis) if:
"xÎA, M = f (x0) ³ f(x)
29
defini@on
m = min f is called minimum of f and x0ÎA
is called minimum point if:
"xÎA, m = f (x0) £ f (x)
example
we consider 𝑦 = √1 − 𝑥 #
max f = 1
min f = 0
maximum points = 0
minimum points = -1 and +1
example: f (x) = [x]+ posi;ve integer point func;on
-
if x = 1,2321 then [x]+ = 1
-
if x = 2.9 then [x]+ = 2
+
f : R - {0} à N x à n(x) =n if n £ x £ n+1
this kind of func;on is also called step-wise
func;on
30
examples: reciprocal func@ons
&
6
In general, if we have f, the reciprocal of f is (pay a|en;on to the denominator! that means f¹0)
INJECTIVE AND SURJECTIVE FUNCTIONS
defini@on
a func;on f is injec;ve if
x1 ¹ x2 Þ f(x1) ¹ f(x2)
"x1, x2ÎA
q
(pÞq)
f(x1) = f(x2) Þ x1 = x2
(ùqÞùp)
equivalently, if
ùq
ùp
31
defini@on
f is surjec;ve if f (A) = B
(in other words, f is surjec;ve if "yÎB, $xÎA s.t. f (x) = y)
remark
in general, f (A) Í B
remark
if we consider f: A à f (A), then f surjec;ve; every ;me that you restrict the func;on f: A à f (A) to
its image, then the func;on is surjec;ve
examples
f: R à R
f (x)
y=x
y = x2
y = x3
y = √𝑥
!
y = √𝑥
y = √1 − 𝑥 #
y=k
y=
&
7
y = n(x)
&
y=
&18
INJECTIVE
YES
NO
YES
YES
YES
NO
NO
YES
SURJECTIVE
YES
NO
YES
NO
YES
NO
NO
NO
NO
NO
NO
NO
32
defini@on
f is bijec;ve if it is injec;ve and surjec;ve
defini@on
if f: A à B is bijec;ve, then we define the INVERSE
MAPPING as:
f -1: B à A
as
f -1(y) = x Û f (x) = y
remark
the inverse mapping is NOT the reciprocal func;on
example
In a monopoly regime, the sale price of produced goods is decided by the producing firm; let us
denote by q(p) the units of good sold as a func;on of the offered price p.
We suppose for simplicity that q(p) is linear; the func;on q(p) can be determined by the maximum
applicable price P which reduces sales to zero and the maximum volume of products Q, feasible
only when the goods are given as a gi}s (the so-called “poten;al market)
given P, Q, the func;on
9
:
q(p) = Q (1 – )
which is the price? how to deduce the price? we have to invert the func;on! P = ….
we have:
p=P–P
9
:
<
p = P (1 – )
;
q = Q (1 – ) q = Q – Q
<
;
9
:
Q
9
=Q–q
:
Qp = PQ – Pq
p=P
;
<
–P
;
;
33
the func;ons q and p are the inverse of the other; the func;on:
q(p) : [0, P] Þ [0, Q]
[0, P] à here there is the p(rice)
[0, Q] à here the is the q(uan;ty)
and
p(q) : [0, Q] Þ [0, P]
examples
1. y =
&
7
hyperbole
Domain = R – {0}
("xÎR, x¹0)
what is f -1 (inverse)? we would like to write x = …
star;ng from y =
Dom = R – {0}
2. y = 3x – 1
&
7
Þ xy = 1 Þ x =
&
=
&
+
intersec;on with x-axis in P ( , 0)
intersec;on with y-axis in Q (0, -1)
what is f -1 ? we start from y = 3x – 1 and we would like to
write x = …
&
+
from y = 3x – 1 Þ -3x = -y – 1 Þ 3x = y + 1 Þ x = (y + 1)
&
+
Þx= y+
&
+
&
+
intersec;ons = A (-1, 0) B (0, )
34
example
we consider y = 𝑥 # (in general, a parabola
has the form y = 𝑎𝑥 # + 𝑏𝑥 + 𝑐, a ¹ 0)
f: R à R is not injec;ve and not surjec;ve;
but if we restrict the domain and the co-domain as f: R+ à R+
then f is injec;ve and surjec;ve Þ bijec;ve
now, since f is bijec;ve, we can invert it: 𝑥 = F𝑦
R+ = [0, +¥[
COMPOSITION OF MAPPING
gof means first we apply f and then we apply g
gof: A à C
z = g (f (x))
remark
must be f (A) Í Domg
35
example
we consider z = h (x) = √2𝑥 − 1 which is the composi;on (gof) of g and f with
y = f (x) = 2x – 1
z = g (y) = F𝑦
Imf = R
Domg = R+
Imf Í Domg
Þ a priori the composi;on is not well-defined;
but, we can require that Imf Í Domg
we must impose that Imf = R
that means:
y ³ 0 Þ 2x – 1 ³ 0 Þ 2x ³ 1 Þ x ³
&
*
&
*
under this assump;on (x ³ ) the composi;on is well-defined
defini@on
if f: A à B and g: B à C are two mappings (func;ons) then we define the composite func;on
(composi;on of func;ons)
gof: A à C
as
(gof) (x) = g (f (x))
"xÎA
remark
if f: A à B is bijec;ve then we have that:
(f o f-1) (y) = y
(f-1 o f) (x) = x
(f-1 o f = idA)
(f-1 o f = idB)
example
let f: R à R g: R à R
be given by f(x) = x + 3
g(x) = 𝑥 $
since f, g have R as domain and co-domain, the composi;on is always well-defined (R Í R obv.)
(gof) (x) = g (f (x)) = g (x + 3) = (x + 3)3
(fog) (x) = f (g (x)) = f (𝑥 $ ) = 𝑥 $ + 3
Þ gof ¹ fog
remark
the composi;on does NOT commute
36
BOUNDED, EVEN, ODD FUNCTIONS
defini@on
a func;on f: A à R is said to be bounded if Imf is bounded, i.e. if its image set f (A) is bounded
examples
&
𝑦 = &1 7!
𝑦 = √1 − 𝑥 #
they’re bounded func;ons
defini@on
let f: Rà R
then, f is said to be:
1. even if f (-x) = f (x) "xÎR
2. odd if f (-x) = -f (x)
"xÎR
examples of even func@ons
1. y = 𝑥 #
𝑥 if 𝑥 ≥ 0
2. y = f (x) = çx ç= G
−𝑥 if 𝑥 < 0
examples of odd func@ons
1. y = 𝑥 $
2. y =
&
7
x¹0
3. y = x
4. y = 𝑥 $ − 5𝑥
MONOTONIC FUNCTIONS
defini@on
consider f: A à R
AÍR
if "x1, x2ÎA:
1. x1 < x2 Þ f (x1) £ f (x2), then f is said to be increasing
2. x1 < x2 Þ f (x1) ³ f (x2), then f is said to be decreasing
1.
3. x1 < x2 Þ f (x1) < f (x2), then f is said to be strictly
increasing
4. x1 < x2 Þ f (x1) > f (x2), then f is said to be strictly
decreasing
2.
37
increasing and decreasing func;ons are called monotonic; strictly increasing and strictly decreasing
func;ons are called strictly monotonic
RELATION BETWEEN MONOTONIC AND INJECTIVE FUNCTIONS
proposi@on
if f is strictly monotonic in an interval I (HYPOTHESIS), then f is injec;ve on I (THESIS)
proof
we can suppose that f is strictly increasing; we have to prove that
f strictly increasing Þ f injec;ve
we have to prove that f is injec;ve, that means that "x1, x2ÎI
we know that f is strictly increasing, i.e. that "x1, x2ÎI
x1 ¹ x2 Þ f(x1) ¹ f(x2)
x1 < x2 Þ f(x1) < f(x2)
then, "x1, x2ÎI
x1 ¹ x2 Þ x1 < x2 Þ f(x1) < f(x2) Þ f(x1) ¹ f(x2) we have to prove that f is injec;ve
(HYPOTHESIS; f is strictly increasing)
remark
38
the proof is similar also if we assume f to be strictly decreasing
we have proven that f strictly monotonic in I Þ f injec;ve in I
ques2ons: is it true the viceversa? is it true that f injec;ve? Þ f strictly monotonic?
NO! IT’S FALSE!
contra-example
given
𝑥 if 𝑥 ∈ 𝑄
f (x) = !
−𝑥 if 𝑥 ∈ 𝑅 − 𝑄
-
Q = ra;onal numbers
R = irra;onal numbers
we have that f is injec;ve, but it is
NOT monotonic
defini@on
f is said to be a constant func;on if "x1, x2Î A.
x1 ¹ x2 Þ f(x1) = f(x2)
remark
1. if f is strictly increasing, then f -1 is strictly increasing
2. if f is strictly decreasing, then f -1 is strictly decreasing
39
TOPIC 8
TRIGONOMETRIC FUNCTIONS
In economic applica;ons the trigonometric func;ons have been used to describe cyclical
phenomena, like the growth of economic system, where periods of developments alternate with
periods of recessions. A historical example (1938) is represented by Samuelson’s accelerator
model.
We consider the unit circle 𝑥 # + 𝑦 # = 1
defini@on
the cosine of x and the sine of x are respec;vely the abscissa and the ordinate of the point Px
which is the intersec;on of rx and the unit circle, where rx is the half-line star;ng from (0, 0) s.t.
the measure of the angle between the x-axis and rx is equal to x; the tangent of x is defined as
tgx =
?@A 7
BC? 7
with cosx ¹ 0
40
remark
the co-tangent is defined as cotg x =
BC? 7
?@A 7
(sinx ¹ 0)
adding to x mul;ples of 2p, the coordinates of Px do
not o}en change a full circle (x + 2p) the point Px
ends up in the same star;ng posi;on
that means that
sin(x + 2p) = sinx
cos(x + 2p) = cosx
and we say that sine and cosine are PERIODIC
func;ons with a period of 2p
defini@on
a func;on f: R à R is sad to be PERIODIC with period T, if T is the smallest posi;ve number s.t.
f (x + T) = f (x)
"xÎDomf
remark
the period of tgx is equal to p
41
RADIANS AND DEGREES
length of circumference = 2pr
r=1
then,
7
E°
=
Þ
*D
+.G°
degrees
radians
0°
0°
E°
𝑥 = +.G° × 2𝜋
90°
π/2
180°
π
270°
3/2π
360°
2π
30°
π/6
45°
π/4
60°
π/3
from Pitagora’s theorem
cos2x + sin2x = 12
the fundamental iden;ty is
cos2x + sin2x = 1
SYMMETRY
the func;on sine is odd the func;on cosine is even, that means:
sin(-x) = -sinx
cos(-x) = cosx
what about tgx? It is odd, because tg(-x) =
/ ?@A 7
= -tgx
BC? 7
42
BOUNDENESS OF SINE AND COSINE
sine and cosine are bounded func;ons between the values -1 and 1
the maximum and minimum points are infinitely many and are respec;vely for the sine:
max. points for sine: x = π/2 + 2kπ kÎZ
min. points for sine: x = -π/2 + 2kπ kÎZ
and for cosine:
max. points for cosine: x = 0 + 2kπ Þ x = 2kπ
kÎZ
min. points for cosine: x = -π + 2kπ kÎZ
remark
the tangent is an UNBOUNDEDN func;on; sup = +¥, inf = -¥ for the tangent
DOMAINS AND ZEROS
sine and cosine have domains R
the tangent has domain R – {π/2 + kπ : kÎZ)
cosine has zeros in π/2 + kπ (kÎZ)
since we have to require that cosx ¹ 0
sine has zeros in kπ, kÎZ
the tangent has the same zeros of sine that means in kπ, kÎZ
radians
0
π/2
π
3/2π
2π
π/6
π/3
π/4
sinx
0
1
0
-1
0
½
√3/2
√2/2
cosx
1
0
-1
0
1
√3/2
1/2
√2/2
tgx
0
/
0
/
0
√3/3
√3
1
43
44
GRAPH OF SINE
the graph of sinx is:
remarks
sin (π – x) = –sin (π + x)
sin (π/2 + x) = sin (π/2 – x)
GRAPH OF COSINE
the graph of cosx is:
remark
cosx = sin (x + π/2), this means that the graph of cosine is shi}ed of π/2 with respect to the graph
of sine
45
GRAPH OF TANGENT
the graph of tangent is:
TRIGONOMETRIC FORMULA
sin (a + b) = sina cosb + cosa sinb
cos (a + b) = cosa cosb – sina sinb
TRIGONOMETRIC INVERSE FUNCTIONS
We consider the sin func;on. To invert it we need that the func;on is bijec;ve; we have to restrict
the domain of sin to [–π/2, π/2] in order to have a BIJECTIVE FUNCTION (Þ we can invert it)
sin÷[ –π/2, π/2] : [–π/2, π/2] Þ [–1, 1]
the inverse of such a func;on is called ARCSINE of x:
arcsin : [–1, 1] Þ [–π/2, π/2]
46
we have:
sinπ/2 = 1 Þ arcsin1 = π/2
- sinπ/2 = we apply arcsin
arcsin (sinπ/2) = arcsin1
- arcsin = f -1
- sin = f
- π/2 = arcsin 1
sinπ/4 = √2/2 Þ arcsin √2/2 = π/4
arcsin (sinπ/4) = arcsin (√2/2)
- arcsin (sinπ/4) = Id
π/4 = arcsin (√2/2)
sin0 = 0 Þ arcsin0 = 0
exercise
calculate the domain of arcsin (𝑥 # – 1)
solu;on:
– 1 £ 𝑥# − 1 £ 1
0 £ 𝑥# £ 2
– √2 £ 𝑥 # £ +√2
We consider cos func;on. To invert it, we need to restrict the domain to [0, π]; in this interval tge
func;on is BIJECTIVE (Þ we can invert it)
cos÷[0, π] : [0, π] Þ [–1, 1]
the inverse func;on is called ARCCOSINE of x:
arccos : [–1, 1] Þ [0, π]
we have:
cos0 = 1 Þ arccos1 = 0
arccos (cos0) = arccos1 = 0
- arccos (cos0) = 0
cosπ/2 = 0 Þ arccos (0) = π/2
- (0) = x
- π/2 = y
cosπ = – 1 Þ arccos (–1) = π
- (–1) = x
- π=y
47
We consider the tg func;on. It is not BIJECTIVE (Þ we cannot invert it);
then, we restrict the domain to ]–π/2, π/2[ in order to invert it
tg÷] –π/2 , π/2[ : ] –π/2, π/2] Þ R
the inverse func;on is called ARCTANGENT:
arctg: R Þ ]– π/2, π/2[
we have that:
tg (0) = 0 Þ arctg (0) = 0
TOPIC 9
CONTINUOUS FUNCTIONS
By the no;on of con;nuity, we can describe a lot of phenomena showing “li|le sensi;vity” to
small variants of quan;;es. In Economics, if in a produc;on process there is a slight varia;on of
the amount of raw material, then the volume of produc;on will undergrow small varia;on.
If we modify by a few Euro the income flow given by a big securi;es por‚olio, the values of the
indexes es;ma;ng the performance of a manager vary in an insignificant way.
the no;on of con;nuity can be described by:
x + Dx Þ f Þ f (x) + E
- Dx = small perturba;on of x
- E = small error
the picture (1) is an example of con;nuous func;on
the picture (2) is an example of a func;on which is NOT con;nuous
48
DEFINITION OF CONTINUITY
defini@on
a func;on f: A £ R à R is said to be con;nuous in a point x0ÎA if one the following equivalent
statements holds:
1. "Vf (x0), $Ux0 s.t.
"xÎUx0ÇA Þ f(x)ÎVf (x0)
1.bis. "Vf (x0), $Ux0 s.t. f(Ux0) £ Vf (x0)
2. "e > 0, $d > 0 s.t. "x : çx – x0 ç< d Þ çf(x) – f(x0) ç< e Þ f(x0) – e < f(x) < f(x0) + e
çx – x0 ç< d = x0 – d < x < x0 + d
xÎUx
Vf (x0) = ]f (x0) – e, f (x0) + e[ (e > 0)
U(x0) = ]x0 – d, x0 + d [ (d > 0)
3. "e > 0, $Ux0 s.t.
"xÎ Ux0 Ç A, $Ux0 Þ çf(x) – f(x0) ç< e
A = Domf
49
we say that f is CONTINUOUS if f is con;nuous in every point
example
let f(x) = 𝑥 # , x0 = 0 and prove that f(x) is con;nuous in x0 = 0
solu;on
we would like to use defini;on (2) of con;nuity in a point; we fix e > 0
AIM: we have to find a d > 0 s.t. "x : çx – x0 ç< d Þ çf(x) – f(x0) ç< e
then,
çf(x) – f(x0) ç< e Þ ç𝑥 # – 0# ç< e
- f(x0) = f(0)
Þ ç𝑥 # ç< e
what is d? d = √𝜀
"x : çx – x0 ç< d Þ çf(x) – f(x0) ç< e
"x : çx – x0 ç< √𝜀 Þ çf(x) – f(x0) ç< e
Þ 𝑥# < e
Þ −√𝜀 < 𝑥 < √𝜀
f(x0) = 0
çx ç< e Þ ç𝑥 # ç< e
defini@on
a func;on f: A Í R à R is said to be
1. leY con@nuous at a point x0ÎA if
"e > 0, $d > 0 s.t. Ax : x0 – d < x < x0 Þ çf(x) – f(x0) ç< e
2. right con@nuous at a point x0ÎA if
"e > 0, $d > 0 s.t. Ax : x0 < x < x0 + d Þ çf(x) – f(x0) ç< e
remark
f is con;nuous in a point x0 if it is le} and right con;nuous in the point x0
THEOREM OF THE SIGN PERMANENCE FOR CONTINUOUS FUNCTION
let f: A Í R à R be con;nuous in a point x0ÎA; if f (x0) > 0, then (HYPOTHESIS)
$Ux0 s.t.
f(x) > 0 "xÎUx0ÇA (THESIS)
50
proof
we start by rewri;ng the hypothesis of con;nuity in x0:
"e > 0, $Ux0 s.t.
"xÎUx0ÇA Þ çf(x) – f(x0) ç< e
then, we fix e > 0 (to be determined), and since f is con;nuous in x0, Ux0 Þ çf(x) – f(x0) ç< e
goal: we have to prove that f(x0) > 0
çf(x) – f(x0) ç< e Þ – e < f(x) – f(x0) < e Þ f(x0) – e < f(x) < f(x0) + e
- f(x0) – e > 0 ?
- f(x0) > 0
+e >0
}>0
if we take
e < f(x0) Þ f(x0) – e Þ f(x) > 0
then, we have concluded the proof
remarks
1. we can choose arbitrarily e > 0 because in the defini;on there is the QUANTIFIER "
2. the theorem is valid also for f(x0) < 0
( Þ $Ux0 s.t. "xÎUx0 we have f(x) < 0)
3. the theorem is NOT valid if f(x0) = 0
4. in general, if f is con;nuous in x0 and f(x0) ¹ 0, then $Ux0 s.t. f(x) has the same sign of f(x0)
"xÎUx0 and f is “for from” zero
5. if f is con;nuous x0, then f is bounded in a neighborhood of x0,
that means $k > 0, $Ux0 s.t. çf(x) ç< k
"xÎUx0
THEOREMS AND CONTINUITY
theorem (con@nuity of the sum)
if f, g: R à R are con;nuous of a point x0 (HYPOTHESIS), then the sum f + g is con;nuous at the
point x0 (THESIS)
proof
we start by rewri;ng the hypothesis:
1. f con;nuous in x0:
"e > 0, $U’x0 s.t.
"xÎU’x0 Þ çf(x) – f(x0) ç< e
2. g con;nuous in x0:
"e > 0, $U’’x0 s.t.
"xÎU’’x0 Þ çg(x) – g(x0) ç< e
we fix e > 0 and we have to prove that $Ux0 s.t. "xÎUx0 Þ ç(f+g)(x) – (f+g)(x0) ç< ke kÎN
we have:
ç(f+g)(x) – (f+g)(x0) ç= çf(x) + g(x) – f(x0) + g(x0) ç
= çf(x) – f(x0) + g(x) – g(x0) ç£ çf(x) – f(x0) ç+ çg(x) – g(x0) ç
£ = ça + b ç£ ça ç+ çb ç
çf(x) – f(x0) ç< e in U’x0 (hypothesis 1)
çg(x) – g(x0) ç< e in U’’x0 (hypothesis 2)
In Ux0 = U’x0 Ç U’’x0 we have that: ç(f+g)(x) – (f+g)(x0) ç£ e + e £ 2e
then, we have concluded.
51
remark
similar theorem for f – g
theorem (con@nuity of the product)
if f, g: R à R are con;nuous in a point x0, then f, g is con6nuous in x0
proof
we rewrite the hypothesis
1. f is con;nuous at x0:
"e > 0, $U’x0 s.t.
"xÎU’x0 Þ çf(x) – f(x0) ç< e
2. g is con;nuous in x0:
"e > 0, $U’’x0 s.t.
"xÎU’’x0 Þ çg(x) – g(x0) ç< e
we fix e > 0 and we have to prove that $Ux0 s.t."xÎUx0 Þ ç(f*g)(x) – (f*g)(x0) ç< ke kÎN N – {0}
we take:
ç(f*g)(x) – (f*g)(x0) ç= çf(x)*g(x) – f(x0)*g(x0) ç
trick: we add and subtract the same quan;ty +/- f(x0)*g(x);
then,
= çf(x)*g(x) – f(x0)*g(x0) + f(x0)*g(x) – f(x0)*g(x) ç
= çg(x)*(f(x) – f(x0)) + f(x0)*(g(x) – g(x0)) ç£ çg(x)*(f(x) – f(x0)) ç+ çf(x0)*(g(x) – g(x0)) ç=
£ = ça + b ç£ ça ç+ çb ç
= çg(x) ç* çf(x) – f(x0) ç+ çf(x0) ç* çg(x) – g(x0)
= à ça * b ç£ ça ç* çb ç
çg(x) ç= < k’ in U’’’x0 (for remark 5 of yesterday)
à thanks to the fact that if g is con;nuous the g is bounded
çf(x) – f(x0) ç< e in U’x0 (hypothesis 1)
çf(x0) ç= k
çg(x) – g(x0) ç< e in U’’x0 (hypothesis 2)
we take Ux0 = U’x0 Ç U’’x0 Ç U’’’x0 and in this Ux0 we have that:
ç(f*g)(x) – (f*g)(x0) ç< k’e + ke = (k’ + k) e
(k’ + k) = k’’
then, we are done.
52
theorem (con@nuity of the reciprocal)
if f: Rà R is con;nuous in a point x0 and f(x0) ¹ 0, then the reciprocal func;on
remark
&
is con;nuous at x0
6
&
6
the reciprocal func;on is completely different from the inverse func;on f –1; indeed, if we think
about f(x) = cos(x) we have that:
1. the reciprocal func;on is
&
(cosx ¹ 0)
BC? (7)
2. the inverse func;on is arccos(x)
and,
proof
we rewrite the hypothesis:
f is con;nuous at x0: "e > 0, $U’x0 s.t.
&
BC? (7)
= arcocos(x)
"xÎU’x0 Þ çf(x) – f(x0) ç< e
&
6
&
6
we fix e > 0 and we have to prove that $Ux0 s.t. "xÎUx0 Þ ç (x) – (x0) ç< ke kÎN
we have:
&
6
&
6
ç (x) – (x0) ç= ç
|6(7" )/6(7)|
&
&
6(7" )/6(7)
–
ç= ç
ç=
|6(7) 6(𝑥0 ) |
6(7) 6(𝑥0 )
6(7) 6(𝑥0 )
'
I
= à ç ç=
=
|6(7" )/6(7)|
|6(7)| | 6(𝑥0 ) |
<
|'|
|I|
&
3
*
|6(7)|
k'
= à ça * b ç= ça ççb ç
çf(x0) ç= k’
&
= for remark 4, we know that if f is con;nuous then f is far from zero, i.e. $k’’ > 0, $U’’x0
|6(7)|
&
&
s.t. çf(x) ç> k’’ > 0
Þ
*
|6(7)|
k''
&
6
&
6
In Ux0 = U’x0 Ç U’’x0 we have: ç (x) – (x0) ç<
then, we have proven the theorem.
&
3
&
*
=
* ε = kε
k''
k'
8LL×8L
corollary
if f, g are con;nuous func;ons in x0 and g(x0) ¹ 0, then
6
N
is con;nuous in x0
proof
6
&
= f * then, we apply the con;nuity of the reciprocal func;on and the theorem of con;nuity of
N
N
the product.
53
theorem (con@nuity of the composite func@on)
if f, g are s.t. gof is well-defined, and f is con;nuous in x0, and g is con;nuous at f(x0), then gof is
con;nuous at x0.
proof
we consider neighborhood W(gof)(x0) of the point (gof)(x0)
since g is con;nuous in f(x0), (using the defini;on of con;nuity 1bis), we have that $Vf(x0) s.t.
g(Vf(x0)) Í W(gof)(x0)
since f is con;nuous in x0, using defini;on 1bis of con;nuity, we have that $Ux0 of x0 s.t.
f(Ux0) Í Vfx0
then,
g(f(Ux0)) Í g(V(fx0)) Í W(gof)(x0)
this is the defini;on of con;nuity for gof
theorem (con@nuity of monotone func@ons)
if f: A Í R à R is monotone, A is an interval and f(A) Is an interval, then f is con;nuous
theorem (con@nuity of the inverse func@on)
if f: A Í R à R is monotone, is STRICTLY monotone and A is an interval, then the inverse f –1 is
con;nuous
examples
1. sinx, cosx are con;nuous func;ons and tgx is con;nuous in each subinterval when it is welldefined
2. arcsin, arcos, arctg are con;nuous func;ons
theorem (Bolzano’s theorem)
if f: [a, b] à R CONTINUOUS and f(a)*f(b) < 0,
then $ x0Î]a, b[
s.t.
f(x0) = 0
54
proof
the proof is based on an interac;ve process
1. define a0 = a and b0 = b and we consider I0 = [a0, b0]
we take the middle point of point of I0, it is a1 =
if f(a1) = 0, we are done!
'" 1I"
*
2. if f(a1) ¹ 0, we consider I1’ = [a0, a1] and I1’’ = [a1, b0]
In one of these two subintervals the func;on f takes values of opposite sign; we call this
interval I1 where the func;on in the external points takes values if opposite sign
3. we consider the middle point of I1 and we call it a2
if f(a2) = 0, we are done!
if f(a2) ¹ 0, we consider the two subintervals I2’ and I2’’
we choose the one where the func;ons f takes values of different sign in the external
points of the interval; call this interval I2
4. we proceed in this manner un;l in a finite number of steps we have that $x0 s.t. f(x0) = 0
if it is not the case, we construct a sequence {In} nÎN of closed and bounded intervals
which is decreasing by the which is decreasing by the inclusion “Í”; from the Cantor’s
theorem and the fact that the length of intervals tends to zero, we have that
*+
L 𝐼) = {𝑥- }
for some x0Î]a, b[
),-
55
5. we have to prove that f(x0) = 0
by contraposi;on, we suppose that f(x0) ¹ 0,
for example f(x0) > 0
since f is con;nuous, thanks to the sign
permanence theorem we have that:
$d > 0 s.t. f(x) > 0 "xÎ ]x0 – d, x0 + d[
this is contradictory with the fact that $𝑛4ÎN s.t. 𝐼)2 Ì ]x0 – d, x0 + d[
then, the func;on f assumes values of different sign in the extremal points of 𝐼)2
but the func;on f was strictly posi;ve in ]x0 – d, x0 + d[
then, this is impossible! Þ it is impossible
that f(x0) ¹ 0 Þ f(x0) = 0
this completes the proof
theorem (intermediate value theorem)
if f: [a, b] à R con;nuous and
g (= gamma) is an y number between
f(a) and f(s), then
$ x0Î]a, b[ s.t.
f(x0) = g
proof
assume f(a) < g < f(b)
we consider an auxiliary func;on
f(x) = f(x) – g
we have that:
f is con;nuous because f is con;nuous
f(a) = f(a) – g < 0
f(b) = f(b) – g > 0
then, f(a)*f(b) < 0
we are in the hypothesis of Bolzano’s theorem, then we can apply Bolzano’s theorem geƒng
$ x0Î]a, b[
s.t.
f(x0) = 0
Þ f(x0) – g = 0 Þ f(x0) = g
this concludes the proof.
CONTINUITY
theorem (of compactness)
if f: k à R is con;nuous and k is a compact set (closed and bounded), then f(k) is a compact
theorem (Weierstrass theorem)
let k be a compact set and f: k à R is a con;nuous func;on; then f(k) has both the maximum and
minimum elements
theorem (Darboux theorem)
if f: [a, b] à R is con;nuous, then f takes all the values between the minf [a, b] and maxf [a, b]
56
TOPIC 10
LIMITS
why do we introduce the no;on of limits? we are interested in the behavior of a func;on f in a
neighborhood of a point x0 which is an ACCUMULATION point;
such a point usually does NOT belong to the domain and also, we don’t know the value f(x0)
example
1. f(x) = x
.
sin( / )
.
2. f(x) = sin( / )
57
defini@on
let f: A Í R à be a func;on and x0ÎA is an accumula;on point of A
we say that lÎR is the limit of f as x approaches to x0 :
lim 𝑓(x) = 𝑙
if one of the following equivalent statements holds:
1. "Vl, $Ux0
s.t.
2. "e > 0, $d > 0
3. "e > 0, $Ux0 s.t.
7→7"
"xÎ Ux0 – {x0} Þ f(x)ÎVl
s.t.
0 < çx – x0 ç< d Þ çf(x) – l ç< e
0 < à x ¹ x0
"xÎ Ux0 – {x0} Þ çf(x) – l ç< e
CHARACTERIZATION OF CONTINUITY BY USING THE NOTION OF LIMIT
f: A à R is con;nuous at x0ÎA if and only if
lim 𝑓(𝑥) = 𝑓(𝑥- )
/→/#
extension of a func@on by con@nuity
aim: to extend a func;on which is not con;nuous in a func;on which is con;nuous
let f: A à R be a func;on and x0 does NOT belong to A
𝑓(𝑥) if 𝑥 ∈ 𝐴
𝑙 if 𝑥 = 𝑥is con;nuous in x0 and it is called extension of con;nuity of f at x0
if $ lim 𝑓(𝑥) = 𝑙 FINITE, then the func;on
/ →/#
𝑓(̅ 𝑥) = !
example
f(x) = x
sin (1/x)
Domf = R – {0} and the func;on is not con;nuous in 0 (see page 53 point 1.)
.
we take lim 𝑥 sin / = 0
/→/#
sin (1/x) = it is a bounded func;on between –1 and 1
𝑓(𝑥) if 𝑥 ≠ 0
𝑓 ̅(𝑥) = G
0 if 𝑥 = 0
the func;on 𝑓 ̅(𝑥) is now con;nuous in 0 and it is the extension by con;nuity of f in 0
the, we can extend f to
58
defini@on
1. if x0 is an accumula;on point of Domf Ç ] – ¥, x0[, then
l is said to be the leY limit of f
as x à x0-
lim 𝑓(𝑥) = 𝑙 Î R
/→/#$
if "e > 0, $d > 0
s.t.
"xÎDomf: x0 – d < x < x0 Þ çf(x) – l ç< e
2. if x0 is an accumula;on point for DomfÇ ] x0, +¥[, then
l is said to be the right limit of f
as x à x0+
lim 𝑓(𝑥) = 𝑙 Î R
/→/#%
if "e > 0, $d > 0
s.t.
"xÎDomf: x0 < x < x0 + d Þ çf(x) – l ç< e
remark
we say that ∃ lim 𝑓(𝑥) = 𝑙 if and only if lim$ 𝑓(𝑥) = lim% 𝑓(𝑥) = 𝑙 = lim 𝑓(𝑥)
7→G
/→/#
example
.
f(x) = sin / , x ¹ 0
/→/#
/→/#
.
$ lim sin( / ) ? NO!
/→-
indeed, are defined
1
𝐴 = { 𝑎) = 𝜋
, 𝑛 ∈ 𝑁*}
2 + 2𝑛𝜋
1
𝐵 = { 𝑏) =
, 𝑛 ∈ 𝑁*}
3
2 𝜋 + 2𝑛𝜋
we have that:
lim 𝑎) =
)→1P
lim 𝑏) =
)→1P
now,
𝑓(𝑎) ) = sin(
lim
) → 1P
lim
) → 1P
&
=0
D
+ 2𝑛𝜋 = + ¥
*
&
=0
+
𝜋 + 2𝑛𝜋 = + ¥
*
#
1*)D
!
%
D1*)D
!
𝜋
+ 2𝑛𝜋) = 1
2
Þ
3
𝑓(𝑏) ) = sin( 𝜋 + 2𝑛𝜋) = −1
2
'
.
𝑓(𝑎) ) = sin ] 1 ^ = sin( ( '
&
)
*#)2
2
) = sin( # + 2𝑛𝜋 )
59
.
Þ ∄ lim sin( / )
lim 𝑓(𝑎) ) = 1 ≠ −1 = lim 𝑓(𝑏) )
1& →-
3& →-
in a similar way, we can prove that
∄ = “does not exist”
/→-
lim sin 𝑥
/→*+
defini@on
in the same hypothesis as before, we define the limit
lim 𝑓(𝑥) = +∞
/ →/#
as
1. ∀V*+ , ∃U𝑥- s.t.
∀𝑥 ∈ U𝑥- − {𝑥- }
2. ∀𝑦4 > 0, ∃𝛿 > 0
s.t.
3. ∀𝑦4 > 0, ∃U𝑥- > 0 s.t.
Þ 𝑓(𝑥) ∈ V*+
∀𝑥 ∶ 0 < |𝑥 − 𝑥- | < 𝛿
Þ 𝑓(𝑥) > 𝑦4
Þ𝑓(𝑥) < − 𝑦4
∀𝑥 ∈ U𝑥- − {𝑥- }
Þ 𝑓(𝑥) > 𝑦4
Þ𝑓(𝑥) < − 𝑦4
defini@on
the lim 𝑓(𝑥) = 𝑙 ∈ 𝑅 (where the domain of f is unbounded) is defined as
/→*+
1. ∀V4 , ∃U*+
s.t.
∀𝑥 ∈ U*+
Þ 𝑓(𝑥) ∈ V4
s.t.
∀𝑥: x > 𝑥̅
Þ|𝑓(𝑥) − 𝑙| < 𝜀
3. ∀𝜀 > 0, ∃U*+ > 0 s.t.
∀𝑥 ∈ U*+
Þ|𝑓(𝑥) − 𝑙| < 𝜀
2. ∀𝜀 > 0, ∃𝑥̅ > 0
examples
D
1. lim 𝑎𝑟𝑐𝑡𝑔𝑥 = *
7→1P
D
2. lim 𝑎𝑟𝑐𝑡𝑔𝑥 = − *
7→/P
60
defini@on
the lim 𝑓(𝑥) = +∞ is defined as
/→*+
Þ 𝑓(𝑥) ∈ V*+
1. ∀V*+ , ∃U*+ s.t.
∀𝑥 ∈ U*+
2. ∀𝑦4 > 0, ∃𝑥̅ > 0
s.t.
∀𝑥: x > 𝑥̅
Þ 𝑓(𝑥) > 𝑦4
3. ∀𝑦4 > 0, ∃U*+ > 0 s.t.
∀𝑥 ∈ U*+
Þ 𝑓(𝑥) > 𝑦4
remark
if lim 𝑓(𝑥) = ±∞ , we have that 𝑥 = 𝑥- is called ver@cal asymptote
/→/#
if lim 𝑓(𝑥) = 𝑙 ∈ 𝑅 FINITE, we have that 𝑦 = 𝑙 is called horizontal asymptote
/→*+
THEOREM (UNIQUENESS OF THE LIMIT)
if ∃ lim 𝑓(𝑥) = 𝑙 ∈ 𝑅,
/ →/#
(HYPOTHESIS)
then the limit is unique (THESIS)
proof
by contradic;on, we suppose that (1) lim 𝑓(𝑥) = 𝑙. and (2) lim 𝑓(𝑥) = 𝑙#
by defini;on (1) is:
by defini;on (2) is:
/→/#
/→/#
∀V4. , ∃U′𝑥- s.t.
∀𝑥 ∈ U 5 𝑥- − {𝑥- } Þ 𝑓(𝑥) ∈ V4.
∀V4# , ∃U′′𝑥- s.t.
∀𝑥 ∈ U 55 𝑥- − {𝑥- } Þ 𝑓(𝑥) ∈ V4#
(with 𝑙. ≠ 𝑙# )
now, we choose V4. and V4# s.t. V4. ÇV4# = Æ (thanks to the quan;fier" in the defini;ons)
since we have fixed V4. from (1) we have that ∃U′𝑥- s.t. ∀𝑥 ∈ U 5 𝑥- Þ 𝑓(𝑥) ∈ V4.
since we have fixed V4# from (2) we have that ∃U′′𝑥- s.t. ∀𝑥 ∈ U 55 𝑥- Þ 𝑓(𝑥) ∈ V4#
61
but, if we consider U𝑥- Ç U′𝑥- ÇU 55 𝑥- , this U𝑥- is not empty because 𝑥- is an accumula;on point
and then we have infinitely many points in U𝑥then, ∀𝑥 ∈ U𝑥- we have that 𝑓(𝑥) ∈ V4. ÇV4# = Æ this is IMPOSSIBLE!
then, the limit is unique!
THEOREM (of the sign permanence for limits)
let f: A Í R à R be a func;on; if lim 𝑓(𝑥) = 𝑙 ≠ 0 , then ∃U𝑥- s.t. ∀𝑥 ∈ U𝑥- − {𝑥- } we have that
/→/#
f(x) has the same sign of the limit 𝑙
proof
we can suppose that 𝑙 > 0; we have to prove that 𝑓(𝑥) > 0 in a U𝑥by defini;on,
∀𝜀 > 0, ∃U𝑥s.t.
∀𝑥 ∈ U𝑥- − {𝑥- } Þ |𝑓(𝑥) − 𝑙| < 𝜀
|𝑓(𝑥) − 𝑙| < 𝜀 Þ −𝜀 < 𝑓(𝑥) − 𝑙 < 𝜀
- 𝑙−𝜀>0
- 𝑙 (> 0) + 𝜀 (> 0) à 𝑙 + 𝜀 > 0
Þ 𝑙 − 𝜀 < 𝑓(𝑥) < 𝑙 + 𝜀
since we can choose the e as we wish, we take 𝜀 > 𝑙 Þ 𝑙 − 𝜀 > 0 Þ 𝑓(𝑥) > 0
remark
the theorem is s;ll valid in the cases lim 𝑓 (𝑥) = ± ∞
/→/#
remark
a func;on with a FINITE limit is locally bounded, that means that if lim 𝑓(𝑥) = 𝑙 < ∞ , then
s.t.
∃𝑘 > 0, ∃U𝑥-
|𝑓(𝑥)| < 𝑘
∀𝑥 ∈ U𝑥- − {𝑥- }
/→/#
THEOREM (of comparison)
consider f, g and let 𝑥- be an accumula;on
point for f and g
assume that:
- lim 𝑓(𝑥) = +∞ and
/→/#
- 𝑔(𝑥) ≥ 𝑓(𝑥) in a neighborhood U𝑥then,
lim 𝑔(𝑥) = +∞
/→/#
proof
by defini;on of lim 𝑓(𝑥) = +∞ , we have
∀𝑦4 > 0, ∃U′𝑥-
/→/#
s.t. ∀𝑥 ∈ U′𝑥- − {𝑥- } Þ 𝑓(𝑥) > 𝑦4
fix 𝑦4 > 0, ∃U 55 𝑥- = U 5 𝑥- ∩ U𝑥-
s.t.
we have
Þ 𝑔(𝑥) > 𝑦4
𝑔(𝑥) ≥ 𝑓(𝑥) > 𝑦4
∀𝑥 ∈ U 55 𝑥- − {𝑥- }
Þ this is the defini;on of lim 𝑔(𝑥) = +∞
/→/#
62
THEOREM (Sandwich theorem)
consider f, g, h and 𝑥- is an accumula;on point
for f, g, h
assume:
- lim 𝑓(𝑥) = 𝑙 = lim ℎ(𝑥)
/→/#
then,
/→/#
𝑓(𝑥) ≤ 𝑔(𝑥) ≤ ℎ(𝑥) in a neighborhood
U𝑥lim 𝑔(𝑥) = 𝑙
/→/#
proof
fix e > 0, by defini;on of lim 𝑓(𝑥) = 𝑙
/→/#
-
∃U′𝑥|𝑓(𝑥) − 𝑙| < 𝜀 Þ
s.t.
∀𝑥 ∈ U′𝑥- − {𝑥- } Þ |𝑓(𝑥) − 𝑙| < 𝜀
𝑙 − 𝜀 < 𝑓(𝑥) < 𝑙 + 𝜀
by defini;on of lim ℎ(𝑥) = 𝑙 ,
/→/#
-
∃U 55 𝑥s.t.
∀𝑥 ∈ U 55 𝑥- − {𝑥- } Þ |ℎ(𝑥) − 𝑙| < 𝜀
|ℎ(𝑥) − 𝑙| < 𝜀 Þ 𝑙 − 𝜀 < ℎ(𝑥) < 𝑙 + 𝜀
we take U 555 = U 5 𝑥- ∩ U 55 𝑥- ∩ U𝑥- :
∀𝑥 ∈ U 55 𝑥- − {𝑥- }
Þ
we have
𝑙 − 𝜀 < 𝑔(𝑥) < 𝑙 + 𝜀
example
?@A 7
;
7→1P 7
take lim
𝑙 − 𝜀 < 𝑓(𝑥) ≤ 𝑔(𝑥) ≤ ℎ(𝑥) < 𝑙 + 𝜀
Þ |𝑔(𝑥) − 𝑙| < 𝜀 Þ this is the defini;on of lim 𝑔(𝑥) = +∞
/→/#
?@A 7
=0
7→1P 7
prove that lim
?@A 7
?@A 7
=
0
since
is even)
7
7→/P 7
(we have also that lim
solu;on
sin 𝑥
1
&
0≤ 𝑥 ≤𝑥
= 0 (𝑥 → ∞)
7
then, thanks to the Sandwich theorem, we have that
sin 𝑥
lim
=0
/→*+ 𝑥
for 𝑥 → +∞ we have that
63
THEOREM (limit of the composiGon of funcGons)
let f: A Í R à R and 𝑥- be an accumula;on point for A
let g: B Í R à R and 𝑦- ∈ 𝑅 be an accumula;on point for B
assume that 𝑓(𝑥) = 𝑦- and lim 𝑔(𝑦) = 𝑙 ∈ 𝑅
6 →6#
if also it holds one of the following condi;ons:
1. ∀𝑥 ∈ 𝐴 − {𝑥- } : 𝑓(𝑥) ≠ 𝑦2. 𝑦- ∈ 𝐵
and 𝑔(𝑦- ) = 𝑙
then,
lim 𝑔l𝑓(𝑥)m = 𝑙
/→/#
OPERATIONS WITH LIMITS
proposi@on
let f, g be two func;ons and let 𝑥- be an accumula;on point; if:
- lim 𝑓(𝑥) = 𝑙. ∈ 𝑅
/→/#
-
lim 𝑓(𝑥) = 𝑙# ∈ 𝑅
/→/#
then,
1. lim (𝑓(𝑥) ± 𝑔(𝑥)) = 𝑙. ± 𝑙#
/→/#
2. lim (𝑓(𝑥) ∗ 𝑔(𝑥)) = 𝑙. ∗ 𝑙#
/→/#
7(/)
4
3. lim 8(/) = 4' (𝑙# ≠ 0)
/→/#
)
the proof is admi|ed since it is similar to the proof for con;nuous func;ons
proposi@on
If lim 𝑓(𝑥) = +∞ (−∞) and g is bounded from below (above) in a U𝑥- , then
/→/#
lim (𝑓(𝑥) ± 𝑔(𝑥)) = +∞ (−∞)
/→/#
proof
fix 𝑦4 > 0, and we know by defini;on of lim 𝑓(𝑥) = +∞ that
∃U′𝑥- s.t.
/→/#
∀𝑥 ∈ U′𝑥- − {𝑥- } Þ 𝑓(𝑥) > 𝑦4
we also know that g is bounded from below, i.e. ∃𝑘 > 0 s.t. 𝑔(𝑥) > 𝑘 > 0 ∀𝑥 ∈ U𝑥- − {𝑥- }
we take U′′𝑥- = U′𝑥- ∩ U𝑥- we have that ∀𝑥 ∈ U′′𝑥- − {𝑥- } one has
𝑓(𝑥) + 𝑔(𝑥) > 𝑦4 + 𝑘
Þ this is the defini;on of lim (𝑓(𝑥) ± 𝑔(𝑥)) = +∞
/→/#
f
g
f+g
𝑙.
𝑙#
𝑙. + 𝑙#
±∞
bounded
±∞
+∞
−∞
?
? = this is an UNDETERMINATE FORM that means that the result will change case by case
64
examples
1. lim (𝑥 # + √𝑥) = +∞
𝑥 # = +∞; √𝑥 = +∞
/→*+
2.
lim (𝑥 # − √𝑥) = +∞ − ∞
à it is NOT DETERMINATE form
/→*+
= lim √𝑥 p
/→*+
.
𝑥#
√𝑥
.
− 1q = lim 𝑥 # r
/→*+
𝑥#
.
#
. s = lim 𝑥 r
𝑥#
/→*+
𝑥#
lim 𝑥
. s = /→*+
𝑥#
.
.
#9
# t𝑥 # − 1u
$
= lim 𝑥 # t𝑥 # − 1u = +∞
/→*+
remark
in the previous proposi;on we can take g bounded from below in the following way:
|𝑔(𝑥)| > 𝑘 > 0
∃𝑘 ∈ 𝑅
s.t.
3.
lim (𝑥 $ + sin 𝑥) = +∞
𝑥 $ = +∞; sin 𝑥 = BOUNDED
/→*+
proposi@on
if lim 𝑓(𝑥) = +∞ (−∞) and g is far from zero and it is posi;ve (nega;ve) in a U𝑥- , then
/→/#
lim (𝑓(𝑥) ∗ 𝑔(𝑥)) = +∞ (+∞)
/→/#
remark
in the product case we have to use the sign rule
proof
since g is far from zero and posi;ve, ∃𝑘 > 0 s.t. 𝑔(𝑥) > 𝑘 > 0
we fix 𝑦4 > 0 , by defini;on lim (𝑓(𝑥) ∗ 𝑔(𝑥)) = +∞ we have
∀𝑥 ∈ U𝑥-
/→/#
∃U′𝑥- s.t.
∀𝑥 ∈ U′𝑥- − {𝑥- }
Þ 𝑓(𝑥) > 𝑦4
now, if we define U 𝑥- = U′𝑥- ∩ U𝑥- we have that ∀𝑥 ∈ U′′𝑥- − {𝑥- } Þ 𝑓(𝑥) ∗ 𝑔(𝑥) > 𝑦4 *k
55
this is the defini;on of lim 𝑓(𝑥) = +∞
/→/#
proposi@on
if lim 𝑓(𝑥) = 0 and g is bounded in a U𝑥- , then
/→/#
lim (𝑓(𝑥) ∗ 𝑔(𝑥)) = 0
/→/#
proof
we know that g is bounded in U𝑥- , then ∃𝑘 > 0 s.t. |𝑔(𝑥)| < 𝑘
fix 𝜀 > 0 , by defini;on of lim 𝑓(𝑥) = 0 we have that
/→/#
-
∃U′′𝑥- s.t.
|𝑓(𝑥) − 0| = |𝑓(𝑥)|
∀𝑥 ∈ U′′𝑥- − {𝑥- }
Þ |𝑓(𝑥) − 0| < 𝜀
if we take U𝑥- = U𝑥- ∩ U′′𝑥- , then we have
∀𝑥 ∈ U′′𝑥- − {𝑥- } Þ |𝑓(𝑥) ∗ 𝑔(𝑥)| = |𝑓(𝑥) ∗ 𝑔(𝑥)| < 𝜀*k
- = à |𝑎 ∗ 𝑏| = |𝑎| ∗ |𝑏|
this is the defini;on of lim (𝑓(𝑥) ∗ 𝑔(𝑥)) = 0
/→/#
65
f
g
f*g
𝑙. (∈ 𝑅)
𝑙# (∈ 𝑅)
𝑙. ∗ 𝑙#
+∞
bounded posi;ve
+∞
0
bounded
0
+∞
+∞
+∞
+∞
−∞
−∞
−∞
−∞
+∞
∞
0
?
thanks to the sign rule
? = this is NOT determinate form
proposi@on
1. if lim 𝑔(𝑥) = ±∞ and f is a bounded func;on in a U𝑥- , then
/→/#
𝑓(𝑥)
=0
/→/# 𝑔(𝑥)
lim
2. if lim 𝑔(𝑥) = 0 and f is a bounded func;on in a U𝑥- , then
/→/#
𝑓(𝑥)
=∞
/→/# 𝑔(𝑥)
lim
note that the sign of the limit is determinate by the sign ruel
proof
we will prove only point 1. since point 2. is similar
fix 𝜀 > 0 , by defini;on of lim 𝑔(𝑥) = +∞ then
/→/#
∃U′𝑥-
s.t.
∀𝑥 ∈ U′𝑥- − {𝑥- }
since f is bounded in U𝑥- , ∃𝑘 > 0 s.t.
|𝑓(𝑥)| < 𝑘
.
Þ |𝑔(𝑥)| > :
∀𝑥 ∈ U𝑥-
if we take U′′𝑥- = U𝑥- ∩ U′𝑥- , we have that
6(7)
&
∀𝑥 ∈ U′′𝑥G − {𝑥G } Þ EN(7)E = |𝑓(𝑥)| ∗ |N(7)| < 𝜀*k
-
!
#
"
"
= à ! ! = |𝑎| ∗ ! !
6(7)
=0
7→7" N(7)
this is the defini;on of lim
66
f
g
𝒇
𝒈
𝑙. (∈ 𝑅)
𝑙# (∈ 𝑅)
𝑙.
(𝑙 ≠ 0)
𝑙# #
bounded
∞
0
bounded
0
∞
∞
∞
?
∞
0
?
? = these are NOT determinate forms
THEOREM (limit of raGonal funcGons)
given 𝑝(𝑥) = 𝑎; 𝑥 ; + 𝑎;9. 𝑥 ;9. + ⋯ + 𝑎and q(𝑥) = 𝑏< 𝑥 < + 𝑏<9. 𝑥 <9. + ⋯ + 𝑏s.t. 𝑎; , 𝑏< ≠ 0 , 𝑎= , 𝑏8 ∈ 𝑅
∀𝑖 = 1, … , 𝑁
∀𝑔 = 1, … , 𝑀
𝑀, 𝑁 ∈ 𝑁 ∗ , we have that
𝑥 ;9.
𝑎𝑝(𝑥)
𝑎; 𝑥 + 𝑎;9. 𝑥
+ ⋯ + 𝑎; + ⋯ + 𝑥;)
𝑥
lim
= lim
= lim
/→+ 𝑞(𝑥)
/→+ 𝑏< 𝑥 < + 𝑏<9. 𝑥 <9. + ⋯ + 𝑏/→+ <
𝑥 <9.
𝑏𝑥 (𝑏< +𝑏<9. < + ⋯ + <
𝑥
𝑥
;
;9.
𝑥 ; (𝑎; +𝑎;9.
∞ if 𝑁 > 𝑀
𝑎;
𝑎; ;9<
if 𝑁 = 𝑀
= lim
𝑥
= ƒ
𝑏<
/→+ 𝑏<
0 if 𝑁 < 𝑀
𝑥 ;9.
𝑎𝑥 <9.
𝑏=
0;
=
0;
= 0; < = 0
;
;
<
𝑥
𝑥
𝑥
𝑥
if N < M Þ N – M < 0 Þ for example we have a power like –2
𝑎; 9#
1
lim
𝑥 = lim #
/→+ 𝑏<
/→+ 𝑥
THEOREM (limit for monotone funcGons)
let f: Aà R be a monotone func;on and 𝑥- = supA, 𝑥- ∉ 𝐴
then,
𝑠𝑢𝑝 𝑓(𝐴) if 𝑓 is increasing
lim$ 𝑓(𝑥) = !
/→/#
𝑖𝑛𝑓 𝑓(𝐴) if 𝑓 is decreasing
67
proof
we assume that f is increasing and we define 𝜆 (lambda) = supf(A) < +∞
we fix 𝜀 > 0 , by defini;on of supremum, we have that
∃𝑥̅ ∈ 𝐴 s.t.
𝑓(𝑥̅ ) > 𝜆 − 𝜀
now, since f is increasing, we have that
∀𝑥 ∈ 𝐴, 𝑥 > 𝑥̅ Þ 𝑓(𝑥) ≥ 𝑓(𝑥̅ ) > 𝜆 − 𝜀
then, ∀𝑥: 𝑥̅ < 𝑥 < 𝑥- , we have found that
𝜆 − 𝜀 < 𝑓(𝑥) ≤ 𝜆 < 𝜆 + 𝜀
so, if we choose 𝛿 = 𝑥- − 𝑥̅ we have that
lim 𝑓(𝑥) = 𝜆
7→7"&
now, we assume that 𝜆 = +∞
we fix 𝑦4 > 0 , since f is not upper bounded (because 𝜆 = +∞), we have that ∃𝑥̅ ∈ 𝐴 ∶ 𝑓(𝑥̅ ) > 𝑦4
since f ia increasing, we have that ∀𝑥 ∈ 𝐴 ∶ 𝑥 > 𝑥̅ we get 𝑓(𝑥) ≥ 𝑓(𝑥̅ ) > 𝑦4
then, we have shown that ∀𝑥 ∈ 𝐴 ∶ 𝑥̅ < 𝑥 < 𝑥- we obtain 𝑓(𝑥) > 𝑦4
if we take 𝛿 = 𝑥- − 𝑥̅ , we infer that
lim 𝑓 (𝑥 ) = 𝜆 = +∞
7→7"&
remarks
1. in the decreasing case the proof is similar
2. the theorem s;ll holds for 𝑥- = 𝑖𝑛𝑓𝐴, 𝑥- ∉ 𝐴
APPENDIX
THE NOTION OF LIMIT
we consider the func;on y = f(x) defined
on A, and we study the behavior of f when
x “is closed” to 𝑥- which is an
accumula;on point for A
if we look at the graph when x is
approaching 𝑥- we have that f(x) is
approaching 𝑙 , BUT in this case 𝑙 does NOT
coincide with 𝑓(𝑥- ) because 𝑥- ∈ 𝐴
for this reason, we introduce the no;on of
limit: “when x is closer to 𝑥- , we have that
f(x) is closer to 𝑙 " and we write
lim 𝑓(𝑥) = 𝑙
/→/#
for example, we consider
2𝑥 # − 6𝑥
𝑥−3
Domf = R – {3}
or
Domf = {xÎR : x ¹ 3)
and we would like to study the behavior of f around the point 𝑥- = 3
𝑦 = 𝑓(𝑥) =
68
x
f(x)
2,9
5,8
2,99
5,98
2,999
5,998
à𝑥- = 3 ß
à𝑙 = 6ß
3,001
6,002
3,01
6,02
3,1
6,2
then, when x approaches 3 we have that f approaches 6
in a formal way
2𝑥 # − 6𝑥
2𝑥 (𝑥 − 3)
= lim
= lim 2𝑥 = 6
/→$ 𝑥 − 3
/→$
/→$
𝑥−3
lim 𝑓(𝑥) = lim
/→$
-
(2*9 – 6*3) / (3 – 3) = 0
2*3 = 6 (result)
remark
when f is con@nuous, we have that
I*F = INDETERMINATE FORM
lim 𝑓(𝑥) = 𝑓(𝑥- )
/→/#
for example:
lim (𝑥 + 3) = 5
/→#
remark
if we mul;ply and divide by the same quan;ty, we are mul;plying y 1, then nothing changes!
remark
?@A /
.
?@A /
.
1. lim / = 0 because − / ≤ / ≤ / and we conclude thank the Sandwich theorem
/→*+
2. theorem on limit for monotone func;ons
69
RELEVANT LIMITS FOR TRIGONOMETRIC FUNCTIONS
1.
sin 𝑥
=1
/→- 𝑥
lim
proof (1.)
first of all we consider
sin 𝑥
/→𝑥
lim%
2
we take 0 < 𝑥 < # and then
sin 𝑥 < 𝑥 < 𝑡𝑔𝑥 =
sin 𝑥
cos 𝑥
tgx = this is well-defined because
2
cos 𝑥 ≠ 0 since 𝑥 ≠ #
we divided by sin 𝑥, because sin 𝑥 ≠ 0 since 𝑥 ≠ 0,
sin 𝑥
sin 𝑥
𝑥
cos
𝑥 Þ1< 𝑥 < 1
<
<
sin 𝑥 sin 𝑥
sin 𝑥
sin 𝑥 cos 𝑥
1 = 𝑥 → 0*
thanks to the Sandwich theorem we have also that
sin 𝑥
lim%
=1
/→𝑥
now, thanks to the theorem on the limit of reciprocal func;ons we have
sin 𝑥
1
lim%
= =1
/→𝑥
1
?@A /
for the case x < 0, there is “nothing” to prove since
is an even func;on then
/
sin 𝑥
sin 𝑥
sin 𝑥
lim$
= lim%
= 1 Þ lim
=1
/→/→- 𝑥
/→𝑥
𝑥
2.
1 − cos 𝑥
1
=
/→𝑥#
2
lim
remark
1 − cos 𝑥 ~𝑥 #
(𝑥~0)
“1 – cosx has the same behavior of 𝑥 # when x is closed to 0”
~ = “approximate”
70
remark
?@A /
for lim / = 1 we can write
/→-
sin 𝑥 ~𝑥
(𝑥~0)
proof (2.)
1 − cos 𝑥
0
1 − cos 𝑥
1 + cos 𝑥
1 − cos # 𝑥
=
I.
F.
Þ
lim
×
=
lim
/→/→/→- 𝑥 # + (1 + cos 𝑥)
𝑥#
0
𝑥#
1 + cos 𝑥
lim
= (A − B)(A + B) = A# − B #
sin# 𝑥
/→- 𝑥 # (1 + cos 𝑥)
= lim
sin# 𝑥 + cos # 𝑥 = 1 Þ 1 − cos # 𝑥 = sin# 𝑥
sin 𝑥 sin 𝑥
1
1
1
×
×
=1×1×
=
/→- 𝑥
𝑥
1 + cos 𝑥
1+1
2
= lim
remark
?@A /
also lim / is an indeterminate form (0/0)
/→*-
3.
tg 𝑥
=1
/→- 𝑥
lim
proof (3.)
sin 𝑥
tg 𝑥 0
cos
𝑥 = lim sin 𝑥 × 1 = 1 × 1 = 1
lim
= I. F. Þ lim
/→- 𝑥
/→/→- 𝑥
0
𝑥
cos 𝑥
4.
arc sin 𝑥
0
=1 Þ
I. F.
/→𝑥
0
lim
proof (4.)
we change the variable
𝑦 = 𝑎𝑟𝑐𝑠𝑖𝑛𝑥 Þ 𝑥 = sin 𝑦 and then
𝑦
=1
6→- sin 𝑦
lim
71
5.
arc tg 𝑥
0
=1 Þ
I. F.
/→𝑥
0
lim
proof (5.)
we change the variable
𝑦 = 𝑎𝑟𝑐𝑡𝑔𝑥 Þ 𝑥 = tg 𝑦 and then
𝑦
=1
6→- tg 𝑦
lim
thanks to the limit (3.)
remark
we can also write limit (3.) as 𝑡𝑔𝑥~𝑥 (𝑥~0)
we can also write limit (4.) ad 𝑎𝑟𝑐𝑠𝑖𝑛𝑥 ~ 𝑥 (𝑥~0)
similar for limit (5.)
TOPIC 11
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
we start by introducing the POWER FUNCTIONS
we consider 𝑎 ∈ 𝑅, 𝑎 > 1:
𝑎) = 𝑎 × 𝑎 × 𝑎 × … × 𝑎
n ;mes, 𝑛 ∈ 𝑁 − {0}
𝑎- = 1
we have that
lim 𝑎) = +∞
)→*+
to prove that we need the Newton’s binomial formula:
)
(𝑎 + 𝑏)) = š ]
B,-
]
𝑛
^ = “from n we choose k”
𝑘
= ]
𝑛
𝑛
𝑛
𝑛
^ 𝑎)9- 𝑏 - + ] ^ 𝑎)9. 𝑏. + ] ^ 𝑎)9# 𝑏 # + ⋯ + ] ^ 𝑎)9) 𝑏 )
0
1
2
𝑛
= 𝑎) + 𝑛𝑎)9. 𝑏 + ]
]
𝑛
^ 𝑎)9B 𝑏 B
𝑘
𝑛 )9# #
^𝑎
𝑏 + ⋯ + 𝑏)
2
𝑛
^ is called binomial coefficient and
𝑘
𝑛!
𝑛(𝑛 − 1)(𝑛 − 2) … × 1
𝑛
] ^=
=
𝑘
(𝑛 − 𝑘)! 𝑘!
(𝑛 − 𝑘)! 𝑘!
(𝑛 − 1)(𝑛 − 2) … × 1 = (𝑛 − 1)!
72
example
4! = 4 × 3 × 2 × 1 = 24
what about 0! ? 0! = 1
𝑛! is called “n factorial”
]
𝑛
^ represents the number of subsets of 𝑘 elements from a set A of 𝑛 element
𝑘
then, to prove
lim 𝑎) = +∞
(𝑎 > 1)
)→*+
we consider
(𝑘 ≤𝑛)
𝑎 = 1 + 𝛿, 𝛿 > 0
since 𝑎 > 1
then,
)
𝑎) = (1 + 𝛿)) = š ]
B,-
𝑛 )9B B
^1
𝛿
𝑘
𝑛
𝑛
𝑛
= ] ^ 1 × 𝛿 C + ] ^ 1 × 𝛿. + ] ^ 1 × 𝛿 # + ⋯ + 𝛿 )
0
1
2
]
𝑛
^=1
0
]
𝑛!
𝑛(𝑛 − 1)!
𝑛
𝑛
^=
=
= =𝑛
1
(𝑛 − 1)! 1!
(𝑛 − 1)! × 1 1
]
𝑛
^ 1 × 𝛿# + ⋯ + 𝛿) ≥ 0
2
1 × 𝛿C = 1
1 × 𝛿. = 𝛿
≥ 1 + 𝑛𝛿
so,
lim 𝑎) ≥ lim (1 + 𝑛𝛿) = +∞
)→*+
)→*+
thanks to the comparison theorem on limits, we have
lim 𝑎) = +∞
)→*+
on the contrary, we have that
lim 𝑎) = 0
)→*+
∀𝑎 ∈ 𝑅, 0 < 𝑎 < 1
now we would like to extend the previous no;on, to the Q powers
in order to do that, we introduce
1
𝑎) = )
∀𝑛 ∈ 𝑁 − {0}
𝑎
if now we consider a ra;onal number
𝑚
𝑟=
𝑛>1
𝑛
73
then we define
'
(
(
𝑎 = √𝑎( Þ
n-th square root of a to the power m
we have then the following proper;es ∀𝑟, 𝑟. , 𝑟# ∈ 𝑄 and ∀𝑎, 𝑏 ∈ 𝑅
𝑎, 𝑏 > 0:
𝑎V) × 𝑎V! = 𝑎V) 1 V!
(𝑎V) )V! = 𝑎V) × V!
(𝑎𝑏)V = 𝑎V × 𝑏 V
𝑎G = 1
𝑎/V =
1
𝑎V
moreover, we have:
1. 𝑓 (𝑟) = 𝑎 V is increasing if 𝑎 > 1
2. 𝑓 (𝑟) = 𝑎 V is decreasing if 0 < 𝑎 < 1
proof
we would like to prove point (1.) +ù
we have to prove that
in other words,
we write
since
then,
if 𝑠 < 𝑡 Þ 𝑓(𝑠) < 𝑓(𝑡)
if 𝑠 < 𝑡 Þ 𝑎 W < 𝑎X
(𝑎 > 1)
𝑎X = 𝑎X/W1W = 𝑎X/W × 𝑎 W
𝑡 − 𝑠 > 0, 𝑎 > 1 Þ 𝑎X/W > 1
𝑎X = 𝑎X/W × 𝑎 W > 1 × 𝑎 W Þ 𝑎X > 𝑎 W
now, we would like to define 𝑎D where 𝛼 ∈ 𝑅 and “a” is any posi;ve real number
THE EXPONENTIAL FUNCTION
we therefore consider the so-called exponen;al func;on
𝑓(𝑥) = 𝑎 /
with 𝑎 > 0 is any fixed basis
the exponen;al func;on usually describes in the economic models, the decay in ;me of
instruments (like machines, …), the produc;vity, the growth of a market store, the con;nuous ;me
dynamics of an invested capital and so on
74
proper@es
1. 𝑎 / is defined in R and it is con;nuous
2. 𝑎 / > 0, ∀𝑥 ∈ 𝑅 and it is increasing when 𝑎 > 1
𝑎 / > 0, ∀𝑥 ∈ 𝑅 and it is decreasing when 0 < 𝑎 < 1
3. for 𝑎 > 1:
lim 𝑎 7 = 0 and
7→/P
lim 𝑎 7 = +∞
7→1P
4. for 0 < 𝑎 < 1:
lim 𝑎 7 = +∞ and
7→/P
lim 𝑎 7 = 0
7→1P
an important basis is represented by Nepero number “e”, 𝑒~2,7183 … and it is defined by
1 )
lim t1 + u = 𝑒
)→*+
𝑛
it could be shown also that
1 /
lim t1 + u = 𝑒
)→±+
𝑥
in the graph, the domain of
𝑓(𝑥) = 𝑒 / is R and the image
of f is ]0, +∞[
the func;on is con;nuous, and
it is increasing
75
THE LOGARITHMIC FUNCTION
is 𝑎 / inver;ble? YES, if 𝑎 > 0, 𝑎 ≠ 1
in other words, if 0 < 𝑎 < 1 or 𝑎 > 1
so, for 𝑎 > 0, 𝑎 ≠ 1 the exponen;al func;on 𝑦 = 𝑓(𝑥) = 𝑎 / admits the inverse func;on
𝑥 = 𝑓 9. (𝑦) = log 1 𝑦
such an inverse func;on is called logarithmic func@on of y with basis “a”
its domain is ]0, +∞[ and the image is R
we have that:
𝑦 = log 1 𝑥 Û x = 𝑎 6 ∀𝑥 > 0, 𝑎 > 0, 𝑎 ≠ 1
proper@es
1. 𝑓(𝑥) = log 1 𝑥 is defined for all 𝑥 > 0 and the image is R; it is also con;nuous func;on
2. log 1 𝑥 is an increasing func;on for 𝑎 > 1
log 1 𝑥 is a decreasing func;on for 0 < 𝑎 < 1
3. for 𝑎 > 1:
lim log ' 𝑥 = −∞ and
7→G*
lim log ' 𝑥 = +∞
7→1P
4. for 0 < 𝑎 < 1:
lim log ' 𝑥 = +∞ and
7→G*
lim log ' 𝑥 = −∞
7→1P
76
the natural logarithm 𝑦 = log F 𝑥 = ln 𝑥 is the logarithmic func;on with basis “e”
the domain of ln 𝑥 is {𝑥 ∈ 𝑅 ∈∶ 𝑥 > 0} and the image is R
example
calculate the domain of 𝑓(𝑥) = ln( ln(ln 𝑥))
we have:
𝑥>0
ln 𝑥 > 0 Þ 𝑒 YA 7 > 𝑒 G Þ 𝑥 > 1 we apply exponen;al func;on to both members
ln(ln 𝑥 ) > 0 Þ 𝑒 YA(YA 7) > 𝑒 G Þ ln 𝑥 > 1 Þ 𝑒 YA 7 > 𝑒 & Þ 𝑥 > 𝑒
the solu;on is: 𝑥 > 𝑒
77
ALGEBRIC PROPPERTIES FOR THE LOGARITHMIC FUNCTIONS
1.
log 1 (𝑥𝑦) = log 1 𝑥 + log 1 𝑦
∀𝑥, 𝑦, 𝑎 > 0,
proof (1.)
it is enough to show that
𝑎≠1
𝑎YCZ+ (7=) = 𝑎(YCZ+ 71YCZ+ =)
since 𝑎 [ is an injec;ve func;on, we have that
𝑥𝑦 = 𝑎YCZ+ (7=)
but,
𝑥 × 𝑦 = 𝑎YCZ+ 7 × 𝑎YCZ+ = = 𝑎YCZ+ 7 1 YCZ+ =
then,
𝑎YCZ+ (7=) = 𝑎YCZ+ 7 1 YCZ+ =
2.
log 1 (𝑥 6 ) = ylog 1 𝑥
∀𝑥, 𝑎 > 0,
proof (2.)
we have that
𝑎≠1
∀𝑦 ∈ 𝑅
𝑦
𝑥 = = 𝑎log𝑎 (𝑥 )
but,
(𝑥)= = (𝑎log𝑎 (𝑥) )= = 𝑎 = YCZ+ (7)
(𝐴J )) = 𝐴J×) à the second “=”
then,
𝑦
𝑎log𝑎 (𝑥 ) = 𝑎 = YCZ+ (7) Þ
log𝑎 (𝑥𝑦 ) = ylog𝑎 𝑥
3.
log 1 𝑐 = log 1 𝑏 × log 3 𝑐
proof (3.) – change of the basis
we have that
but,
𝑐 = 𝑎YCZ+ \
𝑎YCZ+ I×YCZ, \ = 𝑏 YCZ, \ = 𝑐
Þ 𝑎YCZ+ \ = 𝑎YCZ+ I×YCZ, \ Þ log ' 𝑐 = log ' 𝑏 × log I 𝑐
78
LIMITS WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS
1.
𝑎/
= +∞
/→*+ 𝑥
lim
∀𝑎 ∈ 𝑅, 𝑎 > 1
proof (1.)
we start by proving that
𝑎)
= +∞ with 𝑛 ∈ 𝑁
)→*+ 𝑛
lim
since 𝑎 > 1, 𝑎 = 1 + 𝛿 with 𝛿 > 0
then,
𝑛 )9B B
)
𝑎) (1 + 𝛿)) ∑B,- ]𝑘^ 1 𝛿
=
=
𝑛
𝑛
𝑛
we use Newton’s binomial formula
𝑛
𝑛
𝑛
] ^ 1𝛿 - + ] ^ 1𝛿 . + ] ^ 1𝛿 # + ⋯ + 𝛿 )
0
1
2
=
𝑛
+ ⋯ + 𝛿) ≥ 0
𝑛
𝑛(𝑛 − 1) #
1 + 𝑛𝛿 + ] ^ 𝛿 # 1 + 𝑛𝛿 +
𝛿
2
2
≥
=
𝑛
𝑛
𝑛!
𝑛(𝑛 − 1)(𝑛 − 2)! 𝑛(𝑛 − 1)
𝑛
] ^=
=
=
2
(𝑛 − 2)! 2!
(𝑛 − 2)! 2
2
now, we recap that
𝑛(𝑛 − 1) #
𝛿
𝑎) 1 + 𝑛𝛿 +
1 𝑛𝛿 𝑛(𝑛 − 1) #
1
𝑛−1 #
2
≥
= +
+
𝛿 = +𝛿+
𝛿
𝑛
𝑛
𝑛 𝑛
2𝑛
𝑛
2
now, we take the limit:
𝑎)
1
𝑛−1 #
lim
≥ lim t + 𝛿 +
𝛿 u = +∞
)→*+ 𝑛
)→*+ 𝑛
2
1
𝑛 − 1(= +∞) #
= 0; 𝛿 = 𝛿;
𝛿 = +∞
𝑛
2
𝑎)
= +∞
)→*+ 𝑛
lim
remark
lim 𝑘 = 𝑘 (it is constant)
/→/#
79
now, we introduce the integer point 𝑛(𝑥) of 𝑥 (with 𝑥 > 0) as the greatest natural number
smaller or equal than x:
example
x = 2,73
𝑛(𝑥) = 2; 𝑟(𝑥) = 0,73
so, 𝑥 = 𝑛(𝑥) + 𝑟(𝑥) with 0 ≤ 𝑟(𝑥) < 1
then,
𝑎7
𝑎 7 𝑎)(7) 𝑛(𝑥)
= )(7)
𝑥
𝑛(𝑥 ) 𝑥
𝑎
we have already proved the limit for this quan;ty
we consider
𝑎7
≤ 𝑎 7/)(7) = 𝑎V(7)
)(7)
𝑎
now,
𝑎G ≤ 𝑎V(7) < 𝑎& since
a ≤ r(x) < 1
𝑎- = 1; 𝑎. = 𝑎; 𝑎N(/) = is bounded
now, we take
𝑛(𝑥) 𝑥 − 𝑟(𝑥) 𝑥 𝑟(𝑥)
𝑟(𝑥)
=
= −
=1−
𝑥
𝑥
𝑥
𝑥
𝑥
if we take the limit:
𝑛(𝑥)
𝑟(𝑥)
= lim 1 −
=1
/→*+ 𝑥
/→*+
𝑥
lim
now, puƒng all the informa;on together we get:
𝑎7
𝑎 7 𝑎)(7) 𝑛(𝑥)
= lim \ )7 ×
×
] = +∞
7→1P 𝑥
7→1P 𝑎
𝑛(𝑥 )
𝑥
lim
𝑎/
= bounded
𝑎)/
𝑎)(/)
= +∞
𝑛(𝑥)
𝑛(𝑥)
=1
𝑥
80
2.
𝑎/
lim
= +∞
/→*+ 𝑥 O
∀𝑎 > 1, ∀𝑝 > 0, 𝑝 ∈ 𝑅
proof (2.)
we write
I. F. =
& 7 ]
⎡a𝑎] b ⎤
lim ⎢
7→1P ⎢
. /
t𝑎O u
𝑥
∞
∞
𝑥
⎣
⎥
⎥
⎦
= +∞
)
-
and now we apply limit (1.) since 𝑎 > 1 because 𝑎 > 1; then
remark
in par;cular, we have
𝑎7
lim
= +∞
7→1P 𝑥 ]
𝑒7
lim
= +∞
7→1P 𝑥 ]
3.
log 1 𝑥
=0
/→*+ 𝑥 O
lim
∀𝑝 > 0
I. F. =
∀𝑝 > 0
∞
∞
proof (3.)
we change variable
𝑦 = log 1 𝑥
Û 𝑥 = 𝑎6
so,
log ' 𝑥
𝑦
=
lim
=0
7→1P 𝑥 ]
=→1P (𝑎 ] ) =
lim
because
remark
in par;cular, we have
𝑏=
𝑦
lim
= +∞ Þ = = 0
=→1P 𝑦
𝑏
ln 𝑥
= 0 ∀𝑝 > 0
7→1P 𝑥 ]
ln 𝑥
Þ lim
= 0 (𝑝 = 1)
7→1P 𝑥
lim
81
4.
lim 𝑥 O × log 1 𝑥 = 0
/→-*
proof (4.)
we write
∀𝑎 > 1, ∀𝑝 > 0
I. F. = 0 × ∞
1 /&
1
g
h
g
h
log
log
'
'
log
𝑥
log
𝑥
'
'
𝑥
𝑥
𝑥 ] × log ' 𝑥 =
=
=
=
−
1
1 ]
1 ]
1 ]
g h
g h
g h
𝑥]
𝑥
𝑥
𝑥
we change variable
.
.
𝑦=/
lim / = +∞
/→-*
Þ 𝑦 → +∞
1
log ' g𝑥 h
j
lim 𝑥 ] × log ' 𝑥 = lim i−
7→G1
𝑥→0+
1 ]
g h
𝑥
= lim a−
𝑦→+∞
thanks to limit (3.)
remark
in par;cular, we have that
log ' 𝑦
b=0
𝑦]
lim 𝑥 ] × ln 𝑥 = 0
(∀𝑝 > 0)
Þ lim 𝑥 × ln 𝑥 = 0
(𝑝 = 1)
7→G1
𝑥→0+
5.
ln(1 + 𝑥)
=1
/→𝑥
lim
proof (5.)
I. F. =
0
0
&
ln(1 + 𝑥)
1
= lim × ln(1 + 𝑥) = lim ln(1 + 𝑥)7
7→G
7→G 𝑥
7→G
𝑥
lim
=→𝑦 =
1
𝑥
1 =
ln kg1 + h l
𝑦
= lim
=1
=→P
𝑥
1 6
ln §]1 + 𝑦^ ¨
𝑥
=𝑒
82
remark
we already know that
1 6
lim t1 + u = 𝑒
6→±+
𝑦
NEPERO’s number
6.
𝑒/ − 1
=1
/→𝑥
lim
I. F. =
remark
I can write limit (6.) as
remark
we can write limit (5.) as
0
0
𝑒 / − 1~𝑥
Þ 𝑒 / ~1 + 𝑥
(𝑥~0)
(𝑥~0)
ln(1 + 𝑥) ~𝑥
(𝑥~0)
proof (6.)
we change variable
𝑦 = 𝑒 / − 1 Þ 𝑒 / = 𝑦 + 1 (apply ln) Þ ln(𝑎 / ) = ln(1 + 𝑦) = 𝑥 ln 𝑒 (= 1) = ln(𝑦 + 1)
𝑥 = ln(1 + 𝑦)
then,
𝑒7 − 1
𝑦
lim
= lim
=1
7→G
=→G ln(1 + 𝑦)
𝑥
thanks to the limit (5.)
remark (order of infinity)
P
] the exponen;al func;on is “stranger” than every
P
power 𝑥 ] and the logarithm is “weaker” than the power 𝑥 ]
in the case of a conflict like [0 × ∞] or [
then, the exponen;al func;on is an infinity of higher order than any real posi;ve power
of 𝑥 as 𝑥 → +∞ and the logarithm func;on is an infinity of lower than any real posi;ve power
of 𝑥 when 𝑥 → +∞
now, to solve some delicate cases concerning the determina;on of limits of the type
𝑓(𝑥) N(7)
we rewrite it as
𝑓(𝑥) N(7) = 𝑒 YA`6(7)
.(0) a
requiring that 𝑓(𝑥) > 0
remark
the indeterminate forms in this context are:
[1+ ], [∞- ], [0- ]
83
remark
lim 17 = 1
7→1P
because it is a constant func;on
this is different from
1 7
lim a1 + b = 𝑒
7→1P
𝑥
[1P ] = I. F.
remark (order of infinity: graphical insight)
examples of limits of type 𝒇(𝒙)𝒈(𝒙)
1. lim 𝑥 / = - I. F
/→-*
0
lim 𝑥 7 = lim 𝑒 YA(7 ) = lim 𝑒 7 YA 7 = 𝑒 G = 1
7→G1
2.
7→G1
7→G1
𝑥 ln 𝑥 = 0
lim 𝑥 / =
/→*+
0
lim 𝑒 YA(7 ) = lim 𝑒 7 YA 7 = [𝑒 1P ] = +∞
7→1P
7→1P
𝑥 ln 𝑥 = +∞
84
GRAPHICAL INTERPRETATION
1. even func;on 𝑓(−𝑥) = 𝑓(𝑥)
2. odd func;on 𝑓(−𝑥) = −𝑓(𝑥)
85
3. bounded func;on:
4. unbounded func;on:
86
5. ver;cal asymptote
6. horizontal asymptote
remark
lim sin 𝑥 = ∄ Þ we HAVE NOT horizontal asymptotes
/→*+
87
TOPIC 12
DIFFERENTIAL CALCULUS
differen;al calculus is the most appropriate tool for solving op;miza;on problems which are
significant in economics
in the simplest situa;on are interested in finding the values which maximize or minimize a
func;on, which is the target we aim to op;mize; typical examples are the maximiza;on of a
consumer’s u;lity or the minimiza;on of a product cost
to solve problems of this type, we have to analyze how the target func;on varies according to the
func;on to the fluctua;ons of the variable on which it depends
we consider 𝑓: 𝐴 → 𝑅 and let 𝑥- ∈ 𝐴 br an accumula;on point of A
defini@on
6
the incremental ra;o 𝑅7" of f at 𝑥- is defined as
6
𝑅7" (𝑥 ) =
∆𝑓 𝑓 (𝑥 ) − 𝑓(𝑥G )
=
∆𝑥
𝑥 − 𝑥G
∀𝑥 ∈ 𝐴, 𝑥 ≠ 𝑥G
defini@on
if ∃ lim
7(/)97(/# )
/→/#
/9/#
= 𝑓′(𝑥- ) and it is FINITE then we say that f is differen;able at 𝑥- amd f’(𝑥- ) is
called DERIVATIVE of f at 𝑥defini@on
1. if ∃ lim
7(/)97(/# )
2. if ∃ lim
7(/)97(/# )
/→/#%
/→/#$
/9/#
/9/#
= 𝑓′* (𝑥- ), then such a limit is said to be the right deriva@ve of f in 𝑥= 𝑓′9 (𝑥- ), then such a limit is said to be the leY deriva@ve of f in 𝑥88
remark
if f is differen;able at 𝑥- , then lim
/→/#%
7(/)97(/# )
/9/#
= lim
/→/#$
7(/)97(/# )
/9/#
and they are FINITE
example 1
𝑥 if 𝑥 ≥ 0
−𝑥 if 𝑥 > 0
is f differen;able in 𝑥- = 0?
we consider 𝑓(𝑥) = |𝑥| = G
|𝑥| − 0
|𝑥|
𝑓(𝑥) − 𝑓(𝑥- )
𝑥
= lim
= lim
= lim = +1
/→/#%
/→/#% 𝑥 − 0
/→/#% 𝑥
/→/#% 𝑥
𝑥 − 𝑥lim
|𝑥| − 0
|𝑥|
𝑓(𝑥) − 𝑓(𝑥- )
−𝑥
= lim
= lim
= lim
= −1
/→/#$
/→/#$ 𝑥 − 0
/→/#$ 𝑥
/→/#$ 𝑥
𝑥 − 𝑥lim
since we have that 𝑓 5 * (𝑥- ) = +1 and 𝑓 5 9 (𝑥- ) = −1, the func;on f is NOT differen;able in 𝑥- = 0
example 2
.
𝑥 sin ]/^ if 𝑥 ≠ 0
consider 𝑓(𝑥) = ª
0 if 𝑥 = 0
is f differen;able in 𝑥- = 0?
1
1
𝑥 sin ]𝑥^ − 0
𝑥 sin ]𝑥^
𝑓(𝑥) − 𝑓(𝑥- )
1
lim
= lim
= lim
= lim t u = ∄
/→/→/→/→- 𝑥
𝑥 − 𝑥𝑥−0
𝑥
1
t u = +∞, but 𝑥 = 0
𝑥
since the limit of the incremental ra;o does NOT exist, then the func;on f is NOT differen;able
in 𝑥- = 0
89
GEOMETRICAL INTERPETATION OF THE DERIVATIVE
we recall that the straight line is defined as
𝑦 =𝑚×𝑥+𝑞
where 𝑚 us the slope and 𝑞 is the intercept
now, we consider the incremental ra;o
6
𝑅7" (𝑥 ) =
∆𝑓 𝑓 (𝑥 ) − 𝑓(𝑥G )
=
∆𝑥
𝑥 − 𝑥G
with represents the slope of the secant passing through the points l𝑥- , 𝑓(𝑥- )m and l𝑥, 𝑓(𝑥)m
As 𝑥 → 𝑥- , the secant tends to a “limit posi;on” which is represented by the tangent to the graph
of f at the point l𝑥- , 𝑓(𝑥- )m
then, the tangent of a func;on f at 𝑥- defined as:
𝑦 = 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) + 𝑓(𝑥- )
𝑓 5 (𝑥- ) = 𝑚 → slope
𝑓(𝑥- ) = 𝑞 → intercept
example in economic
we consider a factory which has to sustain a total cost C in order to produce a certain quan;ty 𝑞 of
its product
we call C =C(𝑞) the cost as as a func;on of the produced quan;ty 𝑞; if the factory shi}s form a
certain amount of produc;on 𝑞- to another 𝑞, it is useful to es;mate the ra;o between the
increment od the produc;on cost and the increment of the quan;ty which is produced
such a ra;o
is called the average rate of cost varia;on
𝐶(𝑞) − 𝐶(𝑞- )
𝑞 − 𝑞-
the varia;on of C rela;ve to an infinitesimal varia;on of 𝑞, that is the deriva;ve C’(𝑞- ), is called
marginal cost at 𝑞90
THEOREM (“differenGability implies conGnuity”)
if a func;on f is differen;able at point 𝑥- (HYPOTHESIS),
then the func;on is also con;nuous at 𝑥- (THESIS)
proof
since f is differen;able at 𝑥- , we have that
𝑓(𝑥) − 𝑓(𝑥- )
∃ lim
= 𝑓′(𝑥- )
/→/#
𝑥 − 𝑥and it is FINITE
then, we have to prove that f is con;nuous in 𝑥- , which means that:
lim 𝑓(𝑥) = 𝑓(𝑥- )
/→/#
then, we rewrite the previous characteriza;on is the following way
lim l𝑓(𝑥) − 𝑓(𝑥- )m = 0
/→/#
“how to prove that?”
thus,
𝑓(𝑥) − 𝑓(𝑥- )
× (𝑥 − 𝑥- ) = 0
/→/#
𝑥 − 𝑥-
lim l𝑓(𝑥) − 𝑓(𝑥- )m = lim
/→/#
𝑓(𝑥) − 𝑓(𝑥- )
→ 𝑓 5 (𝑥- ) = FINITE
𝑥 − 𝑥(𝑥 − 𝑥- ) = 0
then, we have proven that the func;on is con;nuous in 𝑥remark
differen;ability Þ con;nuity
BUT the vice versa in NOT always true!
the con;nuity does NOT always imply differen;ability!
contra-examples
1. if we consider 𝑓(𝑥) = |𝑥|, then f is con;nuous, but it is not differen;able in zero
.
𝑥 sin ]/^ if 𝑥 ≠ 0
2. if we take 𝑓(𝑥) = ª
then f is con;nuous but is not differen;able in zero
0 if 𝑥 = 0
defini@on
consider 𝑓: 𝐴 → 𝑅; if f is differen;able at any point 𝑥 ∈ 𝐴, then we can consider the DERIVATIVE
FUNCTION 𝑓′(𝑥) as:
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
𝑓 5 (𝑥) = lim
∀𝑥 ∈ 𝐴
S→ℎ
remark
𝑓′(𝑥) is a func;on!
𝑓 5 (𝑥- ) is a value (a number)!
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
𝑓
→ 𝑅𝑥 (ℎ)
ℎ
91
BASIC DERIVATIVES
1.
(𝑘 5 ) = 0
“the deriva;ve of k (constant”
that means the derivate of a constant func;on 𝑓(𝑥) = 𝑘 ∀𝑥 ∈ 𝑅 is 𝑓′(𝑥) = 0 ∀𝑥 ∈ 𝑅
proof (1.)
𝑅/B# (𝑥) =
𝑓(𝑥) − 𝑓(𝑥- ) 𝑘 − 𝑘
=
=0
𝑥 − 𝑥𝑥 − 𝑥-
2.
(𝑥 ) )L = 𝑛𝑥 )/& , 𝑛 ∈ 𝑁 − {0}
proof (2.)
we take 𝑓(𝑥) = 𝑥 ) , then
𝑛
)
)
∑)8cG g h 𝑥 )/8 ℎ8 − 𝑥 )
(
)
(
)
𝑓
𝑥
+
ℎ
−
𝑓(𝑥)
𝑥
+
ℎ
−
𝑥
𝑘
𝑓 L (𝑥 ) = lim
= lim
= lim
b→G
b→G
b→G
ℎ
ℎ
ℎ
𝑛
𝑥 ) + 𝑛𝑥 )/& + g h 𝑥 )/* ℎ* + ⋯ + ℎ) − 𝑥 )
2
= lim
b→G
ℎ
𝑛 )/* *
ℎ
𝑛𝑥 )/& ℎ g2 h 𝑥
ℎ)
= lim {
+
+ ⋯ + | = 𝑛𝑥 )/&
b→G
ℎ
ℎ
ℎ
𝑛
] ^ 𝑥 )9# ℎ#
2
= 0; … =→ 0;
ℎ
ℎ)
=0
ℎ
3.
(sin 𝑥 )L = cos 𝑥
proof (3.)
we take 𝑓(𝑥) = sin 𝑥
then,
𝑓 (𝑥 + ℎ) − 𝑓(𝑥)
sin(𝑥 + ℎ) − sin 𝑥
= lim
b→G
b→G
ℎ
ℎ
𝑓 L (𝑥 ) = lim
sin 𝑥 cos ℎ + cos 𝑥 sin ℎ − sin 𝑥
b→G
ℎ
= lim
cos ℎ − 1
sin ℎ
l = cos 𝑥
= lim ksin 𝑥 a
b + cos 𝑥
b→G
ℎ
ℎ
cos ℎ − 1
sin ℎ
t
u = 0;
= 1
ℎ
ℎ
92
4.
(cos 𝑥)L = −sin 𝑥
proof (4.)
we take 𝑓(𝑥) = cos 𝑥
then,
𝑓 (𝑥 + ℎ) − 𝑓(𝑥)
cos(𝑥 + ℎ) − cos 𝑥
= lim
b→G
b→G
ℎ
ℎ
𝑓 L (𝑥 ) = lim
cos 𝑥 cos ℎ + sin 𝑥 sin ℎ − cos 𝑥
b→G
ℎ
= lim
cos ℎ − 1
sin ℎ
l = −sin 𝑥
= lim kcos 𝑥 a
b − sin 𝑥
b→G
ℎ
ℎ
t
cos ℎ − 1
sin ℎ
u = 0;
= 1
ℎ
ℎ
5.
(𝑒 7 )L = 𝑒 7
proof (4.)
we take 𝑓(𝑥) = 𝑒 /
then,
𝑓 L (𝑥 ) = lim
b→G
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
𝑒 71b − 𝑒 7
𝑒b − 1
7
= lim
= lim 𝑒 \
] = 𝑒7
b→G
b→G
ℎ
ℎ
ℎ
𝑒S − 1
p
q=1
ℎ
𝒇(𝒙)
𝑘
𝑥)
sin 𝑥
cos 𝑥
𝑒7
𝒇′(𝒙)
0
𝑛𝑥 )/&
cos 𝑥
−sin 𝑥
𝑒7
93
DERIVATION RULES
THEOREM (deriva@ve of the sum)
if f, g are differen;able func;ons at the point 𝑥- and we have the following formula:
(𝑓 + 𝑔)5 (𝑥- ) = 𝑓 5 (𝑥- ) + 𝑔′(𝑥- )
proof
we consider
(𝑓 + 𝑔)(𝑥 ) − (𝑓 + 𝑔)(𝑥G )
7→7"
𝑥 − 𝑥G
(𝑓 + 𝑔)L (𝑥G ) = lim
𝑓(𝑥) + 𝑔(𝑥) − }𝑓 (𝑥G ) + 𝑔(𝑥G )~
𝑓 (𝑥 ) − 𝑓(𝑥G ) 𝑔(𝑥 ) − 𝑔(𝑥G )
= lim \
+
]
7→7"
7→7"
𝑥 − 𝑥G
𝑥 − 𝑥G
𝑥 − 𝑥G
= lim
= 𝑓 L (𝑥G ) + 𝑔′(𝑥G )
remark
in a similar way, one can prove that
1. (𝑓 − 𝑔)L (𝑥0 ) = 𝑓′ (𝑥0 ) − 𝑔′(𝑥0 )
2. (𝑘𝑓)L (𝑥0 ) = 𝑘𝑓′(𝑥0 )
where k is a constant
THEOREM (deriva@ve of the product)
if f, g are differen;able func;ons at the point 𝑥- , then 𝑓 × 𝑔 is differen;able at 𝑥- , and one has
(𝑓 × 𝑔)5 (𝑥- ) = 𝑓 5 (𝑥- ) × 𝑔(𝑥- ) + 𝑓(𝑥- ) × 𝑔′(𝑥- )
proof
we consider
(𝑓 × 𝑔)(𝑥 ) − (𝑓 × 𝑔)(𝑥G )
𝑓(𝑥)𝑔(𝑥) − }𝑓(𝑥G )𝑔(𝑥G )~
= lim
7→7"
7→7"
𝑥 − 𝑥G
𝑥 − 𝑥G
(𝑓 × 𝑔)L (𝑥G ) = lim
= lim \
7→7"
= lim p𝑔(𝑥)
/→/#
𝑔(𝑥) = 𝑔(𝑥2 );
𝑓(𝑥 )𝑔(𝑥) − 𝑓 (𝑥G )𝑔(𝑥G ) + 𝑓 (𝑥G )𝑔(𝑥 ) − 𝑓(𝑥)𝑔(𝑥)
]
𝑥 − 𝑥G
𝑓(𝑥) − 𝑓(𝑥- )
𝑔(𝑥) − 𝑔(𝑥- )
+ 𝑓(𝑥- )
q = 𝑓 5 (𝑥- ) × 𝑔(𝑥- ) + 𝑓(𝑥- ) × 𝑔′(𝑥- )
𝑥 − 𝑥𝑥 − 𝑥-
𝑓(𝑥) − 𝑓(𝑥2 )
= 𝑓 3 (𝑥2 );
𝑥 − 𝑥2
𝑓(𝑥2 ;
𝑔(𝑥) − 𝑔(𝑥2 )
= 𝑔′(𝑥2 )
𝑥 − 𝑥2
remark
g is con;nuous in 𝑥- , since g is differen;able in 𝑥94
examples
1. take 𝑓(𝑥) = 𝑥 sin 𝑥 then,
𝑓 5 (𝑥) = 1 × sin 𝑥 + 𝑥 × (sin 𝑥)5 = sin 𝑥 + 𝑥 cos 𝑥
2. take 𝑓(𝑥) = 𝑒 / cos 𝑥
then,
𝑓 5 (𝑥) = (𝑒 / )5 cos 𝑥 + 𝑒 / (cos 𝑥)5 = 𝑒 / cos 𝑥 + 𝑒 / (− sin 𝑥) = 𝑒 / (cos 𝑥 − sin 𝑥)
THEOREM (deriva@ve of the reciprocal)
if f is differen;able in a point 𝑥- and 𝑓(𝑥- ) ≠ 0, then
&
is differen;able at 𝑥- , and we have:
6
1 L
𝑓′(𝑥G )
a b (𝑥G ) = − *
𝑓
𝑓 (𝑥G )
proof
we consider
lim
1
1
t𝑓u (𝑥) − t𝑓 u (𝑥- )
𝑥 − 𝑥-
/→/#
𝑓 (𝑥- ) − 𝑓(𝑥)
1
1
−
𝑓(𝑥) 𝑓(𝑥- )
𝑓(𝑥)𝑓(𝑥- )
= lim
= lim
/→/#
/→/#
𝑥 − 𝑥𝑥 − 𝑥-
𝑓(𝑥2 ) − 𝑓(𝑥)
1
𝑓(𝑥)𝑓(𝑥2 )
1
𝑓′(𝑥0 )
×
= lim 3−
=− 2
4×
4→4!
𝑥 − 𝑥2
𝑓(𝑥)𝑓(𝑥2 ) 4→4!
𝑥 − 𝑥2
𝑓(𝑥)𝑓(𝑥2 )
𝑓 (𝑥0 )
= lim
5−
𝑓(𝑥)𝑓(𝑥2 )
6 = −𝑓 3 (𝑥2 )
𝑥 − 𝑥2
1
→ 𝑓(𝑥) = 𝑓(𝑥2 )
𝑓(𝑥)𝑓(𝑥2 )
remark
f is con4nuous since f is differen4able
THEOREM (deriva@ve of the ra@o)
if f, g are differen;able at a point 𝑥- , 𝑔(𝑥- ) ≠ 0, then
6
is differen;able at 𝑥- , and we have:
N
𝑓 L
𝑓 L (𝑥G )𝑔(𝑥G ) − 𝑓 (𝑥G )𝑔′(𝑥G )
a b (𝑥G ) = −
𝑔
𝑔* (𝑥G )
proof
𝑓 5
1 5
1
1 5
5
t u (𝑥- ) = t𝑓 × u (𝑥- ) = 𝑓 (𝑥- ) t u (𝑥- ) + 𝑓(𝑥- ) t u (𝑥- )
𝑔
𝑔
𝑔
𝑔
theorem: deriva;ve of the product
𝑓′(𝑥- )
𝑔5 (𝑥- )
=
+ 𝑓(𝑥- ) p− #
q
𝑔(𝑥- )
𝑔 (𝑥- )
theorem: deriva;ve of the reciprocal
𝑓 5 (𝑥- )𝑔(𝑥- ) − 𝑓(𝑥- )𝑔5 (𝑥- )
=
𝑔# (𝑥- )
95
example
(𝑡𝑔𝑥)5
sin 𝑥 5 (sin 𝑥)5 cos 𝑥 − sin 𝑥 (cos 𝑥)5 cos # 𝑥 + sin# 𝑥
=t
u =
=
cos 𝑥
cos # 𝑥
cos # 𝑥
cos # 𝑥 + sin# 𝑥 = 1
=
sin# 𝑥
= tg # 𝑥
cos # 𝑥
1
sin# 𝑥
=
1
+
cos # 𝑥
cos # 𝑥
THEOREM (deriva@ve of the composi@on of func@ons)
let f, g be wo func;ons and assume that the composi;on gof exists at 𝑥if f is differen;able at 𝑥- and g is differen;able at 𝑓(𝑥- ), then gof is differen;able at 𝑥- and we
have
(𝑔𝑜𝑓)5 (𝑥- ) = 𝑔5 (𝑓(𝑥- )) × 𝑓 5 (𝑥- )
examples
1. take 𝑓(𝑥) = 𝑒 #/
then,
𝑓 5 (𝑥) = 𝑒 #/ × (2𝑥)5 = 2𝑒 #/
(2𝑥)3 = 2
2. take 𝑓(𝑥) = sin# 𝑥 = (sin 𝑥)#
then,
𝑓 5 (𝑥) = 2(sin 𝑥)#9. × (sin 𝑥)5 = 2 sin 𝑥 × cos 𝑥
(sin 𝑥)3 → we take the derivative
3. take 𝑓(𝑥) = sin 𝑥 #
then,
𝑓 5 (𝑥) = cos(𝑥 # ) × (𝑥 # )5 = 2𝑥cos(𝑥 # )
(𝑥 6 )3 = 2𝑥
4. take 𝑓(𝑥) = ln(𝑥 # + 3)
then,
𝑓 5 (𝑥) =
1
𝑥# + 3
× (𝑥 # + 3)5 =
2𝑥
𝑥# + 3
THEOREM (deriva@ve of the inverse func@on)
let f be strictly monotone; if f is differen;able at 𝑥- and 𝑓 5 (𝑥- ) ≠ 0 then 𝑓 9. is differen;able at
𝑦- = 𝑓(𝑥- ) and
1
(𝑓 9. )(𝑦- ) = 5
𝑓 (𝑥- )
96
proof
we apply the theorem of the limit for the composite func;on, and we have
𝑓 /& (𝑦) − 𝑓 /& (𝑦G )
𝑥 − 𝑥G
lim
=
lim&)
=→="
7→ Y@d 6 (=) 𝑓 (𝑥 ) − 𝑓(𝑥G )
𝑦 − 𝑦G
7→7
"
=→ 𝑦 = 𝑓(𝑥); 𝑥 = 𝑓 89 (𝑦)
lim 𝑓 89 (𝑦) = 𝑥2
:→:!
𝑥 − 𝑥0
1
−
𝑦→𝑦0 𝑓(𝑥) − 𝑓(𝑥0 )
𝑓 ′(𝑥0 )
= lim
remark
lim 𝑓 9. (𝑦) = 𝑓 9. (𝑦- ) = 𝑥6→6#
𝑓 9. is con;nuous
examples
1. (arc sin 𝑥 )$ =
#
(−1 < 𝑥 < 1)
√#&' !
indeed, we consider 𝑦 = arc sin 𝑥 Û 𝑥 = sin 𝑦 for −
(arc sin 𝑥)5 =
D
D
<
𝑦
<
*
*
1
1
1
1
=
=
=
5
(sin 𝑦)
cos 𝑦 F1 − sin# 𝑦 √1 − 𝑥 #
= sin# 𝑦 + cos # 𝑦 = 1 Þ cos 𝑦 = F1 − sin# 𝑦
2. (arc cos 𝑥)$ = −
#
√#&' !
(−1 < 𝑥 < 1)
indeed, we consider 𝑦 = arc cos 𝑥 Û 𝑥 = cos 𝑦 for 0 < 𝑦 < 𝜋
(arc cos 𝑥)5 =
1
1
1
1
=
−
=
−
=
−
(cos 𝑦)5
sin 𝑦
√1 − 𝑥 #
F1 − cos # 𝑦
cos # 𝑦 = 𝑥 #
3. (arc tg 𝑥 )$ =
#
#(' !
indeed, we consider 𝑦 = arc tg 𝑥 Û 𝑥 = tg 𝑦 for −
(arc tg 𝑥)5 =
D
D
<
𝑦
<
*
*
1
1
1
=
=−
5
#
(tg 𝑦)
1 + tg 𝑦
1 + 𝑥#
97
THEOREMS FOR DIFFERENTIABLE FUNCTIONS
defini@on
consider f: Aà R; then, a point 𝑥- ∈ 𝐴 is said to be a point of RELATIVE MINIMUM if ∃U𝑥- s.t.
𝑓(𝑥) > 𝑓(𝑥- ) ∀𝑥 ∈ (U𝑥- − {𝑥- }) ∩ 𝐴
defini@on
consider f: Aà R; then, a point 𝑥- ∈ 𝐴 is said to be a point of RELATIVE MAXIMUM if ∃U𝑥- s.t.
𝑓(𝑥) < 𝑓(𝑥- ) ∀𝑥 ∈ (U𝑥- − {𝑥- }) ∩ 𝐴
98
THEOREM (Fermat’s theorem)
let f: Aà R and let 𝑥- be an interior point of A
assume that the func;on f is differen;able at 𝑥- : if 𝑥- is a point of rela;ve minimum or it is a point
of rela;ve maximum then
𝑓 L (𝑥G ) = 0
proof
we suppose that 𝑥- is a point of rela;ve
maximum
then, ∀𝑥 ∈ U𝑥- ⊊ A (strictly contained)
⊊=
⊂
≠
we have
𝑓(𝑥) − 𝑓(𝑥- )
(≤ 0)
=!
(≥ 0)
𝑥 − 𝑥-
if 𝑥 > 𝑥if 𝑥 < 𝑥-
𝑓(𝑥) − 𝑓(𝑥" )
≤0
𝑥 − 𝑥"
since f is differen;able in 𝑥- (by hypothesis) we get
𝑓(𝑥 ) − 𝑓(𝑥G )
𝑓 (𝑥 ) − 𝑓(𝑥G )
= lim
7→7" 1
7→7" /
𝑥 − 𝑥G
𝑥 − 𝑥G
𝑓 L (𝑥G ) = lim
thus,
𝑓 (𝑥 ) − 𝑓(𝑥G )
𝑓 (𝑥 ) − 𝑓(𝑥G )
≤ 0 ≤ lim
7→7" 1
7→7" /
𝑥 − 𝑥G
𝑥 − 𝑥G
lim
but since they have to coincide, we obtain that
𝑓 L (𝑥G ) = 0
in a similar way, we can prove the thesis for 𝑥- which is a point of rela;ve minimum
example
the managers of a beer factory realize that their cans cost too much; they must maintain the
cylindrical shape, the material and the volume (33cl)
so, they have to act on the propor;ons of the can, trying to minimize the total surface of the can
the problem is: how to choose the height and the base radius in such a way that the total surface is
minimal while maintaining a constant volume of 33cl?
let V be the volume of the can, and h is the height and r its base radius
we recall that
𝑉
V = 𝑟#𝜋 × ℎ Þ ℎ = #
𝑟 𝜋
𝑟 # 𝜋 = base surface area
moreover, the total surface area is
2𝑉
𝐴(𝑟) = 2𝜋𝑟 # + 2𝜋𝑟 × ℎ = 2𝜋𝑟 # +
𝑟
#W
2𝜋𝑟 # = base surface area; 2𝜋𝑟 × ℎ = N = side surface area
99
we observe that the two terms of the sum are in compe;;on: small values of r produce small
values of the base areas but large values of the side area and vice versa
then, it could exist a value r* which balances there two terms and minimize the sum
how to find r*? we consider
𝐴′(𝑟) = 2𝜋(𝑟 * ) + 2V(𝑟 /& )L = 2𝜋(2𝑟 */& ) + 2V(−1𝑟 /&/& )
= 4𝜋𝑟 − 2V𝑟 /* = 4𝜋𝑟 −
2𝑉
𝑟2
to find a point of minimum we impose
𝐴L (𝑟) = 0 Þ 4𝜋𝑟 −
2𝑉
2𝑉
+ − 2V = 0 Þ 4π𝑟 + = 2V Þ 𝑟 + =
Þ
4π𝑟
𝑟*
4𝜋
%
Þ 𝑟∗ = ƒ
𝑉
2𝜋
are the cans on the market “op;mized”?
the common 33cl cans have height equal to 11,5cm and base diameter equal to 6,5cm, thus the
radius is equal to 3,25cm
the volume is:
V = (3,25)# × 𝜋 × (11,5) ⋍ 382cm$
the op;mized radius is:
%
𝑟∗ = ƒ
% 382
𝑉
=ƒ
⋍ 3,93cm
2𝜋
2𝜋
therefore, the op;mized can is a li|le less comfortable to hold but it has lower cost!
100
THEOREM (Rolle’s theorem)
let f: [a, b] à R be a con;nuous func;on and let f be differen;able in ]a, b[ s.t. 𝑓(𝑎) = 𝑓(𝑏)
then,
∃𝜉 ∈ ]𝑎, 𝑏[ s. t. 𝑓 5 (𝜉) = 0
𝜉 = "xi"
proof
thanks to the WEIERSTRASS THEOREM, f has a minimum and a maximum
therefore, we have a point of minimum and a point of maximum
if at least one of these two points is inside the interval ]a, b[, then due to the Fermat’s theorem,
we have that 𝑓 5 (𝜉) = 0 where 𝜉 is a such a point
then, we have concluded the proof
otherwise, the point of minimum and the point of maximum are extremes of the interval [a, b]
since 𝑓(𝑎) = 𝑓(𝑏), the func;on is CONSTANT
then, 𝑓 5 (𝜉) = 0 ∀𝜉 ∈ ]𝑎, 𝑏[
this concludes the proof
THEOREM (Cauchy’s theorem)
let f, g: [a, b] à R be a con;nuous func;on and differen;able in ]a, b[
assume that 𝑔5 (𝑥) ≠ 0 ∀𝑥 ∈ ]𝑎, 𝑏[
then,
𝑓 5 (𝜉) 𝑓(𝑏) − 𝑓(𝑎)
∃𝜉 ∈ ]𝑎, 𝑏[ s. t.
=
𝑔′(𝜉) 𝑔(𝑏) − 𝑔(𝑎)
proof
we consider an auxiliary func;on
𝜙(𝑥) = 𝑓(𝑥)[𝑔(𝑏) − 𝑔(𝑎)] − 𝑔(𝑥)[𝑓(𝑏) − 𝑓(𝑎)]
𝜙 = "phi"
then, we have that:
- 𝜙 is con;nuous in [a, b]
- 𝜙 is differen;able in ]a, b[
- 𝜙(𝑎) = 𝜙(𝑏) à ?
thus, we can apply the Rolle’s theorem having that ∃𝜉 ∈ ]𝑎, 𝑏[ s.t.
𝜙 5 (𝜉) = 0
Þ 𝑓 5 (𝜉)[𝑔(𝑏) − 𝑔(𝑎) − 𝑔5 (𝜉)[𝑓(𝑏) − 𝑓(𝑎) = 0
𝑓 5 (𝜉) 𝑓(𝑏) − 𝑓(𝑎)
Þ 5
=
𝑔 (𝜉) 𝑔(𝑏) − 𝑔(𝑎)
101
THEOREM (Lagrange’s theorem)
let f: [a, b] à R be a con;nuous func;on and let f be differen;able in ]a, b[
then,
𝑓(𝑏) − 𝑓(𝑎)
∃𝜉 ∈ ]𝑎, 𝑏[ s. t. 𝑓 5 (𝜉) =
𝑏−𝑎
proof
we take 𝑔(𝑥) = 𝑥 in the Cauchy’s theorem
then,
- g is con;nuous in [a, b]
- g is differen;able in ]a, b[
- 𝑔5 (𝑥) = 1 ∀𝑥 ∈ ]𝑎, 𝑏[
- 𝑔(𝑏) = 𝑏 and 𝑔(𝑎) = 𝑎
thanks to the Cauchy’s theorem, we have that
𝑓 5 (𝜉) 𝑓(𝑏) − 𝑓(𝑎)
𝑓(𝑏) − 𝑓(𝑎)
∃𝜉 ∈ ]𝑎, 𝑏[ s. t.
=
Þ 𝑓 5 (𝜉) =
1
𝑏−𝑎
𝑏−𝑎
geometrical interpreta@on
we consider
6(I)/6(')
that is the slope of the secant line passing through the points l𝑎, 𝑓(𝑎)m and
I/'
l𝑏, 𝑓(𝑏)m; the Lagrange theorem guarantees the existence of a point ∃𝜉 ∈ ]𝑎, 𝑏[ s.t. the tangent
line to the graph of f in the point l𝜉, 𝑓(𝜉)m is parallel to the secant line
102
consequences of Lagrange’s theorem
1. corollary
let 𝑓: I → R (with I = Interval) s.t. 𝑓 5 (𝑥) > 0 ∀𝑥 ∈ I
then, f is increasing (decreasing) in I
proof (1.)
we assume that 𝑓 5 (𝑥) > 0 ∀𝑥 ∈ I
then, we have to prove that
∀𝑥. , 𝑥# ∈ I,
(𝑓 5 (𝑥) < 0)
if 𝑥. < 𝑥# Þ 𝑓(𝑥. ) < 𝑓(𝑥# )
we apply the Lagrange’s theorem on the interval [𝑥. , 𝑥# ]
then, we have that ∃𝜉 ∈ ]𝑥. , 𝑥# [ s.t.
𝑓(𝑥# ) − 𝑓(𝑥. )
𝑓 5 (𝜉) =
> 0 (by hypothesis)
𝑥# − 𝑥.
since 𝑥. < 𝑥# (Þ 𝑥. − 𝑥# > 0), we have that
𝑓(𝑥# ) − 𝑓(𝑥. ) > 0 Þ 𝑓(𝑥# ) < 𝑓(𝑥. )
remark
in a similar we can exchange the roles of 𝑥. , 𝑥# and moreover we can prove with analogous
argument the case when
𝑓 5 (𝑥) < 0 ∀𝑥 ∈ I
2. corollary
let 𝑓: I → R (with I = Interval) s.t. 𝑓 5 (𝑥) = 0 ∀𝑥 ∈ I
then, f is a constant func;on
proof (2.)
we have to prove that
∀𝑥. , 𝑥# ∈ I,
if 𝑥. < 𝑥# Þ 𝑓(𝑥. ) = 𝑓(𝑥# )
we apply the Lagrange’s theorem on the interval [𝑥. , 𝑥# ]
then, we have that ∃𝜉 ∈ ]𝑎, 𝑏[ s.t.
𝑓(𝑥# ) − 𝑓(𝑥. )
𝑓 5 (𝜉) =
= 0 (by hypothesis)
𝑥# − 𝑥.
since 𝑥. < 𝑥# , we have that
𝑓(𝑥# ) − 𝑓(𝑥. ) = 0 Þ 𝑓(𝑥# ) = 𝑓(𝑥. )
103
3. corollary
let 𝑓, 𝑔: I → R (with I = Interval) s.t. 𝑓 5 (𝑥) = 𝑔5 (𝑥) ∀𝑥 ∈ I
then, 𝑓(𝑥) − 𝑔(𝑥) = 𝑘 (with k = constant)
proof (3.)
we consider an auxiliary func;on
𝜙(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)
we have that
𝜙 5 (𝑥) = 𝑓 5 (𝑥) − 𝑔5 (𝑥) = 0
∀𝑥 ∈ I
at this point, we can apply the previous corollary (2.) and therefore
𝜙(𝑥) = 𝑘 (with k = constant)
Þ 𝑓(𝑥) − 𝑔(𝑥) = 𝑘
THEOREM (De l’Hôpital’s theorem)
-
+
we introduce a powerful tool to solve limits of the form ´-µ and ´+µ
consider f, g differen;able in U𝑥- − {𝑥- } of a point 𝑥- and we assume that
𝑔5 (𝑥) ≠ 0 ∀𝑥 ∈ U𝑥- − {𝑥- }
moreover, if one of the following condi;ons hold:
1. lim 𝑓(𝑥) = lim 𝑔(𝑥) = 0
/→ /#
2.
/→ /#
lim 𝑓(𝑥) = lim 𝑔(𝑥) = ±∞
/→ /#
and if ∃ lim
/→ /#
6< (7)
7→ 7" N< (7)
, then
𝑓(𝑥)
𝑓 5 (𝑥)
= lim 5
/→ /# 𝑔(𝑥)
/→ /# 𝑔 (𝑥)
lim
proof
we will prove the case (1.), that is in the hypothesis
we have that
lim 𝑓(𝑥) = lim 𝑔(𝑥) = 0
/→ /#
/→ /#
now, due to the previous condi;on, we can extend by con;nuity in 𝑥- the func;ons f, g imposing
that 𝑓(𝑥- ) = 𝑔(𝑥- ) = 0
at this point, we can apply the Cauchy’s theorem on the interval
[𝑥- , 𝑥 ] and then ∃𝜉(𝑥) ∈ ]𝑥- , 𝑥[ s.t.
𝑓 5 l𝜉(𝑥)m 𝑓(𝑥) − 𝑓(𝑥- )
𝑓(𝑥)
=
=
𝑓(𝑥)
𝑔5 l𝜉(𝑥)m 𝑔(𝑥) − 𝑔(𝑥- )
𝑓(𝑥2 ) = 0; 𝑔(𝑥2 ) = 0
we no;ce that lim ξ(𝑥) = 𝑥- , then, if we take the limit, we have that
/→ /#
𝑓 L }𝜉 (𝑥)~
𝑓 (𝑥 )
𝑓 L (𝜉 )
lim
= lim L
= lim L
7→ 7" 𝑔(𝑥)
7→ 7" 𝑔 }𝜉 (𝑥 )~
g→ 7" 𝑔 (𝜉 )
= theorem on the limit for composi;on of func;ons
104
examples
1.
𝑥 − sin 𝑥
0
1
=§ ¨=
$
/→𝑥
0
6
1
𝑥 − sin 𝑥 ~ 𝑥 $ (𝑥~0)
6
lim
1
sin 𝑥 ~𝑥 − 𝑥 $ (𝑥~0)
6
indeed,
𝑥 − sin 𝑥
1 − cos 𝑥
1 1 − cos 𝑥 1 1 1
= (Hôpital) = lim
= lim ×
= × =
$
#
/→/→/→- 3
𝑥
3𝑥
𝑥#
3 2 6
lim
1 − cos 𝑥
1
=
6
𝑥
2
2.
𝑒/ − 1 − 𝑥
0
1
=§ ¨=
#
/→𝑥
0
2
1
𝑒 / − 1~ 𝑥 # (𝑥~0)
6
lim
1
𝑒 / ~1 + 𝑥 − 𝑥 # (𝑥~0)
2
indeed,
𝑒/ − 1 − 𝑥
𝑒/ − 0 − 1
1 𝑒/ − 1 1
1
(Hôpital)
=
=
lim
=
lim
×
= ×1=
#
/→/→/→- 2
𝑥
2𝑥
𝑥
2
2
lim
𝑒/ − 1
= 1
𝑥
THEOREM (on the limit of the derivate)
let f be a func;on which is differen;able U𝑥- − {𝑥- } and it is con;nuous 𝑥if ∃ lim 𝑓 5 (𝑥), then
/→ /#
𝑓 5 (𝑥) = lim 𝑓 5 (𝑥)
/→ /#
proof
we apply the Lagrange’s theorem on the interval [𝑥- , 𝑥]
then ∃𝜉(𝑥) ∈ ]𝑥- , 𝑥[ s.t.
𝑓(𝑥) − 𝑓(𝑥- )
𝑓 5 l𝜉(𝑥)m =
𝑥 − 𝑥we no;ce that lim ξ(𝑥) = 𝑥/→ /#
now, we take the limit:
𝑓(𝑥) − 𝑓(𝑥- )
= lim 𝑓 5 l𝜉(𝑥)m = lim 𝑓 5 (𝜉)
/→ /#
/→ /#
X→ /#
𝑥 − 𝑥lim
= theorem on the limit for composi;on of func;ons
example
consider
#
+ 2 if 𝑥 < 0
𝑓(𝑥) = !𝑥 + 2𝑥
/
2𝑒 if 𝑥 ≥ 0
for 𝑥 ≠ 0, f is con;nuous and differen;able
now, we start by checking the con;nuity of f in 𝑥- = 0
lim 𝑓(𝑥) = lim 𝑓(𝑥) = 𝑓(0) = 2
/→ -9
/→ -*
the func;on is also con;nuous in 0
105
what about the differen;ability of f in zero?
lim 𝑓 5 (𝑥) = lim 𝑓 5 (𝑥) = 𝑓 5 (0) = 2
/→ -9
/→ -*
then,
𝑓
5 (𝑥)
2𝑥 + 2 if 𝑥 < 0
= ¹ 2 if 𝑥 = 0
2𝑒 / if 𝑥 > 0
TAYLOR’S FORMULA
basic idea: we would like to approximate smooth func;ons by polynomials
if f is differen;able in a point 𝑥- , its increment 𝑓(𝑥) − 𝑓(𝑥- ) is approximated, when x is close to x0,
by the linear approxima;on
̅
𝑓(𝑥)
= 𝑓 5 (𝑥- )(𝑥 − 𝑥- )
concerning the difference f(𝑥) − 𝑓(𝑥- )
however, this approxima;on does NOT give enough informa;on; think for example about a
maximum problem for a func;on f
according to the Fermat’s theorem, this problem can be tackled by finding the zeros of the
derivate 𝑓 5 , but this condi;on is NOT sufficient: it also holds true for the minimum points, which
are the opposite of what we are looking for
the tangent line “linearizes” the graph near the tangency point and CANNOT capture the difference
between a maximum point and a minimum point, which difference lies in the different way the 2
graphs are “non-linear”
106
to be|er analyze the graph, we can try to es;mate the difference between the increment
∆𝑓 = 𝑓(𝑥) − 𝑓(𝑥- ) and the lineariza;on 𝑓 5 (𝑥- )(𝑥 − 𝑥- )
the difference 𝑓(𝑥) − 𝑓[(𝑥- ) + 𝑓 5 (𝑥)(𝑥 − 𝑥- )] represents the ver;cal difference between the
graph of f and its tangent line at l𝑥- , 𝑓(𝑥- )m
degrees of approxima@on
how can we measure therefore the distance of a graph from its linear approxima;on?
1st step: approxima;on of degree 1 (straight line)
2nd step: approxima;on of degree 2 (parabola)
3rd step: approxima;on of degree 3 (cubic)
we consider the graph of a func;on f and try to approximate it near the point 𝑥- with a curve
which can follow the graph “be|er” than a straight line.
the tangent line at l𝑥- , 𝑓(𝑥- )m has equa;on:
𝑦 = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- )
to do be|er, we can try to use a parabola:
𝑔(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥 − 𝑥- ) + 𝐴(𝑥 − 𝑥- )#
where A has to be determinate
we observe that the parabola passes through the point l𝑥- , 𝑓(𝑥- )m and the tangent to the
parabola at x has slope:
𝑔5 (𝑥) = 𝑓 5 (𝑥- ) + 2𝐴(𝑥 − 𝑥- )
then, at xo we have that 𝑔5 (𝑥- ) = 𝑓 5 (𝑥- ) meaning that it has the same value in the graph of f
then, the graph of f and the parabola g share the 𝑦 − value (since 𝑓(𝑥- ) = 𝑔(𝑥- )) and also the
tangent (since 𝑔5 (𝑥- ) = 𝑓 5 (𝑥- )) line in 𝑥then, to have a be|er approxima;on we have to look at the derivates of higher order!
107
defini@on
the deriva;ve 𝑓 ()) of a func;on f is the deriva;ve of the (𝑛 − 1) − th deriva;ve of f:
𝑓 ()) = l𝑓 ()9.) m!
in par;cular, if 𝑓 5 exists, then we can consider the second deriva;ve 𝑓 55 = (𝑓 5 )5 in case that 𝑓 5 is
differen;able
example
take 𝑓(𝑥) = 𝑥 $ then:
𝑓 5 (𝑥) = 3𝑥 #
5
𝑓 55 (𝑥) = l𝑓 5 (𝑥)m = 6𝑥
5
𝑓 555 (𝑥) = l𝑓 55 (𝑥)m = 6
5
𝑓 (=Y) (𝑥) = l𝑓 555 (𝑥)m = 0
𝑓 (B) (𝑥) = 0 ∀𝑘 ≥ 4
example
take 𝑔(𝑥) = 𝑒 / then:
𝑔5 (𝑥) = 𝑒 /
𝑔55 (𝑥) = 𝑒 /
𝑔555 (𝑥) = 𝑒 /
𝑔(B) (𝑥) = 𝑒 / ∀𝑘 ≥ 1
now, let’s go back to our problem of approxima;on and we require that f (the graph to be
approximated) and g (the parabola) are “similar”
𝑓 55 (𝑥- ) = 𝑔55 (𝑥- )
we recall that:
𝑔5 (𝑥) = 𝑓 5 (𝑥- ) + 2𝐴(𝑥 − 𝑥- )
then,
𝑔55 (𝑥) = 2𝐴 Þ 𝑔55 (𝑥- ) = 2𝐴
now,
𝑓 55 (𝑥- )
55 (𝑥 )
𝑓
- = 2𝐴 Þ 𝐴 =
2
therefore, a be|er approxima;on of f around 𝑥- is represented by:
𝑓 55 (𝑥- )
5
(𝑥)
(𝑥 − 𝑥- )#
P#
= 𝑓(𝑥- ) + 𝑓 (𝑥 − 𝑥- ) +
2
which is called TAYLOR POLYNOMIAL OF SECOND ORDER
the equity of the approxima;on is represented by
R # (𝑥) = 𝑓(𝑥) − P# (𝑥)
called Peano’s reminder
108
we have that:
R # (𝑥) = 𝑜[(𝑥 − 𝑥- )# ]
for 𝑥 → 𝑥 #
𝑜 = small o
that means:
𝑅# (𝑥)
=0
/→/# (𝑥 − 𝑥- )#
lim
remark
we underline that the Taylor’s polynomial of first order (degree 1) is:
P. (𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- )
and
R. (𝑥) = 𝑓(𝑥) − P. (𝑥)
and we have
R. (𝑥) = 𝑜(𝑥 − 𝑥- )
meaning that
𝑅. (𝑥)
lim
=0
/→/# 𝑥 − 𝑥then
𝑓(𝑥) = P. (𝑥) + R. (𝑥)
𝑃1(𝑥) = polynomial; 𝑅1(𝑥) = (error, reminder)
Þ 𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) + 𝑜(𝑥 − 𝑥- )
as 𝑥 → 𝑥 #
we can do be|er!
proposi@on
if f is twice differen;able (this means that “exist 𝑓 5 and 𝑓 55 ) at 𝑥- , then for 𝑥 → 𝑥- we have that
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) +
𝑓 55 (𝑥- )
(𝑥 − 𝑥- )# + 𝑜[(𝑥 − 𝑥- )# ]
2
TAYLOR EXPANSION
what about higher orders of approxima;on?
lemma (Peano’s Lemma)
let f: I à R (I = Interval) with 𝑥- ∈ I
suppose that f is differen;able up to the order 𝑛 (meaning that $𝑓 5 , 𝑓 55 , 𝑓 555 , … , 𝑓 ()) ) and assume
that
𝑓(𝑥- ) = 𝑓 5 (𝑥- ) = 𝑓 55 (𝑥- ) = … = 𝑓 ()) (𝑥- ) = 0
then, we have:
𝑓(𝑥)
lim
=0
/→/# (𝑥 − 𝑥- ))
we can also write
𝑓(𝑥- ) = 𝑜[(𝑥 − 𝑥- )) ]
𝑜 Þ “it goes to zero faster than”
109
proof
we calculate
𝑓(𝑥)
/→/# (𝑥 − 𝑥- ))
lim
G
since we have an indeterminate farm of the type [ ] we can apply De l’Hôpital’s theorem:
G
H
H
5
𝑓(𝑥)
𝑓
(𝑥)
𝑓 55 (𝑥)
=
=
lim
lim
lim
0 /→/# 𝑛(𝑥 − 𝑥- ))9. 0 /→/# 𝑛(𝑛 − 1)(𝑥 − 𝑥- ))9#
/→/# (𝑥 − 𝑥- ))
[ ]
[ ]
0
0
H
𝑓 555 (𝑥)
=
lim
0 /→/# 𝑛(𝑛 − 1)(𝑛 − 2)(𝑥 − 𝑥- ))9$
[ ]
0
if we iterate the previous process we get
𝑓(𝑥)
𝑓 5 (𝑥)
𝑓 ()9.) (𝑥)
H
H
H
lim
lim
…
lim
/→/# (𝑥 − 𝑥- )) = /→/# 𝑛(𝑥 − 𝑥- ))9. =
= /→/# 𝑛! (𝑥 − 𝑥- )
𝑓 ()9.) (𝑥) − 𝑓 ()9.)(𝑥- )
/→/#
𝑛! (𝑥 − 𝑥- )
= lim
𝑓 (=89)(𝑥2 ) = 0
1
𝑓 ()9.)(𝑥) − 𝑓 ()9.)(𝑥- )
=
lim
𝑛! /→/#
𝑛! (𝑥 − 𝑥- )
=
1
lim 𝑓 ())(𝑥- ) = 0
𝑛! /→/#
= à by defini;on of derivate in a point
= 0 à the equal sign refers to “by hypothesis)
THEOREM (Taylor’s theorem with Peano’s reminder)
let f: I à R (I = Interval) with 𝑥- ∈ I
suppose that f is differen;able up to the order 𝑛 (meaning that $𝑓 5 , 𝑓 55 , 𝑓 555 , … , 𝑓 ()) )
then,
𝑓 55 (𝑥- )
𝑓 555 (𝑥- )
𝑓 (@Z) (𝑥- )
5 (𝑥 )(𝑥
#
$
)
)
(𝑥
)
(𝑥
)
(𝑥 − 𝑥- )[
𝑓(𝑥) = 𝑓(𝑥- + 𝑓 − 𝑥- +
− 𝑥- +
− 𝑥- +
2!
3!
4!
𝑓 ()) (𝑥- )
(𝑥 − 𝑥- )) + R ) (𝑥)
+ ⋯+
𝑛!
with
R ) (𝑥)
lim
=0
/→/# (𝑥 − 𝑥- ))
(meaning that R ) (𝑥) = 𝑜[(𝑥 − 𝑥- )) ])
110
𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) Þ approxima;on of order 1
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) +
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) +
7 ,, (/# )
#!
7 ,, (/# )
#!
(𝑥 − 𝑥- )# Þ approxima;on of order 2
(𝑥 − 𝑥- )# +
7 ,,, (/# )
$!
proof
it is sufficient to apply the Peano’s lemma to
𝑓(𝑥) − ¼𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) + ⋯ +
(𝑥 − 𝑥- )$ Þapproxima;on of order 3
𝑓 ()) (𝑥- )
(𝑥 − 𝑥- )) ½
𝑛!
and we have concluded
defini@on
the polynomial
𝑓 55 (𝑥- )
𝑓 ()) (𝑥- )
(𝑥 − 𝑥- )# + ⋯ +
(𝑥 − 𝑥- ))
2!
𝑛!
is called Taylor’s polynomial of degree 𝑛 and the quan;ty R ) (𝑥) is called Peano’s reminder
we have that:
R ) (𝑥)
lim
=0
/→/# (𝑥 − 𝑥- ))
examples
1. consider 𝑓(𝑥) = 𝑒 / and 𝑥- = 0
then,
𝑓 5 (𝑥) = 𝑒 /
𝑓 55 (𝑥) = 𝑒 /
𝑓 555 (𝑥) = 𝑒 /
…
𝑓 ()) (𝑥) = 𝑒 /
Þ 𝑓 (B) (𝑥) = 𝑒 / ∀𝑘 ≥ 1
𝑓 (B) (0) = 1 ∀𝑘
then, using the Taylor’s approxima;on
𝑓 55 (𝑥- )
𝑓 ())(𝑥- )
5
#
(𝑥 − 𝑥- ) + + ⋯ +
(𝑥 − 𝑥- )) + R ) (𝑥)
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 (𝑥- )(𝑥 − 𝑥- ) +
2!
𝑛!
P# (𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- ) ∗ (𝑥 − 𝑥0) +
where
1
1
𝑒 / = 1 + 1(𝑥 − 0) + ! (𝑥 − 0)# + ⋯ + (𝑥 − 0)) + R ) (𝑥)
2
𝑛!
1 # 1 $ 1 [ 1 ]
1
= 1 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + ⋯ + (𝑥)) + R ) (𝑥)
2
3!
4!
5!
𝑛!
R ) (𝑥)
=0
/→/# (𝑥 − 𝑥- ))
lim
111
2. consider 𝑓(𝑥) = sin 𝑥
then,
and
𝑥- = 0
𝑓 5 (𝑥) = cos 𝑥
𝑓 55 (𝑥) = − sin 𝑥
𝑓 555 (𝑥) = − cos 𝑥
𝑓 (@Z) (𝑥) = sin 𝑥
𝑓 (Y) (𝑥) = cos 𝑥
𝑓 5 (0) = 1
𝑓 55 (0) = 0
𝑓 555 (0) = −1
𝑓 (@Z) (0) = 0
𝑓 (Y) (0) = 1
Þ 𝑓 (#)) (0) and
𝑓 (#)*.) (0) = (−1))
therefore, using the Taylor’s approxima;on
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) +
𝑓 55 (𝑥- )
𝑓 ())(𝑥- )
#
(𝑥 − 𝑥- ) + + ⋯ +
(𝑥 − 𝑥- )) + R ) (𝑥)
2!
𝑛!
(−1)
𝑓 (#)*.)(0)
$
(𝑥 − 𝑥- )#)*. + R #)*. (𝑥)
sin 𝑥 = 0 + 1(𝑥 − 0) + 0 +
(𝑥 − 0) + ⋯ +
3!
(2𝑛 + 1)!
𝑓 (6=>9) (0)
= (−1)(=)
𝑛!
(−1)) #)*.
1 $
= 𝑥 − 𝑥 + ⋯+
𝑥
+ R #)*. (𝑥)
3!
𝑛!
3! = 6
where
R *)1& (𝑥 )
=0
7→7" (𝑥 − 𝑥G )*)1&
lim
remark
sin 𝑥
lim
=1
sin 𝑥 ~𝑥
/→- 𝑥
sin 𝑥 − 𝑥
1
lim
=−
$
/→𝑥
6
(𝑥~0)
sin 𝑥 − 𝑥
1
~−
$
𝑥
6
3. consider 𝑓(𝑥) = cos 𝑥 and 𝑥- = 0
we observe that the cos 𝑥 is an even func;on
then,
𝑓 5 (𝑥) = −sin 𝑥
𝑓 55 (𝑥) = − cos 𝑥
𝑓 555 (𝑥) = sin 𝑥
𝑓 (@Z) (𝑥) = cos 𝑥
𝑓 (Y) (𝑥) = −sin 𝑥
1
(𝑥~0) Þ sin 𝑥 ~ 𝑥 − 𝑥 $ (𝑥~0)
6
Þ 𝑓 (#)*.) (0) = 0
𝑓 5 (0) = 0
𝑓 55 (0) = −1
𝑓 555 (0) = 0
𝑓 (@Z) (0) = 1
𝑓 (Y) (0) = 0
112
then,
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) +
𝑓 55 (𝑥- )
𝑓 (#)) (𝑥- )
(𝑥 − 𝑥- )# + + ⋯ +
(𝑥 − 𝑥- )#) + R #) (𝑥)
2!
(2𝑛)!
−1
(−1))
#
(𝑥 − 𝑥- ) + ⋯ +
(𝑥)#) + R #) (𝑥)
cos 𝑥 = 1 + 0 +
2
(2𝑛)!
the general even derivate in zero is
𝑓 #) (0) = (−1))
1
(−1)) #)
cos 𝑥 = 1 − 𝑥 # + ⋯ +
𝑥 + R #) (𝑥)
2
(2𝑛)!
where
R *) (𝑥 )
=0
7→G 𝑥 *)
lim
examples for limits
?@A 7/71*7 ?
&
=−
%
+7
&7→G
1. lim
indeed,
1
1
sin 𝑥 = 𝑥 − 𝑥 $ + 𝑥 ] + 𝑜(𝑥 ] )
6
5!
h(7 ? )
=0
7→G 7 ?
𝑜(𝑥 ] ) = it goes to zero faster than 𝑥 ] meaning lim
]
sin 𝑥 − 𝑥 + 2𝑥
= lim
/→/→3𝑥 $
lim
1
1
p− 6 𝑥 $ + 𝑥 ] + 𝑜(𝑥 ] )q − 𝑥 + 2𝑥 ]
5!
3𝑥 $
1
1 0
− 𝑥+
𝑥
𝑜(𝑥 0 ) 2𝑥 0
6
5!
= lim {
+
+
+ +|
7→G
3𝑥 +
3𝑥 +
3𝑥 +
3𝑥
1 1 *
𝑥
𝑜(𝑥 0 ) 2 *
1
6
5!
|
= lim {
+
+
+
𝑥
=
−
7→G
3
3
3𝑥 +
3
18
−
1
−6
1
= − ;
3
18
1 6
5! 𝑥 = 0;
3
𝑜(𝑥 @ )
= 0;
3𝑥 A
2 6
𝑥 =0
3
h(7 ? )
=0
7→G 7 ?
where lim
𝑜(𝑥 0 )
𝑜(𝑥 0 ) *
𝑜(𝑥 0 ) 1 *
lim
= lim + * 𝑥 = lim 0 × 𝑥 = 0
7→G 3𝑥 +
7→G 3𝑥 𝑥
7→G 𝑥
3
𝑜(𝑥 ] )
= 0;
𝑥]
1 #
𝑥
3
113
?@A(7 ! )17 !
&
=
−
B
7
.
7→G
2. lim
indeed,
1
sin 𝑥 = 𝑥 − 𝑥 $ + 𝑜(𝑥 $ )
6
where
for 𝑥 → 0
𝑜(𝑥 + )
=0
7→G 𝑥 +
lim
thus,
1
sin 𝑥 # = 𝑥 # − 𝑥 ^ + 𝑜(𝑥 ^ )
6
where
for 𝑥 → 0
𝑜(𝑥 . )
lim
=0
7→G 𝑥 .
#)
#
sin(𝑥 − 𝑥
lim
= lim
/→/→𝑥^
1
p𝑥 # − 6 𝑥 ^ + 𝑜(𝑥 ^ )q − 𝑥 #
𝑥^
1
− 6 𝑥^
𝑥^
1
=− ;
6
1
− 6 𝑥 ^ 𝑜(𝑥 ^ )
1
= lim ¾ ^ +
¿=−
^
/→𝑥
𝑥
6
𝑜(𝑥 ^ )
=→ 0
𝑥^
EXERCISES CONCERNING DERIVATES
1. 𝑦 = 𝑥 # cos 𝑥
𝑥 # = 𝑓; cos 𝑥 = 𝑔
then,
2. 𝑦 = 𝑥 ln 𝑥 − sin 𝑥
𝑥 = 𝑓; ln 𝑥 = 𝑔
then,
3. 𝑦 = (2𝑥 + 1) − ln 𝑥
2𝑥 + 1 = 𝑓; ln 𝑥 = 𝑔
(𝑓𝑔)5 = 𝑓 5 𝑔 + 𝑓𝑔5
𝑦 5 = 2𝑥 cos 𝑥 + 𝑥 # (− sin 𝑥) = 2𝑥 cos 𝑥 − 𝑥 # sin 𝑥
(𝑓𝑔)5 = 𝑓 5 𝑔 + 𝑓𝑔5
1
𝑦 5 = ln 𝑥 + 𝑥 − cos 𝑥 = ln 𝑥 + 1 − cos 𝑥
𝑥
(𝑓𝑔)5 = 𝑓 5 𝑔 + 𝑓𝑔5
then,
𝑦 5 = 2ln 𝑥 + (2𝑥 + 1)
1
2𝑥 + 1
= 2ln 𝑥 +
𝑥
𝑥
114
LOCAL CONVEXITY AND CONCAVITY
local “it means in a point 𝑥- ”
defini@on
let f: A à R be differen;able at 𝑥- ∈ 𝐴
then, f is said to be convex at 𝑥- if ∃U𝑥- s.t.
𝑓(𝑥) > 𝑃. (𝑥)
∀𝑥 ∈ (U𝑥- − {𝑥- }) ∩ 𝐴
where 𝑃. (𝑥) is the Taylor’s expansion of degree 1 that is
P. (𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥 − 𝑥- )
defini@on
let f: A à R be differen;able at 𝑥- ∈ 𝐴
then, f is said to be concave if ∃U𝑥- s.t.
𝑓(𝑥) < 𝑃. (𝑥)
∀𝑥 ∈ (U𝑥- − {𝑥- }) ∩ 𝐴
115
defini@on
let f: A à R be differen;able at 𝑥- ∈ 𝐴
then, the point 𝑥- is said to be an inflec@on point
of f if ∃U𝑥- s.t.
if 𝑥 < 𝑥- Þ 𝑓(𝑥) < 𝑃. (𝑥)
𝑖) !
if 𝑥 > 𝑥- Þ 𝑓(𝑥) > 𝑃. (𝑥)
or
if 𝑥 < 𝑥- Þ 𝑓(𝑥) > 𝑃. (𝑥)
𝑖𝑖) !
if 𝑥 > 𝑥- Þ 𝑓(𝑥) < 𝑃. (𝑥)
THEOREM (sufficient condi@on for convexity/concavity)
let f: A à R be differen;able in U𝑥- and it is twice differen;able (∃𝑓 55 ) in 𝑥- ∈ 𝐴 s.t.
𝑓 55 (𝑥- ) > 0
(𝑓 55 (𝑥- ) < 0)
then f is convex (concave) in 𝑥proof
we assume that 𝑓 55 (𝑥- ) > 0 and we have to prove that the func;on f is convex in 𝑥- , that is
∃U𝑥- s.t.
𝑓(𝑥) > 𝑃. (𝑥)
∀𝑥 ∈ U𝑥- − {𝑥- }
then,
we take the Taylor’s expansion up to the second order that is:
𝑓 55 (𝑥- )
(𝑥 − 𝑥- )# + R # (𝑥)
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) +
2
𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) = P. (𝑥)
R # (𝑥) = Peano’s reminder
𝑓 55 (𝑥- )
(𝑥 − 𝑥- )# + R # (𝑥)
= P. (𝑥) +
2
𝑓 55 (𝑥- )
R # (𝑥)
= P. (𝑥) + ¼
+
½ (𝑥 − 𝑥- )#
(𝑥 − 𝑥- )#
2
(𝑥 − 𝑥- )# > 0
if
𝑥 ≠ 𝑥-
116
then,
𝑓 55 (𝑥- )
R # (𝑥)
+
½ (𝑥 − 𝑥- )#
(𝑥 − 𝑥- )#
2
(𝑥 − 𝑥- )# > 0 if 𝑥 ≠ 𝑥-
𝑓(𝑥) − P. (𝑥) = ¼
we consider
𝑓 55 (𝑥- )
R # (𝑥)
𝑓 55 (𝑥- )
lim ¼
+
½=
>0
/→/#
(𝑥 − 𝑥- )#
2
2
R # (𝑥)
= 0;
(𝑥 − 𝑥- )#
> = by hypothesis
therefore, thanks to the sign permanence theorem for limits, ∃U𝑥- s.t.
𝑓 55 (𝑥- )
R # (𝑥)
+
>0
∀𝑥 ∈ U𝑥(𝑥 − 𝑥- )#
2
then, we recall that
𝑓 55 (𝑥- )
R # (𝑥)
𝑓(𝑥) − P. (𝑥) = ¼
+
½ (𝑥 − 𝑥- )#
(𝑥 − 𝑥- )#
2
𝑓 55 (𝑥- )
R # (𝑥)
+
>;
(𝑥 − 𝑥- )#
2
(𝑥 − 𝑥- )# > 0
if
𝑥 ≠ 𝑥-
Þ 𝑓(𝑥) − P. (𝑥) > 0 Þ 𝑓(𝑥) − P. (𝑥) in U𝑥- − {𝑥- }
f is convex in the point 𝑥corollary
let f: [a, b] à R be a con;nuous func;on and twice differen;able in ]𝑎, 𝑏[
then,
1. if 𝑓 55 (𝑥- ) > 0 ∀𝑥 ∈]𝑎, 𝑏[
then f is convex in the interval
2. if 𝑓 55 (𝑥- ) < 0 ∀𝑥 ∈]𝑎, 𝑏[
then f is concave in the interval
117
THEOREM (sufficient condi@on for points of rela@ve minimum/maximum)
assume f: A à R is differen;able in U𝑥- and it is twice differen;able in 𝑥suppose that
𝑓 5 (𝑥- ) = 0
if 𝑓 55 (𝑥- ) > 0 (𝑓 55 (𝑥- ) < 0 ) , then 𝑥- is a point of rela;ve minimum (maximum)
proof
we assume that 𝑓 55 (𝑥- ) > 0
we have to prove that 𝑥- is a point of rela;ve minimum, that means ∃U𝑥- s.t.
𝑓(𝑥) > 𝑓(𝑥- )∀𝑥 ∈ U𝑥- − {𝑥- }
we consider Taylor’s expansion with the Peano’s remained up to the second order
𝑓 55 (𝑥- )
5
(𝑥 − 𝑥- )# + R # (𝑥)
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 (𝑥- )(𝑥 − 𝑥- ) +
2
𝑓 5 (𝑥- )(𝑥 − 𝑥- ) = by hypothesis
𝑓 55 (𝑥- )
R # (𝑥)
Þ 𝑓(𝑥) − 𝑓(𝑥- ) = ¼
+
½ (𝑥 − 𝑥- )#
(𝑥 − 𝑥- )#
2
(𝑥 − 𝑥- )# > 0
if
𝑥 ≠ 𝑥-
we take
𝑓 55 (𝑥- )
R # (𝑥)
𝑓 55 (𝑥- )
+
½
=
>0
/→/#
(𝑥 − 𝑥- )#
2
2
lim ¼
R # (𝑥)
= 0;
(𝑥 − 𝑥- )#
> = by hypothesis
we apply the theorem on the sign permanence for limits and we get that ∃U𝑥- s.t
we recall that
𝑓 55 (𝑥- )
R # (𝑥)
+
>0
(𝑥 − 𝑥- )#
2
𝑓 55 (𝑥- )
R # (𝑥)
)
Þ 𝑓(𝑥) − 𝑓(𝑥- = ¼
+
½ (𝑥 − 𝑥- )#
#
(𝑥
)
2
− 𝑥(𝑥 − 𝑥- )# > 0
if
𝑥 ≠ 𝑥-
𝑥- is a point of rela;ve minimum
118
THEOREM (sufficient condi@on for inflec@on points)
let f: A à R be twice differen;able in U𝑥- and three ;mes differen;able in 𝑥assume that
𝑓 55 (𝑥- ) = 0
if 𝑓 555 (𝑥- ) ≠ 0 , then 𝑥- is an inflec;on point
proof
we can assume for example that
𝑓 555 (𝑥- ) > 0
we take the Taylor’s expansion up to the third order:
𝑓 55 (𝑥- )
𝑓 555 (𝑥- )
(𝑥 − 𝑥- )# +
(𝑥 − 𝑥- )$ + R $ (𝑥)
𝑓(𝑥) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) +
2
3!
𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) = P. (𝑥)
𝑓 55 (𝑥- )
(𝑥 − 𝑥- )# = 0 by hypothesis
2
𝑓 555 (𝑥- )
R # (𝑥)
Þ 𝑓(𝑥) − 𝑃. (𝑥) = ¼
+
½ (𝑥 − 𝑥- )$
(𝑥 − 𝑥- )$
3!
(𝑥 − 𝑥- )# > 0
if
𝑥 ≠ 𝑥-
we take
𝑓 555 (𝑥- )
R # (𝑥)
𝑓 555 (𝑥- )
lim ¼
+
½=
>0
/→/#
(𝑥 − 𝑥- )$
3!
3!
R # (𝑥)
= 0;
(𝑥 − 𝑥- )$
> = by hypothesis
Þ 𝑓(𝑥) − 𝑃. (𝑥) = ¼
𝑓 555 (𝑥- )
R # (𝑥)
+
½ (𝑥 − 𝑥- )$
(𝑥 − 𝑥- )$
3!
then, we recall that
(𝑥 − 𝑥- )# > 0 thanks to the sign permanence theorem for limits
and therefore, in U𝑥- we have
- if 𝑥 > 𝑥- Þ (𝑥 − 𝑥- )$ > 0
Þ 𝑓(𝑥) − 𝑃. (𝑥) > 0 Þ 𝑓(𝑥) > 𝑃. (𝑥)
-
if 𝑥 < 𝑥- Þ (𝑥 − 𝑥- )$ < 0
Þ 𝑓(𝑥) − 𝑃. (𝑥) < 0 Þ 𝑓(𝑥) < 𝑃. (𝑥)
to sum up in U𝑥- − {𝑥- } we have found that:
if 𝑥 > 𝑥- Þ 𝑓(𝑥 ) > 𝑃. (𝑥 )
!
if 𝑥 < 𝑥- Þ 𝑓(𝑥) < 𝑃. (𝑥)
and this implies that 𝑥- is an inflec;on point
119
TOPIC 13
STUDY OF FUNCTION
when we study a func;on f in order to draw a qualita;ve graph, we follow the procedure below:
1. we find the domain of f where it is well-defined
examples
.
i.
𝑓(𝑥) = /
Domf = 𝑅 − {0} = {𝑥 ∈ 𝑅, 𝑥 ≠ 0}
ii.
iii.
𝑓(𝑥) = √𝑥
𝑓(𝑥) = ln 𝑥
Domf = {𝑥 ∈ 𝑅 ∶ 𝑥 ≥ 0}
Domf = {𝑥 ∈ 𝑅 ∶ 𝑥 > 0}
2. we determine the sign of f (where f is posi;ve/nega;ve) and the intersec;ons with the xaxis and the y-axis
example
how to find the intersec;ons with the axes?
- intersec;on with x-axis
𝑥 =?
!
𝑦 = 𝑓(𝑥) = 0
-
intersec;on with the y-axis
𝑥=0
!
𝑦 = 𝑓(𝑥) =?
3. we calculate the limits at the boundaries of the domain (we evaluate the limits in the
points that are excluded from the domain and in ±∞)
examples
i.
lim 𝑓(𝑥) = ±∞ Þ 𝑥 = 𝑥- is a vertical asymptote
/→/#
ii.
iii.
lim 𝑓(𝑥) = 𝐿 (Äinite) Þ 𝑦 = 𝐿 is a horizontal asymptote
/→±+
lim 𝑓(𝑥) = ±∞ Þ there could be an oblique asymptote
/→±+
how to find the oblique asymptote?
𝑓(𝑥)
/→±+ 𝑥
and 𝑚 has to be FINITE and 𝑚 ≠ 0
𝑞 = lim (𝑓(𝑥) − 𝑚𝑥)
𝑚 = lim
/→±+
and 𝑞 has to be FINITE
then, the oblique asymptote is
𝑦 = 𝑚𝑥 + 𝑞
[
for example, we consider 𝑓(𝑥) = / ) *. + 𝑥
first of all, we evaluate
4
lim 𝑓(𝑥) = lim t #
+ 𝑥u = +∞ Þ there could be an oblique aymptote
/→*+
/→*+ 𝑥 + 1
120
then,
𝑓(𝑥)
4
= lim t $
+ 1u = 1
/→*+ 𝑥
/→*+ 𝑥 + 𝑥
4
=0
$
𝑥 +𝑥
𝑚 = lim
now,
𝑞 = lim (𝑓(𝑥) − 𝑚𝑥) = lim (𝑓(𝑥) − 𝑥) = lim t
/→*+
/→*+
/→*+
4
𝑥# + 1
4
𝑥# + 1
+ 𝑥 − 𝑥u = lim
4
/→*+ 𝑥 # + 1
=0
=0
[
then, the oblique asymptote (𝑦 = 𝑚𝑥 + 𝑞) for 𝑓(𝑥) = / ) *. + 𝑥 when 𝑥 → 𝑥- is
𝑦=𝑥
4. es;mate the first derivate 𝑓 5 and we study the sign of 𝑓 5 :
- if 𝑓 5 > 0 in I Þ 𝑓 is increasing in I
- if 𝑓 5 < 0 in I Þ 𝑓 is decreasing in I
example
-
if 𝑓(−1) is the maximum value of the func;on in all the domain, then 𝑥 = 1 is a point of
GLOBAL maximum
if 𝑓(3) is the minimum value of the func;on in all the domain, then 𝑥 = 3 is a point of
GLOBAL minimum
5. we es;mate the second deriva;ve 𝑓 55 and we study its sign:
- if 𝑓 55 > 0 in I Þ 𝑓 is convex in I
- if 𝑓 55 < 0 in I Þ 𝑓 is concave in I
example
6. we draw the qualita;ve graph of f
7. possible ques;on: calculate some bounded areas linked to the graph of f (à integrals)
121
RECAP ON THE SUPPLEMENTARY MATERIAL: PRINICPLE OF INDUCTION
let P(𝑛) be a proposi;on depending on 𝑛 ∈ 𝑁
then, P(𝑛) is true for every 𝑛 ≥ 𝑛- where 𝑛- is a fixed natural number, if the following condi;ons
are verified:
1. P(𝑛- ) is true;
2. P(𝑛) (= induc;on hypothesis) Þ P(𝑛 + 1) ∀𝑛 ≥ 𝑛example
we would like to prove that
)
š𝑘 =
)
B,.
𝑛(𝑛 + 1)
2
š𝑘 → 1 + 2 + 3 + 4 + ⋯+ 𝑛
B,.
.--
š𝑘 =
B,.
100 (101)
= 50 × 101 = 5.050
2
.--
š 𝑘 → 1 + 2 + 3 + 4 + ⋯ + 100
B,.
we employ the principle of induc;on
we start by
1.
.
š𝑘 = 1 =
B,.
1(1 + 1)
2
2. now, we have to show that P(𝑛) ÞP(𝑛 + 1), meaning that we assume that
)
š𝑘 =
B,.
and we have to prove that
is true
)*.
š𝑘 =
B,.
then,
)*.
𝑛(𝑛 + 1)
2
(𝑛 + 1)(𝑛 + 2)
2
)
š 𝑘 = š 𝑘 + (𝑛 + 1)
B,.
B,.
we apply the induc;on hypothesis
=
𝑛(𝑛 + 1)
𝑛(𝑛 + 1) + 2(𝑛 + 1) (𝑛 + 1)(𝑛 + 2)
+ (𝑛 + 1) =
+
∶ ok
2
2
2
122
TOPIC 14
INTEGRALS
the no;on of the integrals is very old
it is linked with important geometric problems, such as “squaring of the circle” (squaring stand for
the possibility of compu;ng a formula for the area) and may be dated back to Archimedes
(287-212 B.C.) and his “method of exhaus;ons”
defini@on
let f: A à R
then, F is said to be a PRIMTIVE of f if
examples
𝒇(𝒙)
𝑘
𝑒/
𝑥 D (𝛼𝜖𝑅, 𝛼 ≠ 1)
sin 𝑥
cos 𝑥
1
1 + 𝑥#
𝐹 5 (𝑥) = 𝑓(𝑥)
𝑭5 (𝒙)
𝑘𝑥 + 𝑐
𝑒/ + 𝑐
𝑥 D*.
+𝑐
𝛼+1
− cos 𝑥 + 𝑐
sin 𝑥 + 𝑐
arc tg 𝑥 + 𝑐
∀𝑥 ∈ 𝐴
𝐹 5 (𝑥) = (𝑘𝑥 + 𝑐)5 = 𝑘 + 0 = 𝑘
𝐹 5 (𝑥) = (𝑒 / + 𝑐)5 = 𝑒 / + 0 = 𝑒 /
5
(𝛼 + 1)𝑥
𝑥 D*.
𝐹
=p
+ 𝑐q =
+ 0 = 𝑥D
𝛼+1
𝛼+1
𝐹 5 (𝑥) = (− cos 𝑥 + 𝑐)5 = −(− sin 𝑥) + 0 = sin 𝑥
𝐹 5 (𝑥) = (sin 𝑥 + 𝑐)5 = cos 𝑥 + 0 = cos 𝑥
1
1
𝐹 5 (𝑥) = (arc tg 𝑥 + 𝑐)5 =
+0=
#
1+𝑥
1 + 𝑥#
5 (𝑥)
consequences of Lagrange
if F, G are two different primi;ves of the same f on I, then
𝐹(𝑥) − 𝐺(𝑥) = 𝑘
k = constant
remark
since F, G are primi;ve of f, we have that
𝐹 5 (𝑥) = 𝑓(𝑥) = 𝐺 5 (𝑥)
defini@on
the INDEFINITE INTEGRAL
Í 𝑓(𝑥) 𝑑𝑥
𝑑𝑥 = differen;al
is the set of all the primi;ves of the func;on f
for example, we consider 𝑓(𝑥) = 𝑒 / then the INDEFINITE INTEGRAL of f is
Í 𝑒 / 𝑑𝑥 = 𝑒 / + 𝑐
remark
.
if 𝑓(𝑥) = / , then
1
Í 𝑑𝑥 = ln|𝑥| + 𝑐
𝑥
123
to help our intui;on, we can think about the integra;on as the “pseudo – inverse” opera;on with
respect to the deriva;on
Íl𝛼𝑓(𝑥) + 𝛽𝑔(𝑥)m 𝑑𝑥 = 𝛼 Í 𝑓(𝑥)𝑑𝑥 + 𝛽 Í 𝑔(𝑥)𝑑𝑥
where 𝛼, 𝛽 ∈ 𝑅
RULES OF INTEGRATION
1. we start by presen;ng the integra;on by parts
theorem
let f, g be two different func;ons defined on an interval
then,
Í 𝑓(𝑥)𝑔5 (𝑥)𝑑𝑥 = 𝑓(𝑥)𝑔(𝑥) − Í 𝑓 5 (𝑥)𝑔(𝑥)𝑑𝑥
∫ 𝑓(𝑥)𝑔5 (𝑥)𝑑𝑥 à very hard to solve
∫ 𝑓 5 (𝑥)𝑔(𝑥)𝑑𝑥 à very easy to calculate
proof
from the deriva;ve of the product of func;ons we have that
5
l𝑓(𝑥)𝑔(𝑥)m = 𝑓 5 (𝑥)𝑔(𝑥) + 𝑓(𝑥)𝑔5 (𝑥)
now, if we integrate from both sides we get
5
Íl𝑓(𝑥)𝑔(𝑥)m 𝑑𝑥 = Í 𝑓 5 (𝑥)𝑔(𝑥)𝑑𝑥 + Í 𝑓(𝑥)𝑔5 (𝑥)𝑑𝑥
Þ 𝑓(𝑥)𝑔(𝑥) = Í 𝑓 5 (𝑥)𝑔(𝑥)𝑑𝑥 + Í 𝑓(𝑥)𝑔5 (𝑥)𝑑𝑥
Þ Í 𝑓 5 (𝑥)𝑔(𝑥)𝑑𝑥 = 𝑓(𝑥)𝑔(𝑥) − Í 𝑓 5 (𝑥)𝑔(𝑥)𝑑𝑥
examples
1.
Í 𝑥𝑒 / 𝑑𝑥 = 𝑥𝑒 / − Í 1𝑒 / 𝑑𝑥 = 𝑥𝑒 / − Í 𝑒 / 𝑑𝑥 = 𝑥𝑒 / − 𝑒 / + 𝑐 = 𝑒 / (𝑥 + 1) + 𝑐
𝑥 = 𝑓; 𝑒 / = 𝑔5
if 𝑔5 (𝑥) = 𝑒 / , then 𝑔(𝑥) = 𝑒 /
if 𝑓(𝑥) = 𝑥, then 𝑓 5 (𝑥) = 1
2.
1
Í ln 𝑥 𝑑𝑥 = Í 1 ln 𝑥 𝑑𝑥 = ln 𝑥 𝑥 − Í 𝑥𝑑𝑥 = 𝑥ln 𝑥 − Í 1𝑑𝑥 = 𝑥ln 𝑥 − 1𝑥 + 𝑐 = 𝑥ln 𝑥 − 𝑥 + 𝑐
𝑥
= 𝑥 (ln 𝑥 − 1) + 𝑐
1 = 𝑔5 ; ln 𝑥 = 𝑓
if 𝑔5 (𝑥) = 1, then 𝑔(𝑥) = 𝑥
.
if 𝑓(𝑥) = ln 𝑥, then 𝑓 5 (𝑥) = /
124
3.
Í 𝑥 # 𝑒 / 𝑑𝑥 = 𝑥 # 𝑒 / − Í 2𝑥𝑒 / 𝑑𝑥 = 𝑥 # 𝑒 / − 2 Í 𝑥𝑒 / 𝑑𝑥 = 𝑥 # 𝑒 / − 2l𝑒 / (𝑥 + 1)m + 𝑐
= 𝑥 # 𝑒 / − 2(𝑥𝑒 / + 𝑒 / ) + 𝑐 == 𝑥 # 𝑒 / − 2𝑥𝑒 / + 2𝑒 / + 𝑐 = 𝑒 / (𝑥 # − 2𝑥 + 2) + 𝑐
𝑥 # = 𝑓; 𝑒 / = 𝑔5
if 𝑔5 (𝑥) = 𝑒 / , then 𝑔(𝑥) = 𝑒 /
if 𝑓(𝑥) = 𝑥 # , then 𝑓 5 (𝑥) = 2𝑥
2. we introduce now a second rule of integra;on; it is called integra;on by subs;tu;on
theorem
given f, g in I, if ∃ fog and F is a primi;ve of f, then we have
Í 𝑓(𝑥)𝑔5 (𝑥)𝑑𝑥 = Í 𝑓 (𝑡)𝑑𝑡 = 𝐹(𝑡) + 𝑐 = 𝐹l𝑔(𝑥)m + 𝑐
= → 𝑡 = 𝑔(𝑥)
proof
from the rule of differen;a;on for the composi;on of func;ons we have that
5
(𝐹𝑜𝑔)5 (𝑥) = (𝐹l𝑔(𝑥)m = 𝐹 5 l𝑔(𝑥)m𝑔5 (𝑥) = 𝑓l𝑔(𝑥)m𝑔5 (𝑥)
𝐹𝑜𝑔 = F is the primi;ve of f
then, if we integrate, we get
5
ÍÑ𝐹l𝑔(𝑥)mÒ 𝑑𝑥 = Í 𝑓l𝑔(𝑥)m𝑔5 (𝑥)𝑑𝑥
𝐹l𝑔(𝑥)m + 𝑐 = Í 𝑓l𝑔(𝑥)m𝑔5 (𝑥)𝑑𝑥
note that if we take 𝑡 = 𝑔(𝑥) we obtain
Í 𝑓l𝑔(𝑥)m𝑔5 (𝑥)𝑑𝑥 = Í 𝑓 (𝑡)𝑑𝑡
which is the meaning of dx?
defini@on
let f be differen;able at 𝑥then, the DIFFERENTIAL of f at 𝑥- , denoted by df, is the change of the linear approxima;on 𝑃. (𝑥)
at 𝑥- , that means
𝑑𝑓 = 𝑃. (𝑥) − 𝑃. (𝑥- )
remark
we have that
𝑑𝑓 = 𝑃. (𝑥) − 𝑃. (𝑥- ) = 𝑓(𝑥- ) + 𝑓 5 (𝑥- )(𝑥 − 𝑥- ) − 𝑓(𝑥- )
(𝑥 − 𝑥- ) = ∆
= 𝑓 5 (𝑥- )∆𝑥
if the increment in x is “small” we have dx, then
𝑑𝑓 = 𝑓 5 (𝑥- )𝑑𝑥
example
if 𝑡 = 𝑔(𝑥), then 𝑑𝑡 = 𝑑𝑔(𝑥) = 𝑔5 (𝑥)𝑑𝑥
125
examples
1.
)
Í 𝑥𝑒 / 𝑑𝑥 = ?
take 𝑡 = 𝑥 # Þ 𝑑𝑡 = 𝑑(𝑥 # ) = (𝑥 # )5 𝑑𝑥 = 2𝑥𝑑𝑥
1
1
1
1 )
)
)
Í 𝑥𝑒 / 𝑑𝑥 = Í 𝑒 / 2𝑥𝑑𝑥 = Í 𝑒 _ 𝑑𝑡 = 𝑒 _ + 𝑐 = 𝑒 / + 𝑐
2
2
2
2
𝑥 # = 𝑡; 2𝑥𝑑𝑥 = 𝑑𝑡
2.
Í tg𝑥 𝑑𝑥 = ?
take 𝑡 = cos 𝑥 Þ 𝑑𝑡 = 𝑑(cos 𝑥) = (cos 𝑥)5 𝑑𝑥 = − sin 𝑥 𝑑𝑥
Í tg𝑥 𝑑𝑥 = Í
sin 𝑥
1
1
1
𝑑𝑥 = Í
sin 𝑥 𝑑𝑥 = − Í
(−sin 𝑥) 𝑑𝑥 = − Í 𝑑𝑡 = − ln|𝑡| + 𝑐
cos 𝑥
cos 𝑥
cos 𝑥
𝑡
= − ln( |cos 𝑥| ) + 𝑐
cos 𝑥 = 𝑡; (−sin 𝑥) 𝑑𝑥 = 𝑑𝑡
why do we take 𝑡 = cos 𝑥?
if we take 𝑡 = sin 𝑥 Þ 𝑑𝑡 = 𝑑(sin 𝑥) = (sin 𝑥)5 𝑑𝑥 = − cos 𝑥 𝑑𝑥
BUT,
sin 𝑥
Í tg𝑥 𝑑𝑥 = Í
𝑑𝑥
cos 𝑥
𝑑𝑡 = cos 𝑥 𝑑𝑥
we cannot subs;tute 𝑑𝑡 and then we cannot go on!
3.
Í
arc sin 𝑥
√1 − 𝑥 #
𝑑𝑥 = ?
take 𝑡 = arc sin 𝑥 Þ 𝑑𝑡 = 𝑑(arc sin 𝑥) = (arc sin 𝑥)5 𝑑𝑥 =
1
=
𝑑𝑥
√1 − 𝑥 #
Í
arc sin 𝑥 = 𝑡;
arc sin 𝑥
√1 − 𝑥 #
1
√1 − 𝑥 #
𝑑𝑥 = Í arc sin 𝑥
1
1
𝑑𝑥 = Í 𝑡𝑑𝑡 = 𝑡 # + 𝑐 = arc sin# 𝑥
2
√1 − 𝑥 #
𝑑𝑥 = 𝑑𝑡
126
DEFINITE INTEGRALS
we start by presen;ng the defini;on of the definite integral, following the construc;on by
Riemann
defini@on
let f be a bounded func;on on [𝑎; 𝑏] for 𝑛 ∈ 𝑁 we consider the par;;on of [𝑎; 𝑏] :
𝑎 = 𝑥- < 𝑥. < 𝑥# <. . . < 𝑥)9. < 𝑥) = 𝑏
we de fine
I= = [𝑥=9. , 𝑥= ]
and
𝑖𝑛𝑓
𝑓
I=
𝑠𝑢𝑝
M= = I 𝑓
=
𝑚= =
now, for every par;;on ∆ we introduce
- lower integral sum
)
𝑠(𝑓, ∆) = š 𝑚= (𝑥=9. , 𝑥= )
=,.
-
upper integral sum
)
𝑐 = š 𝑀= (𝑥=9. , 𝑥= )
=,.
now, we denote by D the set of all possible par;;ons
of [𝑎; 𝑏]
we have
𝑠(𝑓, ∆) ≤ 𝑆(𝑓, ∆) ∀∆∈ 𝐷
𝑠𝑢𝑝
𝑖𝑛𝑓
𝑠(𝑓, ∆) ≤
𝑆(𝑓, ∆)
∆∈ 𝐷
∆∈ 𝐷
127
defini@on
in the case when
𝑠𝑢𝑝
𝑖𝑛𝑓
𝑠(𝑓, ∆) =
𝑆(𝑓, ∆)
∆∈ 𝐷
∆∈ 𝐷
we say that f is integrable on [𝑎; 𝑏] and we indicate the definite integral of f on [𝑎; 𝑏 ] as
3
Í 𝑓(𝑥)𝑑𝑥 =
1
𝑠𝑢𝑝
𝑖𝑛𝑓
𝑠(𝑓, ∆) =
𝑆(𝑓, ∆)
∆∈ 𝐷
∆∈ 𝐷
contra-example
we give an example of a func;on which is NOT integrable (by Riemann)
we consider 𝑓: [0,1] → 𝑅 s.t.
1 if 𝑥 ∈ 𝑄
𝑓(𝑥) = !
0 if 𝑥 ∈ 𝑅 − 𝑄
(Dirichlet func;on)
for every I= = [𝑥=9. , 𝑥= ] we have
𝑠𝑢𝑝
𝑖𝑛𝑓
𝑓 = 0 and I 𝑓 = 1
I=
=
then, ∀∆∈ 𝐷 we get
𝑠(𝑓, ∆) = 0 and 𝑆(𝑓, ∆) = 1
therefore,
𝑠𝑢𝑝
𝑖𝑛𝑓
𝑠(𝑓, ∆) = 0 < 1 =
𝑆(𝑓, ∆)
∆∈ 𝐷
∆∈ 𝐷
GEOMETRICAL INTERPRETATION
OF THE DEFINITE INTEGRAL
#
Í 𝑥 # 𝑑𝑥
9.
is the area below the graph of f (and
above the x-axis) between –1 and 2
proper@es
1. property of linearity
3
Í l𝛼𝑓(𝑥) + 𝛽𝑔(𝑥)m𝑑𝑥
1
3
= 𝛼 Í 𝑓(𝑥)𝑑𝑥
1
3
+ 𝛽 Í 𝑔(𝑥)𝑑𝑥
1
where 𝛼, 𝛽 ∈ 𝑅
2. if 𝑎 < 𝑐 < 𝑏, then
3
`
3
Í 𝑓(𝑥)𝑑𝑥 = Í 𝑓(𝑥)𝑑𝑥 + Í 𝑓(𝑥)𝑑𝑥
1
1
`
128
3.
3
1
Í 𝑓(𝑥)𝑑𝑥 = − Í 𝑓(𝑥)𝑑𝑥
1
3
4. if 𝑓(𝑥) ≤ 𝑔(𝑥) ∀𝑥 ∈ [𝑎, 𝑏] , then
3
3
Í 𝑓(𝑥)𝑑𝑥 ≤ Í 𝑔(𝑥)𝑑𝑥
1
1
THEOREM (sufficient condi@on for the integrability)
if f is con;nuous on [𝑎, 𝑏] , then f is integrable on [𝑎, 𝑏]
THEOREM (mean value theorem)
let f be a con;nuous func;on on [𝑎, 𝑏]
then, ∃𝜉 ∈ [𝑎, 𝑏] s.t.
3
Í 𝑓(𝑥)𝑑𝑥 = 𝑓(𝜉)(𝑏 − 𝑎)
1
proof
since f is con;nuous on [𝑎, 𝑏] , then f is integrable on [𝑎, 𝑏]
moreover, thanks to the Weierstrass theorem f has minimum 𝑚 and maximum 𝑀 on [𝑎, 𝑏]
we consider the trivial par;;on ∆
𝑎 = 𝑥- < 𝑥. = 𝑏
therefore,
𝑠(𝑓, ∆) = 𝑚(𝑏 − 𝑎) → lower integral sum
𝑆(𝑓, ∆) = 𝑀(𝑏 − 𝑎) → upper integral sum
hence, by defini;on
3
𝑚(𝑏 − 𝑎) ≤ Í 𝑓(𝑥)𝑑𝑥 ≤ 𝑀(𝑏 − 𝑎)
1
3
∫ 𝑓(𝑥)𝑑𝑥
Þ 𝑚≤ 1
≤𝑀
𝑏−𝑎
3
∫1 𝑓 (𝑥)𝑑𝑥
=𝛾
𝑏−𝑎
thus, 𝛾 is a value between the minimum of f and the maximum of f
from the intermediate value theorem, ∃𝜉 ∈ [𝑎, 𝑏] s.t.
3
𝑓(𝜉) = 𝛾Û Í 𝑓(𝑥)𝑑𝑥 = 𝑓(𝜉)(𝑏 − 𝑎)
1
129
GEOMETRICAL INTERPRETATION OF THE DEFINITE INTEGRAL
if 𝑓 ≥ 0 on [𝑎, 𝑏] , then ∃𝜉 ∈ [𝑎, 𝑏] s.t. the area of the rectangle with base [𝑎, 𝑏] and height
[0, 𝑓(𝜉)] is equal to the area between the graph of f and the x-axis
remark
f differen;able in [𝑎, 𝑏 ] Þ f con;nuous in [𝑎, 𝑏 ] Þ f integrable in [𝑎, 𝑏] (ALWAYS TRUE)
f integrable in [𝑎, 𝑏] Þ (N.A.T.) Þ f con;nuous in [𝑎, 𝑏] Þ (N.A.T.) f differen;able in [𝑎, 𝑏]
N.A.T. = NOT ALWAYS TRUE!
THEOREM (fundamental theorem of the integral calculus)
let f be a con;nuous func;on on I (consider 𝑥̅ ∈ I) and define the integral func@on as
7
𝐹 (𝑥 ) = • 𝑓 (𝑡) 𝑑𝑡
7̅
then,
where 𝑥 ∈ I
𝐹 5 (𝑥) = 𝑓(𝑥) ∀𝑥 ∈ I
130
proof
we start by observing that 𝐹 is well-defined in I since f is con;nuous (by hypothesis) in I
now, we calculate
𝐹(𝑥 + ℎ) − 𝐹(𝑥)
𝐹 5 (𝑥) = lim
S→ℎ
= à by defini;on
71b
= lim
∫7̅
7
𝑓(𝑡) 𝑑𝑡 − ∫7̅ 𝑓(𝑡)𝑑𝑡
ℎ
b→G
71b
= lim
∫7̅
7̅
𝑓(𝑡) 𝑑𝑡 − ∫7 𝑓(𝑡)𝑑𝑡
ℎ
b→G
= à property of integrals
7̅
71b
∫ 𝑓(𝑡) 𝑑𝑡 − ∫7̅
= lim 7
b→G
ℎ
71b
= lim
∫7
𝑓(𝑡)𝑑𝑡
𝑓(𝑡) 𝑑𝑡
ℎ
b→G
= à property of integrals
at this point, we can apply the mean value theorem and we get
that ∃𝜉(ℎ) ∈ [𝑥, 𝑥 + ℎ] s.t.
71b
lim
∫7̅
𝑓(𝑡) 𝑑𝑡
ℎ
b→G
= 𝑓}𝜉 (ℎ)~
then, if ℎ → 0 we have that 𝜉 (ℎ) → 𝑥
at the end, we have that
71b
𝐹 L (𝑥 ) = lim
b→G
∫7̅
𝑓(𝑡) 𝑑𝑡
ℎ
= lim 𝑓}𝜉 (ℎ)~ = lim 𝑓(𝜉 ) = 𝑓(𝑥)
b→G
g→G
= à f is con;nuous
remark
the previous theorem is important because it highlights the link between differen;a;on and
integra;on
131
THEOREM (Torricelli’s theorem)
now, an important consequence is the Torricelli’s theorem which gives a rule to compute the
definite integrals
defini@on
let f be a con;nuous func;on in [𝑎, 𝑏]
if 𝐺 is a primi;ve of f, then
3
Í 𝑓(𝑥) 𝑑𝑥 = 𝐺(𝑏) − 𝐺(𝑎)
proof
we fix a point 𝑥̅ ∈ [𝑎, 𝑏]
then,
1
I
7̅
I
'
'
7̅
• 𝑓(𝑥) 𝑑𝑥 = • 𝑓 (𝑥 ) 𝑑𝑥 + • 𝑓 (𝑥 ) 𝑑𝑥
= à property of integrals
I
7̅
7̅
'
= • 𝑓 (𝑥 ) 𝑑𝑥 + • 𝑓(𝑥) 𝑑𝑥
I
'
7̅
7̅
= • 𝑓(𝑥) 𝑑𝑥 − • 𝑓(𝑥) 𝑑𝑥
= à property of integrals
= 𝐹 (𝑏) − 𝐹(𝑎)
where 𝐹 is the integral func;on defined in the fundamental theorem of the integral calculus
if now we consider the primi;ve 𝐺, we know that
𝐹 (𝑥 ) − 𝐺 (𝑥 ) = 𝑘
where k is a constant
therefore,
(Þ 𝐹 (𝑥 ) = 𝐺 (𝑥 ) + 𝑘
I
• 𝑓 (𝑥 ) 𝑑𝑥 = 𝐹 (𝑏) − 𝐹 (𝑎)
'
= (𝐺 (𝑏 ) + 𝑘) − (𝐺 (𝑎 ) + 𝑘)
= 𝐺 (𝑏 ) + 𝑘 − 𝐺 ( 𝑎 ) − 𝑘
= 𝐺 (𝑏 ) − 𝐺 (𝑎 )
132
examples
1.
D
• sin 𝑥 𝑑𝑥
G
D
to obtain ∫G sin 𝑥 𝑑𝑥 we calculate
first of all the primi;ve of the sin 𝑥 and
then we subs;tute the values π and 0
in the primi;ve of sin 𝑥
we recall that
• sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝑐
then,
D
• sin 𝑥 𝑑𝑥 = [− cos 𝑥]DG = − cos(𝜋 ) − (− cos(0)) = − cos(𝜋 ) + cos(0)
G
2.
= −(−1) + 1 = 1 + 1 = 2
&
• 𝑥 * 𝑑𝑥
G
• 𝑥 * 𝑑𝑥 =
𝑥+
+𝑐
3
then,
&
&
𝑥+
1+ 0+ 1
1
*
• 𝑥 𝑑𝑥 = “ ” =
−
= −0=
3 G
3
3
3
3
G
133
3.
j
• ln 𝑥 𝑑𝑥 =
&
𝑔5 = 1 Þ g = x; 𝑓 = ln 𝑥
𝑓(𝑔5 ) = 𝑓𝑔 − Í 𝑓 5 𝑔
1
= • 1 ln 𝑥 𝑑𝑥 = 𝑥 ln 𝑥 − • a 𝑥b 𝑑𝑥
𝑥
integra;on by ports
= ln 𝑥 − • 1𝑑𝑥 = ln 𝑥 − 𝑥 + 𝑐
j
• ln 𝑥 𝑑𝑥 = [ln 𝑥 − 𝑥 ]&j = (𝑒 ln(𝑒 ) − 𝑒 ) − (1 ln(1) − 1)
&
ln(𝑒) = −1; ln(1) = 0
= (𝑒 − 𝑒 ) − (0 − 1) = 0 + 1 = 1
134